Even if Ha(s)->H(z), there are two requirements :
1. After the causal transformation is still causally stable
2. The frequency response of a digital filter mimics that of an analog filter
Two methods :
1. Impulse response invariant method: digital frequency and analog frequency have a linear relationship
2. Bilinear transformation method: digital frequency and analog frequency have a nonlinear relationship
Impulse response invariant method (inverse Laplace transform)
The jω imaginary axis of the s plane maps to the unit circle of the z plane
The left half-plane of the s -plane maps to the interior of the unit circle of the z -plane
The right half-plane of the s -plane maps to the outside of the unit circle of the z -plane
The digital filter frequency response completely imitates the analog filter frequency response, which is the biggest advantage of the impulse response invariant method.
Impulse response invariant method is not suitable for designing high-pass and band-stop filters, which is the biggest disadvantage of impulse response invariant method
Advantages :
1. There is a linear relationship between the analog frequency Ω and the digital frequency
If the frequency response band of the analog filter is limited to the folding frequency, the frequency response of the digital filter after transformation can reflect the relationship between the original response and frequency without distortion. For example, a linear-phase Bessel low-pass filter is still a linear-phase low-pass digital filter obtained by the invariant impulse response method.
2. The impulse invariant method makes the unit impulse response of the digital filter completely imitate the unit impulse response waveform of the analog filter, and the time domain characteristics are well approximated . In some occasions, if you want to achieve the imitation of the time domain impulse response, the impulse response invariant method is generally used.
shortcoming:
1. The general analog filter is not band-limited, so the biggest disadvantage of the invariant impulse response method is that it will produce different degrees of spectral aliasing distortion, which makes the frequency response of the digital filter deviate from the frequency response characteristics of the analog filter. It is suitable for low-pass or band-pass filter design.
2. Impulse-invariant method should not be used for high-pass and band-stop filters .
3. If the high-pass and band-stop DF are designed by the invariant impulse response method , a protection filter can be added at this time to filter out the frequency band higher than the folding frequency, and then converted into a digital filter by the invariant impulse response method, which will increase the design The complexity and filter order are only used when it is necessary to meet the frequency linear relationship or maintain the transient response of the network .