Transfer function H(z) in Signal Processing - Step-by-Step Execution

Concept Flow - Transfer function H(z)

Start: Define input signal x[n

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Apply Z-transform to x[n

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Define system difference equation

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Take Z-transform of system equation

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Express output Y(z) in terms of X(z)

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Define Transfer function H(z) = Y(z)/X(z)

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Analyze H(z) for system behavior

The flow shows how an input signal is transformed using Z-transform, then the system's difference equation is converted to Z-domain to define the transfer function H(z), which relates output to input.

Execution Sample

Signal Processing

import numpy as np
z = 0.9 + 0.1j
b = [1, -0.5]
a = [1, -0.8]
H_z = np.polyval(b, z) / np.polyval(a, z)
print(H_z)

Calculate H(z) at a specific z value using numerator and denominator polynomials.

Execution Table

Step	Action	Input	Computation	Result
1	Define numerator coefficients b	[1, -0.5]	Store as polynomial numerator	b = [1, -0.5]
2	Define denominator coefficients a	[1, -0.8]	Store as polynomial denominator	a = [1, -0.8]
3	Choose z value	z = 0.9 + 0.1j	Use for evaluation	z = 0.9 + 0.1j
4	Evaluate numerator polynomial at z	b, z	1*(z^1) + (-0.5)	num = 0.9+0.1j - 0.5 = 0.4+0.1j
5	Evaluate denominator polynomial at z	a, z	1*(z^1) + (-0.8)	den = 0.9+0.1j - 0.8 = 0.1+0.1j
6	Calculate H(z) = num/den	num, den	(0.4+0.1j)/(0.1+0.1j)	H(z) = 2.5 - 1.5j
7	Output H(z)	H(z)	Print result	2.5 - 1.5j

💡 Completed evaluation of H(z) at chosen z value

Variable Tracker

Variable	Start	After Step 4	After Step 5	After Step 6	Final
b	[1, -0.5]	[1, -0.5]	[1, -0.5]	[1, -0.5]	[1, -0.5]
a	[1, -0.8]	[1, -0.8]	[1, -0.8]	[1, -0.8]	[1, -0.8]
z	undefined	0.9+0.1j	0.9+0.1j	0.9+0.1j	0.9+0.1j
num	undefined	0.4+0.1j	0.4+0.1j	0.4+0.1j	0.4+0.1j
den	undefined	undefined	0.1+0.1j	0.1+0.1j	0.1+0.1j
H(z)	undefined	undefined	undefined	2.5-1.5j	2.5-1.5j

Key Moments - 3 Insights

Why do we evaluate polynomials at a complex number z?

Because H(z) is defined in the complex z-plane, evaluating numerator and denominator polynomials at z gives the system's frequency response at that point (see steps 4 and 5 in execution_table).

Why is H(z) a ratio of two polynomials?

The system's difference equation transforms into a ratio of polynomials in z after Z-transform, representing output over input (step 6 shows division of numerator by denominator).

What does the value of H(z) tell us?

It tells how the system modifies the input signal at frequency corresponding to z; the complex value shows magnitude and phase (step 7 outputs this result).

Visual Quiz - 3 Questions

Look at the execution_table step 4, what is the numerator polynomial evaluated at z?

A 2.5 - 1.5j

B 0.1 + 0.1j

C 0.4 + 0.1j

D 1 - 0.5

💡 Hint

Check the 'Result' column at step 4 in execution_table

At which step does the denominator polynomial get evaluated at z?

A Step 5

B Step 6

C Step 3

D Step 7

💡 Hint

Look for 'Evaluate denominator polynomial' in the 'Action' column

If z was changed to 1 + 0j, how would the value of H(z) change at step 6?

A H(z) would be (1 + 0.1j)/(0.1 + 0.1j)

B H(z) would be (1 - 0.5)/(1 - 0.8) = 0.5/0.2 = 2.5

C H(z) would be zero

D H(z) would be undefined

💡 Hint

Evaluate numerator and denominator polynomials at z=1+0j using steps 4 and 5 logic

Concept Snapshot

Transfer function H(z) relates output Y(z) to input X(z) in Z-domain.
H(z) = Y(z)/X(z) = B(z)/A(z), ratio of numerator and denominator polynomials.
Evaluate polynomials at complex z to find system response.
Used to analyze discrete-time system behavior and stability.

Full Transcript

The transfer function H(z) is a key concept in signal processing that relates the output and input of a system in the Z-domain. We start by defining the input signal and applying the Z-transform to get X(z). The system's difference equation is also transformed to Z-domain, resulting in a ratio of polynomials B(z) and A(z). This ratio, H(z) = B(z)/A(z), is the transfer function. To analyze the system, we evaluate these polynomials at a complex number z, which corresponds to a frequency point. The example code shows how to calculate H(z) by evaluating numerator and denominator polynomials at a chosen z value and dividing them. The execution table traces each step, showing how variables change and how the final H(z) value is computed. Understanding this process helps in analyzing system frequency response and stability.