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2 Example 3 Warping |
An analog filter is stable if the poles of its transfer function fall in the negative real half of the complex plane. A digital filter is stable if the poles of its transfer function fall inside the unit circle in the complex plane. The bilinear transform maps the negative real half of the complex plane to the interior of the unit circle. This way, filters designed in the continuous domain can be easily converted to the sampled domain while preserving their stability.
As an example take a simple RC-filter. This filter has a transfer function
If we wish to simulate this filter in a digital simulation, we can apply the bilinear transform by substituting for the formula above, after some reworking, we get the following filter representation:
When transforming a continuous transfer function to a discrete transfer function, one must take two frequencies into acount, namely a continuous frequency and a discrete frequency . The corresponding circular pulsations are
The main advantage of the warping phenomenon is the absence of aliasing distortion of the frequency characteristic, such as observed with the impulse invariant method. It is necessary, however, to pre-warp the given specifications of the continuous system. These pre-warped specifications may then be used in the bilinear transform to obtain the desired discrete system. Note that the warping also occurs in the phase characteristic, as expected.