# Lecture 2: Transfer Functions

Under the assumption of zero initial conditions (the zero-state response) the general LTI difference equation

may be transformed to

as the

**transfer function**. Note that

If the input is , then the output is

**The transfer function is the Z-transform of the impulse response.**

**
Nomenclature
**
. A discrete-time filter which has only a numerator part
(only zeros, except for possible poles at the origin which correspond to delays)
is said to be a
**
finite impulse response
**
(FIR) filter.

A filter with poles is said to be an
**
infinite impulse response
**
(IIR) filter.

Note that there is no practical way of making an FIR filter for continuous time
systems: this is available only for digital filters.