7.6 Filtering: numerical experiment (frequency response of first

Consider the exponential moving average filter

1. Write out a few terms of the sum to show how the filter works.
2. Write x _n as a recursion and discuss the computer memory required to implement the filter.
3. Compute the complex frequency response $H\left(e^{j\theta }\right)$ for the filter.
4. Write a MATLAB program to plot the magnitude and phase of the complex frequency response $H\left(e^{j\theta }\right)$ versus θ for $\theta =-\pi \text{to}+\pi$ in steps of $\frac{2\pi }{64}$ Do this for two values of a , namely, $a=0.98$ and $a=-0.98$ . Explain your findings.
5. Write a MATLAB program to pass the following signals through the filter when $a=0.98$ :
   1. $u_n={\delta }_n$
   2. $u_n={\xi }_n$
   3. $u_n={\xi }_{n}cos\frac{2\pi }{64}n$
   4. $u_n={\xi }_{n}cos\frac{2\pi }{32}n$
   5. $u_n={\xi }_{n}cos\frac{2\pi }{16}n$
   6. $u_n={\xi }_{n}cos\frac{2\pi }{8}n$
   7. $u_n={\xi }_{n}cos\frac{2\pi }{4}n$
   8. $u_n={\xi }_{n}cos\frac{2\pi }{2}n$ .

Plot the outputs for each case and interpret your findings in terms of the complex frequency response $H\left(e^{j\theta }\right)$ . Repeat step 5 for $a=-0.98$ . Interpret your findings.