Digital Signal Processing Basics
Which of the following correctly describes the bilinear transform used to design a digital filter from the transfer function H(s) of an analog filter?
- AIt maps the whole imaginary axis one-to-one onto the unit circle, so no aliasing occurs, but the frequency axis is warped, so important frequencies are corrected by prewarping
- BIt samples the impulse response of the analog filter directly to obtain the coefficients, so high-frequency components fold back and change the response
- CIt is the substitution z = 1 + sT, which approximates a derivative by a forward difference, so a stable analog filter always becomes a stable digital filter
Show answer and explanation
Answer: A. It maps the whole imaginary axis one-to-one onto the unit circle, so no aliasing occurs, but the frequency axis is warped, so important frequencies are corrected by prewarping
The bilinear transform substitutes s = (2/T)(z − 1)/(z + 1); it maps the left half of the s-plane inside the unit circle, so a stable analog filter becomes a stable digital filter. The imaginary axis from 0 to ∞ maps one-to-one onto ω = 0 to π on the unit circle, so there is no aliasing, but the frequency axis is compressed according to Ω = (2/T)tan(ω/2). Important frequencies such as the cutoff frequency are therefore corrected with this relation in advance (prewarping) before the design. Sampling the impulse response is called the impulse-invariance method, and it does suffer from aliasing.
- A. Correct. There is no aliasing, but the frequency warping is corrected by prewarping.
- B. This describes the impulse-invariance method. The bilinear transform causes no aliasing.
- C. With z = 1 + sT (forward difference), a pole of a stable analog filter can map outside the unit circle, so stability is not guaranteed.