7.7 Questions for the reader

7.7 Questions for the reader#

Exercise 35

Computing aliases. A signal is sampled at \(f_s = 8\) kHz. For each of the following pure tones, give the frequency that will actually be heard, and say whether it is aliased:

  1. 3 kHz

  2. 5 kHz

  3. 9 kHz

  4. 16 kHz

  5. -5 kHz

Reveal solution

  1. \(3\) kHz \(\to 3\) kHz (not aliased)

  2. \(5\) kHz \(\to 3\) kHz (aliased)

  3. \(9\) kHz \(\to 1\) kHz (aliased)

  4. \(16\) kHz\( \)\to 0$ Hz (aliased)

  5. \(-5\) kHz \(\to 3\) kHz (aliased)

Exercise 36

Choosing a sample rate. You want to faithfully sample a signal that contains frequency content up to (but not including) 15 kHz.

  1. What is the minimum sample rate required by the Nyquist-Shannon theorem?

  2. If you were restricted to sampling at 24 kHz, what would you need to do to the signal first to prevent aliasing, and why?

Reveal solution

  1. Minimum \(30\) kHz.

  2. To prevent aliasing at \(24\) kHz, you must first low-pass the signal below the \(12\) kHz Nyquist frequency.

Exercise 37

Two signals, same samples. Give the positive frequencies of two different pure sinusoids (other than the one itself) that would be indistinguishable from a 1 kHz tone when sampled at \(f_s = 6\) kHz. Explain using the definition of an alias.

Reveal solution

\(5\) kHz and \(7\) kHz (among many others)

Exercise 38

Decibels. Answer each of the following.

  1. An amplitude is scaled by a factor of 4. By how many dB does it change?

  2. A signal sits at \(-18\) dBFS. By what linear factor must you scale its amplitude to bring it to 0 dBFS?

  3. Multiplication by an amplitude factor of \(0.001\) would be equivalent to an addition of how many decibels?

Reveal solution

  1. About \(+12\) dB

  2. A factor of about \(8\)

  3. \(-60\) dB

Exercise 39

Bit depth and dynamic range. A recording is quantized to 8 bits per sample.

  1. Approximately what dynamic range (in dB) does this provide?

  2. If the quietest sounds you care about are 60 dB below the loudest, is 8-bit quantization sufficient?

  3. How many bits would you choose to comfortably cover a 92 dB range?

Reveal solution

  1. About \(48\) dB

  2. \(8\)-bit is not enough for a \(60\) dB dynamic range

  3. Use \(16\) bits to comfortably cover \(92\) dB.

Exercise 40

Resampling arithmetic. A 4-second clip is sampled at 48 kHz.

  1. How many samples does it contain?

  2. You resample it to 16 kHz, preserving its duration. How many samples does the result contain, and what is its new Nyquist frequency?

Reveal solution

  1. \(192{,}000\) samples

  2. \(64{,}000\) samples with an \(8\) kHz Nyquist frequency

Exercise 41

Designing a PCM protocol. You must encode a mono signal whose highest frequency of interest is \(4\) kHz using linear PCM at a total bitrate of exactly \(100{,}000\) bits per second. Recall that a mono PCM stream’s bitrate is \(f_s \cdot b\), where \(f_s\) is the sample rate and \(b\) is the bit depth. Consider three candidate protocols: (i) \(f_s = 20{,}000\) Hz with \(b = 5\); (ii) \(f_s = 10{,}000\) Hz with \(b = 10\); (iii) \(f_s = 6{,}250\) Hz with \(b = 16\).

  1. Verify that all three hit the target bitrate.

  2. Which of them have a Nyquist frequency high enough to represent the \(4\) kHz content without aliasing?

  3. Among only the protocols that survive both of the previous checks, which maximizes fidelity, and why? (Recall that each additional bit adds roughly \(6\) dB of dynamic range.)

Reveal solution

  1. All three hit \(100{,}000\) bps.

  2. Protocols (i) and (ii) have a high enough Nyquist frequency (\(10\) kHz and \(5\) kHz)

  3. (iii) does not (\(3.125\) kHz). Of the survivors, (ii) maximizes fidelity, since \(10\) bits beat \(5\) bits.