Up to this point in our studies, all components in the system under study, and the resulting models for those components, have been fixed or time-invariant. Linear time-invariant (LTIV) models, and their application to the modeling and simulation of system elements such as filters, were discussed in detail in the preceding chapters. Linear elements in a communication system can be time invariant or time varying in nature. We now remove the restriction of time invariance and turn our attention to the modeling and simulation of time-varying systems. As we will see, there are many examples of time-varying systems. Of particular interest later in our studies, will be the time-varying channel that is encountered in the mobile radio system.
The assumption of time invariance implies that the properties of the system being modeled do not change over time. If we are modeling a time-invariant system by a transfer function, time invariance requires that the transfer function (both magnitude and phase) remain fixed as a function of time. Consider, for example, the model of a third-order Butterworth filter with a 3-dB bandwidth of 1 MHz. Time invariance implies that neither the order nor the bandwidth of the filter changes as a function of time. The filter is only time invariant if the values of all physical components, such as the resistors, capacitors, and the parameters of active components, such as the gain of an operational amplifier, do not change with time. While the time-invariance assumption may hold over a shorter periods of time, component values do change over longer periods of time due to aging. (Recall the role of end-of-life performance predictions discussed in Chapter 1.) From a practical point of view, whether or not a system can be considered time invariant depends not only on the system but also on the nature of the problem being solved.
Whether to use a time-invariant or time-varying system model is usually determined by the rate at which the characteristics of the communication system being modeled are changing in comparison to other parameters of the communication system such as the symbol rate. As an example, if the time constant associated with the system time variations is very large compared to the symbol rate, then a time-invariant model may be justified. On the other hand, if the system parameters are changing at a rate approaching the symbol rate, a time-varying model is appropriate. Thus, there is a notion of “slow” or “fast” variations, compared to the symbol rate, or some other attribute of the system that influences the choice of a time-varying or time-invariant model. We further examine this point using two examples.
Consider a microwave radio communication system in which the transmitting and receiving antennas are located on fixed microwave towers. The “channel” between these antennas is the atmosphere, and the changes in the channel characteristics are due to changes in the atmospheric conditions, which typically have time constants on the order of minutes or hours. If the communication link is operating at a symbol rate of 100 Mbit/s, the time constant associated with the channel variations is very long compared to the symbol time of 10−8s. Indeed, the channel will remain in nearly the same state while billions of symbols flow over the link. If the objective of a system simulation is estimation of the bit error rate (BER), the channel can be assumed to be in a “static” state and a time-invariant model can be used during the time interval over which the BER is estimated. The resulting BER estimate, of course, is valid only for the particular channel state used in the simulation. The long-term behavior of the channel, and its impact on long-term system performance, can be evaluated by analyzing system performance over a series of “snapshots” of static channel conditions, using a different time-invariant model for each snapshot. This is illustrated in Figure 13.1, where three simulations are developed for the three values of channel attenuation shown. From such simulations, one can obtain performance measures such as “outage probabilities,” which describe the portion of time during which the channel performance might fall below some BER threshold. If 10,000 channel snapshots are simulated and 100 of these channel conditions produce a BER corresponding to unsatisfactory system performance, BER> 10−3, for example, then the outage probability is (100/10,000) = 0.01 for this specific BER threshold.
As a second example, consider a mobile communication system consisting of a fixed base station and a mobile user. The characteristics of the communication channel between the transmitter and the receiver will be time varying, since the parameters of the channel, such as the attenuation and delay, are changing due to relative motion between the base station and the mobile user. In addition, changes in atmospheric conditions will also contribute to the time-varying nature of the channel. If the mobile user is rapidly moving and if the symbol rate is of the order of 10,000 symbols per second, the rate at which channel conditions are changing might be comparable to the symbol rate. In this case a time-varying channel model would be required. While a time-varying model may or may not be needed for BER estimation, such a model will be necessary to study the behavior of receiver subsystems such as synchronizers and equalizers.
As with LTIV systems, LTV systems can be modeled and simulated in either the time domain or in the frequency domain. The time-domain approach leads to a model consisting of a tapped delay-line structure with time-varying tap gains. This model is very easy to implement for simulation purposes and is computationally very efficient if the time-varying impulse response is relatively short.
Many of the modeling and simulation concepts previously discussed for LTIV systems apply to LTV systems, but with some important differences. Particular attention must be paid to the sampling rate used in the simulation, since an increase in the sampling rate will be required because of bandwidth expansion resulting from underlying time variations. One source of bandwidth expansion is the “doppler” spreading in a mobile communications system. In addition, caution must be exercised in simplifying the block diagrams of LTV systems, since LTV blocks do not obey commutative properties and hence the order of computations cannot be interchanged between LTV blocks as with LTIV blocks.
However, as long as the time-varying system is linear in nature, superposition and convolution apply, and many of the time-domain and frequency-domain analysis techniques developed for LTIV systems can be used, with slight modifications, to model and simulate LTV systems. We will also use equivalent lowpass signal and system representations as we develop simulation models and techniques for time-varying systems.
In the time domain, a linear time-invariant system is described by a complex envelope impulse response 
, where is defined as the response of the system at time τ to an impulse applied at the input at time t = 0. The variable τ represents the “elapsed time,” which is the difference between the time at which the impulse response is measured and the time at which the impulse is applied at the system input. The complex envelope input-output relationship for a LTIV system is given by the familiar convolution integral
where 
and 
represent the complex envelopes of the system input and output, respectively.
Taking the Fourier transform of (13.1) gives the input-output relationship for the LTIV system in the frequency domain. This is
where 
is the transfer function of the system, and 
and 
are the Fourier transforms of the input and output, respectively. The output is obtained by taking the inverse transform of . This gives
The preceding expressions serve as the starting point for deriving models for time-varying systems.
Time-varying systems are also characterized in the time domain by an impulse response. For time-varying systems the impulse takes the form 
, which is defined as the response of the system measured at time t, to an impulse applied at the input τ seconds earlier. In other words, the impulse is applied at the input at time t − τ and the response is measured at time t, after an “elapsed time” of τ. Since the system is time-varying, the impulse response will change as a function of both the time at which the impulse is applied, t − τ, and the time at which the output is measured, t. For a time-invariant system, the impulse response will strictly be a function of the elapsed time τ and it is therefore represented by . Note that the impulse response of a time-varying system is a function of two arguments, while the impulse response of a time-invariant system has one argument. Figure 13.2 depicts the impulse response of both time-invariant and time-varying systems.
While the impulse response of a LTIV system maintains the same functional form irrespective of when the impulse is applied at the input, the impulse response of a LTV system depends on when the input is applied. In other words, for an LTIV system
but for an LTV system
In addition, the response of an LTIV system to an arbitrary input remains the same irrespective of when the input is applied to the system except for a time delay. If the input produces an output , then the same input applied t0 seconds later, 
, will produce a delayed version, 
, of at the system output. In a time-varying system, this will not be the case, and the responses due to identical inputs may be entirely different if identical inputs are applied to the system at different times.
Note that the important difference between the impulse response of an LTIV system and an LTV system is that the impulse response for a time-invariant system is strictly a function of the elapsed time and not the time at which the input is applied or the time at which the output is observed. The impulse response of a time-varying system, on the other hand, is a function of both the elapsed time, τ, and the observation time, t.
The two time variables τ and t in characterize two different aspects of the system. The variable τ has the same role as the τ variable in the impulse response of time-invariant systems for which the Fourier transform associated with this variable carries the notions of transfer function, frequency response, and bandwidth. For an LTV system, one can develop the notion of a transfer function, although it is a time-varying transfer function 
, by simply taking the Fourier transform of with respect to τ as
If the system is “slowly time-varying,” then the concepts of frequency response and bandwidth can be applied to . Whereas the LTIV system is characterized by a single impulse response function and a single transfer function, the LTV system is characterized by a family of impulse response functions and transfer functions, with one function for each value of t. If 
or equivalently 
, then the system is time invariant.
The variable t in and describes the time-varying nature of the system. Strong dependence on t, and fast changes associated with t, indicate a rapidly time-varying system. Usually, the time-varying nature of the system is modeled as a random phenomenon, and is treated as a random process in t. If the process is stationary, then the time variations can be modeled by an appropriate autocorrelation function in the time domain or by a corresponding power spectral density in the frequency domain. The time constant of the autocorrelation function or the bandwidth of the power spectral density are key parameters that describe whether is slowly or rapidly time-varying.
From the definition of the impulse response of LTV systems, it is easy to see that the input-output relationship can be expressed by the convolution integral
The frequency domain version of the input-output relationship is somewhat more complicated, as shown in the following section.
As a starting point for the frequency domain description of LTV systems, let us define a two-dimensional Fourier transform of as
In defining this two-dimensional Fourier transform, the usual finite energy assumption is made about the impulse response function to ensure existence of the Fourier transform. However, we will see later that for channel modeling, will be treated as a stationary random process in t in which case the Fourier transform of may not exist with respect to t. The autocorrelation function, of course, will exist and the appropriate procedure is to define the autocorrelation of with respect to the t variable, and then take the Fourier transform of the autocorrelation function to obtain the frequency domain representation. The result is the power spectral density of the random process. It should also be noted that if the system is time invariant, then 
, and 
.
Taking the inverse transform of 
gives
Substituting in (13.7), shows that
and
In the two-dimensional “transfer function” defined in (13.8), the frequency variable f1 is associated with the time variable, and it may be viewed as analogous to the frequency variable f in the transfer function H(f) of linear time-invariant systems. However, the input-output relationship for LTV systems in the frequency domain, given in (13.10), involves a convolution in the frequency domain in the second variable of the transfer function . This convolution accounts for the effect of the time-varying nature of the system in the frequency domain.
If the input to an LTIV system is a tone at fc + f0
for which the complex envelope is
the input-output relationship in the frequency domain is given by
and
The system complex envelope output, in the time domain, is defined as
and the time-domain bandpass signal is given by
The relationship given in (13.13) and (13.17) for LTIV systems show that when the input to the system is a complex tone at frequency f0, the system produces an output tone at the same frequency. The amplitude and phase of the output tone are affected by the amplitude and phase response of the system at frequency f0. This is illustrated in Figure 13.3.
We now consider the same situation for a time-varying system. With the same input, (13.13), the output of an LTV system in the frequency domain is
The preceding equation shows that the output does not consist of a single tone but possibly a continuum of tones at all frequencies as dictated by the behavior of the two-dimensional transfer function in f2. An example is shown in Figure 13.4. [Note that if the system is time invariant, then , , and (13.18) produces 
which is the same as (13.15).]
In general, the output of an LTV system responding to an input tone at frequency f0 might be shifted in frequency and also might be spread. In the context of mobile communication channels, this is referred to as the doppler shift and doppler spreading respectively, and is produced by the relative motion (velocity and acceleration) between the transmitting and receiving antennas or by other changes in the channel.
Several properties of LTIV systems are useful in simplifying the simulation model of communication systems. Examples include the combining of the transfer functions of blocks in parallel, blocks in series, and combinations of blocks in series and parallel. These operations are based on the associative, distributive, and commutative properties of LTIV systems. These properties are now examined for LTV systems.
To examine the associative property in detail, consider the system shown in Figure 13.5(a). The output of the first block in Figure 13.5(a) is given by
The output of the second block is
Substituting for 
gives
which can be written
The quantity inside the brackets is the overall impulse response of the system 
. Substituting τ = t − τ1, we obtain the overall impulse response of two cascaded blocks as
In simpler appearing form we have
Using the preceding equation, we can define the associative property for an LTV system as
The overall impulse response is
where
as shown in Figure 13.5(b).
While it is possible to combine the response of two LTV blocks in series according to (13.27), one cannot interchange the order of LTV blocks. Interchanging the two blocks yields the overall impulse response as the convolution of the individual impulse responses. Thus
In general, 
. Hence, interchanging the order of LTV is not, in general, a valid operation. The example given in Figure 13.6 provides a specific example for which the commutative property does not hold for LTV blocks, that is,
for LTV systems. However, it should be noted that the commutative property does hold for LTIV blocks, that is,
for LTIV systems.
The reader can verify that the distributive property
or, equivalently
holds for LTV systems. An example of the simplification of simulation block diagrams of LTV systems, which result from the associative and distributive properties, is shown in Figure 13.7, where
in Figure 13.7(b).
Unfortunately, the one simplification that results in significant computational efficiency in LTIV blocks does not apply directly for LTV blocks. Block diagrams involving feedback cannot be simplified.
Example 13.1.
Assume that the input to a linear time-varying system is defined by a unit amplitude complex exponential having a frequency of 1 kHz. In other words
which, in the frequency domain, is
The system, defined by , is assumed to have a time-varying impulse response
where T is 1 ms and the time-varying attenuations, ai(t), are given by
and
Figure 13.8 illustrates ten “snapshots” of the impulse response taken every 2 ms starting at t = 0 s. We can see that the system is a function of both the elapsed time, τ, and the observation time, t. Diagrams of this type are frequently used to describe the time-varying channel typical of wireless communications systems.
The time-varying frequency response of the channel is found by taking the two-dimensional Fourier transform defined by (13.8). Substituting the impulse response defined by (13.36) into (13.8) gives
and then applying the sifting and modulation Fourier transform theorems, we find the transfer function
whose magnitude response is plotted in Figure 13.9. Note that, in contrast to an LTIV system, the transfer function is dependent on both f1 and f2.
The system output is defined by (13.10). Substitution of (13.35) and (13.41) into (13.10) gives
which, using the shifting property of the delta function, integrates to
The response of the time-varying system to the assumed input is illustrated in Figure 13.10. We see that the input, which is a single tone at 1,000 Hz, has been shifted, as well as spread in frequency, due to the time-varying channel impulse response. The spreading effect will be explained in the following section.
In many time-varying systems, the characteristics of the system change as a function of time in a random manner. Examples include changes in hardware characteristics due to aging of the components and changes in the characteristics of a wireless channel due to changes in the atmospheric conditions. These variations in time are usually modeled as random processes. If the underlying model is linear, the random variations of the system characteristics as a function of time can be handled by treating the impulse response as a random process in t. Such an approach is extensively used to model mobile wireless communication channels. While the details of different types of channel models will be considered in the next chapter, we provide a brief introduction here, in order to develop a generic simulation model for randomly time-varying systems. (In this section, we will use lower-case functions of time to denote random processes even though it is customary to use upper-case letters to denote random variables and random processes. Upper-case letters will be used to denote Fourier transforms.)
As a simple example, consider a system in which the output is an attenuated and delayed version of the input, where the attenuation is randomly changing as a function of time. The system is characterized by an impulse response
where a(t) is the time-varying attenuation and t0 is the delay. The input-output relationship is given by
The attenuation a(t) can be modeled as a stationary or nonstationary random process. The preferred model is a stationary random process, since this leads to the autocorrelation function and power spectral density. Modeling a(t) as a stationary process still permits a time-varying impulse response model.
If the input to the system is a stationary random process, the autocorrelation and the power spectral density of the output can be obtained as
which is
Assuming that the input process and the attenuation ã(t) are uncorrelated, which is a reasonable assumption, the output autocorrelation reduces to the simple form
By taking the Fourier transform of the preceding equation, we can obtain the power spectral density (PSD) of the output. The result is
The convolution in the preceding equation could lead to a spectral shifting and spreading. If the input is a randomly phased complex exponential:
then
from which the input PSD is
The output PSD is
as illustrated in Figure 13.11. An LTIV system will produce an output power spectral density of the form
The essential difference between the LTV system and the LTIV system is the spectral spreading caused by the LTV system.
One manifestation of spectral spreading in simulation is that the sampling rate has to be increased appropriately at least by an amount equal to twice the bandwidth expansion due to the spreading. A rule of thumb is to arrive at a sampling rate treating the system as time invariant and increasing the sampling rate to accommodate the “excess” bandwidth, which is equal to the bandwidth of the random process modeling the time-varying nature of the system.
A more general version of the time-varying model treats as a stationary random process in t with an autocorrelation function
The most commonly used model for is a zero mean stationary Gaussian process that leads to a Rayleigh probability density function for 
. In this model, it is usually assumed that 
and 
are uncorrelated for τ1 ≠ τ2. In other words:
For this case, the autocorrelation of the output of the system can be obtained from
Substituting for yields
Interchanging orders of expectation and integration results in
Assuming and 
independent, which is certainly reasonable, yields
Recognizing that the two expectations are autocorrelation functions and invoking (13.55) provides the simplification
Performing the integration on τ2 using the sifting property gives
which can be expressed by
where
The power spectral density of the output can be obtained by taking the Fourier transform of (13.62), which leads to the convolution
where ⊛, as always, denotes convolution. Note that 
is “averaged” power spectral density and is the Fourier transform of the “averaged” autocorrelation function defined in (13.63).
Note that the output power spectral density is a convolution of the input power spectral density and the “averaged” power spectral density of the random process that models the time variations. In contrast, the power spectral density relationship for an LTIV system is given by
Once again, when the input is a tone, an LTIV system produces an output tone at the same frequency, whereas the output of an LTV system could be shifted and spread in frequency.
Given a description of an LTV system in the form of its impulse response , a simulation model can be derived using the sampling theorem assuming that the input to the channel is bandlimited [1]. We start from the input-output relationship given by the convolution integral
and use the sampling theorem to represent the input in terms of its sampled values.
From the sampling theorem we know that a lowpass signal 
bandlimited to B Hz can be represented in terms of its sampled values as
where 1/T is the sampling rate, which is set equal to the Nyquist rate 2B. The minimum sampling rate of 2B is chosen to minimize the computational burden of the simulation model. Using the representation given above, with 
, we can replace 
in the convolution integral by
which leads to
We can therefore write
where
Equations (13.69) and (13.70) define a simulation model for time-varying systems.
The model given in (13.69) can be implemented in the form of a tapped delay line (TDL) as shown in Figure 13.12, with the tap gain functions specified by (13.70). In general, the tap gain functions will themselves be random processes and they will be correlated, that is, 
and 
will be correlated.
The model shown above can be simplified in many ways using several approximations and assumptions. First, it is often assumed that the tap gain processes are uncorrelated. They are also approximated by
In this approximation, the tap gain functions represent the sampled values of the time-varying impulse response , where the sampling is done in the impulse response variable.
The second approximation involves truncation of the impulse response. If
then the summation in (13.69) can be truncated to 2m + 1 terms as
and the tapped delay line model has only a finite number of taps as shown in Figure 13.12. The total number of taps should be kept to a minimum in order to maximize the computational efficiency of the model.
Finally, if the system is time invariant, then 
, and the tap gains become constants
In other words, the gains are simply the sampled values of the impulse response of the LTIV system, and the tapped delay line model reduces to an impulse invariant model, or an FIR filter performing time domain convolution.
One other aspect of the TDL also deserves additional attention. The TDL model shown in Figure 13.12 has continuous time input and continuous time output . However, in simulation we will use sampled values of and . Sampling should normally be carried out using a sampling frequency 8 to 16 times the bandwidth, where the bandwidth includes the effect of spreading due to the time-varying nature of the system. This effect can be seen in (13.52) and (13.64). Note that the Nyquist rate of 2B was used to derive the TDL model, and the tap spacing is T = 1/2B, which will be >> Ts, where Ts is the sampling time for the input and output signals. It is of course possible to derive a TDL model with a smaller tap spacing of Ts, but such a model will be computationally inefficient and does not necessarily improve the accuracy.
The tap gain processes are stationary random processes with a given probability density function and power spectral density. The simplest model for the tap gain processes assumes them to be uncorrelated, complex, zero mean Gaussian processes with different variances but identical power spectral densities. In this case, the tap gain processes can be generated by filtering white Gaussian processes, as shown in Figure 13.13.
The filter transfer function is chosen such that it produces the desired power spectral density, that is, H(f) is chosen such that
where 
is the power spectral density of the input noise process, which can be set equal to 1, and 
is the specified power spectral density of the tap gain processes. The static gain, ai, in Figure 13.13 accounts for the different power levels or variances for the different taps. If the power spectral density of the tap gains are different, then different filters will be used for the different taps.
There are several ways of implementing the spectral shaping filter (also referred to as the doppler filter in channel models). An FIR filter in the time domain is the most common implementation, since the (doppler) power spectral densities do not lend themselves to spectral factoring and implementation in recursive form.
In generating the tap gain processes, it should be noted that the bandwidth of the tap gain processes of slowly varying systems will be small compared to the bandwidth of the system and the signals that flow through it. In this case, the tap gain filter should be designed and executed at a slower sampling rate. Interpolation can be used at the output of the filter to produce denser samples at a rate consistent with the sampling rate of the signal coming into the tap. Designing the filter at the higher sampling rate will lead to computational inefficiencies as well as stability problems.
We conclude this chapter with two MATLAB examples illustrating the concepts presented in this chapter.
This example illustrates the spectral spreading that takes place in a time-varying system as shown in (13.64). The system simulated is a simple one in which the bandpass input to the system is a “tone” of the form x(t) = cos[2π(f0 + f1) t], which corresponds to a lowpass equivalent signal
The lowpass equivalent impulse response of the system is assumed to be of the form
which is an allpass channel with a delay of τ0 and a complex, time-varying attenuation of ã(t). The attenuation is modeled as a zero-mean Gaussian random process with a power spectral density
where B is the doppler bandwidth. The complex envelope of the output of the time-varying system can be shown to be
Simulation of this model involves generating sampled values of the input tone and multiplying with a filtered complex Gaussian process, where the filter used is chosen to have a transfer function that yields the power spectral density given by (13.76). It is left as an exercise for the reader (Problem 13.6) to show that the required filter has the transfer function
The simulation results illustrated are for parameter values f1 = 512 Hz, B = 64 Hz, τ0 = 0, fs = 8,192 samples/sec and simulation length = 8,192 samples.
Results of the simulation are shown in Figures 13.14 and 13.15. The power spectral density of the filtered Gaussian process and the time-domain values of the correlated Gaussian sequence generated by the filter are shown in the top and bottom frames of Figure 13.14, respectively. The input tone expressed by (13.74) and output of the time-varying system, both in the frequency domain, are shown in the top and bottom frames of Figure 13.15, respectively. The spectral spreading due to the time-varying nature of the system can be seen in the bottom frame of Figure 13.15. The MATLAB code used to develop Figures 13.14 and 13.15 is given in Appendix A. (Note: The MATLAB code given in Appendix A yields four different plots with titles rather than two plots per figure as was done here to save space. The student should experiment with the simulation parameters, especially the tone frequency f1, and observe the results of the parameter changes.)
In this example we illustrate two aspects of the time-varying channel: frequency selectivity and time variation. In the previous example the channel was allpass or frequency nonselective (sometimes also referred to as a flat channel). We now modify the system model so that the complex lowpass equivalent impulse response has the form
It can be shown that this model is frequency selective by considering a time-invariant version with
which leads to the transfer function, with τ1 assumed equal to zero, given by
The transfer function given above has different values at different frequencies and hence is frequency selective.
The complex lowpass equivalent output of the time-varying system given by (13.79) is
If the input to the system is a BPSK signal, the time-varying aspect of the system attenuates and phase rotates the BPSK signal as a function of time. The frequency selective nature of the channel, which is due to the two delayed components of the output [Equations (13.79) and (13.81)], manifests itself in the form of intersymbol interference. Both of these effects are clearly illustrated in the simulation results shown in Figure 13.16. These simulation results were obtained by executing the simulation given in Appendix B with the following parameters: BPSK symbol rate = 512 bits/s, sampling rate = 16 samples per bit, doppler bandwidth for ã1(t), ã2(t) = 16Hz, τ1 = 0, and τ2 = 8 samples (half the bit duration). In Figure 13.16 the top frame illustrates the effect of one time-varying component, and the bottom frame illustrates the effect of two components. Only the first 500 samples are shown. (Note: The program given in Appendix B also generates and displays the PSD of the first component impules response. As in the previous example, the MATLAB code given in Appendix B yields four different plots with titles rather than two plots per figure as was done here to save space. The student should experiment with the simulation parameters and observe the impact of parameter changes.)
Techniques for modeling and simulating linear time-varying systems were discussed in this chapter. These techniques will be used in the next chapter for simulating mobile communication channels.
Linear time-varying systems are usually characterized by a time-varying impulse response , which is the response of the system measured at time t to an input applied at time t − τ. The input-output relationship will be a convolution integral in the time domain
In the frequency domain the input-output relationship is given by
which explains the spectral spreading (sometimes referred to as the doppler spread) that takes place in a time-varying system such as a mobile communication channel.
The convolution integral given in (13.83) can be approximated by a finite convolution, which leads to an FIR simulation model
with time-varying tap gains functions. In the context of mobile communication channels, the time variations will be modeled by complex Gaussian processes with a given autocorrelation or power spectral density function. Sampled values of correlated Gaussian process can be generated by filtering an uncorrelated Gaussian sequence with an appropriate filter.
Linear time-varying components introduce two different forms of distortion. The first one is due to the time-varying nature of the response, and the second one is due to the frequency-selective aspects of the system. Two simulation examples were presented and illustrate the modeling and simulation approaches as well as the effects of time-varying components on two different types of input signals: an unmodulated tone to show the spectral spreading and a BPSK signal to show the effects of the time-varying and frequency-selective aspects.
The tapped delay line model for time-varying systems based on the sampling principle was first derived by Kailath [1]. A general reference for time-varying systems is the book by D’Angelo [2]. Additional details and examples of simulation models may be found in [3].
13.1 | Develop an approach (i.e., write a short paper outlining an approach) for simulating a time-varying system whose behavior is described in the form of an nth order differential equation whose coefficients are functions of time. For example  |
13.2 | Develop an approach (as in Problem 13.1 write a short paper outlining an approach) for simulating a time-varying system in the frequency domain using two-dimensional DFTs—assuming that the system can be represented in the sampled time domain as an FIR approximation given in (13.70).
|
13.3 | The transmitted signal in a mobile communication system is an unmodulated tone x(t) = cos(2πft) where f = f0 + fi. Let f0 = 1 GHz be the nominal carrier and let fi = 512 Hz. Consider simulating the received signal (and its spectrum) at the mobile over a time interval of 2 seconds during which the mobile is moving toward the base station at a constant velocity of 40 miles per hour, starting from a distance of 1 mile from the base station. Note: The received signal can be written as
where
and
In the preceding expressions d(t) is the distance between the base station and the mobile at time t, c is the velocity of light, Δ(t) is the propagation delay, and a(t) is the attenuation of the signal at time t, which can be assumed constant over a 2-second interval. Carry out this simulation using the complex lowpass equivalent model. (The output “spectrum” is the magnitude squared of the DFT.) |
13.4 | In Problem 13.3 assume now that the mobile accelerates steadily from 40 miles per hour to 50 miles per hour over the 2-second interval. What happens to the output spectrum? |
13.5 | Rework Problem 13.3 assuming that the mobile is on a busy highway and its speed, measured at time intervals of 0.01 second, can increase or decrease randomly from 50 mph by 1/10 mph with a probability of 1/2. |
13.6 | Fill in the details of the IIR filter for synthesizing the power spectral density given in (13.76). |
13.7 | In many simulation problems involving time-varying systems, the power spectral density for the random process model will be assumed to be uniform (white) over the doppler bandwidth. Synthesize an FIR filter for generating a complex Gaussian process with a flat power spectral density over a doppler bandwidth of 64 Hz and repeat the simulations given in MATLAB Example 1. |
13.8 | Repeat Problem 13.7 with a doppler power spectral density that has a Gaussian shape defined by  where σ = 64 Hz. |
13.9 | Resimulate MATLAB Example 1 with the two-tone input  |
13.10 | The system described in MATLAB Example 2 is frequency selective. Replace the random binary input in this example with an input of the form  and show via simulation that the two input tones are affected differently by the time-varying system, thus demonstrating the frequency-selective nature of the system. Compare the results with Problem 13.9. |
% File: c13_tiv1.m
%
% Set default parameters}
f1 = 512; % default signal frequency
bdoppler = 64; % default doppler sampling
fs = 8192; % default sampling frequency
tduration = 1; % default duration
%
ts = 1.0/fs; % sampling period
n = tduration*fs; % number of samples
t = ts*(0:n-1); % time vector
x1 = exp(i*2*pi*f1*t); % complex signal
zz = zeros(1,n);
%
% Generate Uncorrelated seq of Complex Gaussian Samples
z = randn(1,n) + i*randn(1,n);
%
% Filter the uncorrelated samples to generate correlated samples
coefft = exp(-bdoppler*2*pi*ts);
h = waitbar(0,'Filtering Loop in Progress'),
for k =2:n
zz(k) = (ts*z(k)) + coefft*zz(k-1);
waitbar(k/n)
end
close(h)
y1 = x1.*zz; % filtered output of LTV system
%
% Plot the results in time domain and frequency domain
[psdzz,freq]=log_psd(zz,n,ts);
figure;
plot(freq,psdzz); grid;
ylabel('Impulse Response in dB')
xlabel('Frequency')
title('PSD of the Impulse Response'),
zzz=abs(zz.^2)/(mean(abs(zz.^2)));
figure;
plot(10*log10(zzz)); grid;
ylabel('Sq. Mag. of h(t) in dB')
xlabel('Time Sample Index')
title('Normalized Magnitude Square of the Impulse Response in dB'),
[psdx1,freq]=log_psd(x1,n,ts);
figure;
plot(freq,psdx1); grid;
ylabel('PSD of Tone Input in dB')
xlabel('Frequency')
title('PSD of Tone Input to the LTV System'),
[psdy1,freq]=log_psd(y1,n,ts);
figure;
plot(freq,psdy1); grid;
ylabel('PSD of Output in dB')
xlabel('Frequency')
title('Spread Output of the LTV System'),
% End of script file.Program log_psd.m is defined in Appendix A of Chapter 7.
% File: c13_tiv2.m
%
% Set default parameters
symrate = 512;
nsamples = 16;
nsymbols = 128;
bdoppler = 16;
ndelay = 8;
%
n = nsymbols*nsamples;
ts = 1.0/(symrate*nsamples);
%
% Generate two uncorrelated seq of Complex Gaussian Samples
z1 = randn(1,n) + i*randn(1,n);
z2 = randn(1,n) + i*randn(1,n);
%
% Filter the two uncorrelated samples to generate correlated sequences
coefft = exp(-bdoppler*2*pi*ts);
zz1 = zeros(1,n);
zz2 = zeros(1,n);
for k = 2:n
zz1(k) = z1(k)+coefft*zz1(k-1);
zz2(k) = z2(k)+coefft*zz2(k-1);
end
%
% Generate a BPSK (random binry wavefrom and compute the output)
M = 2; % binary case
x1 = mpsk_pulses(M,nsymbols,nsamples);
y1 = x1.*zz1; % first output component
y2 = x1.*zz2; % second output component
y(1:ndelay) = y1(1:ndelay);
y(ndelay+1:n) = y1(ndelay+1:n)+y2(1:n-ndelay);
%
% Plot the results
[psdzz1,freq] = log_psd(zz1,n,ts);
figure; plot(freq,psdzz1); grid;
title('PSD of the First Component Impulse Response'),
nn = 0:255;
figure; plot(nn,imag(x1(1:256)),nn,real(y1(1:256))); grid;
title('Input and the First Component of the Output'),
xlabel('Sample Index')
ylabel('Signal Level')
figure; plot(nn,imag(x1(1:256)),nn,real(y(1:256))); grid;
title('Input and the Total Output')
xlabel('Sample Index')
ylabel('Signal Level')
% End of function file.Program log_psd.m is defined in Appendix A of Chapter 7.
% File: mpsk_pulses.m
function [x] = mpsk_pulses(M,nsymbols,nsamples)
% This function genrates a random MPSK complex NRZ waveform of
% length nsymbols; Each symbol is sampled at a rate of nsamples/bit
%
u = rand(1,nsymbols);
rinteger= round ((M*u)+0.5);
phase = pi/M+((rinteger-1)*(2*pi/M));
for m = 1:nsymbols
for n = 1:nsamples
index = (m-1)*nsamples + n;
x(1,index) = exp(i*phase(m));
end
end
% End of function file.