EEL6586: HW#2

Get PDF file Due Friday, Feb 15, 2002 in class. Late homework will lose $e^{\char93  of days late} -1$ percentage points. To see the current late penalty, click on
http://www.cnel.ufl.edu/hybrid/harris/latepoints.html PART A: Noncomputer Problems

A1

Suppose you are estimating a short-term average magnitude function using a 200 point Hamming window for a 10KHz sampled speech waveform. Using Nyquist arguments and some simple assumptions, what is the most you can reasonably shift the window between applications without losing information?

A2

The short-term energy of a sequence $s(n)$ is defined as

$$Q(n)=\sum_{m=-\infty}^\infty [s(m)w(n-m)]^2$$

i

For the particular choice $w(m)=a^m$ for $m \ge 0$ and 0 otherwise, find a recurrence formula for $Q(n)$ in terms of $Q(n-1)$ and the input $s(n)$.

ii

What general property must the window $w(m)$ have in order that it be possible to find a recursive implementation?

A3

Consider the signal

$$s(n)= cos(\omega_o n)$$

i

Find the long-term autocorrelation function $r(k)$ for $s(n)$.

ii

Sketch (by hand) $r(k)$ as a function of $k$. Label important points.

A4

A train of impulses is fed through an all-pole model of $H(z)=1/(1+.25z^{-2})$. Sketch the time domain waveform for a few periods assuming $f_s=3KHz$ and pitch frequency is 300 Hz. Label all important parameters. Show all of your hand calculations, you may check your results with Matlab if you want.

A5

Sketch the magnitude of the Fourier Transform for the all-pole signal created in problem A4. Label all important parameters.

PART B: LPC Example Consider the infinite-length signal

$$x(n)= \ldots+1,0,-1,0,+1,0,-1,\ldots$$

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).

B1

Compute the autocorrelation matrix R (Assume an extremely long window and include a $1/N$ normalization factor for parts B1 and B2).

B2

Compute cross correlation vector $\b{p}$.

B3

Compute the two LPC coefficients for this problem.

B4

What is the resulting error in prediction?

B5

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

PART C: Computer Analysis of Speech You will write a program to segment a recorded sentence into three different components: silence (non-speech), voiced speech and unvoiced speech. You should run your code on the sentence found at
http://www.cnel.ufl.edu/analog/courses/EEL6586/sentence.html.

C1

Describe (in words) your strategy in writing and improving your program. A successful labeling program should utilize at least a short-term energy and a short-term zero crossing measure. However, as usual, you may add whatever you like to further improve the performance of your program. Make sure to turn in all code you use as the appendix of this homework.

C2

Show a plot that, you feel, best depicts the labeling of the test sentence.

C3

Have matlab create a table of the starting location of each labelled segment. For instance, your output should look something like the following:

Sample Number	Type
1	silence
234	unvoiced
578	voiced

C4

Comment on the accuracy of your algorithm. Make sure to run your code on other sentences to see how generally is can be applied. Show the labeling for one other sentence that you have recorded. In what cases does your algorithm tend to have problems?

C5

Using your algorithm, estimate the ratio of the average power in voiced vs. the average power in unvoiced phonemes in normal speech (in dB). You can do this by looking at the average energy per frame. Also estimate the rough percentages of energy in voiced and unvoiced phonemes in normal speech (now you must consider duration and frequency of occurence).