Convolution Demonstration: Flip, Shift, and Integrate.

This page contains some animations that demonstrate the convolution process. They were created using the MathCAD worksheet found at www.ensc.sfu.ca/people/faculty/cavers/ENSC380/convdemo.mcd.

If the animations don't show up correctly in your browser, use these links to download them one at a time:

conv1.avi
filter1.avi
filter2.avi
reson1.avi
reson2.avi

Each of these animations has two parts.

The top graph shows the components of the convolution integral.

  • The blue line is the input signal.
  • The red line is the flipped and shifted impulse response.
  • The black line is the product of the blue and red lines. (ie the term inside the convolution integral)

    The lower graph shows the output signal.

  • The value of the output signal is the area (integral) of the product of the blue and red lines at a given time.

    NOTE: Click on the animations to play/pause them. :)

    RC Response to a Pulse

    In this case, the input signal (blue) is a pulse, and the impulse response (red) is exp(-t). Watch this a few times, stop it in the middle and convince yourself that at a given time the output is the integral of the product of the input and the flipped and shifted impulse response. As you can see, the result is the typical RC output.

    RC Response to a Sine Wave (low frequency)

    Now, the input signal is a little more interesting. Observe how the phase of the output is slightly shifted, and that the magnitude has been decreased. Also, you may notice that there is a transient (non-periodic) component at the beginning of the output (it's a little hard to see, but it's there. It looks like the first crest of the wave has been pushed upwards). Again, stop the animation in the middle a few times and try to figure out *why* the phase is shifted, *why* the magnitude is less, *why* there is a transient, etc..

    RC Response to a Sine Wave (high frequency)

    I've increased the frequency of the input sine wave. Notice that now the magnitude is much less, and the phase is shifted even more! We call this impulse response "low pass" because high frequencies are attenuated, but low frequencies are passed. Again, can you see *why* this is happening?

    Resonance!

    Now, instead of an RC impulse response, let the impulse response be a slowly decaying sine wave itself. If the input function (blue) is a sine wave with the same frequency, the system is in "resonance". See how the output magnitude becomes very large! Can you see why?

    However, if the input frequency is quite different than the impulse response, the output magnitude does not become very large. Also, compare the phases of the input and output in this case to the resonant one. In resonance, the phases are the same.

    Is this stuff cool or what? :)