Introduction to the Fourier Transform

Given a continuous time LTI system with impulse response , recall that the response (if it's well defined) to the input signal is , where is the frequency response of the system. Now let's calculate the response to the input signal in terms of the impulse response:

From this, we see that the integral above in the square brackets must be the frequency response:

The right hand side of the above equation is known as the Fourier Transform of the signal . Thus, the frequency response of an LTI system is the Fourier transform of the system's impulse response.

We know that an LTI systems frequency response completely specifies the system. If the system maps real input signals to real output signals, one way of characterizing the system, in principle, is to do an infinite set of experiments: For each possible frequency , apply the input signal and measure the output signal, which must be a sinusoid of frequency . The amplitide and phase of the output signal gives us the magnitide and angle of the frequency response at frequency , . However, we also know that we can measure the system with only one experiment: apply the unit impulse as the input signal and measure the impulse response. As we will see, the unit impulse contains all frequencies, and in a sense in this case we are really doing the same infinite set of experiments, but all at once.