In the I970s, the development of digital processing of filters led to implementing previously established methods for synthesizing continuous filters. In this section, we will discuss the impulse invariance method for analog to digital data conversion. Then we will present the bilinear transformation, which enables us to use all available techniques for synthesizing continuous filters.
The impulse invariance method for analog to digital filter conversion is based on the fact that the impulse response of a digital filter must correspond to the sampling of the impulse response of a continuous filter.
The method is described below.
– we write the transfer function of the continuous filter in the form of a development in basic elements:
– from this we deduce that
The sampling frequency Ts being fixed, we obtain:
which produces, by using the z-transform:
We see that the poles of H(z) are in the form exp (piTs). If the original continuous-time filter is stable, the poles pi are partly real negative and the poles of the transfer function H(z) are inside the unity circle in the complex z-plane; so we have:
However, this method cannot always be applied. We must respect several conditions:
– on the one hand, the frequency response of the continuous filter must be null or must be considered as null beyond a certain frequency. Moreover, the sampling of the frequency response (see equation (6.22) leads to the relation between the respective transfer functions of the digital or analog filters:
– on the other hand, sampling of the impulse response must be possible at every instant. The response of the continuous filter must not create any discontinuities. We see this means that the denominator of the impulse response must be at least of 2 degrees.
In section 6.2.1, we discussed synthesizing an infinite impulse response digital filter whose impulse response corresponds to the sampled response of a corresponding analog filter. An alternate approach is to conserve the indicial response instead of the impulse response.
So we have:
By using the Laplace transform of the above formula, we obtain:
We then proceed as in section 6.2.1.
Here we must find a transformation that moves from the discrete-time domain to that of continuous-time, and inversely; once this occurs, we can then use nearly all the methods established for synthesizing continuous filters, and then discrete-time filters.
Thus, from a formal point of view, with a given transfer function in s, the bilinear transformation consists 0 replacing s with in the transfer function of the continuous filter. This helps us obtain a digital filter with approximately the same frequency response as that of an analog filter.
In the following sections, we will establish this transformation by using the method of the invariance calculation of a surface with discrete and continuous examples.
We recall that the impulse response of an integrator filter is:
From here, whatever the input e(t), the output of this system is equal to the convolution product of the causal input e(t) of the impulse response h(t), so y(t)=e(t)*h(t).
So, for two successive instants tn and tn+1. we have:
We can then calculate this integral by using the trapeze method so that the increment tn+1 −tn = (n + 1)Ts − nTs = Ts is sufficiently small. We then have:
The z-transform of this last discrete equation gives the transfer function:
This step allows us to establish an equation that leads from the continuous domain to the discrete domain with the following result:
The correspondence on the unity circle between the frequencies In the continuous and discrete domains is:
According to equation (6.35), we see that the bilinear transformation brings about an axial distortion of frequencies. This distortion is called warping. For low values of discrete angular frequency, we observe that:
However, for higher values, equation (6.35) is non-linear. The cut-off frequency of a low-pass filter then undergoes this modification, and we have:
This means we must take this frequency distortion into account.
It is also important to remember that the bilinear transformation conserves the filter's stability. We observe that the transform of the left half-plane of the Laplace s-plane is, by the bilinear transformation, the unity disk in the z-plane.
We can demonstrate that the image of the ordinate axis of the complex s-plane by the bilinear transformation in equation (6.33) is the unity circle in the complex z-plane. Using equation (6.33), we get:
The ordinate axis of the complex s-plane corresponds to the pure-imaginary values for s: s = jα. with α real variant of −∞ to +∞. So using equation (6.38), we have:
To determine another type of filter – that is, a high-pass, passband or cut-off ban d filter – we represent a low-pass filter characterized by its transfer function Hlp(z). Then we carry out the following variable changes by proposing z−1 = f(Z−1) obtain the transfer function of the desired filter Hdesired (Z).