Download List 7 - Rodrigo de Lamare`s website - PUC-Rio

Digital Signal Processing/Processamento Digital de Sinais
CETUC/PUC-Rio - Prof. Rodrigo de Lamare
Tutorial Questions/Lista de Exercícios - 7
1. Consider a Wiener filtering problem characterised as follows: The correlation matrix 𝑹
of the input data vector 𝒙[𝑖] is
1 0.5
𝑹=[
]
0.5 1
The cross-correlation vector 𝒑between 𝒙[𝑖]and the desired signal 𝑑[𝑖] is
0.5
𝒑=[
]
0.25
a) Compute the Wiener filter using both analytical and numerical values.
b) What is the minimum mean-squared error produced by this Wiener filter?
c) Write down a representation of the Wiener filter in terms of eigenvalues of 𝑹 and
associated eigenvectors.
2. A complex-valued linear predictor of discrete-time waveforms can be built by forming
an estimate of a sample 𝑛0 samples later by observing p consecutive data samples.
Consider the estimate of the predictor given by
𝑝
𝑥̂(𝑛 + 𝑛0 ) = ∑
𝑘=1
𝑎𝑝 ∗ (𝑘)𝑥(𝑛 − 𝑘)
The predictor coefficients 𝑎𝑝 (𝑘) should be chosen to minimize
∞
𝜀𝑝 = ∑
|𝑥(𝑛 + 𝑛0 ) − 𝑥̂(𝑛 + 𝑛0 )|2
𝑛=0
a) Derive the equations required to compute the optimal set of coefficients.
b) If 𝑛0 = 0, how is the formulation of the problem different. Please explain.
3. Consider a system identification problem as shown below
𝑛[𝑖]
𝒙[𝑖]
𝑑[𝑖]
𝒘𝒐
𝑒[𝑖]
+
𝑑̂[𝑖]
−
𝒘[𝑖]
where 𝒙[𝑖] is an N x 1 input vector, d[i] is the desired signal, 𝑛[𝑖] is the measurement
noise, 𝒘𝑜 is the system to be identified that can be modelled as an FIR filter with N
coefficients and 𝒘[𝑖] is an adaptive filter with N coefficients used to identify 𝒘𝑜 . Use
complex Gaussian random variables with zero mean and a chosen variance to model 𝒙[𝑖],
𝑛[𝑖] and 𝒘𝑜 , define the signal-to-noise ratio (SNR) as appropriate and employ at least 100
repetitions to obtain well behaved curves.
a) Write a Matlab programme to simulate the mean-square error (MSE) curves that
describe the learning behaviour of an LMS algorithm.
b) Plot curves for different step sizes 𝜇 , SNRs and filter lenghts. What is the effect of large
step sizes 𝜇, high SNRs and large filter lengths on the performance of the LMS algorithm.
c) Compare the simulated MSE curves at steady state with the analytical values available
to predict the MSE.
d) Consider a correlated input signal and observe the effects on the MSE curves.