5.1.1 Digital waveform generation using wave tables

One approach to generate such periodic waveforms is to design a digital filter with an impulse response h(n) corresponding to one period of the waveform, one wishes to generate. The periodicity is generated by exciting this digital filter with a train of impulses separated by the fundamental period of the waveform. This process is shown in Figure 5.2.

A more efficient approach is to pre-compute the samples of the waveform and store them in the system’s memory (RAM or ROM). The data are arranged as a circular buffer and accessed when needed. The period of the waveform can be controlled by either varying the speed of cycling around the table or by accessing a subset of the table at a fixed speed. This approach is called wave table synthesis, which is used very successfully in computer music. Many audio cards available for PCs now use wave table generators.

5.1.2 Sampling frequency

A sinusoidal signal does not necessarily remain periodic when sampled at a given frequency. In order that y(n) remains periodic in the time index n with a period of, say, D samples, it is necessary that one whole period of the sinusoid fit within the D samples. This requires that the sampling frequency is an integral multiple of the analog frequency f. That is,

Due to the periodicity, only the samples for one period of the signal need be calculated and stored. f is now the fundamental frequency of the wave table. A typical sampling frequency for DTMF generation is 8 kHz.

There are sinusoids of 8 different frequencies that need to be generated for DTMF signals. One way to do it is to have 8 wave tables. Since there are only a few entries per table, this approach is not impractical. Another way is to change the fundamental frequency using a single table. This is the approach that we shall examine in more detail.

5.1.3 Generating integer multiples of the fundamental frequency

The frequency f can be changed either by changing the sampling frequency f_s or by changing the effective length D of the basic period. The first approach is not practical in our application since we need to deal with two different frequencies each time. Thus the second approach will be used.

The fundamental frequency generated from a wave table with D entries and a sampling frequency is given by

Replacing D by a smaller value, d will increase the frequency of the sinusoid. For instance, if d=D/2, the frequency is doubled. This also means that only every other entry in the wave table will be accessed for each period. So the new generated frequency will be

Given the desired frequency f_d and a table length D, we will be using only c regularly spaced samples from the full wave table and c is given by

Now we have assumed that d and c are integers. This clearly restricts the choices of frequencies we can generate. We shall now consider how we can generate other frequencies where both d and c are real numbers.

5.1.4 Generating arbitrary frequencies

Any frequency f that we want to generate must be within the nyquist interval

This requires that c satisfies the condition

Negative values of c correspond to negative frequencies. This is useful for introducing 180° phase shifts in waveforms.

Since c is no longer an integer, some truncation or rounding will have to be done in order to get values from the wave table.

Alternatively, we can interpolate between sample values from the wave table for more accurate synthesis. Linear interpolation is usually sufficient. Let w(i) and w(j) be the i and j entries in the wave table and we need a sample with a real-valued index q which is between i and j. The linearly interpolated value is given by

This interpolation is shown in Figure 5.3.

Interpolation naturally produces the most accurate results with rounding coming next. Truncation is the most inaccurate of the three methods. The inaccuracies become smaller as the length of the wave table D increases. In computer music applications, wave table sizes vary between 512 and 32 768.

5.1.5 DTMF example

Coming back to the generation of the DTMF frequencies. Assume that the sampling frequency is 8 kHz and the size of the wave table is 200. The fundamental frequency is therefore 400 Hz.

For the low frequency group, the values of c are given by

Similarly, the values of c for the high frequency groups are found to be 30.225, 33.4, 36.925 and 40.825.

5.2.1 Specifications

All the specifications of the modulation method can be found in

The modulated signal generation process is illustrated in block diagram form in Figure 5.4.

Using this technique, the communication of two data bits is achieved by the transmission of one of four signaling frequencies. The modulated signal is required to have a continuous phase (no sudden jumps in phase). This and the pre-modulation pulse shaping (or filtering) of the data stream constrain the transmitted radio frequency (RF) spectrum. The four signal frequencies are as shown below.

Here f0 is the intended operating frequency. In a data stream, the most significant bit shall be transmitted first. The transmission rates are

The binary signal is filtered by the pre-modulation filter to create a smooth signal rather than one with sudden jumps in levels. The specifications for this filter are shown in Figure 5.5.

The specifications are given in terms of frequency characteristics with upper and lower limits. Apart from the amplitude spectrum, the phase spectrum is also specified as group delay characteristics. Group delay is defined as the negative rate of change of phase. So a constant group delay implies a linear phase function. In most cases for data transmission, this is the ideal. These specifications are derived from a 10th order low-pass Bessel filter with a 3 dB bandwidth of 3.9 kHz.

The rise or fall time for the frequency transition between two successive symbols shall be 88 microseconds with a tolerance of 2 microseconds.

The RF spectrum of the output of the transmitter shall conform to the mask shown in Figure 5.6.

The center frequency of the transmissions shall not exceed 15 Hz either way from the intended operating frequency (f₀). The intended operating frequency may be forced to differ from the nominal channel frequency by up to 185 Hz (frequency offset). The difference between any two adjacent symbol frequencies shall be 3125 ± 15 Hz.

These specifications are very difficult to meet using conventional analog circuitry, especially the center frequency stability and the accuracy of the difference between two adjacent symbol frequencies.

5.2.2 DSP implementation

A classical way to generate a signal of this kind is to use an analog pre-modulation filter followed by an FM modulator. However, this approach presents problems regarding the stability of the parameters. It is very difficult to achieve a precision of 0.1% in the frequency deviation.

In the DSP implementation, the 4-PAM coder and the pre-modulation filter are implemented in the following manner. Since we have four possible symbols in the alphabet, there are a total of 16 possible transitions between symbols. These transitions are being smoothed by the filtering performed by the pre-modulation filter. All these 16 transitions and their corresponding results after filtering can be pre-calculated and stored in a ROM. In this way, the samples of the filtered signal are a function of the current and last symbol and can be looked up from the ROM.

In Figure 5.7, the transition between the symbol 00 and the symbol 10 is shown. The stored transition consists of 64 samples per symbol, 16 bits per sample.

The FM modulator block diagram is shown in Figure 5.8. The frequency samples obtained from the previous process are passed to an integrator, which is simply an accumulator of all past values.

This generates the phase samples corresponding to the intended frequencies. The output from the integrator becomes the arguments of the sine and cosine tables. The cosine table gives the in-phase (I) component of the modulated signal and the sine table gives the quadrature (Q) component. The tables are quantized with 8-bit resolution. These samples are converted to analog signals using two DACs and a quadrature modulator, which can be obtained commercially as a chip, is used to generate the RF signal.

The specification allows frequency offsets that can be introduced intentionally. A very precise offset can be introduced by adding or subtracting a certain constant from the filter output samples.

POCSAG can also be implemented using the same scheme by including a different transition table corresponding to the POCSAG specification. In this way, the transmitter can switch between the two systems by simply getting its samples from a different table.

The symbol rate is low enough to allow us to choose the very high sampling rate of 64 samples per symbol. This high oversampling rate implies that very simple anti-aliasing filters can be used after the DAC. High sampling rate also reduces the time uncertainty of the transitions.

Figure 5.9 shows the analog I and Q signals generated with input symbols (10, 00, 10, 00).

5.2.3 Other advantages

There are some other advantages in using the DSP approach that may not be obvious initially. Two major ones are briefly discussed below:

On the other hand, the most efficient modulation schemes do not produce constant envelope signals. Hence they require linear amplifiers which are much less efficient. By using DSP implementations, we can pre-compensate for the amplifier non-linearity by pre-distorting the discrete-time signal. In this way, the most efficient modulation schemes can be used with the most efficient power amplifiers. This is obviously not possible with analog implementations.

5.3.1 Speech production mechanism

Speech signals consist of a sequence of sounds. These sounds and the transition between them carry the information that needs to be conveyed. These sound sequences obey certain rules. Linguistics is the study of such rules for a certain language. The study of the classification of the basic sounds is in the realm of phonetics.

In order to come up with a model of speech production, we need to have an understanding of the human vocal system. It consists of two main parts: the vocal cords (or glottis), and the vocal tract. The vocal tract in turn consists of three main parts:

The source of energy comes from the air pressure exerted by the lungs, bronchi and trachea. Speech is produced when an acoustic wave is radiated from this vocal system when air is expelled from the lungs and the air flow is perturbed by constrictions somewhere in the vocal tract. When the velum is lowered, the nasal tract is acoustically coupled to the vocal tract to produce nasal sounds.

5.3.2 Classification of sounds

The basic units of speech sounds in the English language are called phonemes. There are two main types of phonemes: vowels and consonants. More detailed classifications are also available. But for our discussions, we shall assume only these two types of sounds.

Vowels are produced when the vocal tract is excited by pulses of air caused by the vibration of the vocal cords. The vibration is periodic in nature and the period is the pitch of that sound. The shape of the vocal tract determines the resonant frequencies of the tract, called formants. For vowels, there are typically three formants between the frequencies 200 Hz and 3 kHz. The exact frequencies of the formants vary from person to person. Figure 5.10 shows a typical frequency spectrum of a vowel.

In the production of consonants, the vocal cord is totally relaxed in general, although there are exceptions. In this way, air flows into the vocal tract without the periodic excitation generated by the vocal cord. Consonants can be broadly classified into:

5.3.3 Speech production model

In order to synthesize speech sounds artificially; we need a model of the speech production system described above. We have looked at one briefly in Chapter 1. Figure 5.11 shows a more detailed model.

The glottal pulse model, the vocal tract model, and the radiation model are linear discrete-time systems. They are therefore essentially discrete-time filters. In order to synthesize speech, the voiced/unvoiced switch will switch to the source for the sound at that particular time. The vocal tract parameters will also need to vary with time.

One of the most successful glottal pulse models is the Rosenberg model. Its impulse response is given by

The vocal tract model is usually a linear predictive model. It is so called because the current speech sample is generated from a number of past samples plus the current

excitation. This can be described in equation form as

Here a_k is the coefficient for the model and it changes from one phoneme to another, and u(n) is the input sample to the vocal tract model. The prediction order, p, is typically 10 to 12.

In most cases, the radiation model is ignored.

In the practical sessions, there will be an opportunity to experiment with the model.

5.4.1 Contrast enhancement

A simple way to improve the contrast or the dynamic range of image pixel intensities is by a technique called gray-scale modification. It applies a transformation T to the original image to produce the enhanced image. This transformation is often represented by a table.

Consider an image of 5×5 pixels represented by 3 bits. There are a total of eight levels with 0 being the darkest to 7 being the brightest. The pixel values are shown in Figure 5.12. We can see that the pixel values are between 2 and 5. So only 4 out of 7 possible levels are used. Making use of the full range of values can produce a better contrast.

The problem is how we can find a suitable transformation that will do a good job. Computing the histogram of the image and studying its characteristics can identify a suitable transformation. The histogram is just a tabulation or a graph of the number of pixels that have specific intensities. The histogram of Figure 5.12 is shown in Figure 5.13.

This histogram showed us that the dynamic range is not well utilized as discussed above. A transformation that will improve the contrast is shown in Figure 5.14 and the resulting output image after the transformation is in Figure 5.15.

The transformation can be obtained automatically by defining a desired histogram. In most cases, the desired histogram is usually a uniform distribution of gray level values within the image. This will make the number of pixels at any one gray level about the same as another. The transformation T must be monotonically non-decreasing like the one in Figure 5.14. It is given by

where N is the total number of pixels in the image, n_i is the number of pixels at gray level i, and M is the total number of gray levels possible.

This simple procedure often produces significant improvements in image quality or intelligibility to the viewer.

5.4.2 Noise reduction

There are two main types of noise in images. One is the uniform random noise similar to those for one-dimensional images. Another type one is known as impulse noise or salt-and-pepper noise. They appear as isolated bright or dark pixels in the image. They can occur due to random bit error during transmission.

The energy of a typical image is primarily in the low frequency region. Therefore, (two-dimensional) low-pass filtering will be quite effective in removing a substantial amount of uniform random noise. This is done at the expense of removing some details of the image. It should be noted that edges that exist in the image produce high frequency components. If these components are removed or reduced in energy, then the edges will become fuzzier.

Median filters are very effective in removing impulse noise while preserving edges. They are non-linear filters however, and therefore the process cannot be reversed. In median filtering, a window or mask slides along the image. This window defines a local area around the pixel being processed. The median intensity value of the pixels within that window becomes the new intensity value of the pixel being processed.

Figure 5.16(a) shows a 3×3 window. The pixel being processed in the middle of this window. The numbers within the window are the intensity levels of the pixels in that window. Figure 5.16(b) indicates the processed output. Note that the intensity of the pixel in the middle is now replaced by the median value.

An important parameter in median filtering is the size of the window. Different results can often be obtained by using different window sizes. The choice also depends on the characteristics of the image and the noise. As a rule of thumb, images with lots of variations require the use of smaller windows while larger windows can be applied to images that have more uniform intensity areas.