Reference no: EM132303377

Adaptive Signal Processing Assignment - Optimal Narrowband Interference Rejection

Background - 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 ouput 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.

Assignment -

Objectives - 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 z_t and the received data y_t;

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 filter 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 filter.

5. To validate the filter performance against the theoretical expectations.

Derivation of the Optimal FIR Filter -

Your first task is to formulate the Wiener-Hopf (WH) equations for the optimal filter. In order to do this, we need knowledge of the signal model, namely the joint statistical properties (covariance functions) of the signals z_t (the "desired" signal) and y_t (the received data).

1. Show that s_t has zero mean for all t.

2. Show that the covariance function for the sinusoidal process s_t is given by R_s(τ) = A²/2 cos(ωτ), for all integers τ.

3. Prove that

R_z(τ) = _m=τ∑^M-1h_mh_m-τ,

for 0 ≤ τ ≤ M-1, R_z(τ) = 0, for all τ ≥ M, and R_z(-τ) = R_z(τ).

4. Derive the matrix-vector Wiener-Hopf (WH) equations for the optimal (MMSE) filter W(z).

Computer Exercises - Simulated Signals -

I have supplied the matlab function gen_sig.m which generates the desired signal z_t, and the received data y_t. This allows you to compare the output of your MMSE filter z^{^}_t to the original input z_t. The function accepts as its input, the following parameters:

T: The total length of the desired signal z_t and the received data sequence y_t.

M: The length M of the signal filter H(z). The channel coefficients h_j are chosen randomly.

The signal-to-noise ratio (SNR) in dB. This is defined as 20 log₁₀ (E{z²_t}/σ²) where σ² is the variance of the additive noise n_t. The function gen_sig.m determines the required additive noise variance σ² based on the realised filter H(z) and the SNR you supply.

The signal-to-interference ratio (SIR). This is defined as 20 log₁₀ (E{z²_t}/E{s²_t}). The function gen_sig.m determines the required amplitude A of the sinusoidal interference based on the realised filter H(z) and the SIR you supply.

The (normalised) frequency 0 < ω < π (rad/s) of the sinusoidal interferer s_t.

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 z_t (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 y_t which is a real column vector of length T, and H is a column vector of length M containing the signal filter coefficients. The computed values of σ² and A for your case are also returned to assist with formulation of the covariance functions.

Please note that it is important to distinguish between system parameters and filter design parameters. System parameters are those specified by the signal model, in this case, the data length T, the filter H(z), the SNR and SIR, and the frequency ω 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 ω). 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 filters will also be illustrative.

Attachment:- Assignment Files.rar