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# From Discrete-Time Fourier Transform to Z-Transform

Forward Z-Transform

The Fourier transform of a discrete signal is defined as:

provided is absolutely summable:

Obviously some signals may not satisfy this condition and their Fourier transform do not exist. To overcome this difficulty, we can multiply the given by an exponential function so that may be forced to be summable for certain values of the real parameter . Now the discrete time Fourier transform becomes:

The result of this summation is a function of a complex variable defined as:

This is the forward Z-transform of the discrete signal :

Inverse Z-Transform

Given the Z transform , the original time signal can be obtained by the inverse Z transform, which can be derived from the corresponding Fourier transform. As shown above, we have:

Now can be obtained by the inverse Fourier transform:

Multiplying both sides by , we get:

To represent the inverse transform in terms of (instead of ), we note

and the inverse Z transform can be obtained as:

Note that the integral with respect to from to becomes an integral with respect to in the complex z-plane, along a circle with a fixed radius and a varying angle from to . Now we have the z-transform pair:

The forward and inverse z-transform pair can also be represented as

In particular, if we let , i.e., , then the Z transform becomes the discrete-time Fourier transform:

This is the reason why sometimes the discrete Fourier spectrum is expressed as a function of .

Different from the discrete-time Fourier transform which converts a 1-D signal in time domain to a 1-D complex spectrum in frequency domain, the Z transform converts the 1D signal to a complex function defined over a 2-D complex plane, called z-plane, represented in polar form by radius and angle .

In particular, when this 2D function is evaluated along the unit circle corresponding to , it becomes a 1D periodic function , the discrete Fourier transform of . Graphically, the periodic spectrum of the signal can be found as the cross section of the 2D function along the unit circle .

Transfer Function of LTI system

The output of a discrete LTI system with input can be found by convolution

where is the impulse response function of the system. In particular, if the input is a complex exponential

then the output can be found to be:

This is the eigenequation with the complex exponential being the eigenfunction of any discrete LTI system, corresponding to its eigenvalue defined as:

which is the z-transform of its impulse response , called the transfer function of the LTI system. In particular, when , i.e., , the transfer function becomes the frequency response function, the Fourier transform of the impulse response:

Next: Conformal Mapping between S-Plane Up: Z_Transform Previous: Z_Transform
Ruye Wang 2014-10-28