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 valuesfor 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):⎥⎦⎤⎢⎣⎡=⎥⎦⎤⎢⎣⎡=25.05.015.05.01P R (a) Suggest a suitable value for the step-size parameter μ that wouldensure 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 forcomputing the elements )(1n w and )(2n w of the tap-weight vector w (n). For this computation, you may assume the initial values0)0()0(21==w w .(c) Investigate the effect of varying the step-size parameter μ on thetrajectory 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 parameterμof 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 onlynatural mode )(n υ.(b) Determine the first derivative of the mean-square error J with respectto the natural mode of the filter.4. Consider an autoregressive (AR) process of order one, described bydifference equation),()1()(n n x n x να+--=where α is the AR parameter of the process and )(n ν is a zero-meanwhite 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 theminimum point of the curve in terms of known parameters.(c) What is the condition on the step-size parameter μ to ensurestability? Justify your answer.5. An autoregressive (AR) process is described by the second-order differenceequation),()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 ofsteepest 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 ARprocess 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 variance2σas 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 theestimation 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 theleaky 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 )(ˆn wis defined by ).()()(ˆ)1()1(ˆn e n n n *+-=+x w wμμα 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 thefigure.(b) Show that this tandem configuration converges in the mean square ifboth adaptive filters 1 and 2 converge in the mean squareindividually.()n e Adapter Filter 1Adapter Filter 2Figure P3.111. Consider the adaptive FIR filter shown in Figure P3.2. The system )(z Cis characterized by the system function19.011)(-+=zz 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。