Frequency response of discrete-time filters

In[15]:=

  y[n_] == 
    (
     Sum[a[k] x[n-k],{k,0,Nmax}] 
     -Sum[b[k] y[n-k],{k,1,Mmax}]
    );

Since this represents a linear shift-invariant system, we know that with input Exp[I 2Pi w n], the output must be of the form H[w] Exp[I 2Pi w n]. Exploiting this knowledge, we can determine H[w], the frequency response, by substituting these values for x[n] and y[n] respectively:

In[16]:=

  H[w] Exp[I 2Pi w n] == 
   (
    Sum[a[k] Exp[I 2Pi w (n-k)],{k,0,Nmax}] 
    -Sum[b[k] H[w] Exp[I 2Pi w (n-k)],{k,1,Mmax}]
   );

We can cancel the factor E[I 2Pi w n] that is common to both sides, and pull H[w] outside the sum, to yield:

In[17]:=

  H[w] == 
   (
    Sum[a[k] Exp[I 2Pi w (-k)],{k,0,Nmax}] 
    - H[w] Sum[b[k] Exp[I 2Pi w (-k)],{k,1,Mmax}]
   );

Now, it is easy to solve for H[w]. Let's let Mathematica do the work:

In[18]:=

  Clear[a,b,H,Nmax,Mmax]; Solve[%,H[w]]

Out[18]=

                   -2 I k Pi w
              Sum[E            a[k], {k, 0, Nmax}]
  {{H[w] -> ----------------------------------------}}
                     -2 I k Pi w
            1 + Sum[E            b[k], {k, 1, Mmax}]

This is the transfer function of a general IIR filter. The transfer function of an FIR filter is the special case b[k]=0; that is, the denominator is unity.

Example: First-order IIR filter

Advanced topic: FIR filter as a truncated Fourier series approximation

Example: FIR approximation to an ideal LPF