Information About Z-transform

	CATEGORIES ABOUT Z-TRANSFORM
	transforms

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The Z-transform and Advanced Z-transform were introduced (under the Z-transform name) by E. I. Jury in 1958 in ''Sampled-Data Control Systems'' (John Wiley & Sons). The idea contained within the Z-transform was previously known as the "generating function method". ''Z-transform'' is a placeholder name, akin to calling the Laplace Transform the "s-transform". More accurate would be "Laurent transform", because it is based on the Laurent Series . The (unilateral) Z-transform is to discrete time domain signals what the one-sided Laplace Transform is to continuous time domain signals. DEFINITION The Z-transform, like many other integral transforms, can be defined as either a ''one-sided'' or ''two-sided'' transform. Bilateral Z-Transform The ''bilateral'' or ''two-sided'' Z-transform of a discrete-time signal ''x {Link without Title} '' is the function ''X(z)'' defined as :$X(z)\; =\; Z\backslash \{x=\; \backslash sum\_\{n=-\backslash infty\}^\{\backslash infty\}\; x[nz^\{-n\}\; \backslash$ where ''n'' is an integer and ''z'' is, in general, a Complex Number : :$z=\; A\; e^\{j\backslash phi\}$ :where ''A'' is the magnitude of ''z'', and φ is the Angular Frequency (in Radians per Sample ). Unilateral Z-Transform Alternatively, in cases where ''x'' {Link without Title} is defined only for ''n'' ≥ 0, the ''single-sided'' or ''unilateral'' Z-transform is defined as :$X(z)\; =\; Z\backslash \{x=\; \backslash sum\_\{n=0\}^\{\backslash infty\}\; x[nz^\{-n\}\; \backslash$ In Signal Processing , this definition is used when the signal is Causal . An important example of the unilateral Z-transform is the Probability-generating Function , where the component ''x {Link without Title} '' 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. INVERSE Z-TRANSFORM The inverse Z-Transform is :$x${Link without Title} = Z^{-1} \{X(z) \}= rac{1}{2 \pi j} \oint_{C} X(z) z^{n-1} dz \ where $C\; \backslash$ is a counterclockwise closed path encircling the origin and entirely in the region of convergence (ROC). The contour or path, $C\; \backslash$, must encircle all of the poles of $X(z)\; \backslash$. A special case of this contour integral that is simply that where $C\; \backslash$ is the unit circle (and can be used when the ROC includes the unit circle) is the inverse Discrete-Time Fourier transform: :$x${Link without Title} = rac{1}{2 \pi} \int_{-\pi}^{+\pi} X(e^{j \omega}) e^{j \omega n} d \omega \ . The 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 Algorithm . The Discrete Fourier Transform (DFT) is a special case of such a Z-transform obtained by restricting ''z'' to lie on the unit circle. 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. :$ROC\; =\; \backslash \{z\; :\; \backslash sum\_\{n=-\backslash infty\}^\{\backslash infty\}x${Link without Title} z^{-n} < \infty\}\ Example 1 (No ROC) Let $x=\; 0.5^n\backslash$. Expanding $x[n\; \backslash$ on $(-\backslash infty,\; \backslash infty)\backslash$ it becomes :$x${Link without Title} = \{..., 0.5^{-3}, 0.5^{-2}, 0.5^{-1}, 1, 0.5, 0.5^2, 0.5^3, ...\} = \{..., 2^3, 2^2, 2, 1, 0.5, 0.5^2, 0.5^3, ...\}\ Looking at the sum :$\backslash sum\_\{n=-\backslash infty\}^\{\backslash infty\}x${Link without Title} z^{-n} < \infty\ There are no such values of $z\backslash$ that satisfy this condition. Example 2 ( Causal ROC) Let $x=\; 0.5^n\; u[n\backslash$ (where $u$ is the Heaviside Step Function ). Expanding $x[n]\backslash$ on $(-\backslash infty,\; \backslash infty)\backslash$ it becomes :$x${Link without Title} = \{..., 0, 0, 0, 1, 0.5, 0.5^2, 0.5^3, ...\}\ Looking at the sum :$\backslash sum\_\{n=-\backslash infty\}^\{\backslash infty\}x${Link without Title} z^{-n} = \sum_{n=0}^{\infty}0.5^nz^{-n} = \sum_{n=0}^{\infty}\left(rac{0.5}{z} ight)^n = rac{1}{1 - 0.5z^{-1}}\

  In Example 2, The Causal System Yields An ROC That Includes <math>\left Z Ight \infty\ </math> while the anticausal system in example 3 yields an ROC that includes <math>\left z ight = 0\ </math>
  In Systems With Multiple Poles It Is Possible To Have An ROC That Includes Neither <math>\left Z Ight \infty\ </math> nor <math>\left z ight = 0\ </math> The ROC creates a circular band For example, <math>x = 05^nu[n - 075^nu </math> has poles at 05 and 075 The ROC will be <math>05 < \left z ight < 075\ </math>, which includes neither the origin nor infinity Such a system is called a Mixed-causality system as it contains a causal term <math>05^nu[n \ </math> and an anticausal term <math>-(075)^nu[-n-1]\ </math>
  The Stability Of A System Can Also Be Determined By Knowing The ROC Alone If The ROC Contains The Unit Circle (ie, <math>\left Z Ight 1\ </math>) then the system is stable In the above systems the causal system is stable because <math>\left z ight > 05\ </math> contains the unit circle



TABLE OF COMMON Z-TRANSFORM PAIRS




  4 <math>n A^n U "n" class="copylinks" target="_blank">{Link without Title} \,</math> <math> rac{az^{-1} }{ (1-a z^{-1})^2 }</math> <math>z > a\,</math>


  6 <math>-n A^n U "-n-1" class="copylinks" target="_blank">{Link without Title} \,</math> <math> rac{az^{-1} }{ (1-a z^{-1})^2 }</math> <math> z < a\,</math>
  7 <math>\cos(\omega 0 N) U "n" class="copylinks" target="_blank">{Link without Title} \,</math> <math> rac{ 1-z^{-1} \cos(\omega_0) }{ 1-2z^{-1}\cos(\omega_0)+ z^{-2} }</math> <math> z >1\,</math>
  8 <math>\sin(\omega 0 N) U "n" class="copylinks" target="_blank">{Link without Title} \,</math> <math> rac{ z^{-1} \sin(\omega_0) }{ 1-2z^{-1}\cos(\omega_0)+ z^{-2} }</math> <math> z >1\,</math>
  9 <math>a^n \cos(\omega 0 N) U "n" class="copylinks" target="_blank">{Link without Title} \,</math> <math> rac{ 1-a z^{-1} \cos( \omega_0) }{ 1-2az^{-1}\cos(\omega_0)+ a^2 z^{-2} }</math> <math> z > a\,</math>
  10 <math>a^n \sin(\omega 0 N) U "n" class="copylinks" target="_blank">{Link without Title} \,</math> <math> rac{ az^{-1} \sin(\omega_0) }{ 1-2az^{-1}\cos(\omega_0)+ a^2 z^{-2} }</math> <math> z > a\,</math>






Zeros and poles

From the Fundamental Theorem Of Algebra the Numerator has M Roots (called Zeros ) and the Denominator has N roots (called Poles ). Rewriting the Transfer Function in terms of poles

:

Where $q\_k\backslash$ is the $k^\{th\}\backslash$ zero and $p\_k\backslash$ is the $k^\{th\}\backslash$ pole. The zeros and poles are commonly complex and when plotted on the complex plane it is called the Pole-zero Plot .

In simple words, zeros are the solutions to the equation obtained by setting the numerator equal to zero, while poles are the solutions to the equation obtained by setting the denominator equal to zero.

In addition, there may also exist zeros and poles at $z=0$ and $z=\backslash infty$. If we take these poles and zeros as well as multiple-order zeros and poles into consideration, the number of zeros and poles are always equal.

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 and the linear constant coefficient difference equation of the system.


Output response

If such a system $H(z)\backslash$ is driven by a signal $X(z)\backslash$ then the output is $Y(z)\; =\; H(z)X(z)\backslash$. By performing Partial Fraction decomposition on $Y(z)\backslash$ and then taking the inverse Z-transform the output $y${Link without Title} \ can be found.


SEE ALSO

• Zeta Function Regularization


• Advanced Z-transform


• Formal Power Series


• Laplace Transform

BIBLIOGRAPHY

• Eliahu Ibrahim Jury, ''Theory and Application of the Z-Transform Method'', Krieger Pub Co, 1973. ISBN 0-88275-122-0.

• Refaat El Attar, ''Lecture notes on Z-Transform'', Lulu Press, Morrisville NC, 2005. ISBN 1-41161-979-X.

EXTERNAL LINKS

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


• Z Transform threads in Comp.DSP