Z-transform Definition Region Of Convergence Example 1 (No ROC)

Sometimes we are only interested in the values of the signal x(n) for non-negative values of n. If such is the case, the Z-transform is defined as

The latter is sometimes called a unilateral Z-transform and the former a bilateral or doubly infinite Z-transform. In signal processing, the latter definition is used when the signal is causal in nature.

An important example of the unilateral Z-transform is the probability generating function, where the component x(n) is the probability that a discrete random variable takes the value n, and the function X(z) is usually written as X(s), in terms of s = z⁻¹. The properties of Z-transforms (below) have useful interpretations in the context of probability theory.

2 Region of Convergence

The region of convergence (ROC) is where the Z-transform of a signal has a finite sum for a region in the complex plane.

2.1 Example 1 (No ROC)

2.2 Example 2 ( Causal ROC)

The last equality arises from the infinite geometric series and the equality only holds if which can be rewritten in terms of as . Thus, the ROC is . In this case the ROC is the complex plane with a disc of radius 0.5 at the origin "punched out".

2.3 Example 3 ( Anticausal ROC)

Using the infinite geometric series, again, the equality only holds if which can be rewritten in terms of as . Thus, the ROC is . In this case the ROC is a disc centered at the origin and of radius 0.5.

2.4 Examples Conclusion

Examples 2 & 3 clearly show that the Z-transform of is unique only when specifying the ROC. Creating the pole-zero plot for the causal and anticausal case show that the ROC for either case does not include the pole that is at 0.5. This extends to cases with multiple poles: the ROC will never contain poles.

In example 2, the causal system yields an ROC that includes while the anticausal system in example 3 yields an ROC that includes .

In systems with multiple poles it is possible to have an ROC that includes neither nor . The ROC creates a circular band. For example, has poles at 0.5 and 0.75. The ROC will be , which includes neither the origin nor infinity. Such a system is called a mixed-causality system as it contains a causal term and an anticausal term .

The stability of a system can also be determined by knowing the ROC alone. If the ROC contains the unit circle (i.e., ) then the system is stable. In the above systems the causal system is stable because contains the unit circle.

If you are provided a Z-transform of a system without an ROC (i.e., an ambiguous ) you can determine a unique provided you desire the following:

If you need stability then the ROC must contain the unit circle. If you need a causal system then the ROC must contain infinity. If you need an anticausal system then the ROC must contain the origin.

3 Properties

Linearity. The Z-transform of the linear combination of two signals is the linear combination of the individual Z-transforms.

Z({a₁x₁(n)+a₂x₂(n)}) = a₁Z({x₁(n)}) + a₂Z({x₂(n)})

Shift. Time-shifting the signal by a distance of k to the right results in multiplying the Z-transform by z^−k.

Z({x(n-k)}) = z^-kZ({x(n)})

Convolution. The Z-transform of the convolution of two sequences is the product of the individual Z-transforms.

Z({x(n)}*{y(n)}) = Z({x(n)})Z({y(n)})

Differentiation .

Z({nx(n)}) = -z dZ({x(n)})/dz

The inverse Z-transform can be computed as follows:

where C is any closed curve around the origin and lying in the region of convergence of X(z).

The (unilateral) Z-transform is to discrete time domain signals what the Laplace transform is to continuous time domain signals.

Z-transform with a finite range of n and a finite number of uniformly-spaced z values can be computed efficiently via Bluestein's FFT algorithmBluestein's FFT algorithm (1968), commonly called the chirp-z algorithm (1969), is a fast Fourier transform (FFT) algorithm that computes the discrete Fourier transform (DFT) of arbitrary sizes (including prime sizes) by re-expressing the DFT as a linear. The discrete Fourier transformIn mathematics, the discrete Fourier transform (DFT sometimes called the finite Fourier transform is a Fourier transform widely employed in signal processing and related fields to analyze the frequencies contained in a sampled signal. It is sometimes deno is a special case of such a Z-transform obtained by restricting z to lie on the unit circle.

4 Relationship to Fourier

The Z-transform is a more general transform than the discrete fourier transform (DFT. The DFT can be found by evaluating the Z-transform at or, in other words, evaluated on the unit circle. In order to determine the frequency responseFrequency response is the measure of any system's response to frequency, but is usually used in connection with electronic amplifiers and similar systems, particularly in relation to audio signals. For example, a high fidelity amplifier may be said to hav of the system the Z-transform must be evaluated on the unit circle.

5 LCCD Equation

Linear constant coefficient difference (LCCD) equation is a representation for a linear system based on the ARMA equation.

Typically and the LCCD equation can be written

This form of the LCCD equation is favorable to make it more explicit that the "current" output is a function of past outputs , current input , and previous inputs . If is not unity then the LCCD equation can be divided by to make the coefficient on be unity. It goes without saying that doing so will change the coefficients on all of the other terms.

5.1 Transfer Function

Taking the Z-transform of the equation yields

and rearranging results in

5.2 Zeros & Poles

From the fundamental theorem of algebraThe fundamental theorem of algebra (now considered something of a misnomer by many mathematicians) states that every complex polynomial of degree n has exactly n zeroes, counted with multiplicity. More formally, if : (where the coefficients a . a can be r the numerator has P roots (called zeros) and the denominator has Q roots (called poles). Rewritting the transfer function in terms of poles

Where is the zero and is the pole. The zeros and poles are commonly complex and when plotted on the complex plane it is called the pole-zero plot .

By factoring the denominator, partial fraction decomposition can be used, which can then be transformed back to the time domain. Doing so would result in the impulse response of the system.

5.3 Output Response

If such a system is driven by a signal then the output is . By performing partial fraction decomposition on and then taking the inverse Z-transform the output can be found.

6 External links

http://mathworld.wolfram.com/Z-Transform.html

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1 Definition