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**Unformatted text preview: **UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN Department of Electrical and Computer Engineering ECE 410 Digital Signal Processing Homework 5 Due Wednesday, October 1, 2008 Prof. Bresler / Prof. Jones 1. The diagram below represents a part of the computation in a 16-point decimation-in-time radix-2 FFT. Indicate the values of the three requested branch weights a, b and c and the indexes W, X, Y, and Z 2. Determine if the following systems are (i) memoryless or with memory (ii) causal or non-causal (iii) time-invariant or time-varying (iv) linear or non-linear Note that no justification is required. (a) (b) (c) (d) (e) 5| ∑ | k, 1 ∑ ∑ sin 1 10 (f) (g) 10 10 4 0 | | 0 0 1 1 1 Note that a system is memoryless if its current output value depends only on the current input value. In other words, the current output value does not depend on either past values or future values of the input. 3. In this problem, let 0.9 . , where , where , where , where 0.7 1 3 and . 10 . 9. 1. (a) Find the DT convolution result: (b) Find the DT convolution result: (c) Find the DT convolution result: (d) Find the DT convolution result: Hint for parts c and d: There’s an easier way to compute this result instead of using the DT convolution formula directly! 4. Compute the linear convolution (a) (b) , 1 | | , 1, | | 1, , given 4, 2 , and | | 1, | | below: 1, 1/ Please use the method of a discrete-time convolution by hand. (No z-transforms) 5. Determine if the systems characterized by the following relations are, with respect to the input, (i) linear or nonlinear, (ii) causal or noncausal, and (iii) shift-invariant or shift-varying. Justify your answers with proofs or counter-examples. (a) y[n] = y[n − 12] + x[n − 2] + x[n − 1] (b) y[n − 1] = 5 y[n] + 3x[n] + y[n 2 ] (c) y[n − 2] + 2 y[n] = nx[n − 1] + 23x[n] 6. Assume that the zero-state response of an LSI system to input x[n] =2-nu[n] is y[n]=(1/3)nu[n]. Use the system properties (linearity and shift-invariance) to find the system’s response h[n] to a unit pulse input. NOTE: Please simplify your answers to all the problems as much as possible! ...

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