Z-Transform and DTFT

by zcd
Tags: dtft, ztransform
 P: 200 1. The problem statement, all variables and given/known data You are given the following pieces of information about a real, stable, discrete-time signal x and its DTFT X, which can be written in the form $$X(\omega)=A(\omega)e^{i\theta_x(\omega)}$$ where $$A(\omega)=\pm|X(\omega|$$. a) x is a finite-length signal b) $$\hat{X}$$ has exactly two poles at z=0 and no zeros at z=0. c) $$\theta_x(\omega)=\begin{cases} \frac{\omega}{2}+\frac{\pi}{2} & 0<\omega<\pi \\ \frac{\omega}{2}-\frac{\pi}{2} & -\pi<\omega<0\end{cases}$$ d) $$X(\omega)\Big|_{\omega=\pi}=2$$ e) $$\int_{-\pi}^\pi e^{2i\omega}\frac{d}{d\omega}X(\omega)d\omega =4\pi i$$ f) The sequence v whose DTFT is V (ω) = Re (X(ω)) satisﬁes v(2) = 3/2. 2. Relevant equations $$\hat{H}(z)=\sum_{n=-\infty}^\infty h(n)z^{-n}$$ $$H(\omega)=\hat{H}(z)\Big|_{z=e^{i\omega}}$$ 3. The attempt at a solution By parts e) and f) I've figured out that x(-2)=4 and x(2)=-1. With that and part a) and b), I've realized that the rightmost endpoint of x is x(2)=-1 and the leftmost endpoint of x is less than n=-2 (the transfer function can't converge at infinity since the system cannot be causal). Part d) gives me $$\sum_{n=-\infty}^\infty (-1)^n x(n) = 2$$, but I'm unsure how to use it at the moment. I do not know how to use part c) at all. Can someone help shed some light on the problem?

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