Contents
Homework 10, ECE438, Fall 2015, Prof. Boutin
Harcopy of your solution due in class, Monday November 23, 2015
Presentation Guidelines
- Write only on one side of the paper.
- Use a "clean" sheet of paper (e.g., not torn out of a spiral book).
- Staple the pages together.
- Include a cover page.
- Do not let your dog play with your homework.
Question 1
Consider a model of the vocal tract consisting of three tubes of equal length l connected to a first tube that is infinitely thin. Assume that l=Tc where c is the speed of sound and T is the period at which you sample the airflow throughout the model.
a) Obtain the transfer function of this model of the vocal tract. (You may use the matrix equations for the tube junction/time delay obtained in class without justification.)
b) How many formants could one create with such a model? Explain.
c) Explain how one would control the location of the formants with such a model.
Question 2
Why do the poles of the transfer function of the vocal tract always come in complex conjugate pairs? Explain.
Question 3
We have seen that the transfer function of the vocal tract for voiced phonemes has poles (which create the formants).
a) What does this imply regarding the difference equation representing the system (in discrete-time)?
b) Could the vocal tract be modeled using an FIR filter? Explain.
Questions 4
Warning: do not confuse the period of the sampling with the period of the pulse train produced by the vocal tract (1/pitch). Use different variables!
A person is pronouncing a phoneme. The pitch of the person's voice is 250Hz. The phoneme has two formants: a large one at 500 Hz, a weak one at 1.25 kHz.
You are given a digital recording of that phoneme. The sampling rate for the recording is 5kHz.
a) From the information given, can you tell the gender of the person?
b) How does the gender of the person influence the location of the local maxima of the magnitude of the frequency response of the vocal tract?
c) Sketch the graph of the magnitude of the CT Fourier transform of the phoneme. (Assume the CTFT is zero for |f|>2.5KHz.) How does it compare to the graph of the magnitude of the DT Fourier transform of the digital recording of the phoneme?
d) Sketch the approximate location of the poles of the transfer function H(z) corresponding to the vocal tract of that person when he/she is pronouncing the phoneme.
Discussion
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