1. Draw the output (impulse response) of a second order resonator given by $$ H(z)=\frac{1}{(1 - z_1 z^{-1})(1 - z_1^{*} z^{-1})} $$ for the case

   a. (z_1 = \cos(\frac{\pi}{4}) + j \sin(\frac{\pi}{4}))

   b. (z_1 = 0.9(\cos(\frac{\pi}{4}) + j \sin(\frac{\pi}{4}))

Assume a sampling rate of 8kHz. What is the resonance frequence. Also draw the pole-zero plot in the z-plane. Which of the two is more likely in a physical system such as a resonator cavity and why?

2. Write a program in Octave or Scilab to synthesize a vowel given the first 3 formants, corresponding bandwidths, the pitch period and sampling rate.