System Characterization

Magnitude of frequency response |H[k]| in response to complex sinusoids at all N frequencies

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 \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.

Impulse response h[n] is system output in response to a unit impulse input

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,

(1)   \begin{align*}             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.

Magnitude of frequency response |H[k]| in response to complex sinusoids at all N frequencies

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?

We answer these questions later in the discussion about convolution.

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