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where N is the window (or frame) size, n is the frame index, and m is the sample index in the frame. We need to solve the following matrix equation to derive the prediction lter coef cients ai : Rn (0) Rn (1) . . . Rn ( p 1) Rn (1) Rn (0) . .. . . . Rn ( p 2) Rn ( p 1) Rn (1) a1 Rn ( p 2) a2 Rn (2) . = . . . . . . . . . Rn (0) Rn ( p) ap
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The left-hand side matrix is symmetric, all the elements on its main diagonal are equal, and the elements on any other diagonal parallel to the main diagonal are also equal. This square matrix is Toeplitz. Several ef cient recursive algorithms have been derived for solving Equation (11.3). The most widely used
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algorithm is the Levinson Durbin recursion summarized as follows:
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a (i 1) Rn (|i j|) j (11.5) (11.6)
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(11.7) (11.8)
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After solving these equations recursively for i = 1, 2, . . . , p, the parameters ai are given by aj = aj
( p)
1 j p.
Example 11.1: Consider the order p = 3 and given autocorrelation coef cients Rn ( j), j = 0, 1, 2, 3, for a frame of speech signal. Calculate the LPC coef cients. We need to solve the following matrix equation:
Rn (0) Rn (1) Rn (2) Rn (1) Rn (0) Rn (1) a1 Rn (2) Rn (1) Rn (1) a2 = Rn (2) . Rn (0) a3 Rn (3)
This matrix equation can be solved recursively as follows: For i = 1:
(0) (0) E n = Rn
k1 =
Rn (1)
(0) En
Rn (1) Rn (0)
(1) a1 = k 1 =
Rn (1) Rn (0)
2 Rn (1) Rn (0). 2 Rn (0)
(1) 2 E n = 1 k1 Rn (0) = 1
(1) (1) For i = 2, E n and a1 are available from i = 1. Thus, we have (1) Rn (2) a1 Rn (1) (1) En 2 Rn (0)Rn (2) Rn (1) 2 2 Rn (0) Rn (1)
k2 =
(2) a2 = k 2 (2) (1) (1) (1) a1 = a1 k2 a1 = (1 k2 ) a1 (1) (2) 2 E n = 1 k2 E n .
(2) (2) (2) For i = 3, E n , a1 , and a2 are available from i = 2. Thus, we get
k3 =
(2) (2) Rn (3) a1 Rn (2) + a2 Rn (1) (2) En
(3) a3 = k 3 (3) (2) (2) a1 = a1 k 3 a2 (3) (2) (2) a2 = a2 k 3 a1 .
Finally, we have
(3) (3) (3) a0 = 1, a1 = a1 , a2 = a2 , a3 = a3 .
We can use MATLAB functions provided in the Signal Processing Toolbox to calculate LPC coef cients. For example, the LPC coef cients can be calculated using the Levinson Durbin recursion as
[a,e] = levinson(r,p)
The parameter r is a deterministic autocorrelation sequence (vector), p is the order of denominator polynomial A(z), a = [a(1) a(2) a( p + 1)] where a(1) = 1, and the prediction error e. The function lpc(x,p) determines the coef cients of forward linear predictor by minimizing the prediction error in the least-square sense. The command
[a,g] = lpc(x,p)
nds the coef cients of a linear predictor that predicts the current value of the real-valued time series x based on past samples. This function returns prediction coef cients a and error variances g.
Example 11.2: Given a speech le voice4.pcm, calculate the LPC coef cients and spectral response using the function levinson( ). Also, compare the contours of speech spectrum with the synthesis lter s frequency response. Assume that the LPC order is 10 and Hamming window size is 256. The partial MATLAB code to calculate the LPC coef cients is listed in Table 11.1. The complete MATLAB program is given in example11_2.m. The magnitude response of the synthesis lter shows the envelope of the speech spectrum as shown in Figure 11.2. Example 11.3: Calculate the LPC coef cients as in Example 11.2 using the function lpc instead of levinson. In this case, g and e are identical if using the same order, and the LPC coef cients are identical to Example 11.2. Using a high-order synthesis lter, the frequency response is closer to the original speech spectrum. Figure 11.3 shows the use of high order (42) to calculate the LPC coef cients using lpc.