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Signals | Systems | System Response | |||||
Frequency Representation | Fourier Series / Transforms | Background Material |
When we have established that a system has the fundamental properties of linearity and time-invariance, we can calculate the unit impulse response, h(t). To calculate the unit impulse response, we set the input to the impulse function δ(t), and set the output to the impulse response function h(t).
Let's see how this works out. If our particular system of interest is linear, then it follows that it should have the following properties:
We can put set the input/output to a ,,
If our system is time invariant, then all of the (h sub #) should be equal to each other. Thus we can sum up this property as h(t).
Once we have h(t), we can calculate the response of the system to ANY input according to the following equation, y(t) = h(t) ** x(t), for continuous signals and y[n] = h[n] ** x[n] (where ** is designated to be the mathematical operation defined by convolution, which can be explored more in the next section of the flow chart).
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