6.9 Questions for the reader

6.9 Questions for the reader#

Exercise 29

Ring-modulation sidebands. Consider the ring-modulated signal \(\cos(440\pi t) \cdot \sin(8\pi t)\).

  1. Identify the carrier and modulating frequencies in Hz.

  2. List the frequencies of all of the sidebands it produces, remembering to include negative frequencies.

  3. Which sidebands would you actually hear, and at what frequencies?

Reveal solution

  1. Carrier \(220\) Hz, modulator \(4\) Hz

  2. Sidebands at \(\pm 216\) and \(\pm 224\) Hz

  3. You hear tones at \(216\) and \(224\) Hz.

Exercise 30

Tremolo or two tones? A 300 Hz carrier is ring-modulated by a modulator at frequency \(f_m\). For which of \(f_m = 3\) Hz, \(f_m = 40\) Hz, and \(f_m = 150\) Hz would you expect to hear a single tone with tremolo, and for which would you expect to hear two distinct tones? Justify your answer in terms of the sideband frequencies.

Reveal solution

\(f_m = 3\) Hz gives tremolo (sidebands at \(297\) and \(303\) Hz); \(f_m = 40\) Hz and \(f_m = 150\) Hz each give two distinct tones.

Exercise 31

Designing a harmonic spectrum with AM. Recall the amplitude-modulation form with a carrier-to-sideband ratio \(r\):

\[\text{AmpMod}(t) = \sin(\omega_c t)\Big[\tfrac{r}{2} + \sin(\omega_m t)\Big].\]

Choose \(\omega_c\), \(\omega_m\), and \(r\) so that the output is a harmonic spectrum at fundamental \(f_0 = 440\) Hz with exactly three equal-amplitude harmonics. (Hint: AM places components at \(\omega_c\) and at \(\omega_c \pm \omega_m\).)

Reveal solution

\(\omega_c = 1760\pi\) (so \(f_c = 880\) Hz), \(\omega_m = 880\pi\) (so \(f_m = 440\) Hz), and \(r = 1\). The three components land at \(440\), \(880\), and \(1320\) Hz — the first three harmonics of \(f_0 = 440\) Hz — each with amplitude \(\tfrac{1}{2}\).

Exercise 32

Negative frequencies and phase. A sine tone is written as \(\sin(-660 \pi t)\).

  1. Rewrite it as a positive-frequency sinusoid, stating both its frequency in Hz and any phase shift.

  2. Would it sound any different from \(\sin(660\pi t)\)? Why or why not?

Reveal solution

  1. \(\sin(-660\pi t) = \sin(660\pi t + \pi)\): a \(330\) Hz tone with a \(\pi\) phase shift.

  2. It sounds identical, since hearing is insensitive to absolute phase for sine tones.

Exercise 33

Why integrate? Suppos you implemented a variable-frequency basic sinusoid by evaluating \(\sin(\omega(t)\cdot t)\) directly, where \(\omega(t)\) sweeps linearly from a low to a high frequency. Explain conceptually why the resulting sound will not match the intended frequency sweep, and describe what must be computed instead.

Exercise 34

FM parameters. An FM tone has carrier frequency \(f_c = 300\) Hz, modulating frequency \(f_m = 100\) Hz, and depth \(D = 400\) Hz.

  1. What is the index of modulation \(I\)?

  2. At which frequencies do the first three sidebands on each side of the carrier appear?

  3. Roughly how many sidebands per side would you expect to be audible?

  4. Is the resulting tone harmonic or inharmonic, and why?

Reveal solution

  1. \(I = D/f_m = 4\)

  2. First three sidebands below the carrier: \(200, 100, 0\) Hz; above: \(400, 500, 600\) Hz.

  3. About \(I + 1 = 5\) per side are audible.

  4. Harmonic, because \(f_c : f_m = 3 : 1\) is an integer ratio.