Exercises

1. A complex number has magnitude one and argument $\pi /4$. What are its real and imaginary parts?
2. A complex number has magnitude one and real part $1/2$. What is its imaginary part? (There are two possible values.)
3. What delay time would you give a comb filter so that its first frequency response peak is at 440 Hz.? If the sample rate is 44100 Hz., what frequency would correspond to the nearest integer delay?
4. Suppose you made a variation on the non-recirculating comb filter so that the delayed signal was subtracted from the original instead of adding. What would the new frequency response be?
5. Derive an explicit formula for the frequency response of a recirculating comb filter with delay time $d$ and feedback gain $g$, as a function of angular frequency $\omega$.

   
   

   $$\mathrm{(answer:} {[ {{(1-g \cos (2 \pi \omega d))}^2} + {{(g \sin (2 \pi \omega d))}^2} ]} ^ {-1/2} \mathrm{)}$$
   
   
6. If you want to make a 6-Hz. vibrato with a sinusoidally varying delay line, and if you want the vibrato to change the frequency by 5%, how big a delay variation would you need? How would this change if the same depth of vibrato was desired at 12 Hz.?