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Department of Electrical and Computer Engineering Digital Speech Processing Homework No. 7 Problem 1 Linear prediction analysis is used to obtain an eleventh-order all-pole model for a segment of voiced speech that was sampled at a rate of FS = 8000 samples/second. The system function of the model is: H (z ) = G = A(z ) G = 11 G 11 k z k 1 k=1 (1 zi z 1 ) i=1 Table 1 shows ve of the roots of the eleventh-order prediction error lter, A(z ). i 1 2 3 4 5 |zi | 0.2567 0.9681 0.9850 0.8647 0.9590 zi (rad) 2.0677 1.4402 0.2750 2.0036 2.4162 Table 1: Root locations of eleventh-order prediction error lter in the z -plane. (a) Determine where the other six zeros of A(z ) are located in the z -plane. If you cannot precisely determine the pole locations, explain where the pole might occur in the z -plane. (b) Estimate the rst three formant frequencies (in Hz) for this segment of speech. (c) Which of the rst three formant resonances has the smallest bandwidth? How is this determined? (d) Plot and label the frequency response due to just the rst three formants of the all-pole model for the (analog) frequency range 0 f FS /2. Problem 2 An unvoiced speech signal segment can be modeled as a segment of a stationary random process of the form: x[n] = w[n] + w[n 1] where w[n] is a zero mean, unit variance, stationary white noise process and | | < 1. (a) What are the mean and variance of x[n]? (b) What system can be used to recover w[n] from x[n]? 1 (c) What is the normalized autocorrelation of x[n] at a delay of 1 sample, i.e., Rx [1] what is rx [1] = ? Rx [0] Problem 3 A causal LTI system has system function: H (z ) = 1 4z 1 1 0.25z 1 0.75z 2 0.875z 3 (a) Use the Levinson recursion to determine whether or not the system is stable. (b) Is the system minimum phase? Problem 4 A speech signal frame (windowed using a Hamming window) has energy: (0) En = s2 [m] = 2000 n m Using the autocorrelation method of analysis on this speech frame, the rst 3 PARCOR coe cients are computed and their values are: k1 = 0.5 k2 = 0.5 k3 = 0.2 e2 [m] that would be n 3 Find the energy of the linear prediction residual, En = m obtained by inverse ltering sn [m] by the optimal third order predictor inverse lter, A3 (z ). Problem 5 Write a MATLAB program to convert from a frame of speech to a set of linear prediction coe cients, using all 3 methods discussed in class, i.e., the Autocorrelation Method, the Covariance Method, and the Lattice Filter Method. Choose a section of a steady state vowel, and a section of unvoiced speech, and plot LPC spectra from the 3 methods along with the normal spectrum from the Hamming window weighted frame. Use N = 300, p = 12, with Hamming Window weighting for the autocorrelation method. Use the same parameters for the Covariance and Lattice Methods. Use the les ah.wav to get a vowel steady state sound beginning at sample 3000, and the le test 16k.wav to get a fricative beginning at sample 3000. (Dont forget that for the covariance and lattice methods, you also need to preserve p samples before the starting sample at n = 3000 for computing correlations, and error signals). 2

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