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Use the Fourier transform synthesis equation (5.8) to determine the inverse Fourier 3W 2 and XX(e 0 Use your answer to determine the values of n for which x[n] = 0

Determine which, if any, of the following signals have Fourier transforms that satisfy each of the following conditions.

(a) Let $x[n]$ be a discrete-time signal with Fourier transform $X\left(e^{j\omega }\right)$, which is illustrated in Figure P5.27. Sketch the Fourier transform of $w[n]=x[n]p[n]$ for each of the following signals $p[n]$ : (i) $p[n]=\cos \pi n$ (ii) $p[n]=\cos (\pi n/2)$ (iii) $p[n]=\sin (\pi n/2)$ (iv) $p[n]={\sum }^{k=-\infty }\delta [n-2k]$ (v) $p[n]={\sum }^{k=-\infty }\delta [n-4k]$ g P5.27 The Discrete-Time Fourier Transform Chap. 5 (b) Suppose that the signal $w[n]$ of part (a) is applied as the input to an LTI system with unit sample response $h[n]=\genfrac{}{}{0ex}{}{\sin (\pi n/2)}{\pi n}.$

A causal LTI system is described by the difference equation $y[n]=y[n-1]+y[n-2]+x[n-1]\text{. }$ (a) Find the system function $H(z)=Y(z)/X(z)$ for this system. Plot the poles and zeros of $H(z)$ and indicate the region of convergence. (b) Find the unit sample response of the system. (c) You should have found the system to be unstable. Find a stable (noncausal) unit sample response that satisfies the difference equation.

(a) Consider a discrete-time system with unit sample response Min) = () un + 3 (?) un). Determine a linear constant-coefficient difference equation relating the in and output of the system. (b) Figure P5.51 depicts a block diagram implementation of a causal LTI system (i) Find a difference equation relating x[n] and y[n] for this system. (ii) What is the frequency response of the system? (iii) Determine the system's impulse response. xin N Fig P5.51