**Homework Assignment 8 **

Due Oct. 18, 2000

- Problem 5.21
- Problem 6.4. Note that the designer also needs some other chips (adders
and multipliers) to complete the job. You should just get an equation where
X(k) can be written in terms of three eight-point DFTsX _{1}(k),X _{2}(k) andX _{3}(k). - Problem 6.11
- Let be a real-valued, bandlimited signal whose Fourier
transform is zero for
. The
sequence is obtained by sampling at 10 kHz. Assume that the
sequence is zero for and .
Let denote the 1000-point DFT of . It is known that and = 5. Determine for as many values of as you can in the region .

- Consider estimating the spectrum of a discrete-time signal using
the DFT with a Hamming window for . A conservative rule of thumb for
the frequency resolution of windowed DFT analysis is that the frequency
resolution is equal to the width of the main lobe of . You
wish to be able to resolve sinusoidal signals that are separated by as little
as in . In addition, your window length is constrained
to be a power of 2. What is the minimum length that will meet
your requirement?
- Let be a discrete-time signal whose spectrum you wish to estimate
using a windowed DFT. You are required to obtain a frequency resolution of at
least and are also required to use a window length . A
safe estimate of the frequency resolution of a spectral estimate is the
main-lobe width of the window used. Which of the windows in Table 8.2 will
satisfy the criteria given for the desired frequency resolution?
- Let be a discrete-time signal obtained by sampling a
continuous-time signal with some sampling period so the
. Assume is bandlimited to 100 Hz, i.e.,
for
. We wish to estimate the continuous-time
spectrum by computing a 1024-point DFT of , . What
is the smallest value of such that the equivalent frequency spacing
between consecutive DFT samples corresponds to 1 Hz or less in
continous time?
- Assume that is a 1000-point sequence obtained by sampling a continous-time signal, at 8 kHz and the is sufficiently bandlimited to avoid aliasing. What is the minimum DFT length such that adjacent samples of correspond to a frequency spacing of 5 Hz or less in the original continuous-time signal?

Note: Problems 4-8 are from **Discrete-Time Signal Processing, 2nd Ed.** by
Oppenheim and Schafer.