System Frequency Response

In the time domain, a system is characterized by an impulse response $h[n]$ which is the system output if a unit impulse signal is given at its input. In the frequency domain, it’s the frequency response that gives us insights into system behavior. There are two methods that lead to the same result.

Method 1

Frequency response is defined as the Fourier transform of its impulse response $h[n]$.

$$H[k]=\sum_{n=0}^{N-1}h[n]e^{-j2\pi\frac{k}{N}n}$$

This expression makes sense because it expresses the system behavior in the frequency domain. However, there’s an interesting interpretation of this idea that gives us a better understanding.

Method 2

Let’s focus now on the impulse input first. What exactly is the Fourier transform of a unit impulse signal? It is an all-ones sequence.

$$X[k]=\sum_{n=0}^{N-1}\delta[n]e^{-j2\pi\frac{k}{N}n}=1$$

When a unit impulse is applied as an input in the time domain, we are implicitly applying an all-ones sequence to the frequency domain. This all-ones sequence represents an equal contribution from all essential sinusoids at frequencies $k/N$.