# System Characterization

After learning some properties of the system, the question is: how should we characterize a system in both time and frequency domains? Impulse response is a tool to characterize a system in time domain and frequency response to do the same in frequency domain. We start with description of impulse response.

## Impulse Response

As the name says it, impulse response is just the output response of a system to a unit impulse as an input. It is a way of characterizing how a system behaves. Impulse response is also a signal sequence and it is usually denoted as .

As an example, when a tuning fork is hit with a rubber hammer, the response it generates is a back and forth vibration of the tines that disturb surrounding air molecules for a certain amount of time: its impulse response. In a very similar manner, when a system is `kicked’ by a unit impulse signal, the whole sequence of samples it generates can be viewed as its impulse response.

If a system delays any input signal by samples, the impulse response will be shifted by samples.

Of course, impulse response of a real-world system is much more complex. An example is illustrated in Figure below.

## Frequency Response

From the definition of impulse response, it is straightforward to guess that frequency response of a system is defined as the DFT of its impulse response.

From the DFT definition,

(1)

Remember from another article on DFT examples that DFT of an all-ones sequence is a single impulse in frequency domain. Owing to the duality of time and frequency domains, the inverse is also true: the DFT of a single impulse in time domain is an all-ones rectangular sequence in frequency domain, which is nothing but complex sinusoids of equal magnitudes.

So when an impulse is input to a system in time domain, a sequence of complex sinusoids with equal magnitudes are input to the system in frequency domain. Figure below shows an example of frequency response magnitude of a system.

Two questions arise at this stage:

1. What happens at the output in frequency domain as a result of a complex sinusoidal input?
2. In fact, any DSP learner notices a dominant use of real and complex sinusoids throughout the DSP literature. What makes such signals so significant?