## The z-Transform

• The unilateral z-Transform of a discrete-time signal x[n] is given by: where z is a complex variable.

• The z-transform maps a discrete-time signal to a function of the complex variable z.

• A convenient property of the z-transform is given by the Shift Theorem, which says that a delay of samples in the time domain corresponds to a multiplication by in the z domain.

• Using the shift theorem, we can easily calculate the z-transform of a digital filter's difference equation. Given the following second-order difference equation,
y[n] = b0 x[n] + b1 x[n-1] + b2 x[n-2] - a1 y[n-1] - a2 y[n-2],
the z-transform can immediately be written (assuming the system is linear)
Y(z) = b0 X(z) + b1 z-1 X(z) + b2 z-2 X(z) - a1 z-1 Y(z) - a2 z-2 Y(z).

• From this expression, we can determine the transfer function, H(z) = Y(z) / X(z), of the filter: • It is convenient to evaluate the z-transform of a system in the complex z-plane, as shown below: • The z-transform is a more general version of the Discrete-Time Fourier Transform, which itself can be viewed as the limiting form of the DFT when its length N is allowed to approach infinity.

• We can determine the frequency response of a system from its z-transform by setting , where is in radians per second and T is the sample period. In the complex z-plane, this is equivalent to evaluating the z-transform on the “unit circle” defined by . ©2004-2020 McGill University. All Rights Reserved.Maintained by Gary P. Scavone.