Sampling, quantization and PCM

Most information that travels today begins as an analog waveform and ends up as bits. A voice, a temperature or a vibration is sampled at regular instants, each sample is rounded to one of a finite set of levels, and each level is written as a binary code word. This process, pulse-code modulation, is what made the digital telephone network possible, and it is the front end of every digital communication system that carries analog information. This lesson reviews the sampling theorem from a communication point of view, analyzes quantization noise and the signal-to-quantization-noise ratio, explains companding, and shows how differential and adaptive methods reduce the bit rate.

The TLR-1 is mostly a data radio, but the city has asked for one analog feature. When a resident complains that an accessible signal is too quiet or too loud, the traffic technician would like to listen to the street sound at the push-button station from the cabinet, without climbing the pole, to check the automatic volume adjustment that ELE-340's APS-2 amplifier performs. Amara has to decide how to digitize a short clip of the microphone signal and whether the radio link can carry it.

Sampling revisited

ELE-250 established the sampling theorem: a signal band-limited to W hertz is completely described by samples taken at a rate f_s > 2W, and it can be recovered by an ideal low-pass filter. Sampling multiplies the signal by an impulse train, so the spectrum of the samples consists of copies of the signal's spectrum repeated at every multiple of f_s. If f_s is less than 2W, the copies overlap and aliasing makes some frequencies impossible to separate. In practice an anti-aliasing filter limits the signal before sampling, and the rate is chosen with a guard band to let a real filter roll off. Telephone speech is filtered to about 3.4 kHz and sampled at 8 kHz, leaving a guard band of about 1.2 kHz between 3.4 kHz and the folding frequency of 4 kHz.

Real samplers do not take instantaneous samples. Pulse-amplitude modulation (PAM) with flat-top pulses, produced by a sample-and-hold circuit, holds each sample for a time τ. Holding is a convolution with a rectangular pulse, so the spectrum is multiplied by a sinc function, which attenuates high frequencies slightly. This aperture effect is small if τ is much less than the sampling period, and an equalizer with the inverse response can correct it. When τ equals the full sampling period, as in a zero-order hold at the output of a digital-to-analog converter, the droop at the top of the band reaches about 3.9 dB at the folding frequency.

Uniform quantization

A quantizer maps each sample to the nearest of L levels. With n bits per sample, L = 2ⁿ. For a uniform quantizer covering the range −V_max to +V_max, the step size is Δ = 2V_max/L. The difference between the true sample and its quantized value is the quantization error. If the signal is busy and covers many steps, the error is equally likely anywhere between −Δ/2 and +Δ/2 and is unrelated to the signal, so it behaves like additive noise with a uniform distribution.

For a full-scale sine wave, of amplitude V_max and power V_max²/2, the signal-to-quantization-noise ratio follows directly. Each added bit halves Δ and quarters the noise power, a gain of 6.02 dB. This is the most quoted rule in data conversion.

Real signals are not full-scale sine waves. Speech has a peak-to-average power ratio, or crest factor, of 10 dB or more, so its average power is well below that of a sine of the same peak, and the SQNR is correspondingly lower. A quiet talker 30 dB below full scale loses 30 dB of SQNR with a uniform quantizer, because the noise Δ²/12 is fixed while the signal shrinks. If the signal exceeds V_max, it clips, causing overload distortion that is far more objectionable than quantization noise. A designer sets the quantizer range, or the gain before it, to balance the two: high enough to avoid frequent clipping, low enough to use most of the levels.

Companding

The quiet-talker problem has an elegant solution: make the steps small for small signals and large for large ones, so the SQNR stays roughly constant over a wide range of levels. This is a non-uniform quantizer, and it is usually built as a compressor, which squeezes large values with a logarithmic curve, followed by a uniform quantizer; the receiver applies the inverse expander. Compressing and then expanding is called companding. North American and Japanese telephone systems use the μ-law with μ = 255; Europe and most other countries use the A-law with A = 87.6. Both give 8-bit telephone channels a usable SQNR of about 35 to 38 dB over a range of about 40 dB of input level, roughly what 12 or 13 uniform bits would give at the top of the range.

import math
def quantize(v, n):                      # uniform mid-rise quantizer on [-1, 1)
    L = 2 ** n; d = 2 / L
    k = min(L - 1, max(0, int((v + 1) / d)))
    return -1 + (k + 0.5) * d
def sqnr(signal, out):
    ps = sum(s * s for s in signal); pn = sum((s - o) ** 2 for s, o in zip(signal, out))
    return 10 * math.log10(ps / pn)
N = 20000
sine = [math.sin(2 * math.pi * 0.0123457 * k) for k in range(N)]   # full-scale test tone
for n in (4, 8, 12):
    q = [quantize(s, n) for s in sine]
    print(f"{n:2d} bits: measured SQNR = {sqnr(sine, q):5.1f} dB, formula = {6.02*n + 1.76:5.1f} dB")
mu = 255
comp = lambda x: math.copysign(math.log1p(mu * abs(x)) / math.log1p(mu), x)
expand = lambda y: math.copysign(((1 + mu) ** abs(y) - 1) / mu, y)
for level_db in (0, -10, -20, -30, -40):
    a = 10 ** (level_db / 20)
    sig = [a * s for s in sine]
    uni = [quantize(s, 8) for s in sig]
    law = [expand(quantize(comp(s), 8)) for s in sig]
    print(f"tone at {level_db:4d} dB: 8-bit uniform {sqnr(sig, uni):5.1f} dB, 8-bit mu-law {sqnr(sig, law):5.1f} dB")

 4 bits: measured SQNR =  25.3 dB, formula =  25.8 dB
 8 bits: measured SQNR =  49.9 dB, formula =  49.9 dB
12 bits: measured SQNR =  74.2 dB, formula =  74.0 dB
tone at    0 dB: 8-bit uniform  49.9 dB, 8-bit mu-law  37.5 dB
tone at  -10 dB: 8-bit uniform  40.8 dB, 8-bit mu-law  39.2 dB
tone at  -20 dB: 8-bit uniform  30.0 dB, 8-bit mu-law  38.4 dB
tone at  -30 dB: 8-bit uniform  19.1 dB, 8-bit mu-law  36.5 dB
tone at  -40 dB: 8-bit uniform   9.3 dB, 8-bit mu-law  34.3 dB

The measured values agree with the formula within a fraction of a decibel. The companding comparison shows the point of μ-law: at full scale, the 8-bit uniform quantizer is better, but every 10 dB drop in level costs it 10 dB of SQNR, while the μ-law quantizer stays near 38 dB until the signal is about 30 dB down, falling only gradually below that. For speech, where levels vary widely, μ-law is the better use of 8 bits.

Pulse-code modulation

In PCM each quantized sample is represented by an n-bit code word, and the code words are sent one after another as a stream of binary pulses. The bit rate is the sampling rate times the bits per sample. For telephony, 8000 samples per second at 8 bits gives the 64 kb/s channel on which the whole digital telephone hierarchy was built. Multiplexing 24 such channels with one framing bit per frame of 24 samples gives the North American T1 or DS1 rate of 1.544 Mb/s; Europe's E1 carries 32 time slots at 2.048 Mb/s. Time-division multiplexing interleaves the code words of several channels in fixed slots, the counterpart in time of frequency-division multiplexing.

The price of PCM is bandwidth. A 3.4 kHz voice channel becomes 64 kb/s, which lesson 6 will show needs at least 32 kHz of baseband bandwidth, nearly ten times the original. The reward is that the bits can be regenerated, error-checked, encrypted, switched and stored. With enough SNR, the only lasting impairment is the quantization noise introduced once at the source.

Differential and adaptive methods

Successive samples of speech, or of most physical signals, are strongly correlated: the next sample is usually close to the last. Differential PCM (DPCM) quantizes the difference between each sample and a prediction formed from previous samples. The difference is much smaller than the sample itself, so fewer bits give the same SQNR. The prediction is made from the decoded values, not the original ones, so the transmitter and receiver predict identically and quantization errors do not accumulate. The improvement over PCM is called the prediction gain, often 5 to 10 dB for speech.

Delta modulation is the extreme case: one bit per sample, saying only whether the signal is above or below the running estimate, which then steps up or down by a fixed amount δ. It needs a sampling rate many times the Nyquist rate. Two kinds of error appear. When the signal changes faster than δ per sample, the estimate cannot keep up, called slope overload. When the signal is nearly constant, the estimate hunts up and down by δ, called granular noise. Choosing δ trades one against the other, and adaptive delta modulation changes δ on the fly. The same adaptive idea in DPCM gives ADPCM, which carries telephone-quality speech at 32 kb/s, half the PCM rate, and is still used in cordless telephones and some radio systems.

Modern speech codecs go much further. Instead of describing the waveform, they model the vocal tract with a short-term filter, transmit its coefficients and a compact description of the excitation, and let the decoder synthesize speech that sounds right. Codecs of this kind carry intelligible speech at 2 to 13 kb/s and are used in cellular telephones and digital land-mobile radio.

The listen-in feature

Amara's first estimate rules out real-time listening. The TLR-1 link, as lessons 8 to 12 will show, will run at about 50 kb/s and is shared by four pole stations, with button presses taking priority. A 64 kb/s μ-law stream does not fit. She proposes a store-and-forward clip instead: the APS-2 records 4 seconds of street sound at 8 kHz with 4-bit ADPCM, 16 kilobytes in total, and sends it in low-priority packets over several seconds. The technician hears it a few seconds late, which is acceptable for checking a volume setting, and button messages are never delayed. The microphone channel's 300 Hz to 3 kHz band, from ELE-250, sampled at 8 kHz with the existing anti-aliasing filter, needs no new hardware.