1. (a) The impulse response h[n] of a linear time-invariant system is known to be zero, except in the interval N0≦n≦N1. The input x[n] is known to be zero, except on the interval N2≦n≦N3. As a result, the output is constrained to be zero, except in the some interval N4≦n≦N5. Determine N4 and N5 in terms of N0, N1, N2, and N3. (b) If x[n] is zero, except for N consecutive points, and h[n] is zero, except for M consecutive points, what is the maximum number of consecutive points for which y[n] can be nonzero? 2. By direct evaluation of the convolution sum, determine the step response of a linear time-invariant system whose impulse response is h[n]=a^(-n) u[-n] 3. Consider an LTI system with frequency response H(e^jw)= (1-e^-j2w)/(1+1/2e^-j4w) -pi<w<pi Determine the output y[n] for all n if the input x[n] for all n is x[n]=sin(n pi/4) 4. Consider a system with input x[n] and output y[n] that satisfy the difference equation y[n]=ny[n-1]+x[n] The system is causal and satisfies initial-rest conditions; i.e., if x[n] = 0 for n < n0, then y[n] = 0 for n < n0. (a) If x[n] = δ[n], determine y[n] for all n. (b) Is the system linear? Justify your answer. (c) Is the system time invariant? Justify your answer. 先謝謝大家 --

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