Analysis: pitch, voicing, amplitudes¶

The model says a frame of speech is a fundamental frequency, a set of per-band voiced/unvoiced flags, and a set of harmonic amplitudes. Analysis is the job of looking at 20 ms of microphone audio and producing those numbers.

It is the hard half. Griffin and Lim are blunt about it: inaccurate pitch or voiced/unvoiced estimates "often introduce very noticeable degradations in the synthesized speech", and in noisy speech the frequency of those degradations "increases dramatically" Griffin & Lim 1988, §I. A perfect model estimated badly sounds worse than a mediocre model estimated well.

Framing and windowing¶

Speech is quasi-stationary, so a window is applied to focus on a short interval — the paper puts the usable range at approximately 10–40 ms Griffin & Lim 1988, §II. The constraint is two-sided: the window must span several pitch periods for the harmonic structure to be resolvable at all, but stay short enough that the fundamental is roughly constant across it.

The 1988 coder used a 25.6 ms Hamming window on 10 kHz speech at a 50 Hz frame rate Griffin & Lim 1988, §V-A. The AMBE-era systems moved to 8 kHz sampling with a frame interval of 20 ms — 160 samples — a 256-point FFT, and a 255-point symmetric Hamming analysis window US 5,701,390 (Griffin & Hardwick, DVSI, filed 1995, issued 1997).

Note the mismatch: a 255-sample window advanced 160 samples at a time. Frames are not independent snapshots but overlapping views of a continuously moving signal, which is what makes the frame-to-frame continuity constraints below meaningful rather than arbitrary.

Drag along the waveform below and watch the window walk. The three-frame inset is drawn from the actual 8 kHz samples, so you can see how little changes between one 20 ms slice and the next, and the panel shows the nine bytes the chip emitted for the slice you are looking at.

Analysis by synthesis¶

Most vocoders estimate excitation and envelope parameters with separate, independent algorithms, each using "some reasonable but heuristic criterion without explicit consideration of how close the synthesized speech will be to the original speech" Griffin & Lim 1988, §III. MBE does not. It defines one error — the squared difference between the original short-time spectrum and the spectrum the model would produce — and estimates every parameter by minimising it:

ε = (1/2π) ∫ ( |S_w(ω)| − |Ŝ_w(ω)| )² dω        over −π to π

Griffin & Lim 1988, eq. 3; a variant using the complex difference, eq. 5, estimates phase as well

A frequency-dependent weighting can be applied first, to emphasise high-SNR regions Griffin & Lim 1988, §III. This is the "analysis by synthesis" idea: you do not measure the pitch, you find the pitch that makes the synthesizer's output match best.

Minimising over all parameters at once would be computationally prohibitive. The saving observation is that for a given pitch period, the best envelope samples have a closed form. Divide the spectrum into intervals one fundamental wide, centred on each harmonic; model the envelope as constant A_m within the interval [a_m, b_m]; the error contributed by that interval is then minimised at

              ∫ |S_w(ω)| · |E_w(ω)| dω        (over a_m … b_m)
   |A_m|  =  ─────────────────────────────
                  ∫ |E_w(ω)|² dω

Griffin & Lim 1988, eq. 7; the complex form, eq. 8, is the same ratio with S_w · E_w* in the numerator

Sum those per-interval minima and you have the total error for an entirely periodic fit at that candidate pitch Griffin & Lim 1988, eq. 9. The multidimensional problem collapses to a one-dimensional search over the pitch period. Everything else in the analyser is downstream of that one move.

Pitch: a search, not a measurement¶

This is where MBE most visibly parts company with textbook pitch detection. A classical autocorrelation detector finds the lag at which the signal most resembles a shifted copy of itself. MBE instead evaluates, for each candidate fundamental, how well the whole spectrum can be explained as harmonics of it, and takes the winner.

Coarse search¶

The error varies slowly with the pitch period, so the initial estimate comes from a coarse grid — in practice, every integer pitch period Griffin & Lim 1988, §III-A. The 1988 coder swept all integers from 20 to 120 samples at 10 kHz sampling Griffin & Lim 1988, §III-D, step 2; the later 8 kHz systems use 22 ≤ P < 115 US 5,216,747 (Hardwick & Lim, DVSI, filed 1991, issued 1993). During this coarse stage the high harmonics cannot be matched well at integer resolution, so the frequency weighting is chosen to de-emphasise high frequencies Griffin & Lim 1988, §III-A.

The bias nobody expects¶

Raw spectral error is systematically biased toward long pitch periods, for a structural reason: a longer period means more closely spaced harmonics, which means the envelope is sampled more densely, which means a better fit — even for pure noise. Uncorrected, the search reliably picks the longest period in range for noisy input.

The fix is an unbiased error criterion: multiply the error by a period-dependent correction factor whose denominator carries a (1 − P · Σ w⁴(n)) term, with the window normalised to unit energy. The result sits near zero for a purely periodic signal and near one for noise, independent of P, and the paper records that this "significantly improves the performance for noisy speech" Griffin & Lim 1988, eq. 10.

Why it is not just autocorrelation¶

Evaluating a full spectral error for every candidate period would be expensive, so the criterion is rearranged into an autocorrelation-domain form. Let φ(m) be the autocorrelation of w²(n)·s(n); then the quantity to maximise is

   Ψ(P) = P · Σ φ(k·P)        summed over integer k

Griffin & Lim 1988, eqs. 11–13

and minimising the unbiased error is equivalent to maximising Ψ(P) Griffin & Lim 1988, §III-A. φ comes from one FFT. The paper's own description of the difference is the sentence to remember: this "is similar to the autocorrelation method, but considers the peaks at multiples of the pitch period instead of only the peak at the pitch period" Griffin & Lim 1988, §III-A.

That summation over all multiples of the lag is what buys robustness. A plain autocorrelation detector that finds a strong peak at lag 2P has no way to know it is wrong. Summing the peaks at P, 2P, 3P, … rewards the candidate that explains the entire periodic structure, not just one coincidence.

Sub-multiples¶

It does not fully solve the octave problem, because integer multiples of the correct period have harmonics at correct frequencies too, so their error is comparable. The standard defence: once the minimising period is found, evaluate its sub-multiples and choose the smallest period whose error is comparable to the minimum Griffin & Lim 1988, §III-A. The patent formalises this into an explicit check of P′, P′/2, P′/3 and P′/4 where those fall in range, with threshold rules comparing their cumulative errors US 5,216,747. An octave error is the most audible failure an MBE encoder can produce — halve someone's pitch and they become a different person mid-syllable — which is why this check reappears in every generation of the patents.

PSEUDOCODE — coarse pitch search (NOT EXECUTABLE)

  φ ← autocorrelation of w²(n)·s(n), computed from one FFT

  best ← none
  for each candidate period P over the allowed integer range:
      Ψ(P) ← P × Σ_k φ(k·P)              # all multiples of the lag,
                                          # not just the first
      ε(P) ← unbiased spectral error implied by Ψ(P), normalised so
             that noise scores near one and periodicity near zero
      if ε(P) < ε(best):  best ← P

  # reject the octave: prefer the shortest period that fits as well
  for each sub-multiple Q of best that lies in range, shortest first:
      if ε(Q) is comparable to ε(best):
          best ← Q
          stop

  return best

Integer resolution is not enough — accurate voiced/unvoiced decisions in the high-frequency bands require better than integer pitch — so the frequency-domain error is minimised locally around the coarse estimate using successively finer evaluation grids. The paper's worked example is persuasive: for a segment of female speech the final estimate is 42.48 samples, and the figure overlaying synthetic and original spectra at 42.48 against the best integer estimate of 42 shows the high harmonics visibly drifting out of alignment in the integer case Griffin & Lim 1988, §III-A, Fig. 2(d)–(e). Later systems refine to ¼ or ⅛ sample, interpolating autocorrelation values at non-integer lag as r(n+d) = (1−d)·r(n) + d·r(n+1) US 5,216,747.

Tracking¶

Pitch does not jump. Exploiting that is worth more than any single-frame improvement — but doing it naively, by smoothing the estimates, costs accuracy even on clean speech Griffin & Lim 1988, §III-B.

The paper's answer is dynamic programming. There are three cases for a pitch track — it starts in this frame, terminates in this frame, or continues through it — and the third turns out to be adequately modelled by one of the first two. So the analyser looks N frames backward and N forward (N = 3, about 60 ms, is typical), allows a frame-to-frame deviation of D samples (D = 2 typical), finds the minimum-error path in each direction by summing the errors along it, and takes the current-frame period from whichever path scored better. The benefit is exactly the asymmetric one you want: it "improves tracking through very low signal-to-noise ratio segments while not decreasing the accuracy in high SNR segments" Griffin & Lim 1988, §III-B and §III-D, step 3.

US 5,216,747 turns this into a shippable form. Look-back tracking constrains the candidate to (1−α)·P₋₁ ≤ P ≤ (1+α)·P₋₁; failing that, look-ahead tracking minimises CE(P) = E(P) + E₁(P₁) + E₂(P₂) over the next two frames under the same fractional constraints, α = β = 0.2 typical. To make that affordable the ~200 candidate periods are grouped into "a small number of non-uniform regions. A reasonable number is 20" — 22 ≤ P < 24, 24 ≤ P < 26, … , 107 ≤ P < 115 — and continuity becomes a permitted change of a fixed number of regions US 5,216,747.

A different front end entirely¶

The later excitation patents attack a specific weakness: when the fundamental changes during the analysis window, the spectral peak at the m-th harmonic broadens more than the peak at ω₀, an effect that "increases with increasing frequency" and so "reduces the effectiveness of higher harmonics in the estimation of the fundamental frequency" US 5,715,365 (Griffin & Lim, DVSI, filed 1994, issued 1998). The harmonics that pin down the pitch most precisely are exactly the ones that smear.

The remedy is a nonlinear operation — absolute value, squared magnitude, log|x|, a fractional power — applied to bandpass-filtered channels. Squaring convolves the spectrum with itself, and the convolution "has spectral peaks at frequencies equal to the differences between the frequencies for which X(ω) has spectral peaks". A channel containing only the 3^rd, 4^th and 5^th harmonics therefore produces a peak at ω₀ — a fundamental that was never present in that channel at all. Sixteen such channels are combined, and ω₀ is chosen to maximise the combined output, refined by parabolic interpolation and octave-checked against 0.5·ω₀ US 5,715,365. The continuation runs two estimators, converts each result into a probability of being correct from its voiced-to-total energy ratio, and keeps whichever is more probably right US 5,826,222 (Griffin, DVSI, filed 1997, issued 1998).

Voicing: the decision you get for free¶

Here is the elegance of analysis by synthesis. You have already fitted the spectrum with a purely periodic model and computed the error in every harmonic interval. The voicing decision is just: where did that fit fail?

For each harmonic, form the normalised error — the error in that interval divided by the energy in that interval:

             ε_m
   ξ_m  =  ─────────────────────────────
           (1/2π) ∫ |S_w(ω)|² dω     (over a_m … b_m)

Griffin & Lim 1988, eq. 14

When ξ_m is below threshold, "this region of the spectrum matches that of a periodic spectrum well and the m-th harmonic is marked voiced". When it is above, "this region of the spectrum is assumed to contain noise-like energy". And then, disarmingly: "A threshold value of 0.2 works well in practice" Griffin & Lim 1988, §III-C.

No separate voicing detector. No zero-crossing rate, no low-band energy ratio, no trained classifier. The per-band voicing map is a by-product of the residual of a fit you had to do anyway.

Because bits are finite, harmonics are grouped into coded bands: given N bits, the spectrum is divided into N equal-width bands and each band's bit comes from an amplitude-weighted sum of its harmonics' normalised errors —

   E_k  =  Σ |A_m| · ξ_m   /   Σ |A_m|        (m in band k)

Griffin & Lim 1988, eq. 18

— which lets loud harmonics dominate their band's decision rather than letting one quiet, noisy harmonic drag a strong voiced band unvoiced. The 1988 coder used 12 bands; the AMBE-rate system uses "eight V/UV decisions ... over eight different frequency bands spaced between 0 and 4 kHz", about 500 Hz each, with every harmonic labelled according to the band its frequency falls in Griffin & Lim 1988, §V-A; US 5,754,974.

PSEUDOCODE — per-band voicing test (NOT EXECUTABLE)

  # ω₀ and the amplitudes A[·] are already known from the pitch fit
  for each harmonic m = 1 … L:
      interval ← one fundamental wide, centred on m·ω₀
      ε_m ← spectral error over interval, using the purely periodic model
      ξ_m ← ε_m / (energy of the original spectrum over interval)
      # ξ ≈ 0  → the band is explained by harmonics
      # ξ ≈ 1  → the band is not periodic at all

  for each coded band k:
      E_k ← Σ A[m]·ξ_m / Σ A[m]      over the harmonics m falling in band k
      voiced[k] ← ( E_k < threshold )

  return voiced[·]

Making the threshold survive the real world¶

A fixed threshold works in a lab and fails on a handheld in a car park. US 5,216,747 makes it adaptive: the threshold becomes a function of pitch and frequency multiplied by a factor derived from running energy statistics. The frame's weighted energy is tracked against a recursively updated average, maximum and minimum — one-pole trackers with deliberately different time constants, so the maximum reacts fast to onsets while the noise-floor estimate crawls — and the decision is biased toward unvoiced in low-relative-energy frames, strongly so in silence. The analyser knows whether this frame is loud or quiet relative to this talker in this environment US 5,216,747, printed constants γ₀ = 0.067, γ₁ = 0.5, γ₂ = 0.01.

US 5,826,222 replaces the single measure with two and a merge. The frequency-domain measure is A_k = 1 − E_v(ω₀)/E_t per band, where the voiced energy sums the power within ±0.25·ω₀ of each harmonic — the width of the spectral peak — and frequencies below 0.5·ω₀ are excluded from the total because "including these frequencies reduces performance". The time-domain measure is a sinusoid detector, B_k = 1 − S_k(1)/S_k(0). They merge as V_k = min(A_k, B_k + α(k)·β(ω₀)) — the minimum, because "a preliminary V/UV parameter having a value close to zero has a higher probability of being correct" — and the result is smoothed across time and frequency US 5,826,222. Note what that is: a continuous degree of voicing, thresholded only at the end. The bit that goes on the air is binary. The analyser that produces it is not.

That continuous measure is exactly what the band-voicing figure in what AMBE actually is plots: each band carries a strength between 0 and 1, and the colour flips where it crosses the threshold. Read it again now, with this section in mind, and the chips showing 0.5-something are the frames this machinery exists to get right.

Amplitudes: measuring the envelope¶

The amplitudes come from the same closed form that made the pitch search tractable — equation 7 above — with the excitation transform substituted according to the band's voicing:

Voiced intervals. Substitute the periodic transform P_w(ω) for E_w(ω). Efficiently: precompute samples of the Fourier transform of the analysis window and centre them on that interval's harmonic frequency — a correlation of the observed spectrum against the window's own lobe shape, normalised by the lobe's energy.
Unvoiced intervals. Substitute idealised white noise, unity across the band, which collapses the ratio to averaging the original spectrum over the interval. Only the magnitude is estimated; the phase is not needed to synthesize noise.

Both: Griffin & Lim 1988, §III-A

What those measured amplitudes look like, one frame at a time, is the figure in turning measurements into bits: dots on a curve, read off a real capture. It is placed there rather than here because the interesting question about an amplitude is not how it was measured but how coarsely it can be rounded before you hear it.

Later systems changed this in a way that matters more than it looks. Using different formulas for voiced and unvoiced amplitudes means that when a band's voicing flips between frames, its amplitude estimate jumps for reasons that have nothing to do with the speech. US 5,754,974 and US 5,701,390 introduce a compensated total energy method: each magnitude is a weighted sum of the spectral energy |S_w(ω)|², with the weighting function offset by the harmonic frequency to compensate for the gap between l·ω₀ and the FFT's sample grid at multiples of 2π/N — and computed "independently of whether the determined frequency is in a frequency band that is voiced or unvoiced" US 5,754,974; US 5,701,390. One estimator, always, so the envelope stays continuous through voicing transitions and does not fluctuate with where the harmonics happen to land between FFT bins.

The count of amplitudes, again, follows from the pitch: L = ⌊α·π/ω₀⌋, with α = 0.925 in the 3.6 kbit/s 8 kHz system for a coded bandwidth of 3700 Hz US 5,701,390; US 5,754,974.

What comes out¶

At the end of analysis, for each 20 ms of speech, the encoder holds:

a refined fundamental frequency ω₀, sub-integer in resolution, tracked for continuity and checked against its own octaves;
a voiced/unvoiced flag for each of eight roughly 500 Hz bands;
L spectral magnitudes, L determined by ω₀, measured the same way regardless of voicing;
and no phases at all.

That is the complete description of a syllable. It is also, at this stage, still a set of real numbers, and there are only 48 or 72 bits to put them in. Getting from one to the other is quantization.

The other direction is worth a look before you go there. A list of harmonic amplitudes with no phases is not obviously a voice, and the clip on synthesis adds those amplitudes up as cosines until a glottal pulse appears out of them.

Next: Turning measurements into bits. How a pitch, eight flags and a few dozen real numbers get forced into fixed-width fields, and which errors that choice hands to the listener. Previously: What AMBE actually is.

Last reviewed: 2026-08-10.