# Signals and Systems

## Some Useful Links to Suppliment or Substitute for a Textbook

### FIR Filters

• Fast Convolution Based on the FFT This reference shows how end effects are dealt with. To use the FFT for convolution, you need to do it in blocks, which leads to end effects, and more latency, but if your blocks are big enough, it speeds up the convolution.

## Articles

### Octave Tutorials

Installing Octave on a Mac (Chris Lau)

Octave and Scilab on a Mac (Ben Henry)

ASN2 - Octave Tutorial (Jodi S. Hodge)

## Homework Assignments

• HW #4 - Given a linear time-invariant system where $\ u(t)$ produces an output $\ w(t)$, find the output due to any function $\ x(t)$ (Chris Lau)
• HW #5: (Chris Lau)
• Part 1 - Find $\mathcal{F}[e^{- \sigma t} x(t)u(t)]$ and relate it to the Laplace Transform. Derive the Inverse Laplace Transform of this from the inverse Fourier Transform.
• Part 2 -
• HW #6 - Pick a property of the Fourier Transform & present it on the Wiki. Make a table with all your properties. Interpret your property. (Ben Henry)(Chris Lau)(Victor Shepherd)
• HW #7 - Finish the practice tests
• HW #8 - Make a page about interpolating FIR filters. Note how many multiply/add operations.(Jodi S. Hodge)(Chris Lau)(Victor Shepherd)
• HW #9 - Add to #8 writeup how to do a decimating filter and figure out how many multiply & adds are needed for a n/2 decimating low pass filter.(Jodi S. Hodge)(Chris Lau)(Victor Shepherd)
• HW #10 - Use Octave (or Mathlab or Silab) to plot the frequency response of low pass filters with cut off frequencies of 1/32T, 1/8T, and 1/4T and compare how many coeffficients are needed with an eye to answer the question "Is it less calculation to decimate and then filter, or better to put the filter in the pre-decimation filter?" (Jodi S. Hodge)(Victor Shepherd)
• HW #11 - Is our method the same as Mark Fowler's? See

Wiki. Same # multiply and adds? See Notes 11/3/10. (Jodi S. Hodge)(Victor Shepherd)

• HW #12 - Experiment with a variety of signals having a 3Khz bandwidth to determine the resolution you can get when doing a cross correlation $\ r(m) = \displaystyle\sum\limits_{n=0}^{N-1} x(n) x(n+m)$. You can generate the signals randomly and filter them to obtain the band-limited signals. (Jodi S. Hodge)
• HW #13 - Derive the following realtions:
• a) $DFT(x(k-l))\!$
• b) $DFT(e^{j2 \pi lk/N}x(k)\!$
• c) $\sum\limits_{k=0}^{N-1} x(k)y(k)^{*}=c\sum\limits_{k=0}^{N-1} X(n)Y(n)^{*}$ (Victor Shepherd)