Z-transform 1

Z-transformIn mathematics and signal processing, the Z-transform converts a discrete time-domain signal, which is a sequenceof real or complex numbers, into a complex frequency-domain representation.It can be considered as a discrete-time equivalent of the Laplace transform. This similarity is explored in the theoryof time scale calculus.

HistoryThe basic idea now known as the Z-transform was known to Laplace, and re-introduced in 1947 by W. Hurewicz asa tractable way to solve linear, constant-coefficient difference equations.[1] It was later dubbed "the z-transform" byRagazzini and Zadeh in the sampled-data control group at Columbia University in 1952.[2][3]

The modified or advanced Z-transform was later developed and popularized by E. I. Jury.[4][5]

The idea contained within the Z-transform is also known in mathematical literature as the method of generatingfunctions which can be traced back as early as 1730 when it was introduced by de Moivre in conjunction withprobability theory.[6] From a mathematical view the Z-transform can also be viewed as a Laurent series where oneviews the sequence of numbers under consideration as the (Laurent) expansion of an analytic function.

DefinitionThe Z-transform, like many integral transforms, can be defined as either a one-sided or two-sided transform.

Bilateral Z-transformThe bilateral or two-sided Z-transform of a discrete-time signal x[n] is the formal power series X(z) defined as

where n is an integer and z is, in general, a complex number:

where A is the magnitude of z, j is the imaginary unit, and is the complex argument (also referred to as angle orphase) in radians.

Unilateral Z-transformAlternatively, in cases where x[n] is defined only for n ≥ 0, the single-sided or unilateral Z-transform is defined as

In signal processing, this definition can be used to evaluate the Z-transform of the unit impulse response of adiscrete-time causal system.An important example of the unilateral Z-transform is the probability-generating function, where the component

is the probability that a discrete random variable takes the value , and the function is usually writtenas , in terms of . The properties of Z-transforms (below) have useful interpretations in the contextof probability theory.

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Geophysical definition

In geophysics, the usual definition for the Z-transform is a power series in as opposed to . This convention isused by Robinson and Treitel and by Kanasewich. The geophysical definition is

The two definitions are equivalent; however, the difference results in a number of changes. For example, the locationof zeros and poles move from inside the unit circle using one definition, to outside the unit circle using the otherdefinition. Thus, care is required to note which definition is being used by a particular author.

Inverse Z-transformThe inverse Z-transform is

where is a counterclockwise closed path encircling the origin and entirely in the region of convergence (ROC). Inthe case where the ROC is causal (see Example 2), this means the path must encircle all of the poles of .A special case of this contour integral occurs when is the unit circle (and can be used when the ROC includes theunit circle which is always guaranteed when is stable, i.e. all the poles are within the unit circle). The inverseZ-transform simplifies to the inverse discrete-time Fourier transform:

The Z-transform with a finite range of n and a finite number of uniformly spaced z values can be computedefficiently via Bluestein's FFT algorithm. The discrete-time Fourier transform (DTFT) (not to be confused with thediscrete Fourier transform (DFT)) is a special case of such a Z-transform obtained by restricting z to lie on the unitcircle.

Region of convergenceThe region of convergence (ROC) is the set of points in the complex plane for which the Z-transform summationconverges.

Example 1 (no ROC)

Let . Expanding on the interval it becomes

Looking at the sum

Therefore, there are no values of that satisfy this condition.

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Example 2 (causal ROC)

ROC shown in blue, the unit circle as a dotted grey circle (appearsreddish to the eye) and the circle is shown as a dashed

black circle

Let (where is the Heaviside stepfunction). Expanding on the interval it becomes

Looking at the sum

The last equality arises from the infinite geometricseries and the equality only holds if

which can be rewritten in terms of as .Thus, the ROC is . In this case the ROC isthe complex plane with a disc of radius 0.5 at the origin"punched out".

Example 3 (anticausal ROC)

ROC shown in blue, the unit circle as a dotted grey circle and thecircle is shown as a dashed black circle

Let (where is theHeaviside step function). Expanding on theinterval it becomes

Looking at the sum

Using the infinite geometric series, again, the equalityonly 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.What differentiates this example from the previousexample is only the ROC. This is intentional to demonstrate that the transform result alone is insufficient.

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Examples conclusion

Examples 2 & 3 clearly show that the Z-transform of is unique when and only when specifying theROC. Creating the pole-zero plot for the causal and anticausal case show that the ROC for either case does notinclude 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 3yields an ROC that includes .

ROC shown as a blue ring

In systems with multiple poles it is possible to have anROC that includes neither nor .The ROC creates a circular band. For example,

has poles at0.5 and 0.75. The ROC will be ,which includes neither the origin nor infinity. Such asystem is called a mixed-causality system as it containsa causal term and an anticausal term

.The stability of a system can also be determined byknowing the ROC alone. If the ROC contains the unitcircle (i.e., ) then the system is stable. In theabove systems the causal system (Example 2) is stablebecause contains the unit circle.If you are provided a Z-transform of a system withoutan ROC (i.e., an ambiguous ) you can determinea unique provided you desire the following:•• Stability•• CausalityIf you need stability then the ROC must contain the unit circle. If you need a causal system then the ROC mustcontain infinity and the system function will be a right-sided sequence. If you need an anticausal system then theROC must contain the origin and the system function will be a left-sided sequence. If you need both, stability andcausality, all the poles of the system function must be inside the unit circle.

The unique can then be found.

Properties

Properties of the z-transform

Time domain Z-domain Proof ROC

Notation ROC:

Linearity At least theintersection of ROC1and ROC2

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Time expansion :

integer

R^{1/k}

Time shifting ROC, except if and

if

Scaling in

the z-domain

Time reversal

Complexconjugation

ROC

Real part ROC

Imaginary part ROC

Differentiation ROC

Convolution At least theintersection of ROC1and ROC2

Cross-correlation At least theintersection of ROCof and

First difference At least theintersection of ROCof X1(z) and

Accumulation

Multiplication -

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Parseval'srelation

•• Initial value theorem

, If causal•• Final value theorem

, Only if poles of are inside the unit circle

Table of common Z-transform pairsHere:

• is the unit (or Heaviside) step function

• is the discrete-time (or Dirac delta) unit impulse function

Both are usually not considered as true functions but as distributions due to their discontinuity (their value on n=0usually does not really matter, except when working in discrete time, in which case they become degenerate discreteseries ; in this section they are chosen to take the value 1 on n=0, both for the continuous and discrete time domains,otherwise the content of the ROC column below would not apply). The two "functions" are chosen together so thatthe unit step function is the integral of the unit impulse function (in the continuous time domain), or the summationof the unit impulse function is the unit step function (in the discrete time domain), hence the choice of making theirvalue on n=0 fixed here to 1.

Signal, Z-transform, ROC

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Relationship to Laplace transformThe Bilinear transform is a useful approximation for converting continuous time filters (represented in Laplacespace) into discrete time filters (represented in z space), and vice versa. To do this, you can use the followingsubstitutions in H(s) or H(z):

from Laplace to z (Tustin transformation), or

from z to Laplace. Through the bilinear transformation, the complex s-plane (of the Laplace transform) is mapped tothe complex z-plane (of the z-transform). While this mapping is (necessarily) nonlinear, it is useful in that it mapsthe entire axis of the s-plane onto the unit circle in the z-plane. As such, the Fourier transform (which is theLaplace transform evaluated on the axis) becomes the discrete-time Fourier transform. This assumes that theFourier transform exists; i.e., that the axis is in the region of convergence of the Laplace transform.

Process of sampling

Consider a continuous time signal . Its one sided Laplace transform is defined as :

If the continuous time signal is uniformly sampled with a train of impulses to get a discrete time signal, then it can be represented as:

where is the sampling interval.Now the Laplace transform of the sampled signal (discrete time) is called Star transform and is given by:

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It can be seen that the Laplace transform of an impulse sampled signal is the star transform and is the same as the Ztransform of the corresponding sequence when . Similar relationship holds when a continuous timesystem is converted into a sampled data system by cascading an actual impulse sampler at the input and a fictitiousimpulse sampler at the output.[7]

Relationship to Fourier transformThe Z-transform is a generalization of the discrete-time Fourier transform (DTFT). The DTFT can be found byevaluating the Z-transform at (where is the normalized frequency) or, in other words, evaluatedon the unit circle. In order to determine the frequency response of the system the Z-transform must be evaluated onthe unit circle, meaning that the system's region of convergence must contain the unit circle. This is the case whereDTFT exists and converges uniformly, if unit circle is not in region of convergence of z-transform, but the signal isfinite energy (not absolutely sumable) DTFT exists but converges only in mean square error, which means Gibbsphenomenon can happen. Also, using Dirac delta function, periodic signals which are not absolutely sumable can berepresented in DTFT form.

Linear constant-coefficient difference equationThe linear constant-coefficient difference (LCCD) equation is a representation for a linear system based on theautoregressive moving-average equation.

Both sides of the above equation can be divided by , if it is not zero, normalizing and the LCCDequation can be written

This form of the LCCD equation is favorable to make it more explicit that the "current" output is a function ofpast outputs , current input , and previous inputs .

Transfer functionTaking the Z-transform of the above equation (using linearity and time-shifting laws) yields

and rearranging results in

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Zeros and polesFrom the fundamental theorem of algebra the numerator has M roots (corresponding to zeros of H) and thedenominator has N roots (corresponding to poles). Rewriting the transfer function in terms of poles and zeros

where is the zero and is the pole. The zeros and poles are commonly complex and when plotted onthe complex plane (z-plane) it is called the pole-zero plot.In addition, there may also exist zeros and poles at and . If we take these poles and zeros as well asmultiple-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 thetime domain. Doing so would result in the impulse response and the linear constant coefficient difference equation ofthe system.

Output response

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

practice, it is often useful to fractionally decompose before multiplying that quantity by to generate a form

of which has terms with easily computable inverse Z-transforms.

References[1] E. R. Kanasewich (1981). Time sequence analysis in geophysics (http:/ / books. google. com/ books?id=k8SSLy-FYagC& pg=PA185) (3rd

ed.). University of Alberta. pp. 185–186. ISBN 978-0-88864-074-1. .[2] J. R. Ragazzini and L. A. Zadeh (1952). "The analysis of sampled-data systems". Trans. Am. Inst. Elec. Eng. 71 (II): 225–234.[3] Cornelius T. Leondes (1996). Digital control systems implementation and computational techniques (http:/ / books. google. com/

books?id=aQbk3uidEJoC& pg=PA123). Academic Press. p. 123. ISBN 978-0-12-012779-5. .[4] Eliahu Ibrahim Jury (1958). Sampled-Data Control Systems. John Wiley & Sons.[5] Eliahu Ibrahim Jury (1973). Theory and Application of the Z-Transform Method. Krieger Pub Co. ISBN 0-88275-122-0.[6] Eliahu Ibrahim Jury (1964). Theory and Application of the Z-Transform Method. John Wiley & Sons. p. 1.[7] Ogata, Katsuhiko. Discrete-Time Control Systems. India: Pearson Education. pp. 75–77,98–103. ISBN 81-7808-335-3.

Further reading• Refaat El Attar, Lecture notes on Z-Transform, Lulu Press, Morrisville NC, 2005. ISBN 1-4116-1979-X.• Ogata, Katsuhiko, Discrete Time Control Systems 2nd Ed, Prentice-Hall Inc, 1995, 1987. ISBN 0-13-034281-5.•• Alan V. Oppenheim and Ronald W. Schafer (1999). Discrete-Time Signal Processing, 2nd Edition, Prentice Hall

Signal Processing Series. ISBN 0-13-754920-2.

External links• Hazewinkel, Michiel, ed. (2001), "Z-transform" (http:/ / www. encyclopediaofmath. org/ index. php?title=p/

z130010), Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4• Z-Transform table of some common Laplace transforms (http:/ / www. swarthmore. edu/ NatSci/ echeeve1/ Ref/

LPSA/ LaplaceZTable/ LaplaceZFuncTable. html)• Mathworld's entry on the Z-transform (http:/ / mathworld. wolfram. com/ Z-Transform. html)• Z-Transform threads in Comp.DSP (http:/ / www. dsprelated. com/ comp. dsp/ keyword/ Z_Transform. php)• Z-Transform Module by John H. Mathews (http:/ / math. fullerton. edu/ mathews/ c2003/ ZTransformIntroMod.

html)

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