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 ^jwn.

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

y(n) = H(w) e ^jwn.

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

a₁ H(w) e ^jwn + a₂ H(w) e ^jw(n-1) + a₃ H(w) e ^jw(n-2)
= b₁ e ^jwn + b₂ e ^jw(n-1) + b₃ e ^jw(n-2).

This can be factored as follows,

H(w) e ^jwn (a₁ + a₂ e ^-jw + a₃ e ^-2jw)
= e ^jwn (b₁ + b₂ e ^-jw + b₃ e ^-2jw).

This can be solved for the frequency response,

H(w) = e ^jwn (b₁ + b₂ e ^-jw + b₃ e ^-2jw)/ e ^jwn (a₁ + a₂ e ^-jw + a₃ e ^-2jw).

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