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ELECENG7015: Adaptive Signal Processing - Optimal Narrowband Interference Rejection - Engineering Assessment Answer
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Code: ELECENG7015
Figure 1: System diagram of the signal generation and processing models.
The signal vt is zero-mean white noise with unit variance. It is passed through a known FIR filter H(z) which models the properties of the signal of interest (SOI) zt. The received data yt consists of the output zt of H(z) to which is added a zero-mean white noise signal nt with known variance 2, and a sinusoidal interference signal st. This received data is processed by an FIR filter W(z) which we will design. Its output ^zt is an estimate of the SOI. We can write the received data as
for t > M ? 1. Here hj ; j = 0; : : : ;M ? 1 are the impulse response coefficients (taps) for the FIR filter H(z). The estimate is given by
for t > N +M - 2. Here the the filter weights w0....;w N-1 are to be designed to minimise the variance of the error signal et = zt - ^zt. The design parameters are (i) the filter order N - 1, and (ii) the lighter weights wj . The sinusoidal interference signal has the form
where A is the known amplitude, 0 < ! < , is the known frequency (rad/sec), and is the phase modelled as a uniformly distributed random variable. The signals vt and uncorrelated for all t; and also are uncorrelated with the phase .
3 The Assignment - ELECENG7015
The objectives of this assignment are :
1. To formulate the optimal (MMSE) design solution for the problem ;
2. To use the provided function to generate the desired signal zt and the received data yt ;
3. Use basic time and frequency domain analysis tools (e.g. power spectrum estimation- in matlab see pwelch) to validate data ;
4. To investigate the performance of the MMSE filter for various values of the signal and fil lter parameters. You should undertake a systematic study of the effects of the choices of these parameters, and draw appropriate conclusions about the design of the optimal fi lter.
5. To validate the fi lter performance against theoretical expectations.
3.2 Derivation of the Optimal FIR Filter
Your f irst task is to formulate the Wiener-Hopf (WH) equations for the optimal fi lter. In order to do this, we need knowledge of the signal model, namely the joint statistical properties (covariance functions) of the signals zt (the \desired" signal) and yt (the received data).
for all integers t .
3. Prove that
3.3 Computer Exercises - Simulated Signals - ELECENG7015
I have supplied the matlab function gen sig.m which generates the desired signal zt, and the received data yt. This allows you to compare the output of your MMSE lter ^zt to the original input zt. The function accepts as its input, the following parameters :
T : The total length of the desired signal zt and the received data sequence yt .
M : The length M of the signal lter H(z). The channel coecients hj are chosen randomly.
The signal-to-noise ratio (SNR) in dB. This is de ned as 20 log10 (Efz2 t g =2) where2 is the variance of the additive noise nt. The function gen sig.m determines the required additive noise variance 2 based on the realised lter H(z) and the SNR you supply.
The signal-to-interference ratio (SIR). This is de ned as 20 log10 (Efz2
t g =Efs2tg).
The function gen sig.m determines the required amplitude A of the sinusoidal interference based on the realised lter H(z) and the SIR you supply.
The (normalised) frequency 0 < ! < (rad/s) of the sinusoidal interferer st.
student id : Please use your student number (don't include the `a' at the start - it has to be a non-negative integer). This is used as a seed for the random number generator and will result in different data and noise realisations for different students.
You'll also have different signal filters H(z). This is so you don't all use the same data sequences, but each person will obtain the same data sequence every time you call the function.
The function returns three arrays: Z is the sequence zt (the output of the signal filter H(z) which is the desired signal) and is a column vector of length T; Y is the received data sequence yt which is a real column vector of length T, and H is a column vector of length M containing the signal later coefficients. The computed values of 2 and A for
your case are also returned to assist with the formulation of the covariance functions. Please note that it is important to distinguish between system parameters and lter design parameters. System parameters are those speci ed by the signal model, in this case, the data length T, the fi filter H(z), the SNR and SIR, and the frequency ! of the sinusoidal interference. The only design parameter in this problem is the optimal filter length N.
You will consider different choices for the system parameters (chosen to illustrate different behaviours e.g. high/low SNR, SIR, short/long channel lengths M, short and long data lengths T, sinusoid frequency !). You'll then examine the performance of the optimal filter as you vary the filter length N. You will estimate the error variance (since you have the actual desired signal) and compare to the theoretical value. Plotting the frequency response of your optimal fi filters will also be illustrative.
3.4 What you need to hand up - ELECENG7015
Your assignment report should be typewritten and include all figures (properly indexed) that you use to support your findings. Please include a brief introduction describing the problem with appropriate references. You should provide a description of your results with a limited number of graphs to support your arguments. You should also append a copy of your matlab code to your assignment report. As always, in recognition of professional engineering practice, your report should contain appropriate references which support your arguments and design choices and should be clear and concise. Please note that I may request you for an electronic copy of your matlab code so I can test it, and con rm your results.
3.5 Assessment Criteria - ELECENG7015
This assignment is worth 40% of assessment for this course. As such you should expect to spend approximately 35 hours on competing it. 1 Assessment will be based on the
following criteria :
1. The suitability of you abstract. Please remember that an abstract is about the report not about what you did ;
2. The clarity of your introduction including appropriate references (e.g. Haykin [1] and/or others).
3. The level of your understanding of the signal processing task presented in this assignment ;
4. The derivation of the optimal fi lter ;
5. The extent to which your report addresses the various design choices such as lter length, and supports your claims about the effects of these choices on the behaviour of the optimal fi lter ;
6. The thoroughness of your study and to what extent you go beyond the speci ed tasks to investigate other aspects of the system performance which might arise during your study ;
7. The presentation of your report including neatness, legibility, figures and references
8. The correctness, modularity, and readability of your matlab code.
Adaptive Signal Processing Engineering Assessment Answer
Assignment Task: 1 Background - ELECENG7015 In many practical problems, we are interested in estimating a signal contaminated with unmeasured narrowband interference (e.g. sinusoidal signals) as well as broadband (i.e. white) noise. Examples include processing of radio signals, including wireless communications signals, and GPS signals. In this assignment, you will consider the design of optimal FIR filters which can perform this task. You will be asked to study systematically the design choices for your filter for several signal and interference scenarios. This problem is an example of the output filtering where we desire to estimate a signal with known spectral properties which is contaminated by both white noise and a sinusoidal interference with known amplitude and frequency. For optimal (MMSE) filtering, we need complete knowledge of the covariance functions of the signals involved. In assignment two, we'll consider an adaptive version of this problem, so it is important to characterise the performance of the optimal filter. 2 Signal Models and Algorithm Design - ELECENG7015 Consider the system model shown in fi figure 1.
Figure 1: System diagram of the signal generation and processing models.
The signal vt is zero-mean white noise with unit variance. It is passed through a known FIR filter H(z) which models the properties of the signal of interest (SOI) zt. The received data yt consists of the output zt of H(z) to which is added a zero-mean white noise signal nt with known variance 2, and a sinusoidal interference signal st. This received data is processed by an FIR filter W(z) which we will design. Its output ^zt is an estimate of the SOI. We can write the received data as
for t > M ? 1. Here hj ; j = 0; : : : ;M ? 1 are the impulse response coefficients (taps) for the FIR filter H(z). The estimate is given by
for t > N +M - 2. Here the the filter weights w0....;w N-1 are to be designed to minimise the variance of the error signal et = zt - ^zt. The design parameters are (i) the filter order N - 1, and (ii) the lighter weights wj . The sinusoidal interference signal has the form
where A is the known amplitude, 0 < ! < , is the known frequency (rad/sec), and is the phase modelled as a uniformly distributed random variable. The signals vt and uncorrelated for all t; and also are uncorrelated with the phase .
3 The Assignment - ELECENG7015
The objectives of this assignment are :
1. To formulate the optimal (MMSE) design solution for the problem ;
2. To use the provided function to generate the desired signal zt and the received data yt ;
3. Use basic time and frequency domain analysis tools (e.g. power spectrum estimation- in matlab see pwelch) to validate data ;
4. To investigate the performance of the MMSE filter for various values of the signal and fil lter parameters. You should undertake a systematic study of the effects of the choices of these parameters, and draw appropriate conclusions about the design of the optimal fi lter.
5. To validate the fi lter performance against theoretical expectations.
3.2 Derivation of the Optimal FIR Filter
Your f irst task is to formulate the Wiener-Hopf (WH) equations for the optimal fi lter. In order to do this, we need knowledge of the signal model, namely the joint statistical properties (covariance functions) of the signals zt (the \desired" signal) and yt (the received data).
for all integers t .
3. Prove that
3.3 Computer Exercises - Simulated Signals - ELECENG7015
I have supplied the matlab function gen sig.m which generates the desired signal zt, and the received data yt. This allows you to compare the output of your MMSE lter ^zt to the original input zt. The function accepts as its input, the following parameters :
T : The total length of the desired signal zt and the received data sequence yt .
M : The length M of the signal lter H(z). The channel coecients hj are chosen randomly.
The signal-to-noise ratio (SNR) in dB. This is de ned as 20 log10 (Efz2 t g =2) where2 is the variance of the additive noise nt. The function gen sig.m determines the required additive noise variance 2 based on the realised lter H(z) and the SNR you supply.
The signal-to-interference ratio (SIR). This is de ned as 20 log10 (Efz2
t g =Efs2tg).
The function gen sig.m determines the required amplitude A of the sinusoidal interference based on the realised lter H(z) and the SIR you supply.
The (normalised) frequency 0 < ! < (rad/s) of the sinusoidal interferer st.
student id : Please use your student number (don't include the `a' at the start - it has to be a non-negative integer). This is used as a seed for the random number generator and will result in different data and noise realisations for different students.
You'll also have different signal filters H(z). This is so you don't all use the same data sequences, but each person will obtain the same data sequence every time you call the function.
The function returns three arrays: Z is the sequence zt (the output of the signal filter H(z) which is the desired signal) and is a column vector of length T; Y is the received data sequence yt which is a real column vector of length T, and H is a column vector of length M containing the signal later coefficients. The computed values of 2 and A for
your case are also returned to assist with the formulation of the covariance functions. Please note that it is important to distinguish between system parameters and lter design parameters. System parameters are those speci ed by the signal model, in this case, the data length T, the fi filter H(z), the SNR and SIR, and the frequency ! of the sinusoidal interference. The only design parameter in this problem is the optimal filter length N.
You will consider different choices for the system parameters (chosen to illustrate different behaviours e.g. high/low SNR, SIR, short/long channel lengths M, short and long data lengths T, sinusoid frequency !). You'll then examine the performance of the optimal filter as you vary the filter length N. You will estimate the error variance (since you have the actual desired signal) and compare to the theoretical value. Plotting the frequency response of your optimal fi filters will also be illustrative.
3.4 What you need to hand up - ELECENG7015
Your assignment report should be typewritten and include all figures (properly indexed) that you use to support your findings. Please include a brief introduction describing the problem with appropriate references. You should provide a description of your results with a limited number of graphs to support your arguments. You should also append a copy of your matlab code to your assignment report. As always, in recognition of professional engineering practice, your report should contain appropriate references which support your arguments and design choices and should be clear and concise. Please note that I may request you for an electronic copy of your matlab code so I can test it, and con rm your results.
3.5 Assessment Criteria - ELECENG7015
This assignment is worth 40% of assessment for this course. As such you should expect to spend approximately 35 hours on competing it. 1 Assessment will be based on the
following criteria :
1. The suitability of you abstract. Please remember that an abstract is about the report not about what you did ;
2. The clarity of your introduction including appropriate references (e.g. Haykin [1] and/or others).
3. The level of your understanding of the signal processing task presented in this assignment ;
4. The derivation of the optimal fi lter ;
5. The extent to which your report addresses the various design choices such as lter length, and supports your claims about the effects of these choices on the behaviour of the optimal fi lter ;
6. The thoroughness of your study and to what extent you go beyond the speci ed tasks to investigate other aspects of the system performance which might arise during your study ;
7. The presentation of your report including neatness, legibility, figures and references
8. The correctness, modularity, and readability of your matlab code.
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