IIR - Bilinear transform - simplifying H(z)

In summary, the speaker is an audio engineer trying to learn IIR filter design through the Bilinear Transform method. They have looked through various resources but are struggling with simplifying H(z) in the final stages. They are specifically confused about the z-2 term and how to arrive at the coefficients. They are seeking help to understand the process and have provided a link to a video they are trying to follow.
  • #1
Scoridd
2
0

Homework Statement



Hello,

I'm an audio engineer with an interest in creating/implementing audio processing techniques and I'm trying to learn IIR filter design via the Bilinear Transform method. I've looked through various books and online resources but I keep on getting stuck when it comes to simplifying H(z) in the final stages of the process. I'm sure the process is relatively straight forward but I cannot figure it out. I can usually 'get' most mathematical concepts but I've had no formal training, so I assume there is something that I'm missing here...

Homework Equations



In the most recent example, I have H(z) as...

[itex]\frac{0.3752^{2}} {\frac{1-z^{-1}}{1+z^{-1}}^{2} + \sqrt{2}(0.3752)\frac{1-z^{-1}}{1+z^{-1}} + (0.3752)^{2}}[/itex]

...and I need to simplify and collect terms of like powers of z in order to get to here...

[itex]\frac{0.0842(1+z^{-1})^2} {1 - 1.0281z^-1 + 0.3651z^-2}[/itex]

The Attempt at a Solution



Unfortunately, I do not know how. I began by calculating the squared and sqroot tems then mutliplying all terms by (1 + z-1). But then I get stuck. Mostly, I'm miffed by the z-2 in the answer, where does this come from? Also, I don't see how I arrive at the coefficients.

I'd be very greatful if someone could explain how I might go about doing this. This is for my own understanding, so I need to know how I'd go about simplifying this - I already have the answer after all! I just need pointing in the right direction.
 
Last edited:
Physics news on Phys.org

1. What is the purpose of the bilinear transform in IIR filters?

The bilinear transform is a mathematical technique used to convert a continuous-time filter into a discrete-time filter. This is necessary because digital signal processing systems operate on discrete-time signals, while most real-world signals are continuous-time. By applying the bilinear transform, we can design and implement IIR filters in digital systems.

2. How does the bilinear transform work?

The bilinear transform maps points from the s-plane (continuous-time) to the z-plane (discrete-time) using the formula: z = (1+Ts/2)/(1-Ts/2), where T is the sampling period. This mapping preserves the frequency response of the original filter, while also ensuring stability and causality in the discrete-time domain.

3. Can the bilinear transform cause distortion in the frequency response of the filter?

Yes, the bilinear transform can cause some distortion in the frequency response, especially at higher frequencies. This is because the transform maps the entire left half of the s-plane onto the unit circle in the z-plane, leading to a non-linear frequency warping. However, this distortion can be minimized by choosing an appropriate sampling rate and/or using pre-warping techniques.

4. How can we simplify the bilinear transform in IIR filters?

The bilinear transform can be simplified by using the impulse invariance method, which directly maps the impulse response of a continuous-time filter to a discrete-time filter. This method does not require any approximations and can result in a more accurate frequency response compared to the bilinear transform.

5. Are there any limitations or drawbacks of using the bilinear transform in IIR filters?

One limitation of the bilinear transform is that it can only be applied to linear time-invariant (LTI) filters. Additionally, the frequency warping caused by the transform can lead to inaccuracies in the frequency response of the filter, especially at high frequencies. As mentioned before, this can be mitigated by using pre-warping techniques or using the impulse invariance method instead.

Similar threads

Engineering Alternating current circuit with transformer
    • Engineering and Comp Sci Homework Help
Replies
12
Views
952
Engineering Z-transform of this Logistic Difference Equation
    • Engineering and Comp Sci Homework Help
Replies
4
Views
1K
Engineering How would I solve this using Laplace transformation?
    • Engineering and Comp Sci Homework Help
Replies
4
Views
1K
Simplifying the Argument of a Complex Number
    • Precalculus Mathematics Homework Help
Replies
19
Views
1K
Response of LTI discrete time causal system
    • Engineering and Comp Sci Homework Help
Replies
1
Views
2K
On weakly singular equations and Frobenius' method
    • Calculus and Beyond Homework Help
Replies
7
Views
554
Equation to give the lookback time as a function of redshift
    • Advanced Physics Homework Help
Replies
2
Views
505
Can the contour integral of z⁷ be simplified using a parameterized expression?
    • Calculus and Beyond Homework Help
Replies
1
Views
608
Engineering Finding the parallel resonance formula
    • Engineering and Comp Sci Homework Help
Replies
10
Views
900
Inverse Z transform - contour integration
    • Engineering and Comp Sci Homework Help
Replies
2
Views
1K

Hot Threads

Recent Insights

Back
Top