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where ai is the short-term LPC coef cient and p is the LPC lter order. The calculation of LPC coef cients can be found in Section 11.4. This all-pole lter can be further decomposed with several second-order sections. If the LPC order p is an even number, it can be written as 1 1 = A(z) (1 + a11 z 1 + a12 z 2 )(1 + a21 z 1 + a22 z 2 ) (1 + aq1 z 1 + aq2 z 2 ) (9.16)
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with q = p/2. If p is an odd number with q = ( p 1)/2, the rst-order component (1 + aq+1 z 1 ) is used and Equation (9.16) can be modi ed as 1 1 = . A(z) (1 + a11 z 1 + a12 z 2 ) (1 + aq1 z 1 + aq2 z 2 )(1 + aq+1 z 1 ) (9.17)
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We assume that we have LPC coef cients and they are shared between a speech coder and a DTMF detector.
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Example 9.1: Compare the similarity of the FFT spectrum of the DTMF digit 5 and the frequency response of a 10th-order LPC synthesis lter. The frequencies used for DTMF digit 5 are f L = 770 Hz and f H = 1336 Hz at sampling rate 8000 Hz, and the DTMF signal can be generated by MATLAB as
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x(1 : N ) = sin (2 f L (1 : N )) + sin (2 f H (1 : N )) . Using MATLAB function levinson, we can compute the LPC coef cients from its autocorrelation function based on Equation (9.15) as follows:
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lpcOrder=10; % LPC order w=hamming(N); % Generate hamming window x=x.*w'; % Windowing m=0; while (m<=lpcOrder); % Calculation of auto-correlation r(m+1)=sum(x(1:(N-m)).*x((1+m):N)); m=m+1; end; a=levinson(r,lpcOrder); % Levinson algorithms
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The generated LCP coef cients are listed as follows:
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a[0] = 1.0000, a[1] = -1.5797, a[2] = 1.4570, a[3] = -0.0021, a[4] = -0.1805, a[5] = 0.1195, a[6] = 0.3082, a[7] = 0.2145, a[8] = 0.0230, a[9] = -0.0556, a[10] = 0.1797
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Figure 9.6 shows the spectrum of DTMF tones for digit 5 and the spectrum from the LPC coef cients estimation. This example demonstrates that the roots of an all-pole lter, which represents the dual frequencies of DTMF tones, can be closely located using the LPC modeling.
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Example 9.2: The roots of LPC synthesis lter coef cients can be computed using MATLAB
function roots(a). The angles of these roots can be converted from frequency in radian to Hz using MATLAB function freq=((angle(r)*8000/(2*pi)). The roots and angles are listed in Table 9.2 and the roots are also plotted in Figure 9.7.
Synthesis filter spectrum response LPC all-pole filter frequency response Original signal spectrum 40
Magnitude (dB)
2000 2500 Frequency (Hz)
Comparison of spectrum estimated by LPC
The roots from Example 9.2 are complex-conjugated pairs. These roots represent ve real secondorder functions in Equation (9.16). The third and fourth pairs of roots are the most important since they are very close to the unit circle, and their frequencies are comparable to two frequencies used for digit 5 . The amplitudes of other roots are smaller since they are located inside the unit circle, and their frequencies are not within the DTMF frequency ranges. The roots with amplitudes close to unity dominate the magnitude response. For the example using DTMF digit 5 , the relative differences in amplitude estimation are 0.0873 % for 770 Hz and 0.1274 % for 1336 Hz. The estimated frequency differences are 0.1799 % for 770 Hz and 0.1223 % for 1336 Hz. Examples 9.1 and 9.2 show that the estimated DTMF frequencies from LPC coef cients are very close to the DTMF frequencies de ned by ITU Q.23 recommendation. Thus, the LPC coef cients from speech coders can be used for DTMF detection.