The Effect of Lowpass Filtering on the Performance

 

ADVANCED DIGITAL SIGNAL PROCESSING

 

 

 

 

The Effect of Lowpass Filtering on the Performance

of a Spread Spectrum System

 

This Assignment carries a maximum of 2 Credits

 

 

Objectives

 

The purpose of this Assignment is for the student to demonstrate an understanding of how to:

 

·      Use MATLAB software for system modelling;

·      Design a lowpass filter to meet a specified performance;

·      Characterise and measure spread spectrum system performance.

 

           

 

 

 

List of Contents

 

1.         Introduction: Matched filters are robust

 

2.         Matched filter performance measures

 

3.         Task 1: Design of a spread-spectrum waveform

 

4.         Task 2: Design of a family of signal postfilters

 

5.         Task 3: Prefiltering the matched filter

 

6.         Task 4: Writing a report

 

 

 

 

 

 

1. Introduction: Matched filters are robust

 

The Lecture material deals roughly equally with three topics:

 

• Frequency-selective filters,

• Matched filters, and

• Fast Fourier Transforms.

 

This Assignment requires an understanding and use of all of these, to investigate the effect on matched filter performance of bandlimiting its spread-spectrum input signal. The degree of bandlimitation to be examined is to varied between rather slight and very severe.

 

Ordinary signal waveforms for use in systems that do not use the spread-spectrum technique are usually quite sensitive to frequency-selective filtering: too much bandlimitation will usually cause severe waveform distortion, and the system performance will deteriorate rapidly as the filter bandwidth is reduced beyond what was originally specified for normal operation.

 

This is not the case with matched filtering. Matched filters are capable of tolerating many types of signal waveform distortion, and perhaps their most fundamental property is that they are still able to detect the signal they were designed to match, even when the signal is invisible, immersed in noise, with a Signal-to-Noise Ratio (SNR) less than 0 dB. Other examples are well-known: the matched filter is still able to operate, with surprisingly small loss of performance when the input signal is coarsely quantised; or, in the case of a binary pseudonoise waveform, when a significant proportion of the signal “chips” are wrong. Yet another example of the various distortions that a matched filter is able tolerate is that of frequency-selective filtering of the received waveform. Even when a significant proportion of the frequency spectrum of the input signal has been removed, the matched filter still manages to detect the filtered waveform when it is immersed in noise.

 

The purpose of this Assignment is to examine quantitatively the way in which various measures of matched filter performance are degraded as the bandwidth of the channel carrying its input signal is decreased.

 

The system scenario is shown in Figure 1.

 

 

 

 

 

                                                

 

Figure 1: Filtering the spread-spectrum waveform in a spread-spectrum system

 

For the purpose of this Assignment, the system is conceptualised as a baseband system, although in practice the communication channel would include a carrier subsystem – and the filter would be bandpass rather than lowpass.

2. Matched filter performance measures

 

The measures of matched filter performance that we will be concerned with are:

 

• Peak-to-Sidelobe Ratio (PSR), and

• Signal-to-Noise Gain (SNG) (also called Processing Gain (PG))

 

The definition of PSR is the easier to understand. Ignoring noise, the output of the matched filter is the autocorrelation, of the bipolar () spread-spectrum signal sequence, consisting of M chips. This will be a sequence of length , symmetric about its midpoint M, where the peak value,  occurs. The sidelobe level is defined as the maximum of the absolute values of the sidelobes:  Straightforwardly, the definition of PSR is:

.

 

The definition of Signal-to-Noise Gain is more complicated:

 

 

where the Signal-to-Noise ratio, SNR, is defined, in the context of matched filtering, to be:

 

Assuming that the channel noise is additive Gaussian white noise (AWGN) with zero mean and RMS value  we have, at the input to the matched filter:

 

 

Similarly, at the output of the matched filter, we have:

 

 

where  is the impulse response of the matched filter. Consequently, the SNG is independent of  and we have:

 

These two measures are both useful in characterising the performance of the matched filter: PSR is more to do with the detectability of the peak when the input signal is noisy: if S is large then the corresponding sidelobe is quite likely to be mistaken for the peak, and a false alarm will occur. By choosing the spread-spectrum waveform to be bipolar we have already ensured that P takes its maximum possible value, M, and the signal waveform design simply consists of making sure that S is reasonably small, so as to obtain a good (large) value of PSR. As the n-th sidelobe () consists of the sum of the sum of n fairly random  values, which by the Central Limit theorem, should be fairly Gaussian with mean zero (provided the spread-spectrum sequence itself is unbiased) and RMS value  Close-in to the main peak, then, it would not be too surprising to find a sidelobe of value around 2 or 3 times  This would imply that  would be in the range  but a clever waveform design might do better than this.

 

The SNG for the bipolar spread-spectrum waveform is easy to evaluate: the peak input signal level is obviously 1; the peak output signal is obviously M, and the square root in the expression for SNG is obviously  Consequently,

 

These values, , with  about 2 or 3, are the values corresponding to an unbiased bipolar spread-spectrum signal. But if that signal is subjected to lowpass filtering, the peak input signal will change, and so will the correlation peak, even though the matched filter’s bipolar coefficients are assumed to be unchanged. So filtering will affect SNG and PSR, and the question is: how much?

 

 

3. Task 1: Design of a spread-spectrum waveform

 

What is required is a bipolar  waveform of M samples, which is unbiased. This means that M needs to be even, and the number of +1s needs to be equal to the number of –1s. This immediately disqualifies the popular binary m-sequences, for which M is confined to be , for some value of k. One way to find a suitable input signal is simply to generate sequences of M (pseudo)random bipolar numbers and reject those with too large an imbalance of +1s and –1s. This can be done easily, for instance, using MATLAB’s Gaussian random number generator randn.m, as follows: s = sign(randn(1,M));; or its uniform random number generator rand.m, as follows: s = sign(rand(1,M)-0.5);. Without too much searching, it is feasible to discover a sequence with zero imbalance, that is, zero bias. This is helpful in reducing the size of the correlation sidelobes.

 

A further design option is to oversample the signal, by a factor K. This can be done using the Kronecker product instruction in MATLAB, kron(.,.). Instead of increasing M, this is an alternative way of increasing the time-bandwidth product of the signal. It can be shown that the time-bandwidth product is equal to the Signal-to-Noise Gain, so this will have a direct impact on SNG, and PSR as well.

 

Each student will be given their own individual values of M and K.

 

 

4. Task 2: Design of a family of signal postfilters

 

For this Assignment, it is required that you use MATLAB’s fir1.m instruction to design the lowpass filter. This filter design instruction is parametrised by normalised cutoff frequency, and also the filter order, N. In addition, the type of window used in the design can be chosen from an extensive repertoire. Obviously, filtering the spread-spectrum waveform removes signal power, and apart from distorting the waveform, there is an overall reduction in the amplitude. To ensure a fair comparison, that is, to make the definitions of input SNR the same, it is necessary to rescale the waveform, increasing its amplitude so that the waveform maximum is unity, as in the original unfiltered case. This is easily done, by measuring the maximum value of the filtered waveform, and renormalizing the entire waveform by dividing by this value.

 

Each student will be given their own individual values of filter order N, and window parameter value b, for a Kaiser window.

 

Calculate and plot the values of SNG in dB and PSR in dB for a range of filter normalised cutoff frequencies:  (The value of  corresponding to an allpass filter, is rejected by MATLAB).

 

The filter impulse response and magnitude of the frequency response should be plotted for the cases  Clearly, for  the values of SNG and PSR should be nearly those for the ideal case discussed in Section 3.

 

5. Task 3: Prefiltering the matched filter

Task 2 should convince you that matched filtering is robust to linear filtering of the spread-spectrum waveform. Now consider the idea that if this is so, then as far as the noiseless signal is concerned, it makes no difference whether the lowpass filter is placed on the far side of the channel in the transmitter following the waveform generator, or on the near side of the channel in the receiver, prior to the matched filter. This is just another way of saying that the PSR is the same in both cases. Formulate a formal proof of this assertion. This new system scenario is shown in Figure 2.

 

 

 

 

 

                                                

 

Figure 2: Filtering the noisy spread-spectrum waveform in a spread-spectrum system

In this alternative scenario, however, the noise also gets filtered by the lowpass filter, so that the SNR at the matched filter input will be better than in the case considered in Task 2. Modify the definition of SNG to model this situation, and repeat the SNG plot of Task 2 for this case. Overplot the two cases for ease of comparison. Is there a case for deliberately reducing the receiver bandwidth to exploit the robustness of the matched filter? Investigate the robustness as a function of the oversampling factor K.

6. Task 4: Writing a report

Write a short well-structured report on your work, summarising and interpreting your results. Include any evidence, such as tables or graphs, on which you have based your conclusions. Include, in an Appendix, a copy of all the software you have developed. If you do use colour in the computer-generated graphics, remember to print the report in colour.

 

You can include a brief statement of further work that could be done to increase your understanding of this subject. For example, there is a theorem in Communication Theory that asserts that channel filtering should be split equally between the transmitter and the receiver: research this issue and include your findings at an appropriate place in the report.

 

Include, taped inside the back cover of the report, an envelope containing a CDROM containing the report and your software.

 

 

Marking Scheme:      Task 1:   15;   Task 2:   25;   Task 3:   35;   Task 4: 25.

 

 

Use of references:     Use of references to support your work is encouraged: but if you do use them, cite them in such a way to allow the assessor to check your use rapidly. For example, use the referencing conventions adopted by IEEE Transactions on Signal Processing. Proper use of references will be awarded bonus marks, and improper use of references will be penalised.