数字信号处理第三章习题

第三章习题

1. Consider a Wiener filtering problem characterized by the following values

for the correlation matrix R of the tap-input vector x (n) and cross-correlation vector p between x (n) and the desired response d(n):

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????=??????=25.05.015.05.01P R (a) Suggest a suitable value for the step-size parameter μ that would

ensure convergence of the method of steepest descent, based on the given value for matrix R .

(b) Using the value proposed in part (a), determine the recursions for

computing the elements )(1n w and )(2n w of the tap-weight vector w (n). For this computation, you may assume the initial values

0)0()0(21==w w .

(c) Investigate the effect of varying the step-size parameter μ on the

trajectory of the tap-weight vector w (n) as a varies from zero to infinity.

2. The error performance of a real-valued filter, using a single tap weight w ,

is defined by

,))(0(20min w w r J J -+=

where r(0) is the autocorrelation function of the tap input x (n) for zero lag,

min J is the minimum mean-square error, and o w is the Wiener solution for the optimum value of the tap weight w .

(a) Determine the bounds on the step-size parameter

μof the steepest-descent algorithm used to recursively compute the optimum solution o w .

(b) Plot the curve for cost function of the filter.

3. Continuing with Problem 2, do the following:

(a) Formulate the learning curve of the filter in the terms of its only

natural mode )(n υ.

(b) Determine the first derivative of the mean-square error J with respect

to the natural mode of the filter.

4. Consider an autoregressive (AR) process of order one, described by

difference equation

),()1()(n n x n x να+--=

where α is the AR parameter of the process and )(n ν is a zero-mean

white noise of variance 2νσ.

(a) Set up a liner predictor of order one to compute the parameter α.

Specifically, use the method of steepest descent for the recursive computation of the Wiener solution for the parameter α.

(b) Plot the error-performance curve for this problem, identifying the

minimum point of the curve in terms of known parameters.

(c) What is the condition on the step-size parameter μ to ensure

stability? Justify your answer.

5. An autoregressive (AR) process is described by the second-order difference

equation

),()2()1(5.0)(n n x n x n x ν+-+--=

where )(n ν is a zero-mean white noise of unit variance. The method of

steepest descent is used for the recursive computation of the optimum weight vector of the forward linear predictor applied to the process x(n). Find the bounds on the step-size parameter μ that ensure stability of the steepest-descent algorithm.

6. Repeat Problem 5 for the backward predictor applied to the second-order AR

process x(n).

7. The LMS algorithm is used to implement a dual-input, single-weight adaptive noise canceller. Set up the equations that define the operation of this algorithm.

8. Consider the use of a white-noise sequence of zero mean and variance

2σas the input to the LMS algorithm. Evaluate

(a) the condition for convergence of the algorithm in the mean square, (b) The excess mean-square error.

9. The leaky LMS algorithm.Consider the time-varying cost function

,)()()(22n w n e n J α+=

where w (n) is the tap-weight vector of a transversal filter, e(n) is the

estimation error, and α is a constant. As usual,

),()()()(n n n d n e H x w -=

where d(n) is the desired response and x (n) is the tap-input vector. In the

leaky LMS algorithm, the cost function J (n) is minimized with respect to the weight vector w (n). Show that the time update for the tap-weight vector )(?n w

is defined by ).()()(?)1()1(?n e n n n *+-=+x w w

μμα 10. The tandem connection of adaptive filters arises in some applications (e.g.,

acoustic echo cancellation). Consider, then, Figure P3.1,which shows a simplified tandem configuration involving a pair of LMS adaptive filters. The input vector x (n) is applied simultaneously to both filters, and the error signal )(1n e produced by filter I serves the purpose of having a desired response for filter II.

(a) Formulate the update equations for the tandem configuration of the

figure.

(b) Show that this tandem configuration converges in the mean square if

both adaptive filters 1 and 2 converge in the mean square

individually.

()

n e Adapter Filter 1

Adapter Filter 2

Figure P3.1

11. Consider the adaptive FIR filter shown in Figure P3.2. The system )(z C

is characterized by the system function

19.011)(-+=z

z C Determine the optimum coefficients of the adaptive FIR filter 110)(-+=z b b z B that minimize the MSE. The additive noise is white with variance 1.02=ωσ

.

Figure P3.2