EEL6586: HW#2

Due Wednesday, March 15, 2000 in class. No electronic submissions, only hardcopies. Late homework will lose $e^{\char93 ~of~days~late} -1$ percentage points. To see the current late penalty, click
http://www.cnel.ufl.edu/analog/harris/latepoints.html

PART A: Noncomputer Problems

A1

(10%) 6-parameter LPC analysis was performed on a voiced phoneme recorded with f_S=10kHz. The resulting pole/zero plot is as shown in the following figure:

Estimate which phoneme was recorded. List the proper symbol of the phoneme and explain your reasoning.

A2

(10%) Compute the complex cepstrum of H(z)=1+z^-2

A3

(30%) Consider the infinite-length signal x(n), a short segment is shown above. Your goal is to derive the LPC coefficients for the prediction of x(n). Assume order p=2 (that is, you will only consider the two previous values in predicting the next one).

(a)

Compute the autocorrelation matrix R and the cross correlation vector $\b{p}$. (Assume an extremely long window and include a 1/N normalization factor).

(b)

Compute the LPC coefficients and the resulting error in prediction.

(c)

Sketch the magnitude response of the all-pole estimator for this signal (H(z)).

PART B: Computer Analysis of Speech

You will write two programs for speech analysis. You should run your code on a sentence that you record as well as on the sentence found at:
http://www.cnel.ufl.edu/analog/courses/EEL6586/sentence.html.

B1

(25%) Write a program that determines the pitch of a signal (F0 in Hz.) We talked about a number of algorithms in class but one of the autocorrelation techniques would be straightforward to implement.

B2

(25%) Write a program that determines the first three formant location (F1, F2, F3 in Hz.) We will talk in class about several techniques but factoring the LPC polynomial is probably the most straightforward.

As always, hand in all of your matlab code. Discuss your algorithms in detail and hand in plots showing the results on the two sentences. Comment on the accuracy of your algorithms.

PART C: Extra Credit

(5%) Assuming that

$$H(z)=\sum_{n=0}^\infty h(n)z^{-n} = \frac{G}{1-\sum_{k=0}^pa(k)z^{-k}}$$

Prove that the complex cepstrum $\hat{h}(n)$ can be derived from the linear prediction coefficients a(k) using the following relation:

$$\hat{h}(n)=a(n) + \sum_{k=1}^{n-1}(k/n) \hat{h}(k)a(n-k)$$

for $n \ge 1$.