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Speech Processing: Theory of LPC Analysis and Synthesis

Module by: Douglas L. Jones, Swaroop Appadwedula, Matthew Berry, Mark Haun, Jake Janovetz, Michael Kramer, Dima Moussa, Daniel Sachs, Brian Wade

Summary: Speech analysis and synthesis with Linear Predictive Coding (LPC) exploit the predictable nature of speech signals. Cross-correlation, autocorrelation, and autocovariance provide the mathematical tools to determine this predictability. If we know the autocorrelation of the speech sequence, we can use the Levinson-Durbin algorithm to find an efficient solution to the least mean-square modeling problem and use the solution to compress or resynthesize the speech.

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 xn x n and yn y n is defined as
r x y l=∑n=-∞∞xnyn-l r x y l n x n y n l (1)
where n n is the sample index, and l l is the lag or time shift between the two signals Proakis and Manolakis (pg. 120). Since speech signals are not stationary, we are typically interested in the similarities between signals only over a short time duration ( <30 30 ms). In this case, the cross-correlation is computed only over a window of time samples and for only a few time delays l=01…P l 0 1 … P .
Now consider the autocorrelation sequence r s s l r s s l , which describes the redundancy in the signal sn s n .
r s s l=lN∑n=0N-1snsn-l r s s l l N n 0 N 1 s n s n l (2)
where sn s n , n=-P-P+1…N-1 n -P -P 1 … N 1 are the known samples (see Figure 1) and the 1N 1 N is a normalizing factor.
correlation.png
Figure 1: Computing the autocorrelation coefficients
Another related method of measuring the redundancy in a signal is to compute its autocovariance
r s s l=1N-1∑n=lN-1snsn-l r s s l 1 N 1 n l N 1 s n s n l (3)
where the summation is over N-l N l products (the samples s-P…s-1 s -P … s -1 are ignored).

Linear prediction model

Linear prediction is a good tool for analysis of speech signals. Linear prediction models the human vocal tract as an infinite impulse response (IIR) system that produces the speech signal. For vowel sounds and other voiced regions of speech, which have a resonant structure and high degree of similarity over time shifts that are multiples of their pitch period, this modeling produces an efficient representation of the sound. Figure 2 shows how the resonant structure of a vowel could be captured by an IIR system.
lpc_model.png
Figure 2: Linear Prediction (IIR) Model of Speech
The linear prediction problem can be stated as finding the coefficients a k a k which result in the best prediction (which minimizes mean-squared prediction error) of the speech sample sn s n in terms of the past samples sn-k s n k , k=1…P k 1 … P . The predicted sample s ^ n s ^ n is then given by Rabiner and Juang
s ^ n=∑k=1P a k sn-k s ^ n k 1 P a k s n k (4)
where P P is the number of past samples of sn s n which we wish to examine.
Next we derive the frequency response of the system in terms of the prediction coefficients a k a k . In Equation 4, when the predicted sample equals the actual signal (i.e., s ^ n=sn s ^ n s n ), we have sn=∑k=1P a k sn-k s n k 1 P a k s n k sz=∑k=1P a k szz-k s z k 1 P a k s z z k
sz=11-∑k=1P a k z-k s z 1 1 k 1 P a k z k (5)
The optimal solution to this problem is Rabiner and Juang a= a 1 a 2 … a P a a 1 a 2 … a P r= r s s 1 r s s 2… r s s PT r r s s 1 r s s 2 … r s s P R= r s s 0 r s s 1… r s s P-1 r s s 1 r s s 0… r s s P-2⋮⋮⋮⋮ r s s P-1 r s s P-2… r s s 0 R r s s 0 r s s 1 … r s s P 1 r s s 1 r s s 0 … r s s P 2 ⋮ ⋮ ⋮ ⋮ r s s P 1 r s s P 2 … r s s 0
a=R-1r a R r (6)
Due to the Toeplitz property of the R R matrix (it is symmetric with equal diagonal elements), an efficient algorithm is available for computing a a without the computational expense of finding R-1 R . The Levinson-Durbin algorithm is an iterative method of computing the predictor coefficients a a Rabiner and Juang (p.115).
Initial Step: E 0 = r s s 0 E 0 r s s 0 , i=1 i 1
for i=1 i 1 to P P.
    Steps
  1. k i =1 E i - 1 r s s i-∑j=1i-1 α j , i - 1 r s s |i-j| k i 1 E i - 1 r s s i j 1 i 1 α j , i - 1 r s s i j
    • α j , i = α j , i - 1 - k i α i - j , i - 1 α j , i α j , i - 1 k i α i - j , i - 1 j=1…i-1 j 1 … i 1
    • α i , i = k i α i , i k i
  3. E i =1- k i 2 E i - 1 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 δn δ n , the unit impulse, to the IIR system with lattice coefficients k i , i=1…P k i , i 1 … P as shown in Figure 3. Applying δn δ n to consecutive IIR systems (which represent consecutive speech segments) yields a longer segment of synthesized speech.
In this application, lattice filters are used rather than direct-form filters since the lattice filter coefficients have magnitude less than one and, conveniently, are available directly as a result of the Levinson-Durbin algorithm. If a direct-form implementation is desired instead, the α α coefficients must be factored into second-order stages with very small gains to yield a more stable implementation.
lattice.png
Figure 3: IIR lattice filter implementation.
When each segment of speech is synthesized in this manner, two problems occur. First, the synthesized speech is monotonous, containing no changes in pitch, because the δn δ n 's, which represent pulses of air from the vocal chords, occur with fixed periodicity equal to the analysis segment 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 the beginning 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 at lag l ≠ 0 l 0 implies the speech segment is periodic (or, more often, approximately periodic) with period l l. In synthesizing these segments, we recreate the periodicity by using an impulse train as input and varying the delay between impulses according to the pitch period. If the speech segment does not have a large peak in the autocorrelation coefficients, then the segment is an unvoiced signal which has no periodicity. Unvoiced segments such 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 k i , and continue with the lattice filter computation.

Additional Issues

  • Spanish vowels (mop, ace, easy, go, but) are easier to recognize using LPC.
  • Error can be computed as aTRa a R a , where R R is the autocovariance or autocorrelation matrix of a test segment and a 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 improve performance.
  • The pitch period for males ( 80 80- 150 150 kHz) is different from the pitch period for females.
  • For voiced segments, r s s T r s s 0≈0.25 r s s T r s s 0 0.25 , where T T is the pitch period.
References
  1. J. G. Proakis and D. G. Manolakis. (1996). Digital Signal Processing: Principles, Algorithms, and Applications. Upper Saddle River, NJ: Prentice-Hall.
  2. L. Rabiner and B. H. Juang. (1993). Fundamentals of Speech Recognition. Englewood Cliffs, NJ: Prentice-Hall.

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