The transfer function of a system is a mathematical function that describes the relationship between the input to the system and the output of the system.
Example transfer functions are shown below. Since in signal processing transfer functions typically make use of the Laplace transform for continuous signals and the Z transform for discrete signals, the following are definitions of the term "transfer function" with these two transforms.
Transfer functions in signal processing
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).
The choice of the Laplace and Z transforms
Simplistically, the Laplace transform converts a function of time into a function of the complex variable s = σ + j ω (σ and ω are both real numbers; in signal processing, ω is typically the angular frequency).
Similarly, the Z transform converts a function of time into a function of the complex variable z = A e j ω (A and ω are real numbers; in signal processing, ω is again the angular frequency).
In both cases, the signal processing system and therefore its transfer function are almost always evaluated where the magnitude of the input signal is 1. It is easier to measure the impact of the system if the input signal is normalized. The Laplace transform is used with σ = 0 and s = j ω. The Z transform is used with A = 1 and z = e j ω. This makes both transforms equivalent to the Fourier transform (the continuous and the discrete-time one respectively).
In short, signal processing uses the Fourier transform, but the Laplace transform and the Z transform largely result in easier notation. In addition, the Laplace transform makes it easy to convert desired magnitude response functions to transfer functions and therefore systems. The time-shifting property of the Z transform, combined with its short notation, makes it easy to convert transfer functions to impulse responses (to actual filter computations).
Linear time-invariant systems
If the inputs to a linear system x1(t), x2(t), …, xn(t) produce the outputs y1(t), y2(t), …, yn(t), then the input a1 x1(t) + a2 x2(t) + … + an xn(t) would produce the output a1 y1(t) + a2 y2(t) + … + an yn(t), where a1, a2, …, an are scalars. This is so because a linear system is additive (i.e., the output of x1(t) + x2(t) is the output of x1(t) plus the output of x2(t), which is y1(t) + y2(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 when 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 examples below are of linear time-invariant systems. In simple terms, such systems allow transfer functions that are independent of the amplitude of the signal. Hence, they can be evaluated with normalized signals and the Fourier transform. Such systems impact signals the same way independently of whether the signals are separate or combined. Finally, such systems act on a constant signal the same way at the beginning of that constant signal, middle, or end.
Not all signal processing systems are linear or time-invariant. Distortion and compression, for example, do not operate linearly on the signal. Compressors with attack and release times are also dependent on time.
Example: transfer function of a feedforward comb filter
A discrete-time feedforward comb filter (a simple delay) 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 (the decay of the delayed signal) and M is an integer (the delay in number of samples). Taking the Z transform of both sides (and minding the linearity and time-shifting property of the Z transform) produces
$$Y(z)=X(z)+b \, z^{-M} \, X(z)=X(z) (1+b \, z^{-M})$$
and the corresponding transfer function is
$$H(z) = 1 + b \, z^{-M}$$
This transfer function is used below to compute the magnitude response and phase response of the comb filter.
Using transfer functions
Suppose that we want the impact of the system on the magnitude of frequencies (the magnitude response of the system). Recall that the transforms of the input and output signals are functions of the frequency. We are interested in the ratio of magnitudes at each frequency
$$\frac{|Y(z)|}{|X(z)|}=|H(z)|$$
As above, we use the Fourier transform, even though we have settled on the Z transform notation. In either case, the transforms convert a function of time into a function of the frequency. The resulting function of the frequency has complex values. We take the magnitude of these complex values at the input and the output stage and check the ratio, which happens to be the magnitude of the transfer function. This is so because of the way the transfer function is defined.
With the feedforward comb filter example, at z = e j ω and expressing the result as a function of the angular frequency, we have the following.
$$|H(\omega)|=|1+b \, z^{-M} |=|1+b \, e^{-j M \omega} |$$ $$=|1+b \, \cos(M\, \omega)-j \, b \, \sin(M \, \omega)|$$ $$=\sqrt{(1+b \, \cos(M \, \omega))^2+(b \, \sin(M \, \omega))^2}$$ $$=\sqrt{1+b^2+2 b \, \cos(M \, \omega)}$$
This function shows us what the comb filter will do to the amplitude of each frequency ω. The periodicity of the cosine term is what makes the graph of this function look like a comb.
If, instead, we want the impact of the system on the phase of frequencies (the phase response), we would take the difference between the phase of the output signal and the phase of the input signal. We do so, again, at z = e j ω
$$\Phi(\omega)=\arg(Y(z))-\arg(X(z))=\arg(\frac{Y(z)}{X(z)})$$ $$=\arg(H(z))=\mathrm{atan2}(Im(H(z)),Re(H(z)))$$ $$=\mathrm{atan2}(b \, \sin(M \, \omega),1+b \, \cos(M \omega)) $$
The computations with transfer functions expressed with the Laplace transform are similar.
Transfer functions and impulse responses
A digital signal processing filter computes its output signal y(k) from some past samples of the input signal x(k) and some past samples of the output signal y(k) in the following way
$$y(k)=\sum_{i=1}^N a_i \, y(k-i)+\sum_{i=0}^M b_i \, x(k-i)$$
where ai and bi are scalars. This is a general function. Finite impulse response filters, for example, are simpler, as they compute the output signal y(k) only from the samples of the input signal x(k) and therefore set ai = 0 for all i.
The function above is the impulse response of the filter. More importantly, this function completely describes the corresponding filter.
If we compute the transfer function of the filter with the Z transform, similarly to the way we computed the transfer function of the comb filter above, we will get
$$H(z)=\frac{\sum_{i=0}^M b_i \, z^{-i}}{1-\sum_{i=1}^N a_i \, z^{-i}}, z=e^{j \omega}$$
This transfer function also completely describes the filter. It is not important whether we describe this filter with its impulse response or its transfer function. The two are equivalent.
Other transfer functions
See All pass filter for the general transfer function of infinite impulse response (IIR) all pass filters and for the computation showing that the magnitude of that transfer function is always 1 (i.e., the all pass filter passes all frequencies with unchanged magnitude).
See Phase response for the general transfer function of finite impulse response (FIR) filters with coefficients that are symmetric around the middle and for the computation showing that the phase response of that transfer function is linear with respect to the angular frequency.
See Butterworth filter for an example Laplace transform transfer function that is derived from a desired magnitude response and for examples of the ways that transfer function can be converted to a digital filter (e.g., with the inverse Laplace transform and the bilinear transformation).
For other transfer function examples, see the descriptions of various filters in Digital music (index).
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