EECS20N: Signals and Systems

Finding the Frequency Response Analytically

Given an LTI difference equation describing a filter, it is easy to analytically find an expression for the frequency response. Recall that if the input x is a complex exponential, then the output y will be the same complex exponential scaled by the frequency response evaluated at the frequency of the complex exponential.

Consider a filter given by

a1 y(n) + a2 y(n-1) + a3 y(n-2) = b1 x(n) + b2 x(n-1) + b3 x(n-2).

Let the input x be given by, for all integers n,

x(n) = e jωn.

Then the output y must be given by, for all integers n,

y(n) = H(ω) e jωn.

where H(ω) is the frequency response evaluated at the frequency ω. Plugging the form of the input and output into the difference equation we get

a1 H(ω) e jωn + a2 H(ω) e jω(n-1) + a3 H(ω) e jω(n-2)
= b1 e jωn + b2 e jω(n-1) + b3 e jω(n-2).

This can be factored as follows,

H(ω) e jωn (a1 + a2 e -jω + a3 e -2jω)
= e jωn (b1 + b2 e -jω + b3 e -2jω).

This can be solved for the frequency response,

H(ω) = e jωn (b1 + b2 e -jω + b3 e -2jω)/ e jωn (a1 + a2 e -jω + a3 e -2jω).

This form of the frequency response can be generalized to LTI difference equations with an arbitrary number of terms.