The bilinear transform approximates the continuous s function obtained by the Laplace transform to the discrete z function.
The relationship between the s function and the z function is as follows.
The bilinear transform formula is as follows. You can convert it to the z function by substituting the following formula for the s function. I will explain it later with a concrete example.
I will explain why it is the above formula. First, the transfer function of the integral is expressed by the s function as follows.
Integral is equivalent to calculating the area, so calculate the area at discrete times.
Area S is trapezoidal
Here, in the above equation, (t-1) means the previous value, and if this is set as Z-1, then
From this, we were able to find the bilinear transform formula.
■Bilinear transform of lowpass filter
As a concrete example, the low-pass filter is expressed in the form of a bilinear transform. The transfer function of the low-pass filter in the s function is as follows.
Substitute equation (1) above. Here, for convenience of calculation, K = 1 is set.
＜Expressed as a block diagram＞
The above formula is expressed as a block diagram as follows.
＜Simulation with Scilab＞
The simulation results by Scilab are as follows. Compared with a continuous time low pass filter.
Be careful, the sample time should be less than the time constant.
The simulation results are as follows. It is rougher than the continuous time low pass filter, but the movement is as intended.
If you want to operate more finely, you need to make it smaller the sample time.
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