Questions tagged [bilinear-transform]
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37 questions
1 vote
0 answers
54 views
Transformation on All-pass transfer functions
I'm currently studying the derivation of the schur-Cohn stability test from the book Digital Signal Processing: A Computer-Based Approach by S.K. Mitra, and I’ve encountered a jump in logic that seems ...
- 11
4 votes
1 answer
130 views
Transformations from $s$ to $z$: what can we gain from higher-order mappings?
Transforming continuous-time (CT) filters to the discrete domain is frequently done via the bilinear transform: $$s=\frac{2}{T}\frac{z-1}{z+1}\tag{1}$$ The constant $2/T$ in $(1)$ can be replaced by ...
- 94.8k
1 vote
1 answer
144 views
For unit step $g(t) = u(t)$, why does $G(z) = \frac{z}{1-z}$, whereas $G(s) = \frac{1}{s}$?
Edit: The source of my confusion is over the existence of multiple s-z mappings. After researching these mappings and where they came from, I couldn't find why $z=e^{sT}$ can be ignored and replaced ...
1 vote
1 answer
681 views
What exactly are the assumptions behind Tustin's formula? Application on state space models
I was parsing the forum when I saw this post surging out of the depths of this forum like an old Kraken. The problem is quite simple. You have a continuous time state space model : $$ \begin{split} \...
- 507
1 vote
0 answers
99 views
Bilinear Transformation in Audio EQ Cookbook
In the Audio EQ CookBook there is $$\omega_0 = 2\pi\cdot\frac{f_0}{f_s}$$so frequency warping will be $$\omega_r = \frac{2}{T}\tan(\frac{\omega_0}{2})$$ Then $$s \longleftarrow \frac{2}{T}\frac{1-z^{-...
- 21
2 votes
2 answers
328 views
Bilinear transform for a system in zero-pole-gain form
MATLAB's bilinear performs the following steps for a system in zero-pole-gain form If fp is present, it prewarps: ...
- 527
2 votes
1 answer
1k views
How to design digital Butterworth filter and return its zeros, poles and gain
I'm implementing digital Butterworth filter and encounter some numerical problem when filter order is high using direct form, so I wonder how to design the digital Butterworth filter and return its ...
- 527
0 votes
0 answers
69 views
Do we use calculator in radians or degree while calculating prewarping analog frequency in Bilinear Transformation?
Say $\omega_s=1.96\;,\omega_p=0.785$ For bilinear transformation-: $\Omega_p=2/T*tan(\omega_p/2)$ $\Omega_s=2/T*tan(\omega_s/2)$ Should I put my calculator in radian mode or degree mode. I believe it ...
- 163
1 vote
2 answers
289 views
Digital IIR LPF Difference Equation from Transfer Function
I want a digital IIR filter with f0=225kHz and fs=53.125GHz. I can come up with the transfer function and plot it using Matlab. ...
- 23
1 vote
1 answer
435 views
States transformation of the bilinear transform
I have used the bilinear (or tustin) transform for a while, have been though the derivation of it and also through the concept of frequency warping. Something that I still not understand that is ...
- 43
2 votes
3 answers
2k views
Why must the order of a Band-Pass and Notch filter always be even?
My professor mentioned that the order of a band-pass and a notch filter must always be even, when showing an example of designing a digital filter using the bilinear transformation. Then he also ...
- 89
0 votes
0 answers
192 views
How to break a second-order filter into two first-order filter
Let's assume the transfer function of a continuous-domain filter consists of two poles and one zero: $H(s) = \frac{k_c (s-\omega_{z_1})}{(s-\omega_{p_1})(s-\omega_{p_2})}$. Let's consider we do the bi-...
- 348
1 vote
1 answer
480 views
Is the sampling period arbitrary for a bilinear transform, and why?
I have been given the specifications for a digital highpass filter (stopband, passband, stopband attenuation and maximum passband ripple). I am expected to design a prototype lowpass filter in the ...
- 11
2 votes
2 answers
2k views
Justification of bilinear transform
I would like to understand the "justification" for the bilinear transform. The basic idea as I understand it is that by integration rule of Laplace transform we have for continuous $y(t)$: $$\mathcal{...
- 348
2 votes
1 answer
4k views
Bilinear Transform (Tustin's Method) applied to the Derivative
I hope that I have not misunderstood something terribly wrong, but the continuous derivative $D=d/dt$ can be considered a transfer function in Laplace space $D(s) = s$, right? So when I try to ...
- 276