(2)
Convolution, as just described using integrals, has a discrete time parallel which is more appropriate to digital signal processing systems.
For the mathematics, the integration is replaced by a summation, hence convolution may be re-expressed as
(2)
The same experiments carried out for the continuous system may be repeated for the discrete case, with similar results. Select a system of 16 sample length unit pulse, and a signal of a sampled ramp. Sliding the signal over the system response, the discrete time integrator can be seen to operate. The product of the two waveforms is displayed on the middle graph, and the sum of the product samples displayed on the lower graph.
Set the system response to be a unit sample at sample number 8, and select any signal that you desire. As you slide the signal through the system response you will observe that the discrete convolution result is a copy of the signal, delayed by 8 samples. This illustrates the discrete time equivalent system response to the impulse in the system response for the continuous time system. Thus the system response is described as the unit pulse response in a discrete time system, while it is described as the impulse response in a continuous time system.
Again, experiment with different system responses, and signals, designing your own on the signal window if you wish to try out signals that are not pre-defined.
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