Also known as: voiced synthesis, sinusoidal synthesis, 6.3 synthesis
MBE voiced synthesis is the IMBE decoder stage — TIA-102.BABA §6.3 — that turns the voiced harmonic amplitudes into audio by summing one sinusoid per voiced harmonic, each with amplitude and phase interpolated smoothly across the frame.1 The multi-band-excitation model represents the periodic part of speech as harmonics of a fundamental ω₀; §6.3 regenerates the corresponding oscillators and mixes them, taking care that every harmonic stays continuous through the boundary between frames so the output has no clicks.
The §6.3 model
Over one output frame of N = 160 samples (20 ms at 8 kHz), harmonic l is a cosine whose
amplitude and phase both move continuously from where they were last frame to where they are this
frame:
a(n) = (1 − n/N)·M_prev[l] + (n/N)·M_curr[l] (eq. 88)
θ(n) = θ_prev[l] + n·l·ω₀_prev + l·(ω₀_curr − ω₀_prev)·n²/(2N)
s_v(n) = Σ a(n)·cos(θ(n)) (eq. 89)
The amplitude term is a straight linear ramp between the previous and current linear amplitudes.
The phase term is quadratic in n: a linear part l·ω₀_prev (the frequency at the start of the
frame) plus a quadratic correction that sweeps the frequency toward l·ω₀_curr by the end, so the
oscillator’s instantaneous frequency glides between the two pitch estimates rather than jumping.
GopherTrunk’s synthVoiced in internal/voice/mbe/synth_voiced.go implements exactly this,
computing the linear coefficient a = l·ω_prev and quadratic coefficient
b = l·(ω_curr − ω_prev)/(2N) once per harmonic and accumulating a(n)·cos(θ₀ + a·n + b·n²).
Cross-frame continuity
The sum runs over every harmonic that is voiced this frame or was voiced last frame. That dual
condition is what makes voicing transitions clean: on an unvoiced→voiced onset M_prev is zero, so
the amplitude ramp fades the harmonic in over the frame; on a voiced→unvoiced offset M_curr is
zero, so it fades out. Neither produces the click a hard on/off switch would. On the very first
frame of a stream (PrevW0 == 0) the previous ω₀ is set equal to the current one, collapsing the
quadratic term to zero so the harmonic simply starts at its target frequency with no synthetic
sweep.
Continuity across frames depends on carrying phase memory forward correctly.
UpdateVoicedState, called after synthesis, advances each harmonic’s stored phase by the
closed-form average-frequency increment Δθ_l = N·l·(ω_prev + ω_curr)/2 (the integral of the
linear-frequency tilt over the frame), wraps it into [0, 2π), and stores the current amplitudes
as next frame’s M_prev. Keeping this memory phase-locked is essential — it is what lets harmonic
l pick up next frame exactly where θ(n=N) left off.
Phase dispersion
Fully phase-coherent synthesis re-aligns every harmonic once per pitch period, which radiates a
buzzy impulse train — the “robotic” artifact of a naive MBE decoder. §6.3 addresses it by
regenerating the phase of the upper harmonics: for l > L/4, GopherTrunk’s
SynthVoicedDispersed starts the frame from the coherent phase plus a bounded random offset
(PHIl = PSIl + offset), scaled by the unvoiced-harmonic fraction so mostly-voiced frames get
near-zero dispersion. The offset perturbs only that frame’s synthesis phase and is deliberately
not folded into the phase memory, so it never accumulates into a random walk — the memory stays
coherent while the audible buzz is broken up.
Relevance to SDR
Voiced synthesis is the half of MBE decoding that carries the pitched, sonorant character of speech — vowels and voiced consonants. Together with the unvoiced synthesis that supplies the noise-like bands, it reconstructs the full P25 Phase 1 voice waveform in pure Go, no vendor DSP chip. The amplitudes it renders are those already sharpened by spectral enhancement and, on weak channels, cleaned by adaptive smoothing.
Sources
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Multi-Band Excitation — Wikipedia, on the harmonic (voiced) model synthesized by §6.3. ↩