Squeezing speech through a wire.

From one mouth to another ear.

Follow your voice on a phone call. It is turned into numbers, squeezed, sent, unsqueezed and turned back into sound:

What is a bit rate?

A digital audio signal is a sequence of numbers: samples, taken \(f_s\) times per second, each stored with \(b\) bits. The bits per second needed to send it raw is simply

The catch: the listener must still understand the words, recognise the speaker, and not be annoyed. Engineers use rough quality classes: toll quality (as good as a traditional phone line), communication quality (clearly intelligible, noticeably degraded) and synthetic quality (intelligible, but robotic). The rest of this page is a race down the bit rate while trying to stay in the best class possible.

Fewer bits per sample, or fewer samples per second.

The formula has two knobs that don't need any knowledge of speech at all. Both work on the waveform directly: they try to reproduce the exact shape of the signal, sample by sample.

Knob 1: fewer bits per sample (quantisation)

With \(b\) bits, each sample must be rounded to one of \(2^b\) levels. Fewer bits means fewer, coarser levels, and the rounding error becomes audible as a hiss, then a crackle, then distortion.

Each extra bit doubles the number of levels, which halves the step size and therefore halves the typical rounding error. Halving an amplitude is −6 dB, so:

Deriving \(\text{SNR} \approx 6.02\,b + 1.76\) dB

Step 1: the step size. A uniform quantiser with \(b\) bits splits the full range \([-1, 1)\) into \(2^b\) equal steps:

$$\Delta = \frac{2}{2^b} = 2^{1-b}$$

Step 2: the noise power. Rounding to the centre of a step leaves an error \(q\) that always lies within \(\pm\Delta/2\) (Figure 2, bottom row). When the step is small compared with the signal's wiggles, every value in that interval is equally likely, so the error is uniformly distributed on \([-\Delta/2, \Delta/2]\) with density \(1/\Delta\). Its power is its mean square:

$$\sigma_q^2 = \int_{-\Delta/2}^{\Delta/2} q^2\,\frac{1}{\Delta}\,dq = \frac{1}{\Delta}\left[\frac{q^3}{3}\right]_{-\Delta/2}^{\Delta/2} = \frac{1}{\Delta}\cdot\frac{2}{3}\cdot\frac{\Delta^3}{8} = \frac{\Delta^2}{12}$$

Step 3: the signal power. A sinusoid that just fills the range, \(x(t) = \sin(\omega t)\), has power equal to its mean square, \(\overline{\sin^2} = \tfrac12\).

Step 4: the ratio.

$$\text{SNR} = \frac{1/2}{\Delta^2/12} = \frac{6}{\Delta^2} = \frac{6}{2^{2-2b}} = \frac{3}{2}\cdot 2^{2b}$$

Step 5: in decibels, the product becomes a sum (see the decibels page):

$$10\log_{10}\text{SNR} = \underbrace{20\log_{10}2}_{6.02}\cdot b \;+\; \underbrace{10\log_{10}\tfrac32}_{1.76} \;\approx\; 6.02\,b + 1.76\ \text{dB}$$

The 6.02 is “halving the step halves the error amplitude”, \(20\log_{10}2\). The 1.76 depends only on the signal's shape: it is the sinusoid's power of \(\tfrac12\) compared with the \(\tfrac{1}{12}\) in the noise. For 16-bit audio this gives the familiar 98 dB, and for 8 bits about 50 dB. A quick simulation with a full-scale sinusoid gives 49.8 dB at 8 bits.

For any signal, Step 3 changes. A signal with rms value \(\sigma_x\) (full scale = 1) has power \(\sigma_x^2\), and the same algebra gives

$$\text{SNR} \approx 6.02\,b + 4.77 + 20\log_{10}\sigma_x\ \text{dB}$$

The last term is the signal's level in dBFS, and it is negative. A full-scale sine has \(\sigma_x = 1/\sqrt2\) (−3.01 dB), which gives back 1.76. Our recording's rms level is only about −19 dBFS, because speech has tall peaks and long quiet stretches. That predicts \(6.02 \times 8 + 4.77 - 18.9 \approx 34.0\) dB at 8 bits, and quantising the recording gives 34.0 dB. That is 15 dB worse than a sinusoid with the same number of bits. The segmental SNR of Figure 6 is lower still (27.9 dB at 8 bits), because the quiet frames sit far below that average level. This penalty for quiet speech is exactly what \(\mu\)-law, in the next section, removes.

The uniform-error assumption fails when the signal is smaller than a few steps (Figure 3). At 1–2 bits, or for very quiet passages, the error becomes correlated with the signal and the formula is only a rough guide.

Knob 2: fewer samples per second (sampling rate)

A signal sampled at \(f_s\) can only contain frequencies up to \(f_s/2\) (the Nyquist limit). Lowering \(f_s\) therefore cuts off the high frequencies: first the hiss of /s/ and /f/, then the upper formants, until the speech sounds muffled, as if through a wall.

Try both knobs

Choose a sampling rate, a number of bits and a quantiser. The recording is processed live in your browser. Watch the bit rate, and listen.

Listening examples

All examples on this page use one recording of an Indian-English speaker, at a range of bit rates. Segmental SNR (average SNR over 20 ms frames, higher is better) is given where the waveform is meant to match the original.

Both knobs work up to a point, and then the price in quality becomes steep. To get from 256 down to 32 kbit/s you must either accept heavy noise (4 bits at 8 kHz) or a badly muffled voice (2 kHz sampling). Neither knob knows anything about speech or about hearing.

\(\mu\)-law: spend the levels where the signal actually is.

A uniform quantiser treats every amplitude the same: its levels are evenly spaced from \(-1\) to \(+1\). Two facts about speech and hearing say that is a waste. We'll take them one at a time.

Fact 1 · Most speech samples are small

Make a histogram of every sample in our recording (Figure 5a). It is a tall, narrow spike around zero: 75 % of all samples lie within \(\pm 0.1\), a tenth of the full range. Loud peaks are rare; quiet consonants, the tails of syllables, pauses and soft passages are common. Now look at where a 4-bit uniform quantiser puts its 16 levels (grey ticks under the histogram): spread evenly across the whole range, so only 2 of the 16 levels fall where nearly all the samples are. Most of the levels sit out at large amplitudes that hardly ever occur.

Fact 2 · The ear judges noise relative to loudness

A uniform quantiser adds noise of the same absolute size (up to \(\pm\Delta/2\)) everywhere. In a loud vowel, a little hiss is masked by the speech. In a quiet consonant or a soft-spoken word, the same hiss is relatively much bigger and clearly audible. What matters to a listener is roughly the ratio of signal to noise in each moment, not the absolute noise level. (This is the same relative sensitivity that makes us measure sound in decibels.)

The fix · fine steps for small values, coarse steps for big ones

Both facts point the same way: put the quantiser levels close together near zero and further apart for large amplitudes. Ideally the step size grows in proportion to the amplitude, so the noise stays a fixed fraction of the signal. That is a logarithmic spacing. The neat trick is that we don't need a new quantiser to get it. We bend the signal first and keep the simple uniform one:

The curve used in North American and Japanese telephony is \(\mu\)-law:

Try it: bend the levels yourself

The histogram of our recording's samples, with the quantiser's levels drawn on top. Raise \(\mu\) from 0 (uniform) and watch the levels migrate towards zero, where the samples are.

This is G.711 (1972): \(8000 \times 8\) bits/s with \(\mu\)-law (North America, Japan) or the very similar A-law (Europe, India and most of the world), giving 64 kbit/s. In the standard the smooth curve is approximated by 8 straight-line segments per polarity, cheap to build in 1970s hardware. G.711 has been the backbone of digital telephone networks for half a century, and it is the reference that “toll quality” is measured against. Compare 8-bit uniform with 8-bit \(\mu\)-law in the listening grid, and 4-bit uniform with 4-bit \(\mu\)-law in section 05: same bit rate, very different noise.

Why \(\mu = 255\)? An engineering compromise

\(\mu\) is a dial between two goals that pull against each other. A telephone network has to serve soft talkers and loud talkers, on short and long lines. Their levels differ by tens of dB, and every one of them should get a decent SNR. Measure the SNR of an 8-bit coder as the input level is swept from very soft to full scale (Figure 7):

So the choice of \(\mu\) answers a practical question: how wide a range of talker levels must be served, and how much peak quality are we willing to give up for it? The history, as usually told:

• 1957: the family. Bernard Smith at Bell Labs analysed “instantaneous companding” and formulated the logarithmic \(\mu\)-law curves, showing how \(\mu\) trades dynamic range against peak SNR, the trade-off of Figure 7.
• 1962: \(\mu = 100\), 7 bits. The first commercial digital telephone system, Bell's T1 carrier, coded speech with 7-bit \(\mu\)-law and \(\mu = 100\).
• Around 1970: \(\mu = 255\), 8 bits. The next generation of equipment moved to 8 bits. With one more bit, a stronger curve could keep an SNR in the mid-30s of dB across the full range of talker levels found in the network. There was also a hardware reason. The smooth curve is built from 8 straight-line segments per polarity, each twice as long as the one before, and with \(\mu = 2^8 - 1 = 255\) the segment boundaries fall on binary-friendly values. A logarithm could then be computed with simple 1970s digital logic: a 3-bit “which segment” (exponent) and a 4-bit “where in the segment” (mantissa), a tiny floating-point number.
• A-law. European engineers chose a slightly different curve with \(A = 87.6\), also built from straight segments. It is very close to \(\mu\)-law in performance.

Note what drove the choice: measured talker levels, SNR targets and what hardware could build cheaply. These were engineering numbers, checked against the listening experience of telephone engineers. Formal, large-scale listening tests with panels of ordinary listeners became the centrepiece of codec selection later.

A waveform coder treats every sample as a surprise. Speech isn't.

PCM, even with \(\mu\)-law, sends every sample in full, as if it were completely new information. But module 4 showed that speech is highly predictable: a weighted sum of the last 10 samples guesses the next one very well. On this recording at 8 kHz, a 10th-order predictor leaves an error 13 dB weaker than the signal itself. That redundancy is pure waste for a waveform coder.

Combine that with the 6-dB-per-bit rule from section 02:

And there is more structure to exploit than that. Module 5 showed that speech is a source (pulses or noise) through a filter (a few formants), both changing slowly, every 10–20 ms. If the receiver already knows that model, maybe we only need to send its settings. The next three sections climb that ladder of ideas, one trick at a time.

Predictive coding (DPCM / ADPCM).

Don't send the sample. Send the prediction error, the part the receiver couldn't have guessed. Because the error is much smaller than the signal, the same quantiser step gives far less noise, or the same noise needs fewer bits.

Listen to four coders at the same 32 kbit/s (\(8000 \times 4\) bits/s), each adding one idea:

Prediction plus an adaptive step turns an unusable 4-bit coder (segmental SNR about 4 dB) into one that comes within about 4 dB of 8-bit uniform PCM, at half the bit rate. This is the idea of G.726 ADPCM (32 kbit/s, also 16–40), long used in cordless phones (DECT) and on international trunk lines. Our implementation is simpler than the standard, and its prediction delivers about 4 dB of the theoretical 13. The gap is the last point in module 4's list: the decoder can only refit the predictor from decoded, noisy samples (with 4 bits, those are quite noisy), and the ideal 13 dB was measured with the weights fitted to the clean original. The principle is the same.

The LPC vocoder: 57 bits every 20 ms.

ADPCM still sends something for every single sample. Module 5 suggested a far more radical idea: speech = source through an all-pole filter, and both change slowly. So, every 20 ms, the encoder just measures the model's settings and sends those. The decoder, which knows the model, rebuilds speech from them: pulses at the right pitch (or noise), through the right filter, at the right loudness.

That is 90 times less than the 256 kbit/s original, and 11 times less than ADPCM. Two small tricks make the filter cheap to send:

• Send reflection coefficients, not the weights. The predictor weights \(a_k\) are fragile: a tiny rounding error can push a pole outside the unit circle and make the decoder's filter unstable (module 5). The equivalent reflection coefficients \(k_i\) (one per joint in the tube model) are always between \(-1\) and \(+1\), and the filter is stable as long as they stay there. So they can be rounded safely.
• Quantise them on a warped scale. The important values of \(k\) crowd near \(\pm 1\), so they are quantised as log-area ratios, \(\ln\frac{1+k}{1-k}\) (literally the log of the area ratio between neighbouring tube sections), which spreads them out. Fewer bits go to the higher, less important coefficients.

The words are clear and the voice is recognisably the same speaker, but it is unmistakably robotic: the real glottal waveform has been replaced by clicks, and voiced/unvoiced is a hard switch. This is “synthetic quality”, the sound of LPC-10 (2.4 kbit/s, US Federal Standard 1015, 1984), of military secure telephones, and of Texas Instruments' Speak & Spell. Note that removing the quantisation (left) changes little: the loss of quality comes from the model's crude source, not from the rounding.

Check your understanding