Transfer Function H(z)

This function completely describes how a linear discrete system processes input signals. You use it to analyze and design digital filters for image and audio processing.

Core Concept

The transfer function forms the core of discrete system analysis. It represents the ratio of the output signal transform to the input signal transform. This assumes all initial conditions are strictly zero.

You extract this function directly from the system difference equation. The roots of the numerator polynomial are called zeros. The roots of the denominator polynomial are called poles.

Plotting poles and zeros on the complex plane gives you immediate insight. You can determine system stability and frequency response purely from their geometric locations within the unit circle.

Key Formula

The transfer function is the complex ratio of output to input.

H(z) = Y(z) / X(z) = sum( h[n] * z^(-n) ). Y(z) represents the output transform. X(z) represents the input transform. h[n] is the discrete impulse response.

Problem: Find H(z) for y[n] - 0.5 y[n-1] = x[n] + x[n-1].\nGiven: Initial conditions are exactly zero.\nSteps:\nTake transform of both sides: Y(z) - 0.5 z^(-1) Y(z) = X(z) + z^(-1) X(z).\nFactor terms: Y(z)[1 - 0.5 z^(-1)] = X(z)[1 + z^(-1)].\nForm the ratio Y(z)/X(z): H(z) = (1 + z^(-1)) / (1 - 0.5 z^(-1)).\nMultiply by z/z to clear negatives: H(z) = (z + 1) / (z - 0.5).\nFinal Answer: H(z) = (z + 1) / (z - 0.5) with zero at -1 and pole at 0.5.

Exam Tip: When given a block diagram, write the difference equation at each summation node before taking the transform. Finding the intermediate node equations prevents sign errors in feedback loops.

Properties Summary

• H(z) is defined only for systems with zero initial conditions.
• It equals the mathematical transform of the system impulse response.
• Zeros are frequencies where the system output drops to zero.
• Poles dictate the transient behavior and output growth rates.
• Cascaded systems multiply their individual transfer functions together.
• Parallel systems add their individual transfer functions together.

Quick Revision

• Transfer function maps input to output directly.
• It converts difference equations to simple algebra.
• Poles are the denominator roots.
• Zeros are the numerator roots.
• Feedback loops alter the denominator polynomial.
• Exam trap: Confusing negative powers of z with positive powers when finding pole locations.