13.9 Questions for the reader

13.9 Questions for the reader#

Exercise 81

Wet/dry mix.

  1. What wet ratio \(w\) corresponds to a wet-to-dry ratio of \(-6\) dB?

  2. What wet-to-dry ratio, in decibels, corresponds to a wet ratio of \(w = 0.25\)?

Reveal solution

  1. \(w \approx 0.334\)

  2. \(\approx -9.54\) dB

Exercise 82

Comb filter, delay, or echo? At \(f_s = 48{,}000\) Hz, consider a feedback delay line with each of the following delays. For each, compute \(\frac{f_s}{M}\), and state whether we would hear comb filtering or distinct echoes.

  1. \(M = 600\) samples

  2. \(M = 4800\) samples

  3. What is the smallest delay \(M\), in samples, at which we would hear distinct echoes?

Reveal solution

  1. \(80\) Hz: comb filtering

  2. \(10\) Hz: distinct echoes

  3. \(M = 2400\) samples

Exercise 83

Echoes and delays.

  1. You shout toward a cliff, and hear the echo \(1.5\) seconds later. Sound travels at \(343\) m/s. How far away is the cliff?

  2. A guitarist plays a note every \(0.3\) seconds, and wants a delay effect whose echoes fall halfway between the notes. At \(f_s = 44{,}100\) Hz, what delay \(M\) should they use?

Reveal solution

  1. \(\approx 257.3\) m

  2. \(M = 6615\) samples

Exercise 84

Flanging. A flanger at \(f_s = 44{,}100\) Hz has a base delay of \(M_c = 2\) ms and a relative deviation of \(A = 0.5\).

  1. What range of delays does it sweep, in samples?

  2. Over what range does the lowest nonzero tooth of its comb, \(\frac{f_s}{M[n]}\), sweep?

  3. If its delay sweeps at a rate of \(f_m = 0.5\) Hz, what is the depth \(D\) of the vibrato that its delay applies to a \(1000\) Hz sinusoid?

Reveal solution

  1. From \(44.1\) to \(132.3\) samples

  2. From \(\approx 333.3\) Hz to \(1000\) Hz

  3. \(D = 2\pi \cdot 0.5 \cdot 0.5 \cdot 0.002 \cdot 1000 = \pi \approx 3.14\) Hz

Exercise 85

Multitap delay lines. A delay line of length \(M' = 9\) samples has taps at \(M_a = 2\) and \(M_b = 5\). What are the delays of the three blocks between the input, the taps, and the end of the delay line?

Reveal solution

\(z^{-2}\), \(z^{-3}\), and \(z^{-4}\)

Exercise 86

All-pass filters. Write down the first seven samples of the impulse response of the all-pass filter \(y[n] = -0.5 \cdot x[n] + x[n - 2] + 0.5 \cdot y[n - 2]\).

Reveal solution

\(h = [-0.5, 0, 0.75, 0, 0.375, 0, 0.1875]\)

Exercise 87

Reverberation time. In a Schroeder reverberator at \(f_s = 44{,}100\) Hz with a reverberation time of \(T_{60} = 2\) seconds, what gain \(g\) should a comb filter with a delay of \(30\) ms use?

Reveal solution

\(g \approx 0.9016\)

Exercise 88

Chebyshev waveshaping. A full-scale sinusoid at \(220\) Hz passes through a waveshaper. What frequency comes out if the transfer function is

  1. \(T_2(x) = 2x^2 - 1\)?

  2. \(T_3(x) = 4x^3 - 3x\)?

Reveal solution

  1. \(440\) Hz

  2. \(660\) Hz

Exercise 89

RMS and compression.

  1. What is the RMS amplitude of a square wave that alternates between \(-1\) and \(1\)?

  2. What is the RMS amplitude of a basic sinusoid with peak amplitude \(1\), in decibels?

  3. A compressor has a threshold of \(-20\) dB and a ratio of \(R = 4\). What output level, and what gain, does it apply to an input level of \(-10\) dB? And to an input level of \(-30\) dB?

  4. At \(f_s = 48{,}000\) Hz, what hop length measures the RMS level \(100\) times per second?

Reveal solution

  1. \(1\)

  2. \(\approx -3.01\) dB

  3. \(-17.5\) dB, with a gain of \(-7.5\) dB. The level of \(-30\) dB passes through unchanged, with a gain of \(0\) dB.

  4. \(N_H = 480\) samples

Exercise 90

Panning and width.

  1. With constant-power panning at \(\alpha = 0.25\), what are the left and right gains?

  2. A stereo sample has \(x_L = 0.8\) and \(x_R = 0.2\). What are \(x_M\) and \(x_S\)? If we double the side signal, what are the new \(x_L\) and \(x_R\)?

Reveal solution

  1. \(\approx 0.924\) (left) and \(\approx 0.383\) (right)

  2. \(x_M = 1.0\) and \(x_S = 0.6\). Then \(x_L = 1.1\) and \(x_R = -0.1\).