For a continuous-linear time-invariant system that takes the input signal x(t) and produces the output signal y(t), the transfer function H(s) relates the output signal to the input signal with the formula

Y(s) = H(s) X(s)

where X(s) and Y(s) are the bilateral Laplace transforms of x(t) and y(t) respectively. For discrete-time linear time-invariant system that takes the input signal x(k) and produces the output signal y(k), the transfer function H(z) relates the output signal to the input signal with the formula

Y(z) = H(z) X(z)

where X(z) and Y(z) are the Z transforms of x(k) and y(k).

## Definitions

In simplest term, if the inputs to a linear system x_{1}(t), x_{2}(t), …, x_{n}(t) produce the outputs y_{1}(t), y_{2}(t), …, y_{n}(t), then the input a_{1} x_{1}(t) + a_{2} x_{2}(t) + … + a_{n} x_{n}(t) would produce the output a_{1} y_{1}(t) + a_{2} y_{2}(t) + … + a_{n} y_{n}(t), where a_{1}, a_{2}, …, a_{n} are scalars. This is so because a linear system is additive (i.e., the output of x_{1}(t) + x_{2}(t) is the output of x_{1}(t) plus the output of x_{2}(t), which is y_{1}(t) + y_{2}(t)) and because a linear system is homogeneous of degree 1 (i.e., the output of b x(t) is b times the output of x(t), which is b y(t)).

A system is time-invariant if the same input produces the same output if given the same input, independently of when the input comes in. In other words, if the output of x(t) is y(t), then the output of x(t – T) is y(t – T), where T is some constant amount of time.

The discrete time transfer function is sometimes called the "pulse transfer function".

## Example: transfer function of a feedforward comb filter

A discrete-time feedforward comb filter computes the output signal y(k) from the input signal x(k) according to the following formula.

y(k) = x(k) + b x(k – M)

Here b is some scalar. Taking the Z transform of both sides produces

Y(z) = X(z) + b z^{-M} X(z) = X(z) (1 + b z^{-M})

and the corresponding transfer function is

H(z) = 1 + b z^{-M}

## Example: transfer function of a continuous-time Butterworth filter

A continuous-time low pass Butterworth filter has the transfer function

where G is some gain, ω_{c} is some cutoff frequency, n is the order of the filter, ω is the angular frequency spanning the frequency spectrum between 0 and π, and k = 1, 2, …, n.

## Example: transfer function of an all pass filters

The transfer function of an infinite impulse response all pass filter is

where M is nonnegative integer and a_{1}, a_{2}, …, a_{N} are scalars.

## Use of transfer functions

It should be easy to see, similarly to the computation of the transfer function of the feedback comb filter above, that the transfer function of any discrete-time filter with the impulse response

is

The magnitude response of a discrete-time system (and of the filter above) with the transfer function H(z) is

G(ω) = |H(e^{jω})|

(The continuous-time equivalent is typically G(ω) = |H(jω)|). Setting z = e^{jω} allows us to compute the Fourier transform, a special case of the Z transform, which in turn allows us to compute the magnitude response. Here e^{jω} = cos(ω) + j sin(ω) according to Euler's formula and

where Re{H} and Im{H} are the real and imaginary parts of the complex number H respectively. For example, the magnitude response of a finite impulse response filter with coefficients w(k) and length N, where k = 1, 2, …, N is

The phase response of a system with transfer function H(s) is

Φ( ω) = arg{H(jω)}

Here the "arg" function returns the angle between the complex number H(jω) and the horizontal axis. For example, when the Re{H(jω)} > 0 then arg{H(jω)} = arctan( Im{H(jω)} / Re{H(jω)} ).

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