Transfer Functions

In this post, we introduce the idea of transfer functions in the context of signal processing. In particular, we distinguish between the case of continuous-time signals (such as an analogue circuit board) and discrete-time signals used in DSP.

Continuous Systems

Laplace transform

The Laplace transform of a function $f(t)$ is given by:
$F(s) = \mathcal{L}(f(t)) = \int_0^{\infty} f(t) e^{-st}dt$
Note that the argument $s$ is a complex number.

Transfer function (in Laplace domain)

Suppose we have a system which maps the continuous time-based signal $x(t)$ to the output signal $y(t)$. (An example could be an analogue circuit which acts on an audio signal.)

Then the transfer function is given by:

$H(s) = \frac{Y(s)}{X(s)}$
where $X(s) = \mathcal{L}(x(t))$ and $Y(s) = \mathcal{L}(y(t))$.

Frequency response

We can use the transfer function to evaluate the response of the system to a given frequency. Suppose we have a frequency $\omega$ (radians per second), then we denote $G(\omega)$ as the response or gain. The function $G(\omega)$ is given by:
$G(\omega) = |H(i\omega)|$

Discrete systems

We now consider a digital system in which the input signal is sampled with a constant time interval $T$.

FIR filter

A discrete FIR filter (Finite Impulse Response) acts on a signal which is sampled digitally at regular time intervals.

Such a filter takes the input $x[n]$ where $n \in \mathcal{Z}$ and produces an output $y[n]$. We shall only consider causal filters where the response depends only on the current and previous input samples.

A general filter $N$th order filter can be written as:
$y[n] = \sum_{i=0}^N b_i x[n-i]$

The output from a $N$th order FIR depends only on the current input sample and the previous $N$ input samples.

IIR filter

A general discrete IIR (infinite impulse response) filter can be written as:

$y[n] = \frac{1}{a_0} \left( \sum_{i=0}^P b_k x[n-i] – \sum_{i=1}^Q a_k y[n-i] \right)$

Z-transform

Given a discrete signal $x[n]$, $n \in \mathcal{Z}$, the Z-transform, is given by:
$X(z) = \mathcal{Z(x)} = \sum_{n=0}^{\infty} x[n]z^{-n}$

Transfer function in the Z-domain

Given a discrete system with input signal $x[n]$ and output $y[n]$, the transfer function in the Z-domain is :

$H(z) = \frac{Y(z)}{X(z)}$
where $X(z)$ is the Z-transform of the input signal and $Y(z)$ is the Z-transform of the output.

For the case of the IIR filter above, we have:
$H(z) = \frac{\sum_{k=0}^P b_kz^{-k}}{\sum_{k=0}^Q a_k z^{-k}}$

Frequency response

As for the continuous case, we can use the transfer function to evaluate the response of the system to a given input frequency. Suppose we have a frequency $\omega$ (radians per second), then the response $G(\omega)$ is given by:

$G(\omega) = | H(e^{i\omega T}) |$
where $T$ is the sampling interval (in seconds).