Synthesis: rebuilding the voice¶

The decoder never receives speech. It receives a description of speech — a pitch, a handful of voiced/unvoiced flags, and a list of harmonic amplitudes — and it has to manufacture a waveform that sounds like the one the encoder was looking at. This page is about how that manufacture works, and about the one thing the decoder is not given and must invent from scratch: phase.

What the decoder is handed¶

Once the channel bits have been corrected and unpacked, a frame of full-rate MBE reduces to three things:

• a fundamental frequency ω₀, from which the harmonic count follows — the harmonics are the multiples of ω₀ that fit below 4 kHz;
• a voiced/unvoiced decision per frequency band, expanded to one flag per harmonic;
• a spectral amplitude for each harmonic.

That arrives once per 20 ms frame — 160 samples at 8 kHz, fifty frames a second. US 5,701,390

There is no waveform in there, and no phase. Everything below follows from those two absences.

Two synthesizers, summed¶

MBE splits the spectrum by voicing, so the decoder runs two independent synthesizers over the same frame and adds their outputs:

    s(n)  =  s_voiced(n)  +  s_unvoiced(n)

The voiced half is built in the time domain from a bank of sinusoidal oscillators. The unvoiced half is built in the frequency domain by shaping noise. Each half is responsible for exactly the frequency regions its voicing decision claims, and each is deliberately silent where the other one is speaking. Griffin & Lim 1988; US 5,754,974

That division is the whole point of the multi-band model. A frame of a voiced fricative — say the z in "zone" — has a periodic low end and a noisy high end at the same instant, and MBE can synthesize both without choosing.

The oscillator bank¶

For the voiced part the decoder assigns "one oscillator to each harmonic labeled voiced" US 5,701,390 and sums them:

    s_voiced(n)  =  SUM over voiced harmonics l of
                        A_l(n) * cos( theta_l(n) )

A_l(n) is that harmonic's amplitude, and theta_l(n) is its running phase — both functions of the sample index n, not constants, for reasons the frame-boundary section below gets to. Griffin and Lim write the phase as the integral of an instantaneous frequency track, so a harmonic that is drifting in pitch traces a smoothly curving phase rather than a straight line. Griffin & Lim 1988

A vowel at 120 Hz has roughly thirty harmonics under 4 kHz, so this is thirty oscillators running concurrently, each with its own amplitude envelope. Their sum is periodic-ish, and their relative phases are what decide whether that sum looks like a sharp glottal pulse or a smeared mush of the same spectrum.

Shaped noise for everything else¶

The unvoiced half is not built oscillator by oscillator. It is built by filtering noise:

"The unvoiced speech component is normally synthesized by filtering a white noise signal with a filter response of zero in voiced frequency bands and with a filter response determined by the spectral magnitudes in frequency bands declared unvoiced." US 5,754,974

In practice the filtering is done as a transform: take a windowed block of white noise, transform it, scale each unvoiced harmonic's band of bins so that band carries the transmitted amplitude, zero every bin belonging to a voiced band, transform back, and overlap-add the result with the previous frame's block. The patent is explicit that this is "performed via a weighted overlap-add procedure which uses a forward and inverse FFT to perform the filtering." US 5,754,974

The overlap-add is only seamless if the synthesis window is built for it. The constraint the patent states is that successive shifted copies of the window must sum to unity across the overlap — ws(n) + ws(n + S) = 1 for -S < n ≤ 0, where S is the frame advance. US 5,754,974 A window that fails this puts a 50 Hz amplitude ripple on every unvoiced sound, which is audible as a buzz sitting underneath the noise.

One step before synthesis¶

Most MBE decoders sharpen the amplitude envelope before they synthesize anything. The idea, from the error-handling patent, is to build a smoothed model of the spectral envelope and then use it to "increase the amplitude of perceptually important frequency regions" while pulling down the ones between them, before restoring the original frame energy. US 5,247,579

This exists because quantization smears formants: a coarsely coded envelope has shallower peaks and shallower valleys than the real one, and the ear hears that as muffled. Re-deepening the peaks costs no bits, and it is one of the larger perceptual differences between two decoders fed identical frames.

The phase problem¶

Here is the thing the decoder is not told.

A sinusoid needs an amplitude, a frequency, and a phase. The frame carries the first two. It carries no phase at all, for the blunt reason the patent gives:

"At low to medium data rates there are not sufficient bits to transmit any phase information between the encoder and the decoder." US 5,701,390

Thirty harmonics at even four bits of phase each would be 120 bits per frame — more than the entire D-STAR voice frame. Phase is simply not affordable, so the encoder measures it, uses it internally, and throws it away.

The decoder therefore has to invent a phase for every voiced harmonic of every frame, fifty times a second, and the two obvious ways of doing that both sound wrong:

• All harmonics phase-locked, advancing coherently from a common fundamental. Perfectly periodic, and it sounds it — mechanical, buzzy, a synthesizer imitating a person.
• All phases random. The spectrum is right and the periodicity is gone, but so is the pulse structure: energy that should have been concentrated into one glottal instant is spread across the whole period. It sounds breathy, distant, and reverberant.

Every MBE phase scheme is a way of sitting between those two.

Generation one: coherent phase plus controlled jitter¶

The first published answer builds the phase in two pieces US 5,081,681:

    theta_k(t)  =  phi_k(t)  +  r_k(t)

phi_k(t) is the coherent part: the harmonic's phase carried forward by integrating the fundamental frequency over time, so a harmonic that was in step at the last frame is still in step at this one. r_k(t) is deliberate disorder — r_k(t) = alpha(t) * u_k(t), where u_k is white noise uniform on [−π, π] and alpha(t) is "the approximate percentage of total harmonics represented by the unvoiced harmonics." US 5,081,681

The scaling is the clever part. A fully voiced frame has alpha = 0 and stays perfectly coherent. A frame that is mostly unvoiced has alpha near one and gets phases that are nearly random. Voicing therefore controls the phase character as well as which synthesizer runs, and the transition between the two is continuous. The stated motivation is that real "speech deviates from a perfect voicing model" US 5,081,681 — a real larynx is not a metronome, and a decoder that is one sounds like a machine.

Generation two: regenerate the phase from the amplitudes¶

The later approach stops treating the missing phase as something to fake and starts treating it as something to derive. The observation is that phase and magnitude are not independent in a physical resonant system: a vocal tract is close enough to a minimum-phase system that the phase response is recoverable, up to a linear term, from the shape of the magnitude response. The patent links "phase to spectral smoothness via linear system theory." US 5,701,390

So the decoder computes each voiced harmonic's phase by running an edge-detection kernel across the log-compressed spectral amplitudes of the frame it already has:

    B_l    =  log2( M_l )                      compressed magnitudes
    phi_l  =  SUM over m of  h(m) * B_(l+m)    regenerated phase

The kernel h(m) is antisymmetric, has only odd-indexed taps (so its centre tap is zero by construction and the kernel is exactly zero-mean), decays as 1/m, and spans roughly nineteen harmonics either side of the one being computed. It is, in effect, a differentiator: where the amplitude envelope is climbing steeply into a formant, the regenerated phase advances; where the envelope is flat, it does not. Amplitudes outside the transmitted range are extrapolated with a downward slope so the kernel has something smooth to chew on at the edges. US 5,701,390, Eqs. 7–9

The stated payoff is that speech synthesized this way "more closely approximates actual speech in peak-to-RMS value" US 5,701,390 — the waveform gets its pulse structure back, giving "improved dynamic range" and fewer "phase related distortions." That is the difference between a decoder that sounds thin and one that sounds present, and it costs zero transmitted bits, because it is computed from data the decoder already has.

What regenerated phase costs¶

It is worth being clear-eyed about what has been given up.

The waveform is not reproduced, and was never going to be. Feed a sine into an MBE codec and the output is a sine of the right frequency and amplitude at an arbitrary phase. Any waveform-domain quality metric — SNR, correlation with the input, a difference plot — measures the phase choice and nothing else. This is why MBE codecs are assessed by listening tests and not by SNR, and why "the decoded file doesn't match the original" is not a bug report.

Sharp transients suffer most. The regeneration derives phase from a spectral envelope averaged over a 20 ms window. A plosive burst, a glottal stop, or a hard consonant onset has its energy concentrated in a couple of milliseconds, and the phase relationship that produced that concentration is not recoverable from the frame's magnitudes. Attacks come out softened.

Non-speech signals come out mangled. The model assumes a harmonic excitation and a resonant tract. Music, DTMF pairs, and modem tones violate that, and phase regeneration cannot rescue what the model misrepresents in the first place.

It does not accumulate gracefully. Decode to audio and re-encode — through a gateway, a repeater link, a transcoder — and the second encoder analyses phases that the first decoder invented. Errors compound in a way they would not if the phase had been transmitted.

Against all that, it works, because human hearing is largely insensitive to the absolute phase of a steady sound and quite sensitive to its spectrum. MBE spends its bits where the ear is looking.

Crossing the frame boundary¶

Fifty frames a second means the decoder is handed a completely new set of parameters every 20 ms. If it simply switched — old oscillator bank off, new one on — the output would contain fifty amplitude steps and fifty phase discontinuities per second, and a 50 Hz buzz on top of the speech. Preventing that is most of the work in a synthesizer.

The decoder pairs each harmonic in the current frame with the same-index harmonic in the previous frame and picks a rule based on how the two frames disagree. US 5,701,390

The transitional cases are handled as an energy handover: "the energy in this region of the spectrum transitions from the voiced synthesis method to the unvoiced synthesis method over the duration of the synthesis interval." US 5,701,390 Because both synthesizers use complementary windows, the total energy in that band stays put while its character changes.

The interesting case is the fourth one. The patent gates it precisely:

"A final synthesis rule is used if the l'th spectral amplitude is voiced for both the current and the previous frame, and if both l < 8 and |ω₀(0) − ω₀(−S)| < .1 ω₀(0)." US 5,701,390

Both conditions, not either. When they hold, the harmonic is synthesized by a single oscillator whose amplitude ramps linearly from the old value to the new one and whose phase follows a second-order polynomial — "a linear phase term and which otherwise meets the desired regenerated phase" US 5,701,390 — chosen so that the phase is correct at both ends of the frame and the instantaneous frequency sweeps smoothly from l·ω₀(old) to l·ω₀(new) in between. A quadratic in n is exactly the minimum you need to satisfy two phase endpoints and two frequency endpoints.

Both gates earn their place:

• l < 8. The harmonic's frequency is l · ω₀, so any error in the pitch estimate is multiplied by l. At the twentieth harmonic a one percent pitch error is a twenty percent frequency error, and forcing a single oscillator to sweep across it produces audible warble. The low harmonics carry the pitch percept and deserve the continuous treatment; the high ones are better off crossfaded.
• Less than ten percent pitch change. A linear frequency sweep is a model of what the larynx did between the two frames. Across a large pitch jump — a pitch-tracking error, or a genuine octave leap — that model is simply false, and sweeping through the intervening frequencies synthesizes a glide that never happened. Crossfading two steady tones is the more honest failure.

The net effect: sustained vowels get genuinely continuous low harmonics, transitions get smooth energy handovers, and nothing in the output announces where one frame stopped and the next began.

The figure below holds the two halves apart so you can see the division of labour. Slide the mix to either end: one end is the oscillator bank alone, the other is the shaped noise alone, and the centre is the split the measurement actually found for that frame. Pick a sibilant frame and the noise generator is doing nearly all the work.

The synthesis loop, in outline¶

The following is conceptual pseudocode. It is a reading aid for the rules above, not a specification and not a program: the interpolation details, window definitions, band-edge arithmetic, gain staging, and every constant that would make it run are deliberately absent. It will not compile in any language and is not intended to.

CONCEPTUAL PSEUDOCODE — not executable, not a specification.

for each received frame:

    prev, cur  <-  previous and current decoded parameter sets
                   (fundamental, per-harmonic voicing, per-harmonic
                   amplitude)

    # ---- error handling comes first -----------------------------------
    if the frame was flagged unrecoverable:
        if we have not repeated too many times already:
            cur <- prev                       # frame repeat
        else:
            output silence; reset state       # mute
            continue

    # ---- optional envelope sharpening ---------------------------------
    cur.amplitudes <- sharpen_formants(cur.amplitudes)   # US 5,247,579
                                                          # energy preserved

    # ---- phases the frame did not carry -------------------------------
    # Either scheme; both produce one phase per harmonic.
    #   gen 1: carry each harmonic's phase forward coherently, then add
    #          jitter scaled by the unvoiced fraction        US 5,081,681
    #   gen 2: convolve an antisymmetric odd-tap kernel across the
    #          log amplitudes                                US 5,701,390
    cur.phases <- regenerate_phases(cur)

    # ---- voiced half: an oscillator per voiced harmonic ---------------
    voiced_out <- zeros(frame_length)
    for each harmonic index l:
        case (prev.voiced[l], cur.voiced[l]):

            (unvoiced, unvoiced):
                nothing                       # the noise half owns it

            (voiced, unvoiced):
                add windowed fade-out of the OLD oscillator
                    at the old frequency, old amplitude, old phase

            (unvoiced, voiced):
                add windowed fade-in of the NEW oscillator
                    at the new frequency, new amplitude, new phase

            (voiced, voiced) and l < 8
                            and pitch changed by less than 10 percent:
                add ONE continuous oscillator whose
                    amplitude ramps old -> new, and whose
                    phase follows a second-order polynomial hitting
                    the old phase at the start of the frame and the
                    new phase at the end        # US 5,701,390

            (voiced, voiced) otherwise:
                add BOTH oscillators, windowed, and let them
                    overlap-add across the frame

    # ---- unvoiced half: noise, shaped, in the transform domain --------
    noise      <- windowed white noise
    spectrum   <- forward transform of noise
    for each harmonic index l:
        if cur.voiced[l]:
            zero the transform bins belonging to harmonic l
        else:
            scale those bins so the band carries cur.amplitude[l]
    unvoiced_out <- inverse transform, then weighted overlap-add
                    with the previous frame's block

    emit  voiced_out + unvoiced_out

The two halves are genuinely independent; the only thing they share is the voicing decision that partitions the spectrum between them, and the requirement that their windows are complementary so a band changing hands does not change loudness.

When the bits are wrong¶

Everything above assumes the parameters are correct. Over the air they frequently are not, and the synthesizer is where that gets managed.

The decoder can tell the difference between a frame it repaired and a frame it could not, because the channel layer is built to make that distinction cheap: the mechanism is the self-checking trick in the previous chapter. What matters here is the outcome it hands the synthesizer. The decoder "can identify severely corrupted frames through error pattern analysis and either repeat previous parameters or mute output rather than synthesize degraded speech." US 5,870,405

The escalation is three-stage:

1. Correct. The error-correcting codes fix what they can. The count of bits they had to change is itself an estimate of the channel error rate, which feeds the next two stages. US 5,870,405; US 5,247,579
2. Smooth, then repeat. As the estimated error rate climbs, the decoder leans on "adaptive smoothers which increase the perceived speech quality in the presence of uncorrectable bit errors" US 5,247,579 — pulling suspicious parameters toward the recent past rather than trusting them. When a frame is judged unrecoverable outright, "the received bits for the current frame are ignored and the model parameters from the previous frame are repeated for the current frame." US 5,247,579
3. Mute. If corruption persists across frames, repeating stops and the decoder outputs silence. US 5,870,405

This is why a digital voice signal fading out does not sound like an analogue one. The repeat holds the last good parameters, so the voice freezes on a vowel — the characteristic smeared, held-open sound — and then the mute cuts it dead. A parametric codec has no graceful way to get quieter: it either has parameters or it does not, and the descent from speech to silence happens over a handful of 20 ms frames.

The alternative is worse. Synthesizing from corrupted parameters means a random fundamental and random amplitudes driving thirty oscillators at full level, which is not degraded speech but a loud, unpredictable squawk. Frame repeat and mute exist to keep that out of anyone's ears.

Sources for this page¶

All of these are US patents or published literature; every one is listed with a link on Sources and method.

The characterisation of what regenerated phase costs perceptually — transients, non-speech signals, tandem coding, and the uselessness of waveform metrics — is this site's own reasoning from the model, not a claim made by any of the sources above.

Next: What is not in the public record. Where the documents stop and the shipping product begins, and what a black box on a bench can and cannot settle. Previously: The D-STAR frame on the air. Or go and listen to what all of this sounds like.

Last reviewed: 2026-08-10.