2.4. Transfer functions and difference equations – Digital Filters Design for Signal and Image Processing

2.4. Transfer functions and difference equations

2.4.1. The transfer function of a continuous system

A continuous linear system whose input is x(t) produces a response y(t). This system is regulated by a linear differential equation with constant coefficients that links x(t) and y(t). The most general expression of this differential equation is in the form:

By assuming that x(t) = y(t) = 0 for t < 0, we will show that if we apply the Laplace transform to the differential equation (2.14), we will obtain an explicit relation between the Laplace transforms of x(t) and y(t).

Since:

and:

we get:

The relation of the Laplace transforms of the input and output of the system gives the system transmittance, or even what we can term the transfer function. It equals:

This means that whatever the nature of the input (unit sample sequence, unit step signal, unit ramp signal), we can easily obtain the Laplace transform of the output:

The frequency transform of the output generated by the system can then be analyzed by using Bode's, Nyquist's or Black's diagrams.

A quick review of a Black's diagram shows it contains two diagrams: one represents amplitude (or gain); the other shows phase. Each one separately plots the module in logarithmic scale format and the argument of the transfer function according to the frequency.

The Nyquist diagram plots the ensemble of points of Hs(jω) by writing as abscissas Re[Hs (jω)] and in ordinates Im[Hs (jω)].

Lastly, Black's diagram gives the ensemble of definite points in abscissas with |Hs(jω)| and in ordinates by Arg [Hs (jω)].

Except in certain limited cases, we can always approximate the transfer function to a product of rational fractions of orders 1 and 2; this will put into cascade several filters of orders 1 and 2.

2.4.2. Transfer functions of discrete systems

We have seen in section 1.4.2 that an invariant linear system of impulse response h(k) whose input is x(k) and output is y(k) verifies the following equation:

The z-transform of the relation in equation (1.34) gives a basic product between the z-transforms of the input and of the impulse response of the system, on the condition that the z-transforms converge on the same, non-empty ROC. We then have the following on the convergence domain intersection:

or:

The transfer function is the z-transform of the impulse response of the system. This filter is excited by an input of the z-transform written as Xz(z) and delivers the output whose z-transform is Yz(z).

With discrete systems, if at the instant k the filter output is characterized by the input states:

and output states:

the most general relation between the samples is the following difference equation:

From there, by carrying out the z-transform of the input and output, the difference equation becomes:

or

Thus, the transfer function is expressed from the polynomials A(z) and B(z), which are completely represented according to the position of their zeros in the complex plane.

Figure 2.6. Representation of the discrete system with an input and an output

COMMENT 2.1.– we also find this kind of representation in modeling signals with parametric models, the most widely used example being the auto-regressive moving average (ARMA).

Let y(t) be a signal that is represented by M samples (y(k) y(k − 1) … y(kM + 1)} that is assumed to be generated by an excitation characterized by its N samples {x(k) … x(k – 1) … x(kN + 1)}. A linear discrete model of the signal is a linear relation between the samples {x(k)} and {y(k)} that can be expressed as follows:

This kind of representation constitutes an ARMA model, of the order (M-1, N-1). The coefficients and are termed transverse parameters. In general, we adopt the convention a0 = 1. We then have:

The ARMA model can be interpreted as a filtering function of the transfer Hz(z).

Figure 2.7. Realization of a second-order autoregressive process

In the case of a model termed autoregressive (AR), the are null, except b0, and the model is reduced to the following expression:

Figure 2.8. Realization of a second-order autoregressive process

In this way, the polynomial B(z) is reduced to a constant B(z) = b0 and the transfer function Hz(z) now only consists of poles. For this reason, this model is called the all-pole model.

We can also use the moving average or MA so that are null, which reduces the model to:

Here, A(z) equals 1. The model is then characterized by the position of its zeros in the complex plan, so it also is called the all-zero model: