# System Characterization

In wireless communications and other applications of digital signal processing, we often want to modify a generated or acquired signal. A device or algorithm that performs some prescribed operations on an input signal to generate an output signal is called a system.

In another article about transforming a signal, we saw how a signal can be scaled and time shifted, or added and multiplied with another signal. These are all examples of a system. Amplifiers in communication receivers and filters in image processing applications are some systems that we interact with in daily lives. A communication channel is also a system about which we need to learn for successful transmission of data. Our main focus in these articles will be on a particular class of systems which are linear and time-invariant.

After learning about a 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 $\delta[n]$ 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 $h[n]$.

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 $3$ samples, the impulse response $h[n]$ will be $\delta[n]$ shifted by $3$ samples.
\begin{equation*}
h[n] = \delta[n-3]
\end{equation*}

Of course, impulse response $h[n]$ 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.
\begin{align*}
h[n] &~\xrightarrow{\text{\large{F}}}~ H[k]
\end{align*}

From the DFT definition,
\begin{align}\label{eqIntroductionFreqResponse}
H_I[k]\: &= \sum \limits _{n=0} ^{N-1} h_I[n] \cos 2\pi\frac{k}{N}n + h_Q[n] \sin 2\pi\frac{k}{N}n \\
H_Q[k] &= \sum \limits _{n=0} ^{N-1} h_Q[n] \cos 2\pi\frac{ k}{N}n – h_I[n] \sin 2\pi\frac{k}{N}n
\end{align}

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 $N$ complex sinusoids of equal magnitudes.

So when an impulse is input to a system in time domain, a sequence of $N$ 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?