3.6 Questions for the reader

3.6 Questions for the reader#

Exercise 8

Angular frequency conversion. A sinusoid has angular frequency \(\omega = 1000\pi\) \(\frac{\text{radians}}{\text{second}}\).

  1. What is its frequency in Hertz?

  2. What is its period in seconds?

Reveal solution

  1. \(f = \omega / 2\pi = 500\) Hz

  2. Period \(T = 1/f = 0.002\) s (2 ms).

Exercise 9

Phase periodicity. Is the instantaneous phase \(\theta(t) = \omega t + \phi\) a periodic function of \(t\)? Why or why not?

Reveal solution

No. It grows without bound as \(t\) increases, so it never repeats.

Exercise 10

Waveform identification. Given a periodic waveform whose Fourier coefficients are \(a_k = 0\) for even \(k\) and \(a_k \propto 1/k\) for odd \(k\), identify which classic waveform shape this most closely resembles. What would change perceptually if the amplitudes were instead \(a_k \propto 1/k^2\) for odd \(k\)?

Reveal solution

A square wave. With \(a_k \propto 1/k^2\) it would resemble a triangle wave, sounding mellower with weaker high harmonics.

Exercise 11

Harmonic frequencies. A tone has fundamental frequency \(f_0 = 330\) Hz. What are the frequencies of its first five harmonics?

Reveal solution

\(330, 660, 990, 1320, 1650\) Hz.

Exercise 12

Fundamental frequency of a sum. Consider the periodic waveform \(x(t) = \sin(8\pi t) + \tfrac{1}{2}\cos(16\pi t) + \tfrac{1}{4}\sin(24\pi t)\). Give the frequency in Hz of each of the three components. Then state the fundamental frequency \(f_0\) of the combined waveform, and explain your reasoning. (Hint: \(f_0\) is the largest frequency for which every component is a harmonic, that is, an integer multiple, of \(f_0\).)

Reveal solution

Harmonics at \(4\), \(8\), and \(12\) Hz; the fundamental is \(f_0 = 4\) Hz.

Exercise 13

Phase increment. A wavetable has \(M = 2048\) entries and you want to synthesize a tone at \(f_0 = 261.63\) Hz (middle C) with sample rate \(f_s = 44{,}100\) Hz.

  1. What is the phase increment \(\Delta m\)?

  2. After 100 output samples, at what table index would you be reading?

Reveal solution

  1. \(\Delta m = M f_0 / f_s \approx 12.15\) table indices per sample

  2. After 100 samples you are near index \(1215\).

Exercise 14

Wavetable complexity. Suppose you need to synthesize 10 different notes simultaneously, each using a sawtooth waveform with \(K = 8\) harmonics. Compute the total number of sin evaluations needed to synthesize one second of audio at \(f_s = 1000\) Hz using (1) direct additive synthesis, and (2) wavetable synthesis with a table size of \(512\) (assuming all notes share the same table).

Reveal solution

Direct additive: \(80000\). Wavetable synthesis: \(4096\)