Bilinear transformation method in Signal Processing - Step-by-Step Execution

Concept Flow - Bilinear transformation method

Start with analog filter H(s)

↓

Substitute s = 2/T * (z-1)/(z+1)

↓

Obtain digital filter H(z)

↓

Analyze frequency warping

↓

Apply frequency pre-warping if needed

↓

Use H(z) for digital filtering

The bilinear transformation converts an analog filter H(s) into a digital filter H(z) by substituting s with a function of z, adjusting frequencies to avoid distortion.

Execution Sample

Signal Processing

s = 2/T * (z - 1) / (z + 1)
H_z = H_s.subs(s, 2/T * (z - 1) / (z + 1))
# T is sampling period
# H_s is analog filter transfer function
# H_z is digital filter transfer function

This code shows the substitution step of s in H(s) with the bilinear transform formula to get H(z).

Execution Table

Step	Operation	Expression	Result/Value
1	Define sampling period T	T	0.1 (example)
2	Write analog variable s	s	s
3	Substitute s with bilinear formula	s = 2/T * (z-1)/(z+1)	s = 20 * (z-1)/(z+1)
4	Replace s in H(s) with substitution	H(z) = H(s) with s replaced	Digital filter H(z) obtained
5	Check frequency warping	Omega_analog vs Omega_digital	Nonlinear mapping observed
6	Apply frequency pre-warping	Omega_prewarped = tan(Omega_desired*T/2)*2/T	Corrected frequency mapping
7	Use H(z) for digital filtering	Filter input signal	Filtered digital output
8	End	-	Transformation complete

💡 Transformation ends after obtaining digital filter H(z) and applying frequency corrections.

Variable Tracker

Variable	Start	After Step 3	After Step 6	Final
T	undefined	0.1	0.1	0.1
s	s	20*(z-1)/(z+1)	20*(z-1)/(z+1)	20*(z-1)/(z+1)
H(s)	Analog filter	Substituted s	Substituted s with pre-warping	Digital filter H(z)
Omega_prewarped	undefined	undefined	Computed	Used for frequency correction

Key Moments - 3 Insights

Why do we substitute s with 2/T*(z-1)/(z+1) instead of a simpler formula?

Because this substitution maps the entire analog frequency axis to the digital frequency axis without aliasing, preserving stability and causality as shown in step 3 of the execution_table.

What is frequency warping and why does it happen?

Frequency warping is the nonlinear distortion of frequencies caused by the bilinear transform mapping, as seen in step 5. It happens because the transformation is a nonlinear function of frequency.

How does frequency pre-warping fix the warping problem?

Pre-warping adjusts the desired analog cutoff frequencies before transformation (step 6), so after warping, the digital filter matches the intended frequency response.

Visual Quiz - 3 Questions

Look at the execution_table at step 3, what is the expression for s after substitution?

A s = 2/T * (z-1)/(z+1)

B s = 2/T * (z+1)/(z-1)

C s = T/2 * (z-1)/(z+1)

D s = (z-1)/(z+1)

💡 Hint

Refer to the 'Expression' column in step 3 of the execution_table.

At which step does the method address frequency warping?

A Step 2

B Step 5

C Step 4

D Step 7

💡 Hint

Check the 'Operation' column for frequency warping in the execution_table.

If the sampling period T is changed, which variable_tracker row shows its updated value?

A s

B H(s)

C T

D Omega_prewarped

💡 Hint

Look at the variable_tracker row for T and how it changes after steps.

Concept Snapshot

Bilinear transformation converts analog filter H(s) to digital H(z) by substituting s = 2/T*(z-1)/(z+1).
This maps analog frequencies to digital frequencies without aliasing.
Frequency warping occurs and can be corrected by pre-warping frequencies.
Use the transformed H(z) for stable digital filtering.

Full Transcript

The bilinear transformation method converts an analog filter transfer function H(s) into a digital filter H(z) by substituting the complex frequency variable s with a function of z and the sampling period T. This substitution is s = 2/T times (z-1) over (z+1). This mapping preserves stability and causality but causes frequency warping, a nonlinear distortion of frequencies. To correct this, frequency pre-warping is applied before the substitution. The process starts by defining the sampling period T, then substituting s in H(s), analyzing frequency warping, applying pre-warping, and finally using the digital filter H(z) for filtering signals. Variables like T, s, and the filter functions change step-by-step as shown in the execution table and variable tracker. Key points include why the substitution formula is used, what frequency warping is, and how pre-warping fixes it. The visual quiz tests understanding of these steps and variable changes.