Exercises

1. A sound has fundamental 440. How could it be ring modulated to give a tone at 110 Hertz with only odd partials? How could you then fill in the even ones if you wanted to?

2. A sinusoid with frequency 400 and unit peak amplitude is squared. What are the amplitudes and frequencies of the new signal's components?

3. What carrier and modulation frequencies would you give a two-operator FM instrument to give frequencies of 618, 1000, and 2618 Hertz? (This is a prominent feature of Chowning's Stria [DJ85].)

4. Two sinusoids with frequency 300 and 400 Hertz and peak amplitude one (so RMS amplitude $\approx$0.707) are multiplied. What is the RMS amplitude of the product?

5. Suppose you wanted to make FM yet more complicated by modulating the modulating oscillator, as in:
   
   $$\cos( \omega_c n + a \cos( \omega_m n + b \cos( \omega_p n )))$$
   
   
   How, qualitatively speaking, would the spectrum differ from that of the simple two-modulator example (Section 5.5)?

6. A sinusoid at a frequency $\omega$ is ring modulated by another sinusoid at exactly the same frequency. At what phase differences will the DC component of the result disappear?