Frequency response of a system

H[f] and H[w] are often called the "frequency response" of the system. Let's now see why. They are complex-valued functions. Consider the continuous-time case. It is convenient to write this function in polar coordinates,

In[22]:=

  H[f] = A[f] * Exp[I p[f]]

where A[f] is called the amplitude frequency response of the system and p[f] is called the phase frequency response. The origin of the terminology will follow if we return to the sinusoidal input input signal, from which we can infer the output signal as follows. Assume the input is

In[23]:=

  x[t] = Cos[2Pi f t] 
       = Re[Exp[I 2Pi f t]]

Then the output must be the real part of H[f] Exp[I 2Pi f t], or

In[24]:=

  y[t] = Re[A[f] Exp[I p[f]] Exp[I 2Pi f t]]
       = A[f] Cos[I (2Pi f t + p[f])]

Thus the output is a sinusoid of the same frequency, with amplitude multiplied by A[f] and phase shifted by p[f].

You can see why H[f] might be called a frequency response. It tells us, for each frequency f, the amount by which the system shifts the amplitude and phase. It is complex-valued because it has to tell us two things: amplitude shift and phase shift, both as a function of frequency.

Example: Continuous-time differentiator system

Example: Discrete-time first difference system

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