Cepstral Analysis in Source-Filter Separation for TTS

Cepstral analysis is a foundational signal processing technique in parametric Text-to-Speech (TTS) synthesis that separates a speech signal into its two core constituents: the vocal tract filter and the glottal excitation source. By transforming the multiplicative combination of source and filter into an additive relationship through logarithmic computation, the cepstrum isolates slowly varying vocal tract resonances from rapidly oscillating pitch harmonics in the quefrency domain. This distinct separation enables parametric TTS models to independently extract, model, and modify vocal tract shapes (timbre) and fundamental frequency (pitch) without mutual interference.

The Source-Filter Convolution Problem

According to the classical source-filter model of speech production, the acoustic speech signal \(s(n)\) is generated by convolving an excitation signal \(e(n)\)—representing airflow pulses from the vocal folds or noise from constriction—with the impulse response of the vocal tract \(h(n)\):

\[s(n) = e(n) * h(n)\]

In the frequency domain via the Fourier Transform, convolution becomes multiplication:

\[S(\omega) = E(\omega) \cdot H(\omega)\]

Directly separating \(E(\omega)\) and \(H(\omega)\) from the composite signal \(S(\omega)\) using standard linear filtering is impossible because both components span overlapping frequency bands across the entire audible spectrum.

Homomorphic Deconvolution via the Log Spectrum

Cepstral analysis solves this problem using homomorphic deconvolution. By calculating the complex or real magnitude spectrum of the speech segment and applying a logarithm, the multiplication operation is converted into a linear summation:

\[\log |S(\omega)| = \log |E(\omega)| + \log |H(\omega)|\]

This mathematical transformation changes how the components behave:

• \(\log |H(\omega)|\) (The Filter): The vocal tract consists of resonant cavities (formants) that produce smooth, broad peaks across the frequency spectrum, corresponding to low-frequency variations along the frequency axis.
• \(\log |E(\omega)|\) (The Source): For voiced speech, the glottal excitation consists of an impulse train, which produces fine, closely spaced harmonic ripples across the frequency spectrum, corresponding to high-frequency variations along the frequency axis.

Separation in the Quefrency Domain

To separate these additive components, an Inverse Discrete Fourier Transform (IDFT) is applied to the log-magnitude spectrum. This maps the signal into the "quefrency" domain (the cepstrum):

\[c(n) = \text{IDFT}\{\log |S(\omega)|\}\]

Because the vocal tract envelope \(\log |H(\omega)|\) varies slowly across frequency, its energy concentrates almost entirely at the lower end of the quefrency scale (low quefrency). Conversely, the periodic harmonic peaks of \(\log |E(\omega)|\) manifest as a sharp, isolated peak at the pitch period (\(T_0\)) along with its harmonics at higher quefrencies.

Because the two signals occupy distinct quefrency regions, they can be disentangled using a linear filtering process known as "liftering":

• Low-pass liftering eliminates the pitch peak and retains only the low-quefrency coefficients, yielding an accurate representation of the vocal tract's spectral envelope.
• High-pass liftering removes the vocal tract formants and isolates the pitch periodicity and excitation characteristics.

Value to Parametric Text-to-Speech

In parametric TTS architectures (such as HMM-based synthesis or vocoders like STRAIGHT and WORLD), high-quality speech generation requires parameters that can be manipulated independently:

1. Independent Control of Prosody and Identity: By isolating the vocal tract envelope into compact representations—such as Mel-Cepstral Coefficients (MCEPs)—the system can modify pitch (\(F_0\)) and duration arbitrarily during inference without altering the speaker's vocal tract characteristics or vowel identities.
2. Compact Dimensionality for Acoustic Models: Low-quefrency cepstral coefficients capture the most perceptually relevant acoustic features with very few parameters (typically 25 to 60 coefficients per frame), making statistical or neural modeling computationally efficient and stable.
3. Artifact-Free Synthesis: Recombining predicted vocal tract envelopes with independently generated excitation signals avoids the phase distortion and buzziness that occur when excitation information leaks into the spectral envelope model.