What is z-transform?

A method that converts a discrete-time signal x[n] into a function of the complex number z, X(z) = Σx[n]z^(−n). A delay of one sample corresponds to multiplying by z⁻¹, so a difference equation can be handled as a ratio of polynomials in z (the transfer function H(z)). Values of z that make the denominator of H(z) zero are called poles and those that make the numerator zero are called zeros; a causal system is stable when all its poles lie inside the unit circle. Evaluating on the unit circle with z = e^(jω) gives the frequency response. It is the discrete-time counterpart of the Laplace transform.

Japanese name: z変換

z-transform appears in 1 practice question across 1 topic, including Digital Signal Processing Basics.

Field: Digital Signal Processing Basics

From a difference equation to a transfer function

For example, applying the z-transform to both sides of y[n] = x[n] + 0.5x[n − 1] + 0.25y[n − 1] gives Y(z) = X(z) + 0.5z⁻¹X(z) + 0.25z⁻¹Y(z), which rearranges to H(z) = (1 + 0.5z⁻¹)/(1 − 0.25z⁻¹). The numerator represents the input side and the denominator the output feedback, so the filter structure and the transfer function can be converted into each other just by reading off the coefficients.

Poles, zeros and frequency response

The gain magnitude at a given frequency is proportional to the product of the distances from the point e^(jω) on the unit circle to each zero, divided by the product of the distances to each pole. A zero on the unit circle therefore makes the gain at that frequency zero (a notch), and a pole close to the unit circle raises the gain sharply near that frequency. The rough shape of a filter response can be seen just from the pole-zero layout.

Judging stability

The impulse response of a causal system contains a component of the form pⁿ for each pole p, which shrinks with time if |p| < 1 and keeps growing if |p| > 1. Stability requires all poles to lie inside the unit circle; a system with a pole on the unit circle (such as an accumulator at z = 1) is not BIBO stable, because a bounded input can make its output grow without limit.

Try it: questions about z-transform

Which of the following is the transfer function H(z) = Y(z)/X(z) of the system described by the difference equation y[n] = x[n] + 0.5x[n − 1] + 0.25y[n − 1]?

A(1 + 0.5z⁻¹)/(1 + 0.25z⁻¹)
B(1 − 0.25z⁻¹)/(1 + 0.5z⁻¹)
C(1 + 0.5z⁻¹)/(1 − 0.25z⁻¹)

Show answer and explanation

Answer: C. (1 + 0.5z⁻¹)/(1 − 0.25z⁻¹)

A delay of one sample corresponds to multiplying by z⁻¹ in the z-transform, so transforming both sides gives Y(z) = X(z) + 0.5z⁻¹X(z) + 0.25z⁻¹Y(z). Collecting the Y(z) terms on the left gives (1 − 0.25z⁻¹)Y(z) = (1 + 0.5z⁻¹)X(z), so H(z) = (1 + 0.5z⁻¹)/(1 − 0.25z⁻¹). The numerator holds the coefficients on the input side and the denominator those of the output feedback; z = 0.25, which makes the denominator 0, is a pole, and z = −0.5, which makes the numerator 0, is a zero.

FAQ

How is the z-transform related to the Fourier transform?

Setting z = e^(jω) (on the unit circle) in the z-transform gives the discrete-time Fourier transform (DTFT). By considering z more broadly, inside and outside the unit circle as well, the z-transform makes it possible to discuss stability and frequency response from the positions of poles and zeros. The DFT of a finite-length signal corresponds to the DTFT evaluated at N equally spaced points on the unit circle.

Why does z⁻¹ mean "a delay of one sample"?

The z-transform of x[n − 1] is Σx[n − 1]z^(−n), and substituting m = n − 1 turns it into z⁻¹Σx[m]z^(−m) = z⁻¹X(z). For this reason a delay element is written "z⁻¹" in block diagrams, and a digital filter is expressed as a combination of multiplications, additions and z⁻¹.

z-transform in the explanations

A delay of one sample corresponds to multiplying by z⁻¹ in the z-transform, so transforming both sides gives Y(z) = X(z) + 0.5z⁻¹X(z) + 0.25z⁻¹Y(z).
Which of the following is the transfer function H(z) = Y(z)…

Questions that mention z-transform (1)

Which of the following is the transfer function H(z) = Y(z)/X(z) of the system described by the difference equation y[n] = x[n] + 0.5x[n − 1] + 0.25y[n − 1]?

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