Revision as of 10:02, 29 October 2011

Homework 6, ECE438, Fall 2011, Prof. Boutin

Question 1

a)

$ y[n]= \frac{x[n]+x[n-1]}{2} $

Applying Z-transform on both sides and grouping terms, we can obtain the transfer function

$ \begin{align} Y[z]&= \frac{X[z]+X[z].z^{-1}}{2} \\ \frac{Y[z]}{X[z]}&= \frac{1+z^{-1}}{2} \\ H[z] &= \frac{1+z^{-1}}{2} \\ \end{align} $

Frequency Response H_1(ω),
$ \begin{align} H[e^{j\omega }] &= \frac{1+e^{-j\omega }}{2} \\ &= e^{-j\frac{\omega }{2}} \left( \frac{e^{j\frac{\omega }{2}}+e^{-j\frac{\omega }{2}}}{2} \right) \\ &= e^{-j\frac{\omega }{2}} cos \left( \frac{\omega }{2} \right) \\ \end{align} $

b)

$ y_2[n]= \frac{x[n]-x[n-1]}{2} $

Applying Z-transform on both sides and grouping terms, we can obtain the transfer function

$ \begin{align} Y_2[z]&= \frac{X[z]-X[z].z^{-1}}{2} \\ \frac{Y_2[z]}{X[z]}&= \frac{1-z^{-1}}{2} \\ H_2[z] &= \frac{1-z^{-1}}{2} \\ \end{align} $

Frequency Response H_2(ω),
$ \begin{align} H_2[e^{j\omega }] &= \frac{1-e^{-j\omega }}{2} \\ &= e^{-j\frac{\omega }{2}} \left( \frac{e^{j\frac{\omega }{2}}-e^{-j\frac{\omega }{2}}}{2} \right) \\ &= je^{-j\frac{\omega }{2}} \left( \frac{e^{j\frac{\omega }{2}}-e^{-j\frac{\omega }{2}}}{2j} \right) \\ &= je^{-j\frac{\omega }{2}} sin \left( \frac{\omega }{2} \right) \\ \end{align} $

Question 2

File:HW6Q2fig1.jpg

Question 3

Question 4

File:HW6Q4fig2.jpg

Question 5

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