EEL6586: HW#3

Due Friday, March 23, 2001 in class. Late homework will lose $e^{\char93 ~of~days~late} -1$ percent. See http://www.cnel.ufl.edu/analog/harris/latepoints.html for penalty.

PART A: Noncomputer Problems

A1

Compute the complex cepstrum of H(z)=1/(1+.25z^-2)

A2

Let x₁(n) and x₂(n) denote two sequences and $\hat{x}_1(n)$ and $\hat{x}_2(n)$ their corresponding complex cepstra. If ${x}_1(n) * {x}_2(n) = \delta(n)$ determine the relationship between $\hat{x}_1(n)$ and $\hat{x}_2(n)$.

A3

Suppose the complex cepstrum of of y(n) is $\hat{y}(n)=\hat{s}(n)+2\delta(n)$. Determine y(n) in terms of s(n).

A4

Why is the predictor in a DPCM scheme often placed in a feedback loop around the quantizer?

A5

Why is a higher-order predictor more often used in DPCM with an adaptive scheme then in one without time adaptation?

PART B: LPC Hand Analysis of Simplfied Unvoiced Phoneme

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 part, you will use LPC to derive an all-pole approximation to H(z).

B1

Derive the difference equation for s(n). Make sure you get this difference equation correct because you will use it in the next two parts.

B2

Compute the autocorrelation function r(0) for the speech signal s(n).

B3

Compute the autocorrelation function r(1) and r(2) for the speech signal s(n).

B4

Compute the first two LPC coefficients (p=2).

B5

Derive $\hat H(z)$, the all-pole approximation to H(z). Does your answer make sense?

PART C: Adaptive Quantization Coder Problem

Consider a waveform quantization system with step size d(n), input x(n), and output $\hat x(n)$, where

$$\hat x(n) = \left\{ \begin{array}{ll} 3d(n)/2 & {\rm for~~} ... ...0 \\ -3d(n)/2 & {\rm for~~} x(n) < -d(n) \end{array}\right.$$

$$d(n) = \left\{ \begin{array}{ll} Bd(n-1)+ D & {\rm if~} \ver... ...,n-2, n-3 \\ Bd(n-1)+ E & {\rm otherwise} \end{array}\right.$$

(i.e. D is used if, in two of the last three samples, the input exceeded the step size.). Assume that D>E>0, that d(n)=0 for $n \le 0$, and the the sampling rate of x(n) is 10kHz.

C1

What is the bit rate if this quantizer codes x(n)?

C2

Is it necessary to transmit the step size as well as a coded version of $\hat x(n)$ each sample instant? Explain.

C3

What is the maximum value that d(n) could attain? What sequence of x(n) would lead to such a d(n)? Explain.

C4

What is the minimum value that d(n) could attain? What sequence of x(n) would lead to such a d(n)? Explain.

C5

Assume that D=6, E=0, B=0.8 and x(n)=20[u(n-3)-u(n-12)], where u(n) is a step function. Make a table of values of x(n), $\hat x(n)$, and d(n) for $0 \le n \le 23$.

PART D: Computer Analysis of Speech

You will write two programs for speech analysis. You should run your code on the two sentences found at:
http://www.cnel.ufl.edu/analog/courses/EEL6586/sentences.html.

D1

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.

D2

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. Make sure that your code is your own, don't download anything from other students or the internet without proper acknowledgements. Discuss your algorithms in detail and hand in plots showing the results on the two sentences. Comment on the accuracy of your algorithms.