EEL6586: Automatic Speech Processing HW#1
EEL 6586: HW#1
Assignment is due Friday, January 16, 2004 in class. Late home-
work loses e#of days late −1percentage points. See the current late
penalty at http://www.cnel.ufl.edu/hybrid/harris/latepoints.html
This is a short review of some DSP topics relevant for speech processing. You
should not use Matlab to do any of these problems however you are welcome
(and encouraged) to use Matlab to check your answers.
1. In speech processing, all-pole filters are commonly used to model the
human vocal tract response with the following equation:
H(z) = 1
1−Pp
i=1 aiz−i
Suppose the vocal tract frequency response is crudely modelled with
the following second-order, causal equation:
H(z) = 1
1 + .9z−1+.81z−2,
(a) Sketch the complex plane including the unit circle with all poles
and zeros of this system. Is the system stable? Explain.
(b) Derive the inverse z-transform, h[n].
2. Rectangular and Hamming windows are widely used in speech signal
processing. The rectangular window is defined as follows:
hπ[n] = (1,0≤n < N (1)
0, otherwise (2)
(a) Use the DTFT to derive the frequency response of the rectangular
window with N=9. Show all your work.
(b) Calculate the width of the main lobe of the rectangular window
(hint: find the first zero).
(c) The Hamming window is defined as:
hh[n] =
(1 −.46) −.46 cos( 2πn
M),0≤n < M (3)
0, otherwise (4)
J.G. Harris January 9, 2004 1
