EEL6586: HW#3

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EEL 6586: HW#3

Assignment is due Friday, Feb 20, 2004 in class. Late homework loses $e^{\char93  of days late} -1$ percentage points. See the current late penalty at http://www.cnel.ufl.edu/hybrid/harris/latepoints.html This assignment includes both matlab and textbook questions. PART A: Textbook problems

A1

Prove the following property of autocorrelation functions: $r[0] \ge r[k]$ for any value of k.

A2

Compute the autocorrelation function $r[k]$, of the following discrete-time signal:

$$x[n] = \sin(\omega n)$$

A3

Assume that $r[k]$ is the autocorrelation function of $x[n]$ and that $X(\omega)$ and $R(\omega)$ are the Fourier transforms of $x[n]$ and $r[k]$ respectively. Prove that:

$$R(\omega) = \vert X(\omega)\vert^{2}$$

A4

Assume that an infinite impulse train

$$\sum_k \delta[n+kP]$$

is filtered by a vocal-tract model given by $H(z)=1/(1+.9z^{-1}+.81z^{-2})$ to produce a speech signal $s[n]$. For simplicity, assume that the excitation (the impulse train) is uncorrelated and answer the following:

1. Derive the difference equation for $s[n]$.
2. Compute the autocorrelation function $r[0]$ for the signal $s[n]$.
3. Compute the autocorrelation function $r[1]$ for the signal $s[n]$.
4. Compute the single LPC coefficient ($p=1$) for this system.
5. How does this coefficient compare to the first coefficient when $p=2$? Explain.

A5

Assume that white noise excitation $w[n]$ is filtered by an all-pole vocal-tract model $H(z)=1/(1+.25z^{-2})$ to produce a speech signal $s[n]$. $w[n]$ is defined:

$$E\{w[n]w[m]\}= \left\{ \begin{array}{ll} 1 & m=n \\ 0 & m\neq n \end{array} \right.$$

In this problem you will use LPC to derive an all-pole approximation to $H(z)$.

1. Derive the difference equation for $s[n]$.
2. Compute the autocorrelation function $r[0]$ for the speech signal $s[n]$.
3. Compute the autocorrelation function $r[1]$ and $r[2]$ for the speech signal $s[n]$.
4. Compute the first two LPC coefficients ($p=2$).
5. Derive $\hat H(z)$, the all-pole approximation to $H(z)$. Does your answer make sense?

PART B: Short Answer

B1

Give an example of a voiced fricative and also suggest an English word that contains that voiced fricative.

B2

Pre-emphasis filters are usually of the form $(1-\alpha z^{-1})$ with typical values of $\alpha$ of 0.95 or 0.97. Why is the value of $\alpha=0.7$ a poor choice?

B3

Explain what happens to your speech when you breath helium into your vocal tract and try to speak.

B4

Explain why humans have no problem determining the pitch of voices through the telephone, even though the cutoff frequency (about 300Hz)is larger than typical pitch frequencies.

B5

Why are overlapping windows preferred over non-overlapping windows in speech processing?

PART C: Computer Analysis of Speech In this part you will write a program for automatic pitch analysis. You will run your program on three recorded sentences at
http://www.cnel.ufl.edu/hybrid/courses/EEL6586/sentence.html for an adult male (sentence 1), adult female (sentence 2) and a child (sentence 3). Through the following steps, you will develop a pitch algorithm that processes the autocorrelation of the LPC residue for each window of speech:

C1

Break the sentence into overlapping windows. Describe how you choose the window type, length and overlap for this pitch estimation algorithm.

C2

Compute the LPC coefs for each window and inverse filter the signal in that window to get the residue. Show a typical example of the windowed signal, with plots of the time domain signal, its power spectrum, the smooth envelope from the LPC coefs and the residue (in the time domain).

C3

Run an autocorrelation on the residue signal and show an example plot using the residue from [C2].

C4

Write a procedure that automatically computes the pitch by finding the ``first biggest peak" after the lag zero peak. What pitch is detected for your example window from [C3]?

C5

Put all of the pieces together and write an algorithm to compute pitch for each window Write a program that determines the pitch of each window of a sentence (if the pitch exists). Show a plot of F0 (in Hz) vs. time (in seconds). You may need to add an additional filtering step to smooth out the pitch values. Indicate unvoiced regions and silence with a pitch of zero. Hand in plots showing the results on the three sentences. Compute the average pitch of each sentence making sure to only consider the voiced regions.

As always, hand in all of your matlab code as the appendix of your homework. Discuss your algorithms in detail and comment on the accuracy of your algorithms.