Key points you have learned

The transfer function or frequency response is one way to characterize a filter. The impulse response is an alternative description.

Causal filters do not respond to their input before they arrive. In terms of the impulse response, they are characterized by h[n]=0 for n<0.

A causal finite-impulse response (FIR) filter also has h[n]=0 for n>N.

A causal infinte-impulse response (IIR) filter is not FIR.

An FIR filter can be implemented using a delay line with coefficients and summation. An IIR filter can be implemented by adding a feedback FIR filter. An equivalent mathematical description is a difference equation with constant coefficients.

The frequency response of a filter implemented using a difference equation is easily determined. It is the ratio of two polynomials in Exp[I 2Pi w].

Any desired filter transfer function can be approximated by an FIR filter, where the approximation improves as the number of coefficients increases.