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In this module we examine convolution for continuous time signals. This willresult in the convolution integral and its properties . These concepts are very important in ElectricalEngineering and will make any engineer's life a lot easier if the time is spent now to truly understand what is going on.
In order to fully understand convolution, you may find it useful to look at the discrete-time convolution as well. Accompanied to this module there is a fully worked out example with mathematics and figures. It will also be helpful to experiment with the Convolution - Continuous time applet available from Johns Hopkins University . These resources offers different approaches to this crucial concept.
The derivation used here closely follows the one discussed in the motivation section above. To begin this, it is necessary to state theassumptions we will be making. In this instance, the only constraints on our system are that it be linear andtime-invariant.
As mentioned above, the convolution integral provides an easy mathematical way to express the output of an LTI system basedon an arbitrary signal, $x(t)$ , and the system's impulse response, $h(t)$ . The convolution integral is expressed as
Taking a closer look at the convolution integral, we find that we are multiplying the input signal by the time-reversedimpulse response and integrating. This will give us the value of the output at one given value of $t$ . If we then shift the time-reversed impulse response by a small amount, we getthe output for another value of $t$ . Repeating this for every possible value of $t$ , yields the total output function. While we would never actually do thiscomputation by hand in this fashion, it does provide us with some insight into what is actually happening. We find that weare essentially reversing the impulse response function and sliding it across the input function, integrating as we go.This method, referred to as the graphical method , provides us with a much simpler way to solve for the outputfor simple (contrived) signals, while improving our intuition for the more complex cases where we rely on computers. Infact Texas Instruments develops Digital Signal Processors which have special instruction sets for computations such as convolution.
Convolution is a truly important concept, which must be well understood.
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