Keywords

Software defined radio; Polyphase filter bank; Polyphase channelizer; Digital up converter; Digital down converter; Digital filter; Nyquist pulse; Perfect reconstruction property; MMDM; Multirate signal processing; Resampling

1.07.1 Introduction

Signal processing is the art of representing, manipulating, and transforming wave shapes and the information content that they carry by means of hardware and/or software devices. Until 1960s almost entirely continuous time, analog technology was used for performing signal processing. The evolution of digital systems along with the development of important algorithms such as the well known fast Fourier transform (FFT), by Cooley and Tukey in 1965, caused a major shift to digital technologies giving rise to new digital signal processing architectures and techniques. The key difference between analog processing techniques and digital processing techniques is that while the first one processes analog signals, which are continuous functions of time; the second one processes sequences of values. The sequences of values, called digital signals, are series of quantized samples (discrete time signal values) of analog signals.

The link between the analog signals and their sampled versions is provided by the sampling process. When sampling is properly applied it is possible to reconstruct the continuous time signal from its samples preserving its information content. Thus the selection of the sampling rate is crucial: an insufficient sampling frequency causes, surely, irrecoverable loss of information, while a more than necessary sampling rate causes useless overload on the digital systems.

The string of words multirate digital signal processing indicates the operation of changing the sample rate of digital signals, one or multiple times, while processing them. The sampling rate changes can occur at a single or multiple locations in the processing architecture. When the multirate processing applied to the digital signal is a filtering then the digital filter is named multirate filter; a multirate filter is a digital filter that operates with one or more sample rate changes embedded in the signal processing architecture. The opportunity of selecting the most appropriate signal sampling rate at different stages of the processing architecture, rather than having a single (higher) sampling rate, enhances the performance of the system while reducing its implementation costs. However, the analysis and design of multirate systems could result complicated because of the fact that they are time-varying systems. It is interesting to know that the first multirate filters and systems were developed in the context of control systems. The pioneer papers on this topic were published in the second half of 1950s [1–3]. Soon the idea spilled in the areas of speech, audio, image processing [4,5], and communication systems [6]. Today, multirate structures seem to be the optimum candidates for cognitive and software defined radio [7–11] which represents the frontier of innovation for the upcoming communication systems.

One of the well known techniques that find its most efficient application when embedded in multirate structures is the polyphase decomposition of a prototype filter. The polyphase networks, of generic order M, originated in the late 1970s from the works by Bellanger et al. [12,13]. The term polyphase is the aggregation of two words: poly, that derives from the ancient greek word polys, which means many, and phase. When applied to an M-path partitioned filter these two words underline the fact that, on each path, each aliased filter spectral component, experiences a unique phase rotation due to both their center frequencies and the time delays, which are different in each path because of the way in which the filter has been partitioned. When all the paths are summed together the undesired spectral components, having phases with opposite polarity, cancel each other while only the desired components, which experience the same phase on all the arms of the partitions, constructively add up. By applying appropriate phase rotators to each path we can arbitrarily change the phases of the spectral components selecting the spectral component that survives as a consequence of the summation. It is interesting to know that the first applications of the polyphase networks were in the areas of real-time implementation of decimation and interpolation filters, fractional sampling rate changing devices, uniform DFT filter banks as well as perfect reconstruction analysis/synthesis systems. Today polyphase filter banks, embedded in multirate structures, are used in many modern DSP applications as well as in communication systems where they represent, according to the author’s belief, one of the most efficient options for designing software-based radio.

The processing architecture that represents the starting point for the derivation of the standard polyphase down converter channelizer is the single channel digital down converter. In a digital radio receiver this engine performs the operations of filtering, frequency translation and resampling on the intermediate frequency (IF) signals. The resampling is a down sampling operation for making the signal sampling rate commensurate to its new reduced bandwidth. When the output sampling rate is selected to be an integer multiple of the signal’s center frequency, the signal spectrum is shifted, by aliasing, to base-band and the complex heterodyne defaults to unity value disappearing from the processing path. The two remaining operations of filtering and down sampling are usually performed in a single processing architecture (multirate filter). After exchanging the positions of the filter and resampler, by applying polyphase decomposition, we achieve the standard M-path polyphase down converter channelizer which shifts a single narrowband channel to base-band while reducing its sampling rate. By following a similar procedure the standard up converter channelizer can also be obtained. It performs the operation of up converting while interpolating a single narrowband channel.

When the polyphase channelizer is used in this fashion the complex phase rotators can be applied to each arm to select, by coherent summation, a desired channel arbitrarily located in the frequency domain. Moreover, the most interesting and efficient application of this engine is in the multichannel scenario. When the inverse discrete Fourier transform block is embedded, the polyphase channelizer acquires the capability to simultaneously up and down convert, by aliasing, multiple, equally spaced, narrowband channels having equal bandwidths. Due to its computational efficiency, the polyphase channelizer, and its modified versions, represents the best candidate for building up new flexible multichannel digital radio architectures [14] like software defined radio promises to be.

Software defined radio represents one of the most important emerging technologies for the future of wireless communication services. By moving radio functionality into software, it promises to give flexible radio systems that are multi-service, multi-standard, multi-band, reconfigurable, and reprogrammable by software. The goal of software defined radio is to solve the issue of optimizing the use of the radio spectrum that is becoming more and more pressing because of the growing deployment of new wireless devices and applications [15–18].

In the second part of this chapter we address the issue of designing software radio architectures for both the transmitter and the receiver. The novel structures are based on modified versions of the standard polyphase channelizer, which provides the system with the capability to optimally adapt its operating parameters according to the surrounding radio environment [7,19–21]. This implies, on the transmitter side, the capability of the radio to detect the available spectral holes [22,23] in the spanned frequency range and to dynamically use them for sending signals having different bandwidths at randomly located center frequencies. The signals could be originated from different information sources or they could be spectral partitions of one signal, fragmented because of the unavailability of free space in the radio spectrum or for protecting it from unknown and undesired detections.

The core of the proposed transmitter is a synthesis channelizer [24]. It is a variant of the standard M-path polyphase up converter channelizer [6,8] that is able to perform 2-to-M up sampling while shifting, by aliasing, all the base-band channels to the desired center frequencies. Input signals with bandwidths wider than the synthesis channelizer bandwidth are pre-processed through small down converter channelizers that disassemble their bandwidths into reduced bandwidth sub-channels. The proposed transmitter avoids the need to replicate the sampled data section when multiple signals have to be simultaneously transmitted and it also allows partitioning of the signal spectra, when necessary, before transmitting them. Those spectral fragments will be reassembled in the receiver after being shifted to base-band without any loss of energy because perfect reconstruction filters are used as low-pass prototype in the channelizer polyphase decompositions.

On the other side of the communication chain, a cognitive receiver has to be able to simultaneously detect multiple signals, recognize, when necessary, all their spectral partitions, filter, down convert and recompose them without energy losses [7] independently of their bandwidths or center frequencies. An analysis channelizer is the key element of the proposed receiver [7]. This engine is able to perform M-to-2 down sampling while simultaneously demodulating, by aliasing, all the received signal spectra having arbitrary bandwidths residing at arbitrary center frequencies [9]. Post-processing up converter channelizers are used for reassembling, from the analysis channelizer base-line channels, signal bandwidths wider than the analysis channelizer channel bandwidth.

In the transmitter complex frequency rotators apply the appropriate frequency offsets for arbitrary center frequency positioning of the spectra. When the frequency rotators are placed in the proposed receiver, they are responsible for perfect DC alignment of the down converted signals.

This chapter is composed of 10 main sections. The first four sections are dedicated to the basics of digital signal processing, multirate signal processing, and polyphase channelizers. These sections contain preliminary concepts necessary for completely understanding the last sections of this work in which actual research issues on software radio (SR) architecture design are discussed and innovative results are presented. In particular, Section 1.07.2 recalls basic concepts on the resampling process of a discrete time signal as opposed to the sampling process of an analog, continuous time, signal. Section 1.07.3 introduces the readers to digital filters providing preliminaries on their design techniques. Section 1.07.5 presents the standard single path architectures for down sampling and up sampling digital signals. Section 1.07.6 introduces the standard version of the M-path polyphase down and up converter channelizers. Both of them are derived, step by step, by the single path structures that represent the current state of the technology. In Section 1.07.7 we present the modified versions of these engines. These are the structures that give form to the proposed digital down and up converters for SDRs which are presented in Section 1.07.9 which also provides the simulation results. Section 1.07.8 introduces the readers to the concept of software radio while Section 1.07.10 gives concluding remarks along with suggestions for future research works in this same area.

1.07.2 The Sampling process and the “Resampling” process

In the previous section we mentioned that the sampling process is the link between the continuous time world and the discrete time world. By sampling a continuous time signal we achieve its discrete time representation. When sampling frequency is properly selected the sampling process becomes invertible and it is possible to re-shift from the discrete time representation to the continuous time representation of a signal maintaining its information content during the process.

It is well known that, when the signal is bandlimited and has no frequency components above it can be uniquely described by its samples taken uniformly at frequency

(7.1)

Equation (7.1) is known as Nyquist sampling criterion (or uniform sampling theorem) and the sampling frequency is called the Nyquist sampling rate. This theorem represents a theoretically sufficient condition for reconstructing the analog signal from its uniformly spaced samples.

Even though sampling is practically implemented in a different way, it is convenient, in order to facilitate the learning process, to represent it as a product of the analog waveform, , with a periodic train of unit impulse functions defined as

where is the sampling period. By using the shifting property of the impulse function we obtain

(7.2)

A pictorial description of Eq. (7.2) is given in Figure 7.2.

For facilitating the readers’ understanding of the uniform sampling effects on the bandlimited continuous time signal, , we shift from time domain to the frequency domain where we are allowed to use the properties of the Fourier transform. The product of two functions in time domain becomes the convolution of their Fourier transforms in the frequency domain. Thus, if is the Fourier transform of and is the Fourier transform of , then the Fourier transform of is

where indicates linear convolution and

Note that the Fourier transform of a impulse train is another impulse train with the values of the periods reciprocally related to each other. Then in the frequency domain Eq. (7.2) becomes

(7.3)

whose pictorial view is shown in Figure 7.3.

From Eq. (7.3) we conclude that the spectrum of the sampled signal, in the original signal bandwidth , is the same as the continuous time one (except for a scale factor, ) however, as a consequence of the sampling process, this spectrum periodically repeats itself with a period of . We can easily recognize that it should be possible to recover the original spectrum (associated to the spectral replica which resides in , by filtering the periodically sampled signal spectrum with an appropriate low-pass filter.

Notice, from Eq. (7.3) that the spacing, , between the signal replicas is the reciprocal of the sampling period . A small sampling period corresponds to a large space between the spectral replicas.

A fundamental observation, regarding the selection of the uniform sampling frequency, needs to be done at this point: when the sampling rate is selected to be less than the maximum frequency component of the bandlimited signal the periodic spectral replicas overlap each other. The amount of the overlap depends of the selected sampling frequency. Smaller the sampling frequency, larger the amount of overlap experienced by the replicas. This phenomenon is well known as aliasing. When aliasing occurs it is impossible to recover the analog signal from its samples. When the spectral replicas touch each other without overlapping and it is theoretically (but not practically) possible to recover the analog signal from its samples; however a filter with infinite number of taps would be required. As a matter of practical consideration, we need to specify here that the signals (and filters) of interest are never perfectly bandlimited and some amount of aliasing always occurs as effect of sampling however some techniques can be used to limit the phenomena making it less harmful.

After the sampling has been applied the amplitude of each sample is one from an infinite set of possible values. This is the reason for which the samples are not compatible with a digital system yet. A digital system can, in fact, only deal with a finite number of values. The digital samples need to be quantized before being sent to the digital data system. The quantization process limits the amplitude of the samples to a finite set of values. After the quantization process the signal can still be recovered, however some additional imprecision is added to it. The amount of imprecision depends of the quantization levels used in the process and it has the same effect on the signal as white noise. This is the reason for which it is referred to as quantization noise. The sampling of a continuous time signal and the quantization of its discrete time samples are both performed with devices called analog-to-digital converters (ADCs).

The selection of the appropriate sampling rate is a fundamental issue faced when designing digital communication systems. In order to preserve the signal, the Nyquist criterion must be always satisfied. Also, it is not difficult to find tasks for which, at some points in the digital data section of the transmitter and the receiver, it is recommended to have more than two samples per signal bandwidth. On the other hand, large sample rates cause an increase in the total workload of the system. Thus the appropriate sampling rate must be selected according to both the necessities: to have the required number of samples and to minimize the total workload. Most likely the optimum sampling frequency changes at different points in the digital architecture. It would be a huge advantage to have the option of changing the signal sampling rate at different parts in the systems while processing it, thus optimizing the number of samples as a function of the requirements. The process of changing the sampling rate of a digital signal is referred to as resampling process. The resampling process is the key concept in multirate signal processing.

After the brief previous discussion about the sampling process of a continuous time signal, we address now the process of resampling an already sampled signal. Notice that when a continuous time signal is sampled there are no restrictions on the sample rate or the phase of the sample clock relative to the time base of the continuous time signal. On the other hand, when we resample an already sampled signal, the output sample locations are intimately related to the input sample positions. A resampled time series contains samples of the original input time series separated by a set of zero valued samples. The zero valued time samples can be the result of setting a subset of input sample values to zero or the result of inserting zeros between existing input sample values. Both options are shown in Figure 7.4. In the first example shown in this figure, the input sequence is resampled 2-to-l, keeping every second input sample starting at sample index 0 while zeroing the interim samples. In the second example, the input sequence is resampled 1-to-2, keeping every input sample but inserting a zero valued sample between each input samples. These two processes are called down sampling and up sampling respectively.

The non-zero valued samples of two sequences having the same sample rate can occur at different time indices (i.e., the same sequence can have different initial time offset) as in Figure 7.5 in which the two 2-to-1 down sampled sequences, and , have different starting time index, 0 and 1, which gives equal magnitude spectral components with different phase profiles. This is explicitly shown in Figures 7.6 and 7.7, in which the time domain and the frequency domain views of the resampled sequences and are depicted respectively. Remember, in fact, the time shifting property of the discrete Fourier transform (DFT) for which a shift in time domain is equivalent to a linear phase shift in the frequency domain. The different phase profiles play a central role in multirate signal processing.

The M zero valued samples inserted between the signal samples create M periodic replicas of the original spectrum at the frequency locations k/M, with in the normalized domain. This observation suggests that resampling can be used to affect translation of spectral bands, up and down conversion, without the use of sample data heterodynes.

In the following paragraphs we clarify this concept showing how to embed the spectral translation of narrowband signals in resampling filters and describe the process as aliasing.

A final comment about the resampling process is that it can be applied to a time series or to the impulse response of a filter, which, of course, is simply another time series. When the resampling process is applied to a filter, the architecture of the filter changes considerably and the filter is called multirate filter.

1.07.3 Digital filters

Filtering is the practice of processing signals which results in some changes of their spectral contents. Usually the change implies a reduction or filtering out some undesired input spectral components. A filter allows certain frequencies to pass while attenuating others. While analog filters operate on continuous-time signals, digital filters operate on sequences of discrete sampled value (see Figure 7.8) although digital filters perform many of the same functions as analog filters, they are different!

The two classes of filters share many of the same or analogous properties. In the standard order of our educational process we first learn about analog filters and later learn about digital filters. To ease the entry into the digital domain we emphasize the similarities of the two classes of filters. This order is due to fact that an analog filter is less abstract than a digital filter. We can touch the components of an analog filter; capacitors, inductors, resistors, operational amplifiers and wires. On the other hand a digital filter doesn’t have the same physical form because a digital filter is merely a set of instructions that perform arithmetic operations on an array of numbers. The operations can be weighted sums or inner products (see Figure 7.9). It is convenient to visualize the array as a list of uniformly spaced sample values of an analog waveform. The instructions to perform these operations can reside as software in a computer or microprocessor or as firmware in a dedicated collection of interconnected hardware elements.

Let us examine the similarities that are emphasized when we learn about digital filters with some level of prior familiarity with analog filters and the tools to describe them. Many analog filters are formed by interconnections of lumped linear components modeled as ideal capacitors, inductors, and resistors while others are formed by interconnections of resistors, capacitors, and operational amplifiers. Sampled data filters are formed by interconnections of registers, adders, and multipliers. Most analog filters are designed to satisfy relationships defined by linear time invariant differential equations. The differential equations are recursive which means the initial condition time domain response, called the homogeneous or undriven response, is a weighted sum of exponentially decaying sinusoids. Most sampled data filters are designed to satisfy relationships defined by linear time invariant difference equations. Here we find the first major distinction between the analog and digital filters: the difference equations for the sampled data filters can be recursive or non recursive. When the difference equations are selected to be recursive the initial condition response is, as it was for the analog filter, samples of a weighted sum of exponentially decaying sinusoids. On the other hand, when the difference equation is non recursive the initial condition response is anything the designer wants it to be and is limited only by her imagination but is usually designed to satisfy some frequency domain specifications.

When we compare the two filter classes, analog and digital, we call attention to the similarities of the tools we use to describe, analyze, and design them. They are described by differential and difference equations which are weighted sums of signal derivates or weighted sums of delayed sequences. They both have linear operators or transforms, the Laplace transform , and the z transform , that offer us insight into the internal structure of the differential or difference equations (see Figure 7.10). They both have operator descriptions that perform the equivalent functions. These are the integral operator, denoted by , and the delay operator, denoted by , in which reside the system memories or state of the analog and digital filters respectively. Both systems have transfer functions H(s) and H(z), ratios of polynomials in the operator variable s and z. The roots of the denominator polynomial, the operator version of the characteristic equation, are the filter poles. These poles describe the filter modes, the exponentially damped sinusoids, of the recursive structure. The roots of numerator polynomial are the filter zeros. The zeros describe how the filter internal mode responses are connected to the filter input and output ports. They tell us the amplitude of each response component, (called residues to impress us) in the output signal’s initial condition response. The transforms of the two filter classes have companion transforms the Fourier transform and the sampled data Fourier series which describe, when they exist, the frequency domain behavior, sinusoidal steady state gain and phase, of the filters. It is remarkable how many similarities there are in the structure, the tools, and the time and frequency responses of the two types of filters. It is no wonder we emphasize their similarities.

We spend a great deal of effort examining recursive digital filters because of the close relationships with their analog counterparts. There are digital filters that are mappings of the traditional analog filters. These include maximally flat Butterworth filters, equal ripple pass-band Tchebychev-I filters, equal ripple stop-band Tchebychev-II filters, and both equal ripple pass-band and equal ripple stop-band Elliptic filters. The spectra of these infinite impulse response (IIR) filters are shown in Figure 7.11. All DSP filter design programs offer the DSP equivalents of their analog recursive filter counterparts.

In retrospect, emphasizing similarities between the analog filter and the equivalent digital was a poor decision. We make this claim for a number of reasons. The first is there are limited counterparts in the analog domain to the non-recursive filter of the digital domain. This is because the primary element of the non recursive filter is the pure delay, represented by the delay operator . We cannot form a pure delay response, represented by the analog delay operator , in the analog domain as the solution of a linear differential equation. To obtain pure delay, we require a partial differential equation which offers the wave equation whose solution is a propagating wave which we use to convert distance to time delay. The propagation velocity of electromagnetic waves is too high to be of practical use in most analog filter designs but the velocity of sound waves in crystal structures enables an important class of analog filters known as acoustic surface wave (SAW) devices. Your cell phone contains one or more such filters. In general, most of us have limited familiarity with non-recursive analog filters so our first introduction to the properties and design techniques for finite duration impulse response (FIR) filters occurs in our DSP classes.

The second reason for not emphasizing the similarities between analog and digital filters is that we start believing the statement that “a digital filter is the same as an analog filter” and that all the properties of one are embedded in the other. That seemed like an ok perspective the first 10 or 20 times we heard that claim. The problem is, it is not true! The digital filter has a resource the analog filter does not have! The digital filter can perform tasks the analog filter cannot do! What is that resource? It is the sampling clock! We can do applied magic in the sampled data domain by manipulating the sample clock. There is no equivalent magic available in the analog filter domain! Analog designers shake their heads in disbelief when they see what a multirate filter can do. Actually digital designers also shake their heads in disbelief when they learn that manipulating the clock can accomplish such amazing things.

We might ask how can the sampling clock affect the performance or more so, enhance the capabilities of a digital filter? Great question! The answer my friend is written in the way define frequency of a sampled data signal. Let us review how frequency as we understand it in the analog world is converted to frequency in the digital world. We now present a concise review of complex and real sinusoids with careful attention paid to their arguments.

The exponential function has the interesting property that it satisfies the differential equation shown in Eq. (7.4). This means the exponential function replicates under the differential operator, an important property inherited by real and complex sinusoids. It is easy to verify that the Taylor series shown in Eq. (7.5) satisfies Eq. (7.4). When we replace the argument “x” of Eq. (7.5) with the argument “” we obtain the series shown in Eq. (7.6) and when we gather the terms corresponding to the even and odd powers respectively of the argument “” we obtain the partitioned series shown in Eq. (7.7). We recognize that the two Taylor series of the partitioned Eq. (7.7) are the Taylor series of the cosine and sine which we show explicitly in Eq. (7.8). We note that the argument of both the real exponential and of the complex exponential must be dimensionless otherwise we would not be able to sum the successive powers of the series. We also note that the units of the argument “” are radians, a dimensionless unit formed by the ratio of arc length on a circle normalized by the radius of the circle. When we replace the “” argument of Eq. (7.8) with a time varying angle “” we obtain the time function shown in Eq. (7.9). The simplest time varying angle is one the changes linearly with time, , which when substituted in Eq. (7.9) we have Eq. (7.10). All of this discussion leads us to the next statement. Since the argument of a sinusoid has the dimensionless units radians and the units of “t” has units of seconds, then the units of must be rad/s, a velocity. In other words, the frequency is the time derivative of the linearly increasing phase angle

(7.4)

(7.5)

(7.6)

(7.7)

(7.8)

(7.9)

(7.10)

We now examine the phase argument of the sampled data sinusoid. The successive samples of a sampled complex sinusoid are formed as shown in Eq. (7.11) by replacing the continuous argument t with the sample time positions nT as was done in Eq. (7.12). We now note that while the argument of the sampled data sinusoid is dimensionless, the independent variable is no longer “t” with units of seconds but rather “n” with units of sample. This is consistent with the dimension of “T” which of course is s/sample. Consequently the product has units of (rad/s)  (s/smpl) or rad/smpl, thus digital frequency is rad/smpl or emphatically the angle change per sample! A more useful version of  Eq. (7.12) is obtained by replacing with as done in Eq. (7.13) and then replace T with as was done in Eq. (7.14). We finally replace with as was done in Eq. (7.15). Here we clearly see that the parameter has units of rad/smpl, which described the fraction of the circle the argument traverses per successive sample

(7.11)

(7.12)

(7.13)

(7.14)

(7.15)

Now suppose we have a sampled sinusoid with sample rate 8-times the sinusoid’s center frequency so that there are 8-samples per cycle. This is equivalent to the sampled sinusoid having a digital frequency of . We can visualize a spinning phasor rotating 1/8th of the way around the unit circle per sample. Figure 7.12 subplots 1 and 2 show 21 samples of this sinusoid and its spectrum. Note the pair of spectral lines located at on the normalized frequency axis. We can reduce the sample rate of this time series by taking every other sample. The spinning rate for this new sequence is or . The down sampled sequence has doubled its digital frequency. The newly sampled sequence and its spectrum are shown in Figure 7.12 subplots 3 and 4. Note the pair of spectral lines are now located at on the normalized frequency axis. We can further reduce the sample rate of by again taking every other sample (or every 4th sample of the original sequence). The spinning rate for this new sequence is or . The down sampled sequence has again doubled its digital frequency. The newly sampled sequence and its spectrum are shown in Figure 7.12 subplots 5 and 6. Note the pair of spectral lines are now located at on the normalized frequency axis. We once again reduce the sample rate of by taking every other sample (or every 8th sample of the original sequence). The spinning rate for this new sequence is or , but since a rotation is equivalent to no rotation, the spinning phasor appears to be stationary or rotating at 0 rad/smpl. The newly sampled sequence and its spectrum are shown in Figure 7.12 subplots 7 and 8. Note the pair of spectral lines are now located at which is equivalent to 0 on the normalized frequency axis. Frequency shifts due to sample rate changes are attributed to aliasing. Aliasing enables us to move spectrum from one frequency location to another location without the need for the complex mixers we normally use in a digital down converter to move a spectral span to base-band.

Speaking of the standard digital down converter Figure 7.13 presents the primary signal flow blocks based on the DSP emulation of the conventional analog receiver. Figure 7.14 shows the spectra that will be observed at successive output ports of the DDC. We can follow the signal transformations of the DDC by following the spectra in the following description. The top figure shows the spectrum at the output of the analog to digital converter. The second figure shows the spectrum at the output of the quadrature mixer. Here we see the input spectrum has been shifted so that the selected channel band resides at zero frequency. The third figure shows the output of the low-pass filter pair. Here we see that a significant fraction of the input bandwidth has been rejected by the stop band of the low pass filter. Finally, the last figure shows the spectrum after the M-to-1 down sampling that reduced the sample rate in proportion to the reduction in bandwidth performed by the filter. We will always reduce sample rate if we are reducing bandwidth. We have no performance advantage for over satisfying the Nyquist criterion, but incur significant penalties by operating the DSP processors at higher than necessary sample rates.

Figure 7.15 presents the primary signal flow blocks based on aliasing the selected band pass span to base-band by reducing the output sample rate of the band pass filter. Figure 7.16 shows the spectra that will be observed at successive output ports of the aliasing DDC. We can follow the signal transformations of the aliasing DDC by following the spectra in the following description. The top figure shows the spectrum at the output of the analog to digital converter. The second figure shows the spectrum of the complex band pass filter and the output spectrum from this filter. The band pass filter is formed by up-converting, by a complex heterodyne, the coefficients of the prototype low pass filter. We note that this filter does not have a mirror image. Analog designers sigh when they see this digital option. The third figure shows the output of the band pass filter pair following the M-to-1 down sampling. If the center frequency of the band is a multiple of the output sample rate the aliased band resides at base band. For the example shown here the center frequency was above k and since all multiples of alias to base-band, our selected band has aliased to above zero frequency. Finally, the last figure shows the spectrum after the output heterodyne from to base-band. Note that the heterodyne is applied at the low output rate rather than at the input at the high input rate as was done in Figure 7.13.

Our last section of this preface deals with the equivalency of cascade operators. Here is the core idea. Suppose we have a cascade of two filters, say a low pass filter followed by a derivative filter as shown in Figure 7.17. The filters are linear operators that commute and are distributive. We could reorder the two filters and have the same output from their cascade or we could apply the derivative filter operator to the low pass filter to form a composite filter and obtain the same output as from the cascade. This equivalency is shown in Figure 7.18. The important idea here is that we can filter the input signal and then apply an operator to the filter output or we can apply the operator to the filter and have the altered filter perform both operators to the input signal simultaneously.

Here comes the punch-line! Let us examine the aliasing digital down converter of Figure 7.15 which contains a band pass filter followed by an M-to-1 resampler. This section is redrawn in the upper half of Figure 7.19. If we think about it, this almost silly: Here we compute one output sample for each input sample and then discard M-1 of these samples in the M-to-1 down sampler. Following the lead of the cascade linear operators, the filter and resampler, being equivalent to a composite filter containing the resampler applied to the filter we replace the pair of operators with the composite operator as shown in the lower half of Figure 7.19.

Embedding the M-to-1 resampler in the band pass filter is accomplished by the block diagram shown in Figure 7.20. This filter accepts M inputs, one for each path, and computes 1 output. Thus we do not waste operations computing output sample scheduled to be discarded. An interesting note here is that the complex rotators applied to the prototype low pass filter in Figure 7.15 have been factored out of each arm and are applied once at the output of each path. This means that if the input data is real, the samples are not made complex till they leave the filter. This represents a 2-to-1 savings over the architecture of Figure 7.13 in which the samples are made complex on the way into the filter which means there are 2 filters, one for each path of the complex heterodyne. The alias based DDS with resampler embedded in the filter is not a bad architecture! It computes output samples at the output rate and it only uses one partitioned filter rather than 2. Not bad is understatement! The cherry on top of this dessert is the rotator vector applied to the output of the path filters. This rotator can extract any band from the aliased frequency spans that have aliased to base-band by the resampling operator. If it can extract any band then we have the option of extracting more than one band, and in fact we can extract every band from the output of the partitioned filter. That’s amazing! This filter can service more than one output channel simultaneously; all it needs is additional rotator vectors. As we will see shortly, the single filter structure can service all the channels that alias to base-band and the most efficient bank of rotators is implemented by an M-point IFFT. More to come! Be sure to read on! We have only scratched the surface and the body of material related to multirate filters if filled with many wonderful properties and applications.

1.07.4 Windowing

As specified in the previous section, digital filters can be classified in many ways, starting with the most general frequency domain characteristics such as low pass, high pass, band pass, band stop and finishing with, secondary characteristics such as uniform and non-uniform group delay. We now also know that an important classification is based on the filter’s architectural structure with a primary consideration being that of finite impulse response and infinite impulse response filter. Further sub-classifications, such as canonic forms, cascade forms, lattice forms are primarily driven by consideration of sensitivity to finite arithmetic, memory requirements, ability to pipeline arithmetic, and hardware constraints.

The choice to perform a given filtering task with a recursive or a non-recursive filter is driven by a number of system considerations, including processing resources, clock speed, and various filter specifications. Performance specifications, which include operating sample rate, pass band and stop band edges, pass band ripple, and out-of-band attenuation, all interact to determine the complexity required of the digital filter.

In filters with infinite impulse response each output sample depends on previous input samples and on previous filter output samples. That is the reason for which their practical implementation always requires a feedback loop. Thus, like all feedback based architectures, IIR filters are sensitive to input perturbations that could cause instability and infinite oscillations at the output. However infinite impulse response filters are usually very efficient and require far few multiplications than an FIR filter per output sample. Notice that an IIR filter could have an infinite sequence of output samples even if the input samples become all zeros. It is this characteristic which gives them their name.

Finite impulse response filters use only current and past input samples and none of the filter’s previous output samples to obtain a current output sample value (remember that they are also sometimes referred to as non-recursive filters). Given a finite duration of the input signal, the FIR filter will always have a finite duration of non-zero output samples and this is the characteristic which gives them their name.

The procedure by which the FIR filters calculate the output samples is the convolution of the input sequence with the filter impulse response which is shown in Eq. (7.16)

(7.16)

where is the input sequence, is the filter impulse response and M is the number of filter taps. The convolution is nothing but a series of multiplications followed by the addition of the products while the impulse response of a filter is nothing but what the name tells us: it is the time domain view of the filter’s output values when the input is an impulse (a single unity-valued sample preceded and followed by zero-valued samples).

In the following paragraphs we introduce the basic window design method for FIR filters. Because low pass filtering is the most common filtering task, we will introduce the window design method for low pass FIR filters. However the relationships between filter length and filter specifications and the interactions between filter parameters remain valid for other filter types. Many other techniques can be also used to design FIR filters and, at the end of this section, we will introduce the Remez algorithm, which is one of the most used filter design technique, as well as one of its modified versions which allows us to achieve 1/f type of decay for the out of band side-lobes.

The frequency response of a prototype low pass filter is shown in Figure 7.21. The pass band is seen to be characterized by an ideal rectangle with unity gain between the frequencies Hz and zero gain elsewhere.

The attraction of the ideal low pass filter H(f) as a prototype is that, from its closed form inverse Fourier transform, we achieve the exact expression for its impulse response h(t) which is shown in Eq. (7.17)

(7.17)

The argument of the function is always the product of , half the spectral support , and the independent variable t. The numerator is periodic and becomes zero when the argument is a multiple of . The impulse response of the prototype filter is shown in Figure 7.22. The filter shown in Figure 7.22 is a continuous function which we have to sample to obtain the prototype sampled data impulse response. To preserve the filter gain during the sampling process we scale the sampled function by the sample rate . The problem with the sample set of the prototype filter is that the number of samples is unbounded and the filter is non-causal. If we had a finite number of samples, we could delay the response to make it causal and solve the problem. Then our first task is to reduce the unbounded set of filter coefficients to a finite set. The process of pruning an infinite sequence to a finite sequence is called windowing. In this process, a new limited sequence is formed as the product of the finite sequence w(n) and the infinite sequence as shown in Eq. (7.18) where the.^∗ operator is the standard MATLAB point-by-point multiply

(7.18)

Applying the properties of the Fourier transforms we obtain the expression for the spectrum of the windowed impulse response which is the circular convolution of the Fourier transform of h(n) and w(n).

The symmetric rectangle we selected as our window abruptly turns off the coefficient set at its boundaries. The sampled rectangle weighting function has a spectrum described by the Dirichlet kernel which is the periodic extension of the transform of a continuous time rectangle function. The convolution between the spectra of the prototype filter with the Dirichlet kernel forms the spectrum of the rectangle windowed filter coefficient set. The contribution to the corresponding output spectrum is seen to be the stop band ripple, the transition bandwidth, and the pass band ripple. The pass band and stop band ripples are due to the side-lobes of the Dirichlet kernel moving through the pass band of the prototype filter while the transition bandwidth is due to the main lobe of the kernel moving from the stop band to the pass band of the prototype. A property of the Fourier series is that their truncated version forms a new series exhibiting the minimum mean square (MMS) approximation to the original function. We thus note that the set of coefficients obtained by a rectangle window exhibits the minimum mean square approximation to the prototype frequency response. The problem with MMS approximations in numerical analysis is that there is no mechanism to control the location or value of the error maxima. The local maximum errors are attributed to the Gibbs phenomena, the failure of the series to converge in the neighborhood of a discontinuity. These errors can be objectionably large. A process must now be invoked to control the objectionably high side-lobe levels. We have two ways to approach the problem. First we can redefine the frequency response of the prototype filter so that the amplitude discontinuities are replaced with a specified tapering in the transition bandwidth. In this process, we trade-off the transition bandwidth for side-lobe control. Equivalently, knowing that the objectionable stop band side-lobes are caused by the side-lobes in the spectrum of the window, we can replace the rectangle window with other even symmetric functions with reduced amplitude side-lobe levels. The two techniques, side-lobe control and transition-bandwidth control are tightly coupled. The easiest way to visualize control of the side-lobes is by destructive cancelation between the spectral side-lobes of the Dirichlet kernel associated with the rectangle and the spectral side-lobes of translated and scaled versions of the same kernel. The cost we incur to obtain reduced side-lobe levels is an increase in main lobe bandwidth. Remembering that the window’s two-sided main lobe width is an upper bound to the filter’s transition bandwidth, we can estimate the transition bandwidth of a filter required to obtain a specified side-lobe level. The primary reason we examined windows and their spectral description as weighted Dirichlet kernels was to develop a sense of how we trade the main lobe width of the window for its side-lobe levels and in turn filter transition bandwidth and side-lobe levels. Some windows perform this trade-off of bandwidth for side-lobe level very efficiently while others do not.

The Kaiser-Bessel window is very effective while the triangle (or Fejer) window is not. The Kaiser-Bessel window is in fact a family of windows parameterized over the time-bandwidth product, , of the window. The main lobe width increases with while the peak side-lobe level decreases with . The Kaiser-Bessel window is a standard option in filter design packages such as Matlab and QED-2000.

Other filter design techniques include the Remez algorithm [6], sometimes also referred to as the Parks-McClellan or P-M, the McClellan, Parks and Rabiner or MPR, the Equiripple, and the Multiple Exchange algorithm, which found large acceptance in practice. It is a very versatile algorithm capable of designing FIR filters with various frequency responses, including multiple pass band and stop band frequency responses with independent control of ripple levels in the multiple bands. The desired pass band cut off frequencies, the frequencies where the attenuated bands begin and the desired pass band and stop band ripple are given as input parameters to the software which will generate N time domain filter coefficients. N is the minimum number of taps required for the desired filter response and it is selected by using the harris approximation or the Hermann approximation (in Matlab). The problem with the Remez algorithm is that it shows equiripple side-lobes which is not a desirable characteristic. We would like to have a filter frequency response which has a l/f out-of-band decay rate rather than exhibit equiripple. The main reasons for desiring that characteristic are related to system performance. We often build systems comprising a digital filter and a resampling switch. Here the digital filter reduces the bandwidth and is followed by a resampling switch that reduces the output sample commensurate with the reduced output bandwidth. When the filter output is resampled, the low level energy residing in the out-of-band spectral region aliases back into the filter pass band. When the reduction in sample rate is large, there are multiple spectral regions that alias or fold into the pass band. For instance, in a 16-to-l reduction in sample rate, there are 15 spectral regions that fold into the pass band. The energy in these bands is additive and if the spectral density in each band is equal, as it is in an equiripple design, the folded energy level is increased by a factor of sqrt(15). The second reason for which we may prefer FIR filters with l/f side-lobe attenuation as opposed to uniform side-lobes is finite arithmetic. A filter is defined by its coefficient set and an approximation to this filter is realized by a set of quantized coefficients. Given two filter sets and , the first with equiripple side-lobes, the second with l/f side-lobes, we form two new sets, and , by quantizing their coefficients. The quantization process is performed in two steps: first we rescale the filters by dividing with the peak coefficient. Second, we represent the coefficients with a fixed number of bits to obtain the quantized approximations. The zeros of an FIR filter residing on the unit circle perform the task of holding down the frequency response in the stop band. The interval between the zeros contains the spectral side-lobes. When the interval between adjacent zeros is reduced, the amplitude of the side-lobe between them is reduced and when the interval between adjacent zeros is increased the amplitude of the side-lobe between them is increased. The zeros of the filters are the roots of the polynomials and . The roots of the polynomials formed by the quantized set of coefficients differ from the roots of the non-quantized polynomials. For small changes in coefficient size, the roots exhibit small displacements along the unit circle from their nominal positions. The amplitude of some of the side-lobes must increase due to this root shift. In the equiripple design, the initial side-lobes exactly meet the designed side-lobe level with no margin for side-lobe increase due to root shift caused by coefficient quantization. On the other hand, the filter with 1/f side-lobe levels has plenty of margin for side-lobe increase due to root shift caused by coefficient quantization.

More details on the modified Remez algorithm can be found in [6].

1.07.5 Basics on multirate filters

Multirate filters are digital filters that contain a mechanism to increase or decrease the sample rate while processing input sampled signals. The simplest multirate filter performs integer up sampling of 1-to-M or integer down sampling of Q-to-1. By extension, a multirate filter can employ both up sampling and down sampling in the same process to affect a rational ratio sample rate change of M-to-Q. More sophisticated techniques exist to perform arbitrary and perhaps slowly time varying sample rate changes. The integers M and Q may be selected to be the same so that there is no sample rate change between input and output but rather an arbitrary time shift or phase offset between input and output sample positions of the complex envelope. The sample rate change can occur at a single location in the processing chain or can be distributed over several subsections.

Conceptually, the process of down sampling can be visualized as a two-step progression indicated in Figure 7.23. There are three distinct signals associated with this procedure. The process starts with an input series that is processed by a filter to obtain the output sequence with reduced bandwidth. The sample rate of the output sequence is then reduced Q-to-1 to a rate commensurate with the reduced signal bandwidth. In reality the processes of bandwidth reduction and sample rate reduction are merged in a single process called multirate filter. The bandwidth reduction performed by the digital filter can be a low-pass or a band-pass process.

In a dual way, the process of up sampling can be visualized as a two-step process as indicated in Figure 7.24. Here too there are three distinct time series. The process starts by increasing the sample rate of an input series by resampling it 1-to-M. The zero-packed time series with M-fold replication of the input spectrum is processed by a filter to reject the spectral replicas and output the sequence with the same spectrum as the input sequence but sampled at M times higher sample rate. In reality the processes of sample rate increase and selected bandwidth rejection are also merged in a single process again called multirate filtering.

The presented structures are the most basic forms of multirate filters; in the following we present more complicated multirate architecture models. In particular we focus on the polyphase decomposition of a prototype filter which is very efficient when embedded in multirate structures. We also propose the derivation, step by step, of both the polyphase down converter and up converter channelizers. These are the two standard engines that we will modify for designing a novel polyphase channelizer that better fits the needs of the future software defined radio.

1.07.6 From single channel down converter to standard down converter channelizer

The derivation of the standard polyphase channelizer begins with the issue of down converting a single frequency band, or channel, located in a multi channel frequency division multiplexed (FDM) input signal whose spectrum is composed of a set of M equally spaced, equal bandwidth channels, as shown in Figure 7.25. Note that this signal has been band limited by analog filters and has been sampled at a sufficiently high sample rate to satisfy the Nyquist criterion for the full FDM bandwidth.

We have many options available for down converting a single channel; The standard processing chain for accomplishing this task is shown in Figure 7.26. This structure performs the standard operations of down converting a selected frequency band with a complex heterodyne, low pass filtering to reduce the output signal bandwidth to the channel bandwidth, and down sampling to a reduced rate commensurate with the reduced bandwidth. The structure of this processor is seen to be a digital signal processor implementation of a prototype analog I-Q down converter. We mention that the down sampler is commonly referred to as a decimator, a term that means to destroy one sample every tenth. Since nothing is destroyed and nothing happens in tenths, we prefer, and will continue to use, the more descriptive name, down sampler.

The output data from the complex mixer is complex, hence it is represented by two time series, and . The filter with real impulse response is implemented as two identical filters, each processing one of the quadrature time series. The convolution process between a signal and a filter is often performed by simply multiply and sum operations between signal data samples and filter coefficients extracted from two sets of addressed memory registers. In this form of the filter, one register set contains the data samples while the other contains the coefficients that define the filter impulse response. By using the equivalency theorem, which states that the operations of down conversion followed by a low-pass filter are equivalent to the operations of band-pass filtering followed by a down conversion, we can exchange the positions of the filter and of the complex heterodyne achieving the block diagram shown in Figure 7.27.

Note here that the up converted filter, , is complex and as such its spectrum resides only on the positive frequency axis without a negative frequency image. This is not a common structure for an analog prototype because of the difficulty of forming a pair of analog quadrature filters exhibiting a 90° phase difference across the filter bandwidth. The closest equivalent structure in the analog world is the filter pair used in image-reject mixers and even there, the phase relationship is maintained by a pair of complex heterodynes.

Applying the transformation suggested by the equivalency theorem to an analog prototype system does not make sense since it doubles the required hardware. We would have to replace a complex scalar heterodyne (two mixers) and a pair of low-pass filters with a pair of band-pass filters, containing twice the number of reactive components, and a full complex heterodyne (four mixers). If it makes no sense to use this relationship in the analog domain, why does it make sense in the digital world? The answer is found in the fact that we define a digital filter as a set of weights stored in the coefficient memory. Thus, in the digital world, we incur no cost in replacing the pair of low-pass filters required in the first option with the pair of band-pass filters and required for the second one. We accomplish this task by a simple download to the coefficient memory.

An interesting historical perspective is worth noting here. In the early days of wireless, radio tuning was accomplished by sliding a narrow band filter to the center frequency of the desired channel to be extracted from the FDM input signal. These radios were known as tuned radio frequency (TRF) receivers. The numerous knobs on the face of early radios adjusted reactive components of the amplifier chain to accomplish the tuning process. Besides the difficulty in aligning multiple tuned stages, shifting the center frequency of the amplifier chain often initiated undesired oscillation due to the parasitic coupling between the components in the radio. Edwin Howard Armstrong, an early radio pioneer, suggested moving the selected channel to the fixed frequency filter rather than moving the filter to the selected channel. This is known as the superheterodyne principle, a process invented by Armstrong in 1918 and quickly adopted by David Sarnoff of the Radio Corporation of America (RCA) in 1924. Acquiring exclusive rights to Armstrong’s single-tuning dial radio invention assured the commercial dominance of RCA in radio broadcasting as well the demise of hundreds of manufacturers of TRF radio receivers. It seems we have come full circle. We inherited from Armstrong a directive to move the desired spectrum to the filter and we have readily applied this legacy to DSP-based processing. We are now proposing that, under appropriate conditions, it makes more sense to move the filter to the selected spectral region. We still have to justify the full complex heterodyne required for the down conversion at the filter output rather than at the filter input, which is done in the subsequent paragraphs.

Examining Figure 7.27, we note that following the output down conversion, we perform a sample rate reduction in which we retain one sample out of M-samples. Because we find it useless to down convert the samples we discard in the next down sample operation, we move the down sampler before the down converter achieving the demodulator scheme shown Figure 7.28. We note in this figure that also the time series of the complex sinusoid has been down sampled. The rotation rate of the sampled complex sinusoid is and radians per sample at the input and output of the M-to-1 resampler respectively. This change in observed rotation rate is due to aliasing. When aliased, a sinusoid at one frequency or phase slope appears at another phase slope due to the resampling.

We now invoke a constraint on the sampled data center frequency of the down converted channel. We choose center frequencies , which will alias to DC as a result of down sampling to . This condition is assured if is congruent to , which occurs when , or more specifically, when . The modification to Figure 7.28 to reflect this provision is seen in Figure 7.29. The constraint, that the center frequencies be limited to integer multiples of the output sample rate, assures aliasing to base-band by the sample rate change. When a channel aliases to base-band because the resampling operation the corresponding resampled heterodyne defaults to a unity-valued scalar, which consequently is removed from the signal processing path.

Note that if the center frequency of the aliased signal is offset by from a multiple of the output sample rate, the aliased signal will reside at an offset of from zero frequency at base-band and a complex heterodyne, or base-band converter, is needed to shift the signal by the residual offset. This base-band mixer operates at the output sample rate rather than at the input sample rate for a conventional down converter. We can consider this required final mixing operation as a post conversion task and allocate it to the next processing block.

The spectral effect of the signal processing performed by the structure in Figure 7.29 is shown in Figure 7.30. The savings realized by this form of down conversion is due to the fact that we no longer require a quadrature oscillator or the pair of input mixers to effect the required frequency translation.

Applying again the idea that it is useless to filter those samples that will be lately discarded by the down sampler we apply the noble identity to exchange the operations of filtering and down sampling. The noble identity states that a filter processing every Mth input sample followed by an output M-to-1 down sampler is the same as an input M-to-1 down sampler followed by a filter processing every input sample. Its interpretation is that the M-to-l down sampled time series from a filter processing every input sample presents the same output by first down sampling the input by M-to-1, to discard the samples not used by the filter when computing the retained output samples, and then operating the filter on only the retained input samples. The noble identity works because samples of delay at the input clock rate is the same interval as one-sample delay at the output clock rate.

At this point, with only a filter followed by a resampler, we can apply the polyphase decomposition to the filter and separate it into M-parallel paths for achieving the standard M-path polyphase down converter channelizer. In order to make the understanding process easier for the reader, we first perform the polyphase decomposition of a low-pass prototype filter and then we extend the results to the band-pass filter case. Equation (7.19) represents the z transform of a digital low-pass prototype filter

(7.19)

In order to achieve the polyphase partition of , we rewrite the sum in Eq. (7.19), as shown in Eq. (7.20), partitioning the one-dimensional array of weights in a two-dimensional array. In this mapping we load an array by columns but process it by rows. The partition forms columns of length M containing M successive terms in the original array, and continues to form adjacent M-length columns until we account for all the elements of the original one-dimensional array

(7.20)

We note that the first row of the two-dimensional array is a polynomial in , which we will denote a notation to be interpreted as an addressing scheme to start at index 0 and increment in strides of length M. The second row of the same array, while not a polynomial in , is made into one by factoring the common term and then identifying this row as . It is easy to see that each row of the two-dimensional array described by Eq. (7.8) can be written as achieving the more compact form as shown in Eq. (7.9) where r is the row index which is coincident with the path index

(7.21)

The block diagram depicting the M-path polyphase decomposition of the resampled low-pass filter of Eq. (7.21) is depicted in Figure 7.31.

At this point, in the structure depicted in Figure 7.31, we can pull the resampler on the left side of the adder and down sample the separate filter outputs performing the sum only for the retained filter output samples. With the resamplers at the output of each filter we can invoke again the noble identity and place them at the input of each filter.

The resamplers operate synchronously, all closing at the same clock cycle. When the switches are closed, the signal delivered to the filter on the top path is the current input sample. The signal delivered to the filter one path down is the content of the one stage delay line, which, of course, is the previous input sample. Similarly, as we traverse the successive paths of the M-path partition, we find upon switch closure, that the path receives a data sample delivered k samples ago. We conclude that the interaction of the delay lines in each path with the set of synchronous switches can be likened to an input commutator that delivers successive samples to successive arms of the M-path filter. This interpretation is shown in Figure 7.32 which depicts the final structure of the polyphase decomposition of a low-pass prototype filter.

At this point we can easily achieve the band-pass polyphase partition from the low-pass partition by applying the frequency translation property of the z-Transform that states the following.

If

and

then

By using this property and replacing in Eq. (7.21) each with , where , we achieve

(7.22)

The complex scalars attached to each path of the M-path filter can be placed anywhere along the path. We choose to place them after the down sampled path filter segments . This change is shown in Figure 7.33 that depicts the standard polyphase filter bank used for down converting a single channel. The computation of the time series obtained from the output summation in Figure 7.33 is shown in Eq. (7.23). Here, the argument reflects the down sampling operation, which increments through the time index in strides of length M, delivering every Mth sample of the original output series. The variable is the nMth sample from the filter segment in the rth path, and is the nMth time sample of the time series from the center frequency. Remember that the down converted center frequencies located at integer multiples of the output sample frequency alias to zero frequency after the resampling operation. Note that the output is computed as a phase coherent summation of the output series . This phase coherent sum is, in fact, an inverse discrete Fourier transform of the M-path outputs, which can be likened to beam forming the output of the path filters

(7.23)

The beam forming perspective offers an interesting insight to the operation of the resampled down converter system we have just examined; the reasoning proceeds as follows: the commutator delivering consecutive samples to the M input ports of the M-path filter performs a down sampling operation. Each port of the M-path filter receives data at 1/Mth of the input rate. The down sampling causes the M-to-1 spectral folding, effectively translating the multiples of the output sample rate to base-band. The alias terms in each path of the M-path filter exhibit unique phase profiles due to their distinct center frequencies and the time offsets of the different down sampled time series delivered to each port. These time offsets are, in fact, the input delays shown in Figure 7.31 and Eq. (7.3). Each of the aliased center frequency experiences a phase shift shown in Eq. (7.24) equal to the product of its center frequency and the path time delay.

(7.24)

The phase shifters of the IDFT perform phase coherent summations, very much like that performed in narrowband beam forming, extracting from the myriad of aliased time series, the alias with the particular matching phase profile. This phase sensitive summation aligns contributions from the desired alias to realize the processing gain of the coherent sum while the remaining alias terms, which exhibit rotation rates corresponding to the M roots of unity, are destructively canceled in the summation. The inputs to the M-path filter are not narrowband, and phase shift alone is insufficient to cause the destructive cancelation over the full bandwidth of the undesired spectral contributions. Continuing with our beam-forming perspective, to successfully separate signals with unique phase profiles due to the input commutator delays, we must perform the equivalent of time-delay beam forming. The M-path filters, obtained by M-to-1 down sampling of the prototype low-pass filter supply the required time delays. The M-path filters are approximations to all-pass filters, exhibiting, over the channel bandwidth, equal ripple approximation to unity gain and the set of linear phase shifts that provide the time delays required for the time-delay beam-forming task. The filter achieves this property by virtue of the way it is partitioned. Each of the M-path filters, , for instance, with weights , is formed by starting with an initial offset of r samples and then incrementing in strides of M samples. The initial offsets, unique to each path, are the source of the different linear phase-shift profiles. It is because of the different linear phase profiles, that the filter partition is known as polyphase filter. An useful perspective is that the phase rotators following the filters perform phase alignment of the band center for each aliased spectral band while the polyphase filters perform the required differential phase shift across these same channel bandwidths.

When the polyphase filter is used to down convert and down sample a single channel, the phase rotators are implemented as external complex products following each path filter. When a small number of channels are being down converted and down sampled, appropriate sets of phase rotators can be applied to the filter stage outputs and summed to form each channel output. Therefore the most advantageous applications of that structure are those in which the number of channels to be down converted becomes sufficiently large. Sufficiently large means on the order of . Since the phase rotators following the polyphase filter stages are the same as the phase rotators of an IDFT, we can use the IDFT to simultaneously apply the phase shifters for all the channels we wish to extract from the aliased signal set. This is reminiscent of phased-array beam forming. For computational efficiency, the IFFT algorithm implements the IDFT.

The complete structure of a standard M-path polyphase down converter channelizer is shown in Figure 7.34.

To summarize: in this structure the commutator performs an input sample rate reduction by commutating successive input samples to selected paths of the M-path filter. Sample rate reduction occurring prior to any signal processing causes spectral regions residing at multiples of the output sample rate to alias to base-band. The partitioned M-path filter performs the task of aligning the time origins of the offset sampled data sequences delivered by the input commutator to a single common output time origin. This is accomplished by the all-pass characteristics of the -path filter sections that apply the required differential time delay to the individual input time series. The IDFT performs the equivalent of a beam-forming operation; the coherent summation of the time-aligned signals at each output port with selected phase profiles. The phase coherent summation of the outputs of the -path filters separates the various aliases residing in each path by constructively summing the selected aliased frequency components located in each path, while simultaneously destructively canceling the remaining aliased spectral components.

We conclude the reasoning on the standard M-path down converter channelizer by stating that it simultaneously performs the following three basic operations: sample rate reduction, due to the input commutator; bandwidth reduction, due to the M-path partitioned filter weights and Nyquist zone selection, due to the IFFT block.

In the following, in order to facilitate the understanding process for the reader, we show some figures, which are results of Matlab simulation, to support the theoretical reasoning of the previous paragraphs. The example concerns a 6-path down converter channelizer that performs 6-to-1 down sampling. In particular Figure 7.35 shows the impulse response and the frequency response of the low-pass prototype filter used for the polyphase partition.

Figure 7.36 shows the impulse responses of the zero-packed filter on each arm of the 6-path polyphase partition while Figure 7.37 shows the spectra corresponding to the filters of Figure 7.36. It is evident from this figure that the zero packing process has the effect of producing spectral copies of the filter frequency response.

In Figure 7.38 we show the phase of each spectral copy of the zero packed prototype filter on each arm of the partition. In this figure the phases of the spectral copies of each arm are overlaid and each line of the plot corresponds to a specific arm in the partition.

In Figure 7.39 the same phase profiles of Figure 7.38 are de-trended, zoomed, and made causal.

In Figure 7.40 the 3-D view of the filter spectral copies, with their phase profiles, is shown for each path, while, in Figure 7.41, the final result of the channelization process is shown, again in a 3-D fashion.

1.07.7 Modifications of the standard down converter channelizer—M:2 down converter channelizer

In the standard polyphase down converter channelizer shown in Figure 7.34 the channel bandwidth , the channel spacing , and the sampling frequency are fixed to be equal.

This configuration could represent a good choice when the polyphase channelizer is used for communication systems. In these kind of applications, in fact, an output sample rate matching the channel spacing is sufficient to avoid adjacent channel cross talk since the two-sided bandwidth of each channel is less than the channel spacing. An example of a signal that would require this mode of operation is the Quadrature Amplitude Modulation (QAM) channels of a digital cable system. In North America, the channels are separated by 6 MHz centers and operate with square-root cosine tapered Nyquist-shaped spectra with 18% or 12% excess bandwidth, at symbol rates of approximately 5.0 MHz. The minimum sample rate required of a cable channelizer to satisfy the Nyquist criterion would be 6.0 MHz (The European cable plants have channel spacing of 8.0 MHz and symbol rates of 7.0 MHz). The actual sample rate would likely be selected as a multiple of the symbol rate rather than as a multiple of the channel spacing. Systems that channelize and form samples of the Nyquist-shaped spectrum often present the sampled data to an interpolator to resample the time series collected at bandwidth-related Nyquist rate to a rate offering two samples per symbol or twice symbol rate. For the TV example just cited, the 6 Ms/s, 5 MHz symbol signal would have to be resampled by 5/3 to obtain the desired 10 Ms/s. This task is done quite regularly in the communication receivers and it may represent a significant computational burden. It would be appropriate if we could avoid the external interpolation process by modifying the design of the standard polyphase channelizer for directly providing us the appropriate output sampling frequency.

Many others applications desire channelizers in which the output sampling frequency is not equal to the channel spacing and the channel bandwidth. The design of software defined radio receiver and transmitter is only one of them. Another one can be found in [8].

We have concluded the previous section mentioning that the channel spacing, the channel bandwidth and the output sampling frequency of the polyphase channelizer are completely independent of each other and that they can be arbitrary selected based on the application. Figure 7.46 shows some possible options in which the channel bandwidth, the output sampling rate and the channel spacing of the channelizer are not equal. In this chapter, for the purpose of designing flexible radio architectures, the third option, when the selected low-pass prototype filter is a Nyquist filter (perfect reconstruction filter), results to be the most interesting one.

In this section of the chapter we derive the reconfigured version of the standard down converter channelizer that is able to perform the sample rate change from the input rate to the output sampling rate maintaining both the channel spacing and the channel bandwidth equal to . Similar reasoning can be applied for implementing different selections.

Note that the standard down converter channelizer is critically sampled when the channel spacing and the channel bandwidth are both equal to and when the output sample rate is also equal to . This particular choice, in fact, causes the transition band edges of the channelizer filters to alias onto itself which would prevent us from further processing the signals when they are arbitrarily located in the frequency domain which is the case of software radio. This problem can be visualized in the upper example depicted in Figure 7.46.

For the record, we remind that a polyphase filter bank can be operated with an output sample rate which can be any rational ratio of the input sample rate. With minor modifications the filter can be operated with totally arbitrary ratios between input and output sample rates. This is true for the sample rate reduction imbedded in a polyphase receiver as well as for the sample rate increase embedded in a polyphase modulator.

We have control on the output sampling rate of the down converter channelizer by means of the input commutator that delivers input data samples to the polyphase stages. We normally deliver M successive input samples to the M-path filter starting at port M-1 and progressing up the stack to port 0 and by doing so we deliver M inputs per output which means to perform an M-to-1 down sampling operation. To obtain the desired (M/2)-to-1 down sampling, we have to modify the input commutator in a way that it delivers M/2 successive input samples starting at port (M/2)-1 and progressing up the stack to port 0. We develop and illustrate the modifications with the aid of Figures 7.47–7.50.

Figure 7.47 represents the polyphase M-path filter partition shown in Eq. (7.9) with an M/2-to-1 rather than the conventional M-to-1 down sample operation after the output summing junction. We need it in order to perform the desired sampling rate change.

In Figure 7.48 we apply the noble identity to the polyphase paths by pulling the M/2-to-1 down sampler through the path filters which converts the polynomials in operating at the high input rate to polynomials in operating at the lower output rate.

Figure 7.49 shows the second application of the noble identity in which we again take the M/2-to-1 down samplers through the parts of the input path delays for the paths in the second or bottom half of the path set.

In Figure 7.50 the M/2-to-1 down samplers switch and their delays are replaced with a two pronged commutator that delivers the same sample values to path inputs with the same path delay. Here we also merged the delays in the lower half of filter bank with their path filters.

Figure 7.51 shows and compares the block diagrams of the path filters in the upper and lower half of this modified polyphase partition.

When the input commutator is so designed, the M/2 addresses to which the new M/2 input samples are delivered have to be first vacated by their former contents, the M/2 previous input samples. All the samples in the two-dimensional filter undergo a serpentine shift of M/2 samples with the M/2 samples in the bottom half of the first column sliding into the M/2 top addresses of the second column while the M/2 samples in the top half of the second column slide into the M/2 addresses in the bottom half of the second column, and so on. This is equivalent to performing a linear shift through the prototype one-dimensional filter prior to the polyphase partition. In reality, we do not perform the serpentine shift but rather perform an addressing manipulation that swaps two memory banks. This is shown in Figure 7.52.

After each M/2-point data sequence is delivered to the partitioned M-stage polyphase filter, in the standard channelizer configuration, the outputs of the M stages are computed and conditioned for delivery to the M-point IFFT. What we need to do at this point is the time alignment of the shifting time origin of the input samples in the M-path filter with the stationary time origin of the phase rotator outputs of the IFFT. We can understand the problem by visualizing, in Figure 7.53, a single cycle of a sine wave extending over M samples being inserted in the input data register, the first column of the polyphase filter in segments of length M/2. We can assume that the data in the first M/2 addresses are phase aligned with the first M/2 samples of a single cycle of the sine wave offered by the IFFT.

When the second M/2 input samples are delivered to the input data register the first M/2 input samples shift to the second half of the M-length array. Its original starting point is now at address M/2 but the IFFT’s origin still resides at address 0. The shift of the origin causes the input sine wave in the register to have the opposing phase of the sine wave formed by the IFFT; in fact the data shifting into the polyphase filter stages causes a frequency dependent phase shift of the form shown in Eq. (7.30). The time delay due to shifting is nT where n is the number of samples, and T is the time interval between samples. The frequencies of interest are integer multiple k of 1/Mth of the sample rate . Substituting these terms in Eq. (7.30) and canceling terms, we obtain the frequency dependent phase shift shown in Eq. (7.31). From this relationship we see that for time shifts n equal to multiples of M, as demonstrated in Eq. (7.32), the phase shift is a multiple of and contributes zero offset to the spectra observed at the output of the IFFT. The M-sample time shift is the time shift applied to the data in the normal use of the polyphase filter. Now suppose that the time shift is M/2 time samples. When substituted in Eq. (7.31) we find, as shown in Eq. (7.33), as frequency dependent phase shift of from which we conclude that odd-indexed frequency terms experience a phase shift of radians for each successive N/2 shift of input data

(7.30)

(7.31)

(7.32)

(7.33)

This radiants phase shift is due to the fact that the odd-indexed frequencies alias to the half sample rate when the input signal is down sampled by M/2. What we are observing is the sinusoids with an odd number of cycles in the length M array alias to the half sample rate when down sampled M/2-to-1. Note that, when down sampled M/2-to-1, the sinusoids with an even number of cycles in the length M array alias to DC. We can compensate for the alternating signs in successive output samples by applying the appropriate phase correction to the spectral data as we extract successive time samples from the odd-indexed frequency bins of the IFFT. The phase correction here is trivial, but for other down-sampling ratios, the residual phase correction would require a complex multiply at each transform output port. Alternatively, since time delay imposes a frequency dependent phase shift, we can use time shifts to cancel the frequency dependent phase shifts. We accomplish this by applying a circular time shift of samples to the vector of samples prior to their presentation to the IFFT. As in the case of the serpentine shift of the input data, the circular shift of the polyphase filter output data is implemented as an address-manipulated data swap. This data swap occurs on alternate input cycles and a simple two-state machine determines for which input cycle the output data swap is applied. This is shown in Figure 7.54.

The complete structure of the modified version of the M-to-2 down converter channelizer with the input data buffer and the circular data buffer is shown in Figure 7.55.

For brevity of notation, we avoid reporting here all the dual mathematical derivations that led at the final structure of the modified up converter channelizer. We briefly explain its block diagram in the next subsection.

1.07.8 Preliminaries on software defined radios

The 20th century saw the explosion of hardware defined radio as a means of communicating all forms of data, audible and visual information over vast distances. These radios have little or no software control. Their structures are fixed in accordance with the applications; the signal modulation formats, the carrier frequencies and bandwidths are only some of the factors that dictate the radio structures. The smallest change to one of these parameters could imply a replacement of the entire radio system. A consequence of this is, for example, the fact that a television receiver purchased in France does not work in England. The reason, of course, is that the different geographical regions employ different modulation standards for the analog TV as well as for digital TV. Then, the citizens cannot use the same TV for receiving signals in both countries; they need to buy a new television for each country in which they decide to live. Sometimes, even if the communication devices are designed for the same application purposes and they work in the same geographical area, they are not able to communicate between each other. One of the most evident examples of this is that the city police car radio cannot communicate with the city fire truck radio, or with the local hospital ambulance radio even if they have the common purpose of helping and supporting the citizen. Also, the city fire truck radio cannot communicate with the county fire truck radio, or with the radios of the fire truck operated by the adjacent city, or by the state park service, or the international airport. None of these services can communicate with the National Guard, or with the local Port Authority, or with the local Navy base, or the local Coast Guard base, or the US Border Patrol, or US Customs Service. In an hardware defined radio, if we decide to change one of the parameters of the transmitted signal, like bandwidth or carrier frequency (for example because the carrier frequency we want to use is the only one available at that particular moment), we need to change the transmitter. On the other side, every time we want to receive a signal having different bandwidth or center frequency, we need to change the receiver. Hardware defined transmitter and receiver devices are not flexible at all; we must modify their structure every time we change even one of the transmitting and receiving signal parameters.

In 1991, Joe Mitola coined the term software defined radio. It was referred to a class of reprogrammable (and reconfigurable) devices. At that time it was not clear at which level the digitization should occur to define a radio as software but the concept sounded pretty interesting, and the dream of building a completely reconfigurable radio device involved scientists from all over the world. Today the exact definition of software defined radio is still controversial, and no consensus exists about the level of reconfigurability needed to qualify a radio as software.

Figure 7.64 shows, in a simple block diagram, all the possible places in which the digitization can occur in a radio receiver. The exact dual block diagram can be portrayed for the radio transmitter. Current radios, often referred to as digital but sometimes referred as software defined radios (depending on their particular structure), after shifting the signals to intermediate frequency, digitize them and assign all the remaining tasks to a digital signal processor. One of the main reasons for shifting the signals to intermediate frequency, before digitizing them, is to reduce their maximum frequency so that a smaller number of samples can be taken for preserving the information content.

Implementation of ideal software radios requires digitization at the antenna, allowing complete flexibility in the digital domain. Then it requires both the design of a flexible and efficient DSP-based structure and the design of a completely flexible radio frequency front-end for handling a wide range of carrier frequencies, bandwidths and modulation formats. These issues have not been exploited yet in the commercial systems due to technology limitations and cost considerations.

As pointed out in the previous section, in a current software defined radio receiver the signals are digitized in intermediate frequency bands. The receiver employs a super heterodyne frequency down conversion, in which the radio frequency signals are picked up by the antenna along with other spurious, unwanted signals (noise and interferences), filtered, amplified with a low noise amplifier and mixed with a local oscillator to shift it to intermediate frequency. Depending on the application, the number of stages of this operation may vary. Digitizing the signal in the IF range eliminates the last analog stage in the conventional hardware defined radios in which problems like carrier offset and imaging are encountered. When sampled, digital IF signals give spectral replicas that can be placed accurately near the base-band frequency, allowing frequency translation and digitization to be carried out simultaneously. Digital filtering and sample rate conversion are often needed to interface the output of the ADC to the processing hardware to implement the receiver. Likewise, on the transmitter side, digital filtering and sample rate conversion are often necessary to interface the digital hardware, that creates the modulated waveforms, to the digital to analog converter. Digital signal processing is usually performed in radio devices using field programmable gate arrays (FPGAs), or application specific integrated circuits (ASICs).

Even if the dream of building a universal radio is still far away, current software defined radio architectures are quite flexible, in the sense that they usually down convert to IF a collection of signals and, after sampling, they are able to shift these signals to base-band via software. Changes in the signal bandwidths and center frequencies are performed by changing some parameters of the digital data section. The flexibility of such a structure can be improved by moving the analog to digital converter closest to the receiver antenna. By digitizing the signals immediately after (and before in the transmitter) the antenna, which is the ideal software radio case, the down conversion (and up conversion) processes are performed completely in software and the radio acquires the capability of changing its personality, possibly in real-time, guaranteeing a desired quality of service (QoS). The digitization after the receiver antenna, in fact, allows service providers to upgrade the infrastructure and market new services quickly. It promises multi-functionality, global mobility, ease of manufacture, compactness and power efficiency. The flexibility in hardware architectures combined with flexibility in software architectures, through the implementation of techniques such as object oriented programming, can provide software radio also with the ability to seamlessly integrate itself into multiple networks with widely different air and data interfaces.

1.07.9 Proposed architectures for software radios

A digital transmitter accepts binary input sequences and outputs radio frequency amplitude and phase modulated wave shapes. The digital signal processing part of this process starts by accepting b-bit words from a binary source at input symbol rate. These words address a look-up table that outputs gray coded ordered pairs, i.e., I-Q constellation points, that control the amplitude and phase of the modulated carrier. The I-Q pair is input to DSP-based shaping filters that form 1-to-4 up sampled time series designed to control the wave shape and limit the base-band modulation bandwidth. The time series from the shaping filter are further up sampled by a pair of interpolating filters to obtain a wider spectral interval between spectral replicas of the sampled data. The interpolated data are then heterodyned by a digital up converter to a convenient digital intermediate frequency and then moved from the sampled data domain to the continuous analog domain by a digital to analog converter and analog IF filter. Further analog processing performs the spectral transformations required to couple the wave shape to the channel.

On the other side of the communication chain, a digital receiver has to perform the tasks of filtering, spectral translation and analog to digital conversion to reverse the dual tasks performed at the transmitter. The receiver must also perform a number of other tasks absent in the transmitter for estimating the unknown parameters of the received signal such as amplitude, frequency and timing alignment. It samples the output of the analog IF filter and down converts the intermediate frequency centered signal to base-band with a digital down converter (DDC). The base-band signal is down sampled by a decimating filter and finally processed in the matched filter to maximize the signal-to-noise ratio (SNR) of the samples presented to the detector. The DSP portion of this receiver includes carrier alignment, timing recovery, channel equalization, automatic gain control, SNR estimation, signal detection, and interference suppression blocks. In order to suppress the undesired artifacts introduced by the analog components, the receiver also incorporates a number of digital signal processing compensating blocks.

Figure 7.65 shows the first tier processing block diagrams of a typical digital transmitter-receiver chain. The depicted transmitter is designed for sending, one per time, signals having fixed bandwidths at precisely located center frequencies. On the other hand, the depicted receiver is also designed for down converting to base-band fixed bandwidth signals having fixed center frequencies. Therefore, the goal of a cognitive radio is quite different: cognitive transmitter and receiver should be completely flexible. The transmitter has, in fact, to be able to simultaneously up convert, at desired center frequencies, which are selected based on the temporary availability of the radio spectrum, a collection of signals having arbitrary bandwidths. It should also be able to decompose a signal in spectral fragments when a sufficiently wide portion of the radio spectrum is not available for transmission. Of course no energy loss has to occur in the partitioning process. A cognitive receiver has to be able to simultaneously down convert the transmitted signal spectra wherever they are positioned in the frequency domain and whatever bandwidths they have. Also it should have the capability of recognizing spectral segments belonging to the same information signal and recompose them, after the base-band translation, without energy losses. It could be possible to reach this goal by using the current technology but the cost would be very high: for every signal we want to transmit we have to replicate the digital data section of both the transmitter and the receiver. The more signals we have to simultaneously up and down convert, the more sampled data sections we need to implement! Also, it is not possible, by using the current technology, to fragment a single signal, transmit its spectral partitions using the temporary available spectral holes and to perfectly recombine them at the receiver. Today the transmission happens under the constraint that the available space in the radio spectrum is large enough to accommodate the entire bandwidth of the signal to be transmitted.

In the next sections of this chapter, we present novel channelizers to simultaneously up and down convert arbitrary bandwidth signals randomly located in the frequency domain. Differently from the previous contents of this document, from this point onwards the presented material reproduces the results of the authors’ research on software defined radio design, thus it is new. The proposed architectures can be used for the transmitter and for the receiver devices of a cognitive radio respectively. By using them we avoid the need to replicate the sampled data sections for simultaneously transmitting and receiving more signals with arbitrary bandwidths over multiple desired center frequencies. We also gain the capability to partition the signal spectra before transmitting them and to perfectly reassemble them in the receiver after the base-band shifting.

The core of the proposed transmitter structure is the variant of the standard M-path polyphase up converter channelizer that is able to perform 2-to-M up sampling while shifting, by aliasing, all the channels to desired center frequencies. It has been presented in Section 1.07.5 of this same chapter. In the following we use the term synthesis channelizer for referring to this structure. This name has been given for the tasks that this structure accomplishes when embedded in a SDR transmitter. When the input signals have bandwidths wider than the synthesis channelizer bandwidth they are pre-processed through small down converter channelizers that disassemble the signals’ bandwidths into reduced bandwidth sub-channels which are matched to the base-line synthesis channelizer bandwidths. Complex sampled frequency rotators are used to offset the signal spectra by arbitrary fractions of the sample frequency before processing them through the channelizers. This is necessary for the completely arbitrary center frequency positioning of the signals.

On the other side the variation of the standard M-path down converter channelizer, presented in Section 1.07.6, represents the key element of the proposed receiver. It is able to perform M-to-2 down sampling while simultaneously down converting, by aliasing, all the received signal spectra having arbitrary bandwidths. In the following, we refer to this structure as analysis channelizer. Post-processing channelizers, that are a smaller version of the synthesis channelizer composing the transmitter, are used for reassembling, from the analysis channelizer base-line channels, signal bandwidths wider than the analysis channelizer channel bandwidth. Possible residual frequency offsets can be easily solved with the aid of a digital complex heterodyne while a post-analysis block performs further channelization, when it is required, for separating signal spectra falling in the same channelizer channel after the analysis processing.

1.07.10 Closing comments

This chapter had the intention of providing the basic concepts on multirate signal processing and polyphase filter banks to which, when an IDFT block and a commutator are applied, are known as polyphase channelizer because of the processing task that they perform on the input signals. This chapter also had the intention of presenting innovative results on software and cognitive radio designs. While the polyphase filter banks and polyphase channelizers are well known topics in the DSP area, the software radio design is still an open research topic.

The document started with preliminaries, in Section 1.07.2, on the resampling process of a digital signal as opposed to the sampling process of an analog signal. In Section 1.07.3 an introduction on digital filters has been provided and the differences with the analog filters have been explained. In Section 1.07.4 an introduction on the window method for digital filter design has been provided. Multirate filters have been defined and explained in Section 1.07.5 while the polyphase decomposition of a prototype filter has been introduced in Section 1.07.6 along with the standard up sampler and down sampler polyphase channelizers. In Section 1.07.7 the modifications of the standard polyphase channelizer that allow us to change the output sampling rate have been presented. These engines are the basic components of the proposed architectures (presented in Sections 1.07.8 and 1.07.9): the synthesis and analysis channelizers, that are suitable for being used as software defined transmitter and receiver respectively.

The synthesis channelizer, in fact, when supported by small analysis channelizers, is able to simultaneously up convert multiple signals having arbitrary bandwidths to randomly located center frequencies. In this engine small analysis channelizers pre-process wider bandwidth signals into segments acceptable to the following synthesis channelizer that up samples, translates to the proper center frequency, and reassembles them. A channel selector block, connected with a channel configuration block, is used to properly deliver the analysis channelizer outputs to the synthesis channelizer.

On the other side of the communication chain, the analysis channelizer, that is thought for being embedded in a SDR receiver, when supported by small synthesis channelizers, is able to simultaneously demodulate these signals. A channel configuration block, inserted in the receiver, communicates with a selector block that properly delivers the analysis channelizer outputs to the synthesizer up converters that follow it. These synthesizers up sample, translate and reassemble the spectral fragments when they belong to the same source signal. Nyquist prototype low-pass filters, which are perfect reconstruction filters, are used to allow the reconstruction of the signal fragments without energy losses.

Complex frequency rotators are used for compensating residual frequency offsets and arbitrary interpolators provide us exactly two samples per symbol required for the following processing tasks of the receiver.

Theoretical reasoning and simulation results are presented, for both the receiver and the transmitter, in order to demonstrate their correct functionality.

Note that because in both of the proposed DUC and DDC structures, the IFFT sizes of the small synthesis and analysis channelizers are chosen in order to give us at least two samples per symbol for every processed bandwidth, they result to be very efficient, from a computational point of view, when the received signal is composed of spectra with widely varying bandwidths.

Slightly different versions of channelizers can be used based on different input signals and/or for adding functionalities to the proposed architectures. Channelizers are, in fact, highly efficient and very flexible structures. Among their most common applications we find spectral analysis, modem synchronization (phase, frequency and time) and channel equalization.

Glossary

Filter calculation procedure that transforms the input signals into others

Digital filter filter that operates on digital signals

Multirate filter digital filter that contains a mechanism to increase or decrease the sampling rate while processing input signals

Polyphase filter digital filter partitioned by following Eq. (7.3)

Polyphase channelizer flexible digital device which can arbitrarily change the sample rate and the bandwidth of the input signal and can also select randomly located Nyquist zones

Software radio radio device in which the digitization of the signal occurs before the intermediate frequency translation

Cognitive radio software radio with the capability to adapt its operating parameters according to the interactions with the surrounding radio environment

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