Turning measurements into bits¶

The analysis stage hands us a description of 20 milliseconds of speech: a fundamental frequency, a voiced/unvoiced verdict for each region of the spectrum, and a set of harmonic amplitudes. All of it is continuous. Real numbers, as many decimal places as you care to compute.

The radio has 72 bits.

Everything in this chapter is about that collision. Quantization is the part of a codec where an honest measurement of a voice becomes an integer index that fits in a fixed-width field, and where the designer decides, irrevocably, which kinds of error the listener will hear.

The budget is fixed; the voice is not¶

US 5,754,974 describes the AMBE coder that D-STAR uses at the level of the budget: "In the 3.6 kbps system 72 bits are used to quantize the model parameters for each 20 ms frame. Seven (7) bits are used to quantize the fundamental frequency, and 8 bits are used to code the V/UV decisions in 8 different frequency bands... The remaining 57 bits per frame are used to quantize the spectral magnitudes for each frame."²

Hold on to that shape — 7 + 8 + 57 = 72 — but not to those exact numbers. That allocation is the patent's example for a system in which all 72 bits carry parameters; the D-STAR variant spends part of the same 3600 bps on forward error correction (the D-STAR frame on the air), leaving a parameter budget roughly two-thirds this size. What survives is the structure: a small field for pitch, a smaller one for voicing, and everything left over for the spectral envelope. The envelope is where the bits go, because the envelope is what makes a voice sound like a voice. Those same three families are the MBE model itself — US 5,649,050 describes it in passing as representing speech by its "fundamental frequency, spectral envelope, and voiced character"⁴.

Scalar quantization: one number, one index¶

The simplest way to spend bits is one parameter at a time. Choose a set of reproduction levels, find the nearest one to your measurement, transmit its index. With n bits you get 2^n levels, and the error you inject is roughly half the spacing between them.

The interesting decision is where to put the levels. Uniform spacing minimises absolute error, and it is the wrong choice for almost every quantity in a vocoder, because hearing is not linear in anything. Three examples from the patent record:

Pitch is coded uniformly in period, which is non-uniform in frequency. US 5,630,011 describes the IMBE approach as "The parameter P is uniformly quantized using 8 bits and a step size of 0.5. This corresponds to a pitch period accuracy of one half sample."¹ Even steps in period mean the step in hertz grows as the pitch falls — which is what you want, because pitch discrimination is roughly proportional rather than absolute. A half-semitone error at 100 Hz and at 300 Hz sound about equally wrong; a 2 Hz error at those two pitches does not.

Amplitudes are coded in the log domain. US 5,754,974 puts it plainly: "A differential block Discrete Cosine Transform (DCT) method is applied to the log spectral magnitudes."² A fixed number of quantizer steps then buys a fixed number of decibels, everywhere in the spectrum and at every input level — and since loudness perception is approximately logarithmic, that makes the error approximately uniform perceptually, which is the only kind of uniformity worth having.

Gain gets its own non-uniform quantizer. US 5,870,405 allocates the overall level term "6 bit non-uniform quantizer" and codes the remaining block-level terms with "uniform scalar quantizers ... step sizes dependent upon L, the number of harmonics."³ Note that last clause. We will come back to it.

Vector quantization: many numbers, one index¶

Scalar quantization treats each number as if the others did not exist. In a speech spectrum that is a poor assumption — adjacent harmonics are strongly correlated, formants move as units, and the shape of a vowel lives in the relationship between amplitudes rather than in any one of them.

Vector quantization exploits that directly. Instead of 2^n levels on a line, you build a codebook: a stored list of 2^n candidate vectors, each one a complete little chunk of spectrum shape. The encoder searches the codebook for the entry closest to the measured vector and sends only that entry's index. The decoder looks the index up.

Three things follow, and they explain why every serious low-rate vocoder does this:

1. Correlation becomes free. If harmonics 3 and 4 usually rise together, the codebook simply contains no entry where one rises and the other falls. Impossible shapes cost nothing, because they are never addressable.
2. Bit cost decouples from dimensionality. A 9-bit index selects one of 512 vectors whether each holds three numbers or thirty — fractional bits per parameter, which scalar quantization cannot do.
3. The design work moves into the table. The codebook is the model of what speech spectra look like, trained offline and encoding the designer's priorities permanently.

That last point is why this chapter will not show you a codebook. The tables used by D-STAR's AMBE variant have never been published — not in the JARL specification, and not in the DVSI patents, which give the mechanism and the vector sizes without listing entries. They exist, they are searched by nearest-neighbour distance, they are the reason the codec sounds the way it does, and their contents are not public. What is not in the public record has more.

The envelope, coded as a residual¶

The largest field in the budget is the spectral envelope, so it gets the most structure. Three ideas stack on top of each other.

Start with the raw material. The dots below are the harmonic amplitudes of one frame of a real capture, and the slider rounds them to a coarser and coarser step. Push it to 6 dB and then to 12 dB, and listen to the number in the header rather than the picture: that is the error you are choosing to inject, in decibels, before a single bit has been allocated.

1. Predict this frame from the last one¶

Speech is slow. Twenty milliseconds is short enough that a vowel's spectrum barely moves, so transmitting the amplitudes afresh each frame wastes bits describing something the decoder could have guessed. US 5,630,011 predicts the current frame's log amplitudes from the previous frame's and codes only the difference — the prediction residual.

The prediction is scaled by a decay factor. The patent gives it as γ, and says it "is typically equal to 0.7, however any value in the range 0≦γ≦1 can be used."¹ That scaling is error containment, not tuning. With γ = 1 the predictor is a pure integrator and one corrupted frame biases every frame after it forever; with γ < 1 the influence of any single frame decays geometrically, so a hit from the channel fades out over a few frames. You pay in efficiency — the smaller γ is, the less of the previous frame you get to reuse, and the bigger the residual you must code.

The patent goes further and makes the factor adaptive, tying it to how many harmonics the frame has: "For a relatively low number of spectral harmonics ... the decay factor α will be a relatively small number ... so that any errors decay away quickly, at the expense of a less purely differential coding method," while for higher counts "the decay factor is high ... which may result in a more persistent error, but which requires fewer bits to encode the differential values."¹ Few harmonics means few bits and more riding on each one, so the design leans toward robustness; many harmonics means more redundancy to exploit, so it leans toward efficiency.

2. Handle the fact that the frames are different lengths¶

Here is the wrinkle that makes MBE prediction harder than it sounds. The number of harmonics is not a constant. It falls out of the pitch:

"the total number of spectral magnitudes, L, is inversely related to the estimated fundamental frequency for the current frame and is typically computed as follows: L = floor(απ/ω₀) where 0≤α<1. A 3.6 kbps system which uses an 8 kHz sampling rate has been designed with α=0.925 giving a bandwidth of 3700 Hz."²

At an 8 kHz sampling rate that reduces to something you can do in your head: L = floor(3700 / F₀), with F₀ in hertz. A speaker at 100 Hz produces 37 harmonics below 3700 Hz. The same speaker's spectrum, coded a few frames later at 105 Hz, produces 35. Across the range of adult voices the count swings by a factor of five or so — a low male voice near 70 Hz yields around 52 magnitudes, a high voice near 400 Hz yields 9.

So the predictor is being asked to predict a 35-element vector from a 37-element one. US 5,630,011 resolves this by resampling: a relative index maps each current-frame harmonic onto a fractional position in the previous frame's spectrum — "If the ratio of the current to the previous fundamental frequencies is ⅓ ... k₁ is equal to ⅓·1, for each index number 1"¹ — and linear interpolation between the two neighbouring previous-frame amplitudes supplies the prediction. The two spectra are treated as samples of the same underlying envelope function, and that envelope is what is really being predicted. The codec is not tracking harmonics; it is tracking the shape of the vocal tract, sampled wherever the current pitch happens to put the harmonics.

One more subtlety. The patent subtracts the mean of the predicted spectrum before coding — "establishes ... the average of the interpolated spectral log amplitudes ... and then subtracts this average from the vector of predictions"¹ — so that a frame-average level error cannot leak through the predictor into later frames. The overall level is then sent separately, which is what the gain field is for. Level and shape travel on different paths precisely so a mistake in one cannot contaminate the other.

3. Split the residual into blocks and transform each one¶

The residual vector — L numbers, however many that is this frame — is then divided up. US 5,630,011: "this vector is divided into blocks, for instance six, and ... a transform, such as a DCT is performed."¹ The DCT of each block concentrates that block's energy into a few coefficients, so most of the block can be coded coarsely or not at all.

Then comes the move that gives the scheme its name. The DC coefficient of each block — the block's average residual level — is pulled out, and those DC values are collected into a vector of their own:

"The output of the DCT transform is organized in two groups: a set of D.C. values, associated into a vector referred to as the Prediction Residual Block Average (PRBA); and the remaining, higher order coefficients, both of which are quantized."¹

The PRBA vector is the coarse tilt of the spectrum: is this frame bass-heavy, bright, flat. US 5,870,405 transforms it once more — the gain vector formed from each block's first DCT coefficient receives "a six point DCT"³ — and it is this small, perceptually dominant vector that gets vector-quantized against a codebook and given the best error protection. The higher-order coefficients within each block are fine detail, and are quantized more cheaply.

The structure is a hierarchy: level, then tilt, then detail, in decreasing order of how much a listener would miss it.

Why the bit allocation has to adapt¶

Everything above collides with the fact that L is not constant. A frame with 9 magnitudes and a frame with 52 arrive at the quantizer with the same bits available and wildly different amounts of information to carry.

So the allocation is a function, computed identically at both ends from a value the decoder already has. That is the load-bearing trick: the decoder derives L from the received pitch index before it unpacks anything else, so both ends agree on the field widths without a single bit spent describing them. Get the pitch index wrong and the frame does not merely sound wrong — it unpacks wrong, because every subsequent field boundary moves.

PSEUDOCODE — NOT EXECUTABLE, ILLUSTRATIVE ONLY.
Shape of the allocation. Codebook contents omitted.

  given pitch_index (received, or chosen by the encoder):

      F0 = dequantize_pitch(pitch_index)          # log-spaced levels
      L  = floor(3700 / F0)                       # US 5,754,974, alpha=0.925
                                                  # decoder derives this too:
                                                  # never transmitted

  # partition the L residuals into a fixed number of blocks whose
  # lengths depend on L, longer blocks toward the high end
  block_lengths = partition(L)                    # sums to L

  # fixed-size fields: the perceptually dominant path
  spend( gain_index )                             # scalar, non-uniform
  spend( prba_codebook_index )                    # vector quantized

  # variable-size fields: detail, scaled to what is left
  for each block:
      k = number_of_higher_order_coeffs(block)    # grows with block length
      spend( hoc_codebook_index[block] )          # coarser for small blocks

  # invariant that makes the scheme work at all:
  #   total bits spent is the same for every L
  #   both ends compute the same partition from the same L

Two consequences worth stating out loud.

Low-pitched voices get less resolution per harmonic. A 70 Hz speaker has five times as many magnitudes to describe as a 400 Hz speaker, out of the same budget. The codec compensates by coding shape rather than individual harmonics — at 52 magnitudes the envelope is heavily oversampled, so per-harmonic precision matters less than it does at 9. The tradeoff is real, though, and audible if you listen for it.

Step sizes move with L, not just bit counts. US 5,870,405 describes the block-level quantizers as having "step sizes dependent upon L", and gives the higher-order coefficient allocation in units of standard deviation — for example "4 bits allocated gives step size 0.40σ"³. The quantizers are normalized to the expected statistics of each coefficient rather than to a fixed absolute range, so a 4-bit field covers the same fraction of the likely spread wherever in the spectrum it sits.

None of that structure is visible from outside. The nine bytes a real chip emits for one frame are in the figure on the D-STAR frame on the air, and the honest thing to say about them is that no public document maps any of those 72 positions onto any of the fields described above.

What you should take away¶

The bits are not divided evenly, and were never meant to be. The scheme spends them in strict priority order — pitch first, because every other field's meaning depends on it; then overall level; then the coarse tilt of the spectrum, vector-quantized against a trained codebook; then whatever detail is left over. Prediction from the previous frame recovers what the redundancy of speech allows, and a decay factor below one stops that borrowing from turning one bad frame into a long bad noise.

The D-STAR frame on the air picks the same ordering up from the other side: the channel coding uses it too.

Sources for this chapter¶

The derivation L = floor(3700 / F₀) is arithmetic on US 5,754,974's formula at an 8 kHz sampling rate, not a separately sourced claim. The illustrative harmonic counts for 70 Hz and 400 Hz voices follow from the same formula; the exact range of fundamental frequencies the D-STAR variant actually codes is not stated in any public source available to this project. Neither are the codebook contents, the number of blocks used by the D-STAR variant, or the exact widths of its parameter fields. See what is not in the public record.

Next: The D-STAR frame on the air. What happens to those indices on the way out of the radio: unequal error protection, a descrambling key chained to the most important field, and 24 bits of data riding alongside every voice frame. Previously: Analysis: pitch, voicing, amplitudes.

Last reviewed: 2026-08-10.