Speech processing: theory of lpc analysis and synthesis

Introduction

Linear predictive coding ( LPC ) is a popular technique for speech compression and speech synthesis.The theoretical foundations of both are described below.

Correlation coefficients

Correlation, a measure of similarity between two signals, is frequently used in the analysis of speech and other signals.The cross-correlation between two discrete-time signals $x(n)$ and $y(n)$ is defined as

Now consider the autocorrelation sequence $r_{ss}(l)$ , which describes the redundancy in the signal $s(n)$ .

Another related method of measuring the redundancy in a signal is to compute its autocovariance

Linear prediction model

Linear prediction is a good tool for analysis of speech signals. Linear prediction models the human vocaltract as an infinite impulse response ( IIR ) system that produces the speech signal. For vowel sounds and other voiced regions of speech, whichhave a resonant structure and high degree of similarity over time shifts that are multiples of their pitch period, thismodeling produces an efficient representation of the sound. [link] shows how the resonant structure of a vowel could be captured by an IIR system.

The linear prediction problem can be stated as finding the coefficients $a_k$ which result in the best prediction (which minimizes mean-squared prediction error) of the speech sample $s(n)$ in terms of the past samples $s(n-k)$ , $k=\{1, \dots , P\}$ . The predicted sample $\hat{s}(n)$ is then given by Rabiner and Juang

Next we derive the frequency response of the system in terms of the prediction coefficients $a_k$ . In [link] , when the predicted sample equals the actual signal ( i.e. , $\hat{s}(n)=s(n)$ ), we have $$s(n)=\sum_{k=1}^P a_{k}s(n-k)$$ $$s(z)=\sum_{k=1}^P a_{k}s(z)z^{-k}$$

Initial Step: $E_0=r_{ss}(0)$ , $i=1$

for $i=1$ to $P$ .

Steps

1. $k_i=\frac{1}{E_{i-1}}(r_{ss}(i)-\sum_{j=1}^{i-1} {\alpha }_{j,i-1}()r_{ss}(\left|i-j\right|))$
2. • ${\alpha }_{j,i}={\alpha }_{j,i-1}-k_i{\alpha }_{i-\mathrm{j},i-1}$ $j=\{1, \dots , i-1\}$
   • ${\alpha }_{i,i}=k_i$
3. $E_i=(1-k_i^2)E_{i-1}$

Lpc-based synthesis

It is possible to use the prediction coefficients to synthesize the original sound by applying $\delta (n)$ , the unit impulse, to the IIR system with lattice coefficients $k_i,i=\{1, \dots , P\}$ as shown in [link] . Applying $\delta (n)$ to consecutive IIR systems (which represent consecutive speech segments) yields a longer segment of synthesizedspeech.

In this application, lattice filters are used rather than direct-form filters since the lattice filter coefficientshave magnitude less than one and, conveniently, are available directly as a result of the Levinson-Durbinalgorithm. If a direct-form implementation is desired instead, the $\alpha$ coefficients must be factored into second-order stages with very small gains to yield a more stableimplementation.

When each segment of speech is synthesized in this manner, two problems occur. First, the synthesized speech ismonotonous, containing no changes in pitch, because the $\delta (n)$ 's, which represent pulses of air from the vocal chords, occur with fixed periodicity equal to the analysissegment length; in normal speech, we vary the frequency of air pulses from our vocal chords to change pitch. Second,the states of the lattice filter ( i.e. , past samples stored in the delay boxes) are cleared at thebeginning of each segment, causing discontinuity in the output.

To estimate the pitch, we look at the autocorrelation coefficients of each segment. A large peak in the autocorrelation coefficient atlag $l\neq 0$ implies the speech segment is periodic (or, more often, approximately periodic) with period $l$ . In synthesizing these segments, we recreate the periodicity by using an impulse train as input and varyingthe delay between impulses according to the pitch period. If the speech segment does not have a large peak in theautocorrelation coefficients, then the segment is an unvoiced signal which has no periodicity. Unvoiced segmentssuch as consonants are best reconstructed by using noise instead of an impulse train as input.

To reduce the discontinuity between segments, do not clear the states of the IIR model from one segment to the next.Instead, load the new set of reflection coefficients, $k_i$ , and continue with the lattice filter computation.

Additional issues

• Spanish vowels (m o p, a ce, ea sy, g o , b u t) are easier to recognize using LPC.
• Error can be computed as $a^TRa$ , where $R$ is the autocovariance or autocorrelation matrix of a test segment and $a$ is the vector of prediction coefficients of a template segment.
• A pre-emphasis filter before LPC, emphasizing frequencies of interest in the recognition or synthesis, can improveperformance.
• The pitch period for males ( $80$ - $150$ kHz) is different from the pitch period for females.
• For voiced segments, $\frac{r_{ss}(T)}{r_{ss}(0)}\approx 0.25$ , where $T$ is the pitch period.