|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{about||standard z-score in statistics|Standard score|Fisher z-transformation in statistics|Fisher transformation}}
| | I'm Xiomara (24) from Ingersoll, Canada. <br>I'm learning French literature at a local university and I'm just about to graduate.<br>I have a part time job in a university.<br><br>My web site ... [http://Www.dgjianyuan.com/plus/guestbook.php How To Get Free Fifa 15 Coins] |
| | |
| In [[mathematics]] and [[signal processing]], the '''Z-transform''' converts a [[discrete signal|time domain signal]], which is a [[sequence]] of [[real number|real]] or [[complex number]]s, 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 theory of [[time scale calculus]].
| |
| | |
| == History ==
| |
| The basic idea now known as the Z-transform was known to [[Laplace]], and re-introduced in 1947 by [[Witold Hurewicz|W. Hurewicz]] as a tractable way to solve linear, constant-coefficient [[difference equation]]s.<ref>{{cite book | title = Time sequence analysis in geophysics | edition = 3rd | author = E. R. Kanasewich | publisher = University of Alberta | year = 1981 | isbn = 978-0-88864-074-1 | pages = 185–186 | url = http://books.google.com/books?id=k8SSLy-FYagC&pg=PA185}}</ref> It was later dubbed "the z-transform" by [[John R. Ragazzini|Ragazzini]] and [[Lotfi A. Zadeh|Zadeh]] in the sampled-data control group at Columbia University in 1952.<ref> {{cite journal | journal = Trans. Am. Inst. Elec. Eng. | title = The analysis of sampled-data systems | author = J. R. Ragazzini and L. A. Zadeh | volume = 71 | issue = II | publisher = | pages = 225–234 | year = 1952 }}</ref><ref> {{cite book | title = Digital control systems implementation and computational techniques | edition = | author = Cornelius T. Leondes | publisher = Academic Press | year = 1996| isbn = 978-0-12-012779-5 | page = 123 | url = http://books.google.com/books?id=aQbk3uidEJoC&pg=PA123 }}</ref>
| |
| | |
| The modified or [[advanced Z-transform]] was later developed and popularized by [[Eliahu I. Jury|E. I. Jury]].<ref>
| |
| {{cite book
| |
| | title = Sampled-Data Control Systems
| |
| | author = Eliahu Ibrahim Jury
| |
| | publisher = John Wiley & Sons
| |
| | year = 1958
| |
| }}</ref><ref>
| |
| {{cite book
| |
| | title = Theory and Application of the Z-Transform Method
| |
| | author = Eliahu Ibrahim Jury
| |
| | publisher = Krieger Pub Co
| |
| | year = 1973
| |
| | isbn = 0-88275-122-0
| |
| }}</ref>
| |
| | |
| The idea contained within the Z-transform is also known in mathematical literature as the method of [[generating function]]s which can be traced back as early as 1730 when it was introduced by [[Abraham de Moivre|de Moivre]] in conjunction with probability theory.<ref>
| |
| {{cite book
| |
| | title = Theory and Application of the Z-Transform Method
| |
| | author = Eliahu Ibrahim Jury
| |
| | publisher = John Wiley & Sons
| |
| | year = 1964
| |
| | page = 1
| |
| }}</ref>
| |
| From a mathematical view the Z-transform can also be viewed as a [[Laurent series]] where one views the sequence of numbers under consideration as the (Laurent) expansion of an analytic function.
| |
| | |
| ==Definition==
| |
| The Z-transform, like many [[integral transform]]s, 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[n]'' is the [[formal power series]] ''X(z)'' defined as
| |
| | |
| :<math>X(z) = \mathcal{Z}\{x[n]\} = \sum_{n=-\infty}^{\infty} x[n] z^{-n} </math>
| |
| | |
| where ''n'' is an integer and ''z'' is, in general, a [[complex number]]:
| |
| :<math>z = A e^{j\phi} = A(\cos{\phi}+j\sin{\phi})\,</math>
| |
| where ''A'' is the magnitude of ''z'', ''j'' is the [[imaginary unit]], and ɸ is the ''[[complex argument]]'' (also referred to as ''angle'' or ''phase'') in [[radian]]s.
| |
| | |
| === Unilateral Z-transform ===
| |
| Alternatively, in cases where ''x[n]'' is defined only for ''n'' ≥ 0, the ''single-sided'' or ''unilateral'' Z-transform is defined as
| |
| | |
| :<math>X(z) = \mathcal{Z}\{x[n]\} = \sum_{n=0}^{\infty} x[n] z^{-n}.</math>
| |
| | |
| In [[signal processing]], this definition can be used to evaluate the Z-transform of the [[Finite_impulse_response#Impulse_response|unit impulse response]] of a discrete-time [[causal system]].
| |
| | |
| 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''<sup>−1</sup>. The properties of Z-transforms (below) have useful interpretations in the context of probability theory.
| |
| | |
| ===Geophysical definition===
| |
| In geophysics, the usual definition for the Z-transform is a power series in ''z'' as opposed to ''z''<sup>−1</sup>. This convention is used, for example, by Robinson and Treitel<ref name=robinson/> and by Kanasewich.<ref name=kanasewich/> The geophysical definition is:
| |
| | |
| :<math>X(z) = \mathcal{Z}\{x[n]\} = \sum_{n} x[n] z^{n}.</math>
| |
| | |
| The two definitions are equivalent; however, the difference results in a number of changes. For example, the location of zeros and [[Pole (complex analysis)|poles]] move from inside the [[unit circle]] using one definition, to outside the unit circle using the other definition.<ref name=robinson>
| |
| {{cite book
| |
| | title = Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
| |
| Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
| |
| | author = Enders A. Robinson, Sven Treitel
| |
| | publisher = SEG Books
| |
| | year = 2008
| |
| | isbn = 9781560801481
| |
| | pages = 163, 375–376
| |
| | url = http://books.google.com/books?id=IH-Pu3PlJgAC&pg=PA375&dq=%22all+the+poles+lie+outside+the+unit+circle%22&hl=en&sa=X&ei=jQDcUo24MM7eoASQ-IHADQ&ved=0CC8Q6AEwAA#v=onepage&q=%22all%20the%20poles%20lie%20outside%20the%20unit%20circle%22&f=false
| |
| }}</ref><ref name=kanasewich>
| |
| {{cite book
| |
| | title = Time Sequence Analysis in Geophysics
| |
| | author = E. R. Kanasewich
| |
| | publisher = University of Alberta
| |
| | year = 1981
| |
| | isbn = 9780888640741
| |
| | page = 186, 249
| |
| | url = http://books.google.com/books?id=k8SSLy-FYagC&pg=PA249&dq=inauthor:Kanasewich++poles+stability&hl=en&sa=X&ei=igLcUqeXFMmxoQTxzIHoAg&ved=0CC8Q6AEwAA#v=onepage&q=inauthor%3AKanasewich%20%20poles%20stability&f=false
| |
| }}</ref>
| |
| Thus, care is required to note which definition is being used by a particular author.
| |
| | |
| ==Inverse Z-transform==
| |
| The ''inverse'' Z-transform is
| |
| | |
| :<math> x[n] = \mathcal{Z}^{-1} \{X(z) \}= \frac{1}{2 \pi j} \oint_{C} X(z) z^{n-1} dz</math>
| |
| | |
| where ''C'' is a counterclockwise closed path encircling the origin and entirely in the [[Radius of convergence|region of convergence]] (ROC). In the case where the ROC is causal (see [[#Example 2 (causal ROC)|Example 2]]), this means the path ''C'' must encircle all of the poles of ''X(z)''.
| |
| | |
| A special case of this [[contour integral]] occurs when ''C'' is the unit circle (and can be used when the ROC includes the unit circle which is always guaranteed when ''X(z)'' is stable, i.e. all the poles are within the unit circle). The inverse Z-transform simplifies to the [[Discrete-time_Fourier_transform#Inverse_transform|inverse discrete-time Fourier transform]]:
| |
| | |
| :<math> x[n] = \frac{1}{2 \pi} \int_{-\pi}^{+\pi} X(e^{j \omega}) e^{j \omega n} d \omega.</math>
| |
| | |
| 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-time Fourier transform]] (DTFT)—not to be confused with 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 [[Radius of convergence|region of convergence]] (ROC) is the set of points in the complex plane for which the Z-transform summation converges.
| |
| | |
| :<math>ROC = \left\{ z : \left|\sum_{n=-\infty}^{\infty}x[n]z^{-n}\right| < \infty \right\} </math>
| |
| | |
| ===Example 1 (no ROC)===
| |
| Let ''x[n]'' = (0.5)<sup>''n''</sup>. Expanding ''x[n]'' on the interval (−∞, ∞) it becomes
| |
| | |
| :<math>x[n] = \left \{\cdots, 0.5^{-3}, 0.5^{-2}, 0.5^{-1}, 1, 0.5, 0.5^2, 0.5^3, \cdots \right \} = \left \{\cdots, 2^3, 2^2, 2, 1, 0.5, 0.5^2, 0.5^3, \cdots \right\}.</math>
| |
| | |
| Looking at the sum
| |
| | |
| :<math>\sum_{n=-\infty}^{\infty}x[n]z^{-n} \to \infty.</math>
| |
| | |
| Therefore, there are no values of ''z'' that satisfy this condition.
| |
| | |
| ===Example 2 (causal ROC)===
| |
| [[Image:Region of convergence 0.5 causal.svg|thumb|250px|ROC shown in blue, the unit circle as a dotted grey circle (appears reddish to the eye) and the circle |''z''| = 0.5 is shown as a dashed black circle]]
| |
| | |
| Let <math>x[n] = 0.5^n u[n]\ </math> (where ''u'' is the [[Heaviside step function]]). Expanding ''x[n]'' on the interval (−∞, ∞) it becomes
| |
| | |
| :<math>x[n] = \left \{\cdots, 0, 0, 0, 1, 0.5, 0.5^2, 0.5^3, \cdots \right \}.</math>
| |
| | |
| Looking at the sum
| |
| | |
| :<math>\sum_{n=-\infty}^{\infty}x[n]z^{-n} = \sum_{n=0}^{\infty}0.5^nz^{-n} = \sum_{n=0}^{\infty}\left(\frac{0.5}{z}\right)^n = \frac{1}{1 - 0.5z^{-1}}.</math>
| |
| | |
| The last equality arises from the infinite [[geometric series]] and the equality only holds if |0.5''z''<sup>−1</sup>| < 1 which can be rewritten in terms of ''z'' as |''z''| > 0.5. Thus, the ROC is |''z''| > 0.5. In this case the ROC is the complex plane with a disc of radius 0.5 at the origin "punched out".<br clear="all" />
| |
| | |
| ===Example 3 (anticausal ROC)===
| |
| [[Image:Region of convergence 0.5 anticausal.svg|thumb|250px|ROC shown in blue, the unit circle as a dotted grey circle and the circle |''z''| = 0.5 is shown as a dashed black circle]]
| |
| | |
| Let <math>x[n] = -(0.5)^n u[-n-1]\ </math> (where ''u'' is the [[Heaviside step function]]). Expanding ''x[n]'' on the interval (−∞, ∞) it becomes
| |
| | |
| :<math>x[n] = \left \{ \cdots, -(0.5)^{-3}, -(0.5)^{-2}, -(0.5)^{-1}, 0, 0, 0, 0, \cdots \right \}.</math>
| |
| | |
| Looking at the sum
| |
| | |
| :<math>\sum_{n=-\infty}^{\infty}x[n]z^{-n} = -\sum_{n=-\infty}^{-1}0.5^nz^{-n} = -\sum_{m=1}^{\infty}\left(\frac{z}{0.5}\right)^{m} = 1-\frac{1}{1 - 0.5^{-1}z} =\frac{1}{1 - 0.5z^{-1}}</math>
| |
| | |
| Using the infinite [[geometric series]], again, the equality only holds if |0.5<sup>−1</sup>''z''| < 1 which can be rewritten in terms of ''z'' as |''z''| < 0.5. Thus, the ROC is |''z''| < 0.5. In this case the ROC is a disc centered at the origin and of radius 0.5.
| |
| | |
| What differentiates this example from the previous example is ''only'' the ROC. This is intentional to demonstrate that the transform result alone is insufficient.
| |
| <br clear="all" /> | |
| | |
| ===Examples conclusion===
| |
| Examples 2 & 3 clearly show that the Z-transform ''X(z)'' of ''x[n]'' is unique when and 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 |''z''| = ∞ while the anticausal system in example 3 yields an ROC that includes |''z''| = 0.
| |
| | |
| [[Image:Region of convergence 0.5 0.75 mixed-causal.svg|thumb|250px|ROC shown as a blue ring 0.5 < |''z''| < 0.75]]
| |
| In systems with multiple poles it is possible to have an ROC that includes neither |''z''| = ∞ nor |''z''| = 0. The ROC creates a circular band. For example,
| |
| | |
| :<math>x[n] = 0.5^nu[n] - 0.75^nu[-n-1]</math>
| |
| | |
| has poles at 0.5 and 0.75. The ROC will be 0.5 < |''z''| < 0.75, which includes neither the origin nor infinity. Such a system is called a [[mixed-causality system]] as it contains a causal term (0.5)<sup>ν</sup>[''n''] and an anticausal term <math>-(0.75)^nu[-n-1]\ </math>.
| |
| | |
| The [[Control theory#Stability|stability]] of a system can also be determined by knowing the ROC alone. If the ROC contains the unit circle (i.e., |''z''| = 1) then the system is stable. In the above systems the causal system (Example 2) is stable because |''z''| > 0.5 contains the unit circle.
| |
| | |
| If you are provided a Z-transform of a system without an ROC (i.e., an ambiguous ''x[n]'') you can determine a unique ''x[n]'' provided you desire the following:
| |
| * Stability
| |
| * Causality
| |
| | |
| If you need stability then the ROC must contain the unit circle. If you need a causal system then the ROC must contain infinity and the system function will be a right-sided sequence. If you need an anticausal system then the ROC must contain the origin and the system function will be a left-sided sequence. If you need both, stability and causality, all the poles of the system function must be inside the unit circle.
| |
| | |
| The unique ''x[n]'' can then be found.
| |
| | |
| ==Properties==
| |
| {| class="wikitable"
| |
| |+ '''Properties of the z-transform'''
| |
| !
| |
| ! Time domain
| |
| ! Z-domain
| |
| ! Proof
| |
| ! ROC
| |
| |-
| |
| ! Notation
| |
| | <math>x[n]=\mathcal{Z}^{-1}\{X(z)\}</math>
| |
| | <math>X(z)=\mathcal{Z}\{x[n]\}</math>
| |
| |
| |
| |<math>r_2<|z|<r_1</math>
| |
| |-
| |
| ! [[Linearity]]
| |
| | <math>a_1 x_1[n] + a_2 x_2[n]</math>
| |
| | <math>a_1 X_1(z) + a_2 X_2(z)</math>
| |
| | <math>\begin{align}X(z) &= \sum_{n=-\infty}^{\infty} (a_1x_1(n)+a_2x_2(n))z^{-n} \\
| |
| &= a_1\sum_{n=-\infty}^{\infty} x_1(n)z^{-n} + a_2\sum_{n=-\infty}^{\infty}x_2(n)z^{-n} \\
| |
| &= a_1X_1(z) + a_2X_2(z) \end{align} </math>
| |
| | Contains ROC<sub>1</sub> ∩ ROC<sub>2</sub>
| |
| |-
| |
| ! [[Upsampling|Time expansion]]
| |
| | <math>x_K[n] = \begin{cases} x[r], & n = rK \\ 0, & n \not= rK \end{cases}</math>
| |
| ''r'': integer
| |
| | <math>X(z^K)</math>
| |
| | <math>\begin{align} X_K(z) &=\sum_{n=-\infty}^{\infty} x_K(n)z^{-n} \\
| |
| &= \sum_{r=-\infty}^{\infty}x(r)z^{-rK}\\
| |
| &= \sum_{r=-\infty}^{\infty}x(r)(z^{K})^{-r}\\
| |
| &= X(z^{K}) \end{align}</math>
| |
| | <math>R^{\frac{1}{K}}</math>
| |
| |-
| |
| ! [[Downsampling|Decimation]]
| |
| | <math>x[nK]</math>
| |
| | <math>\frac{1}{K} \sum_{p=0}^{K-1} X\left(z^{\tfrac{1}{K}} \cdot e^{-i \tfrac{2\pi}{K} p}\right)</math>
| |
| | [http://www2.ece.ohio-state.edu/~schniter/ee700/handouts/multirate.pdf ohio-state.edu] or [http://www.ee.ic.ac.uk/hp/staff/dmb/courses/DSPDF/01100_Multirate.pdf ee.ic.ac.uk]
| |
| |
| |
| |-
| |
| ! Time shifting
| |
| | <math>x[n-k]</math>
| |
| | <math>z^{-k}X(z)</math>
| |
| | <math>\begin{align} Z\{x[n-k]\} &= \sum_{n=0}^{\infty} x[n-k]z^{-n}\\
| |
| &= \sum_{j=-k}^{\infty} x[j]z^{-(j+k)}&& j = n-k \\
| |
| &= \sum_{j=-k}^{\infty} x[j]z^{-j}z^{-k} \\
| |
| &= z^{-k}\sum_{j=-k}^{\infty}x[j]z^{-j}\\
| |
| &= z^{-k}\sum_{j=0}^{\infty}x[j]z^{-j} && x[\beta] = 0, \beta < 0\\
| |
| &= z^{-k}X(z)\end{align} </math>
| |
| | ROC, except ''z'' = 0 if ''k'' > 0 and ''z'' = ∞ if ''k'' < 0
| |
| |-
| |
| ! Scaling in
| |
| | |
| the z-domain
| |
| | <math>a^n x[n]</math>
| |
| | <math>X(a^{-1}z)</math>
| |
| | <math>\begin{align}\mathcal{Z} \left \{a^n x[n] \right \} &= \sum_{n=-\infty}^{\infty} a^{n}x(n)z^{-n} \\
| |
| &= \sum_{n=-\infty}^{\infty} x(n)(a^{-1}z)^{-n} \\
| |
| &= X(a^{-1}z)
| |
| \end{align} </math>
| |
| | <math>|a|r_2 < |z|< |a|r_1</math>
| |
| |-
| |
| ! Time reversal
| |
| | <math>x[-n]</math>
| |
| | <math>X(z^{-1})</math>
| |
| | <math>\begin{align} \mathcal{Z}\{x(-n)\} &= \sum_{n=-\infty}^{\infty} x(-n)z^{-n} \\
| |
| &= \sum_{m=-\infty}^{\infty} x(m)z^{m}\\
| |
| &= \sum_{m=-\infty}^{\infty} x(m){(z^{-1})}^{-m}\\
| |
| &= X(z^{-1}) \\
| |
| \end{align} </math>
| |
| | <math>\tfrac{1}{r_1}<|z|<\tfrac{1}{r_2}</math>
| |
| |-
| |
| ! [[Complex conjugation]]
| |
| | <math>x^*[n]</math>
| |
| | <math>X^*(z^*)</math>
| |
| | <math>\begin{align} \mathcal{Z} \{x^*(n)\} &= \sum_{n=-\infty}^{\infty} x^*(n)z^{-n}\\
| |
| &= \sum_{n=-\infty}^{\infty} \left [x(n)(z^*)^{-n} \right ]^*\\
| |
| &= \left [ \sum_{n=-\infty}^{\infty} x(n)(z^*)^{-n}\right ]^*\\
| |
| &= X^*(z^*)
| |
| \end{align} </math>
| |
| |
| |
| |-
| |
| ! [[Real part]]
| |
| | <math>\operatorname{Re}\{x[n]\}</math>
| |
| | <math>\tfrac{1}{2}\left[X(z)+X^*(z^*) \right]</math>
| |
| |
| |
| |
| |
| |-
| |
| ! [[Imaginary part]]
| |
| | <math>\operatorname{Im}\{x[n]\}</math>
| |
| | <math>\tfrac{1}{2j}\left[X(z)-X^*(z^*) \right]</math>
| |
| |
| |
| |
| |
| |-
| |
| ! Differentiation
| |
| | <math>nx[n]</math>
| |
| | <math> -z \frac{dX(z)}{dz}</math>
| |
| | <math>\begin{align} \mathcal{Z}\{nx(n)\} &= \sum_{n=-\infty}^{\infty} nx(n)z^{-n}\\
| |
| &= z \sum_{n=-\infty}^{\infty} nx(n)z^{-n-1}\\
| |
| &= -z \sum_{n=-\infty}^{\infty} x(n)(-nz^{-n-1})\\
| |
| &= -z \sum_{n=-\infty}^{\infty} x(n)\frac{d}{dz}(z^{-n}) \\
| |
| &= -z \frac{dX(z)}{dz}
| |
| \end{align} </math>
| |
| |
| |
| |-
| |
| ! [[Convolution]]
| |
| | <math>x_1[n] * x_2[n]</math>
| |
| | <math>X_1(z)X_2(z)</math>
| |
| | <math>\begin{align} \mathcal{Z}\{x_1(n)*x_2(n)\} &= \mathcal{Z} \left \{\sum_{l=-\infty}^{\infty} x_1(l)x_2(n-l) \right \} \\
| |
| &= \sum_{n=-\infty}^{\infty} \left [\sum_{l=-\infty}^{\infty} x_1(l)x_2(n-l) \right ]z^{-n}\\
| |
| &=\sum_{l=-\infty}^{\infty} x_1(l) \left [\sum_{n=-\infty}^{\infty} x_2(n-l)z^{-n} \right ]\\
| |
| &= \left [\sum_{l=-\infty}^{\infty} x_1(l)z^{-l} \right ] \! \!\left [\sum_{n=-\infty}^{\infty} x_2(n)z^{-n} \right ] \\
| |
| &=X_1(z)X_2(z)
| |
| \end{align} </math>
| |
| | Contains ROC<sub>1</sub> ∩ ROC<sub>2</sub>
| |
| |-
| |
| ! [[Cross-correlation]]
| |
| | <math>r_{x_1,x_2}=x_1^*[-n] * x_2[n]</math>
| |
| | <math>R_{x_1,x_2}(z)=X_1^*(\tfrac{1}{z^*})X_2(z)</math>
| |
| |
| |
| | Contains the intersection of ROC of <math>X_1(\tfrac{1}{z^*})</math> and <math>X_2(z)</math>
| |
| |-
| |
| ! First difference
| |
| | <math>x[n] - x[n-1]</math>
| |
| | <math> (1-z^{-1})X(z)</math>
| |
| |
| |
| | Contains the intersection of ROC of ''X<sub>1</sub>(z)'' and ''z'' ≠ 0
| |
| |-
| |
| ! Accumulation
| |
| |<math>\sum_{k=-\infty}^{n} x[k]</math>
| |
| |<math> \frac{1}{1-z^{-1} }X(z)</math>
| |
| |<math>\begin{align}
| |
| \sum_{n=-\infty}^{\infty}\sum_{k=-\infty}^{n} x[k] z^{-n}&=\sum_{n=-\infty}^{\infty}(x[n]+\cdots + x[-\infty])z^{-n}\\
| |
| &=X[z] \left (1+z^{-1}+z^{-2}+\cdots \right )\\
| |
| &=X[z] \sum_{j=0}^{\infty}z^{-j} \\
| |
| &=X[z] \frac{1}{1-z^{-1}}\end{align}</math>
| |
| |
| |
| |-
| |
| ! [[Multiplication]]
| |
| | <math>x_1[n]x_2[n]</math>
| |
| | <math>\frac{1}{j2\pi}\oint_C X_1(v)X_2(\tfrac{z}{v})v^{-1}\mathrm{d}v</math>
| |
| |
| |
| | At least <math>r_{1l}r_{2l}<|z|<r_{1u}r_{2u}</math> |-
| |
| |}
| |
| | |
| '''[[Parseval's theorem]]'''
| |
| :<math>\sum_{n=-\infty}^{\infty} x_1[n]x^*_2[n] \quad = \quad \frac{1}{j2\pi}\oint_C X_1(v)X^*_2(\tfrac{1}{v^*})v^{-1}\mathrm{d}v</math>
| |
| | |
| '''[[Initial value theorem]]''': If ''x''[''n''] causal, then
| |
| :<math>x[0]=\lim_{z\to \infty}X(z).</math>
| |
| | |
| '''[[Final value theorem]]''': If the poles of (''z''−1)''X''(''z'') are inside the unit circle, then
| |
| :<math>x[\infty]=\lim_{z\to 1}(z-1)X(z).</math>
| |
| | |
| ==Table of common Z-transform pairs==
| |
| Here:
| |
| | |
| :<math>u : n \mapsto u[n] = \begin{cases} 1, & n \ge 0 \\ 0, & n < 0 \end{cases}</math>
| |
| | |
| is the [[Heaviside step function|unit (or Heaviside) step function]] and
| |
| | |
| :<math>\delta : n \mapsto \delta[n] = \begin{cases} 1, & n = 0 \\ 0, & n \ne 0 \end{cases}</math>
| |
| | |
| is the [[Dirac delta function|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'' = 0 usually does not really matter, except when working in discrete time, in which case they become degenerate discrete series ; 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 that the unit step function is the [[integral]] of the unit impulse function (in the continuous time domain), or the summation of the unit impulse function is the unit step function (in the discrete time domain), hence the choice of making their value on ''n'' = 0 fixed here to 1.
| |
| | |
| {| class="wikitable"
| |
| |-
| |
| ! !! Signal, <math>x[n]</math> !! Z-transform, <math>X(z)</math> !! ROC
| |
| |-
| |
| | 1 || <math>\delta[n]</math> || 1 || all ''z''
| |
| |-
| |
| | 2 || <math>\delta[n-n_0]</math> || <math> z^{-n_0}</math> || <math> z \neq 0</math>
| |
| |-
| |
| | 3 || <math>u[n] \,</math> || <math> \frac{1}{1-z^{-1} }</math> || <math>|z| > 1</math>
| |
| |-
| |
| | 4 ||<math>e^{-\alpha n} u[n] </math> || <math> 1 \over 1-e^{-\alpha }z^{-1}</math> || <math> |z| > |e^{-\alpha}| \,</math>
| |
| |-
| |
| | 5 ||<math> -u[-n-1]</math> || <math> \frac{1}{1 - z^{-1}}</math> ||<math>|z| < 1</math>
| |
| |-
| |
| | 6 ||<math> n u[n]</math> || <math> \frac{z^{-1}}{( 1-z^{-1} )^2}</math> || <math>|z| > 1</math>
| |
| |-
| |
| | 7 ||<math> - n u[-n-1] \,</math> || <math> \frac{z^{-1} }{ (1 - z^{-1})^2 }</math> ||<math> |z| < 1</math>
| |
| |-
| |
| | 8 ||<math>n^2 u[n]</math> || <math> \frac{ z^{-1} (1 + z^{-1} )}{(1 - z^{-1})^3} </math> || <math>|z| > 1\,</math>
| |
| |-
| |
| | 9 ||<math> - n^2 u[-n - 1] \,</math> || <math> \frac{ z^{-1} (1 + z^{-1} )}{(1 - z^{-1})^3} </math> || <math>|z| < 1\,</math>
| |
| |-
| |
| | 10 ||<math>n^3 u[n]</math> || <math> \frac{z^{-1} (1 + 4 z^{-1} + z^{-2} )}{(1-z^{-1})^4} </math> || <math>|z| > 1\,</math>
| |
| |-
| |
| | 11 ||<math>- n^3 u[-n -1]</math> || <math> \frac{z^{-1} (1 + 4 z^{-1} + z^{-2} )}{(1-z^{-1})^4} </math> || <math>|z| < 1\,</math>
| |
| |-
| |
| | 12 ||<math>a^n u[n]</math> || <math> \frac{1}{1-a z^{-1}}</math> ||<math> |z| > |a|</math>
| |
| |-
| |
| | 13 ||<math>-a^n u[-n-1]</math> || <math> \frac{1}{1-a z^{-1}}</math> ||<math>|z| < |a|</math>
| |
| |-
| |
| | 14 ||<math>n a^n u[n]</math> || <math> \frac{az^{-1} }{ (1-a z^{-1})^2 }</math> || <math>|z| > |a|</math>
| |
| |-
| |
| | 15 ||<math>-n a^n u[-n-1]</math> || <math> \frac{az^{-1} }{ (1-a z^{-1})^2 }</math> ||<math> |z| < |a|</math>
| |
| |-
| |
| | 16 ||<math>n^2 a^n u[n]</math> || <math> \frac{a z^{-1} (1 + a z^{-1}) }{(1-a z^{-1})^3} </math> || <math>|z| > |a|</math>
| |
| |-
| |
| | 17 ||<math>- n^2 a^n u[-n -1]</math> || <math> \frac{a z^{-1} (1 + a z^{-1}) }{(1-a z^{-1})^3} </math> || <math>|z| < |a|</math>
| |
| |-
| |
| | 18 ||<math>\cos(\omega_0 n) u[n]</math> || <math> \frac{ 1-z^{-1} \cos(\omega_0)}{ 1-2z^{-1}\cos(\omega_0)+ z^{-2}}</math> ||<math> |z| >1</math>
| |
| |-
| |
| | 19 ||<math>\sin(\omega_0 n) u[n]</math> || <math> \frac{ z^{-1} \sin(\omega_0)}{ 1-2z^{-1}\cos(\omega_0)+ z^{-2} }</math> ||<math> |z| >1</math>
| |
| |-
| |
| | 20 ||<math>a^n \cos(\omega_0 n) u[n]</math>||<math>\frac{1-a z^{-1} \cos( \omega_0)}{1-2az^{-1}\cos(\omega_0)+ a^2 z^{-2}}</math>||<math>|z|>|a|</math>
| |
| |-
| |
| | 21 ||<math>a^n \sin(\omega_0 n) u[n]</math>||<math> \frac{ az^{-1} \sin(\omega_0) }{ 1-2az^{-1}\cos(\omega_0)+ a^2 z^{-2} }</math> ||<math>|z|>|a|</math>
| |
| |}
| |
| | |
| ==Relationship to discrete-time Fourier transform (DTFT)==
| |
| The Z-transform is a generalization of the [[discrete-time Fourier transform]] (DTFT). The DTFT can be found by evaluating the Z-transform ''X(z)'' at ''z'' = ''e<sup>jω</sup>'' (where ω is a [[Normalized_frequency_(digital_signal_processing)#Alternative_normalizations|normalized frequency]]) or, in other words, evaluated on the unit circle.
| |
| | |
| In order to determine the [[frequency response]] of the system the Z-transform must be evaluated on the unit circle, meaning that the system's region of convergence must contain the unit circle. This is the case where DTFT exists and converges uniformly. If the unit circle is not in region of convergence of z-transform, but the signal is finite energy (not absolutely summable), then DTFT exists but converges only in mean square error, which means [[Gibbs phenomenon|Gibbs phenomena]] can happen. Also, using Dirac delta function, periodic signals, which are not absolutely summable, can be represented in DTFT form.
| |
| | |
| ==Relationship to Fourier series==
| |
| Let X(f) be the [[continuous Fourier transform|Fourier transform]] of any function, x(t), whose samples at some interval, T, equal the x[n] sequence. Then the [[discrete-time Fourier transform]] (DTFT) of the x[n] sequence is the [[Fourier series]] representation of a [[periodic summation]] of X(f)''':'''
| |
| | |
| {{NumBlk|:|<math>\underbrace{
| |
| \sum_{n=-\infty}^{\infty} \overbrace{x(nT)}^{x[n]}\ e^{-j 2\pi f nT}
| |
| }_{\text{DTFT}} = \frac{1}{T}\sum_{k=-\infty}^{\infty} X(f-k/T).</math>|{{EquationRef|Eq.1}}}}
| |
| | |
| When T has units of seconds, <math>\scriptstyle f</math> has units of [[hertz]].
| |
| | |
| For values of z constrained to the form ''e<sup>jω</sup>'', the Z-transform is the same Fourier series on a [[Normalized_frequency_(digital_signal_processing)#Alternative_normalizations|normalized frequency]] scale. By comparison with {{EquationNote|Eq.1}}, we deduce''':'''
| |
| | |
| :<math>
| |
| \sum_{n=-\infty}^{\infty} x[n]\ z^{-n} = \sum_{n=-\infty}^{\infty} x[n]\ e^{-j\omega n} = \frac{1}{T}\sum_{k=-\infty}^{\infty} \underbrace{X\left(\tfrac{\omega}{2\pi T} - \tfrac{k}{T}\right)}_{X\left(\frac{\omega - 2\pi k}{2\pi T}\right)}.
| |
| </math>
| |
| | |
| The units of ω are ''radians per sample''. The value ω=2π corresponds to <math>\tfrac{1}{T}</math> Hz.
| |
| | |
| ==Relationship to Laplace transform==
| |
| ===Bilinear transform===
| |
| {{Main|Bilinear transform}}
| |
| The '''bilinear transform''' is a useful approximation for converting continuous time filters (represented in Laplace space) into discrete time filters (represented in z space), and vice versa. To do this, you can use the following substitutions in ''H(s)'' or ''H(z)'':
| |
| | |
| :<math>s =\frac{2}{T} \frac{(z-1)}{(z+1)}</math>
| |
| | |
| from Laplace to z (Tustin transformation), or
| |
| | |
| :<math>z =\frac{2+sT}{2-sT}</math>
| |
| | |
| from z to Laplace. Through the bilinear transformation, the complex s-plane (of the Laplace transform) is mapped to the complex z-plane (of the z-transform). While this mapping is (necessarily) nonlinear, it is useful in that it maps the entire ''j''Ω axis of the s-plane onto the [[unit circle]] in the z-plane. As such, the Fourier transform (which is the Laplace transform evaluated on the ''j''Ω axis) becomes the discrete-time Fourier transform. This assumes that the Fourier transform exists; i.e., that the ''j''Ω axis is in the region of convergence of the Laplace transform.
| |
| | |
| ===Star transform===
| |
| {{Main|Star transform}}
| |
| Given a continuous time function ''x(t)'' and its one sided [[Laplace transform]]''':'''
| |
| | |
| :<math>X(s) = L\{x(t)\}\ \stackrel{\mathrm{def}}{=}\ \int_0^{\infty}{x(t)e^{-st}dt},</math>
| |
| | |
| a periodic summation of X(s) can be constructed from just a discrete set of samples of x(t), taken at multiples of a sampling interval, T. This function is known as the '''star transform''' of the discrete sequence''':'''
| |
| | |
| :<math>X^*(s) = \frac{1}{T}\sum_{k=-\infty}^\infty X\left(s-j\tfrac{2\pi}{T}k\right)+\frac{x(0)}{2} = \sum_{n=0}^\infty x(nT)\cdot e^{-nTs}.</math>
| |
| | |
| This can be written as a Laplace transform as follows''':'''
| |
| | |
| :<math>
| |
| \begin{align}
| |
| \sum_{n=0}^\infty x(nT)\cdot e^{-nTs} &= \sum_{n=0}^\infty x(nT)\cdot \int_0^{\infty}\delta(t-nT)\cdot e^{-st}dt\\
| |
| &= \sum_{n=0}^\infty \int_0^{\infty}x(t)\cdot \delta(t-nT)\cdot e^{-st}dt\\
| |
| &= \int_0^{\infty}\underbrace{\left(\sum_{n=0}^\infty x(t)\cdot \delta(t-nT)\right)}_{x^*(t)}\cdot e^{-st}dt\\
| |
| &\ \stackrel{\mathrm{def}}{=}\ L\left\{x^*(t)\right\}.
| |
| \end{align}
| |
| </math>
| |
| | |
| <math>x^*(t)</math> is a purely mathematical concept called an ''impulse sampled'' function. Thus, the [[Laplace transform]] of an impulse sampled function is the [[star transform]].
| |
| | |
| <math>X^*(s)</math> is also equivalent to the [[Z transform]] when ''s'' = ln(''z'')/''T''''':'''
| |
| | |
| :<math>
| |
| \begin{align}
| |
| \left.\sum_{n=0}^\infty x(nT)\cdot e^{-nTs}\right|_{s = \frac{\ln(z)}{T}} &= \sum_{n=0}^\infty x(nT)\cdot e^{-n\ln(z)}\\
| |
| &= \sum_{n=0}^\infty x(nT)\cdot z^{-n}\ \ \stackrel{\mathrm{def}}{=}\ \ Z\{x(nT)\}.
| |
| \end{align}
| |
| </math>
| |
| | |
| Similar relationship holds when a continuous time system is converted into a sampled data system by cascading an actual impulse sampler at the input and a fictitious impulse sampler at the output.<ref name=ogata_dtcs>{{cite book|last=Ogata|first=Katsuhiko|title=Discrete-Time Control Systems|publisher=Pearson Education|location=India|isbn=81-7808-335-3|pages=75–77,98–103}}</ref>
| |
| | |
| ==Linear constant-coefficient difference equation==
| |
| The linear constant-coefficient difference (LCCD) equation is a representation for a linear system based on the
| |
| [[Autoregressive moving average model|autoregressive moving-average]] equation.
| |
| | |
| :<math>\sum_{p=0}^{N}y[n-p]\alpha_{p} = \sum_{q=0}^{M}x[n-q]\beta_{q}</math>
| |
| | |
| Both sides of the above equation can be divided by α<sub>0</sub>, if it is not zero, normalizing α<sub>0</sub> = 1 and the LCCD equation can be written
| |
| | |
| :<math>y[n] = \sum_{q=0}^{M}x[n-q]\beta_{q} - \sum_{p=1}^{N}y[n-p]\alpha_{p}.</math>
| |
| | |
| This form of the LCCD equation is favorable to make it more explicit that the "current" output ''y[n]'' is a function of past outputs ''y[n−p]'', current input ''x[n]'', and previous inputs ''x[n−q]''.
| |
| | |
| ===Transfer function===
| |
| Taking the Z-transform of the above equation (using linearity and time-shifting laws) yields
| |
| | |
| :<math>Y(z) \sum_{p=0}^{N}z^{-p}\alpha_{p} = X(z) \sum_{q=0}^{M}z^{-q}\beta_{q}</math>
| |
| | |
| and rearranging results in
| |
| | |
| :<math>H(z) = \frac{Y(z)}{X(z)} = \frac{\sum_{q=0}^{M}z^{-q}\beta_{q}}{\sum_{p=0}^{N}z^{-p}\alpha_{p}} = \frac{\beta_0 + z^{-1} \beta_1 + z^{-2} \beta_2 + \cdots + z^{-M} \beta_M}{\alpha_0 + z^{-1} \alpha_1 + z^{-2} \alpha_2 + \cdots + z^{-N} \alpha_N}.</math>
| |
| | |
| ===Zeros and poles===
| |
| From the [[fundamental theorem of algebra]] the [[numerator]] has ''M'' [[root of a function|roots]] (corresponding to [[Zero (complex analysis)|zeros]] of H) and the [[denominator]] has N roots (corresponding to [[Pole (complex analysis)|poles]]). Rewriting the [[transfer function]] in terms of poles and zeros
| |
| | |
| :<math>H(z) = \frac{(1 - q_1 z^{-1})(1 - q_2 z^{-1})\cdots(1 - q_M z^{-1}) } { (1 - p_1 z^{-1})(1 - p_2 z^{-1})\cdots(1 - p_N z^{-1})}</math>
| |
| | |
| where ''q<sub>k</sub>'' is the ''k''-th zero and ''p<sub>k</sub>'' is the ''k''-th pole. The zeros and poles are commonly complex and when plotted on the complex plane (z-plane) it is called the [[pole-zero plot]].
| |
| | |
| In addition, there may also exist zeros and poles at ''z'' = 0 and ''z'' = ∞. 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)'' is driven by a signal ''X(z)'' then the output is ''Y(z)'' = ''H(z)X(z)''. By performing [[partial fraction]] decomposition on ''Y(z)'' and then taking the inverse Z-transform the output ''y[n]'' can be found. In practice, it is often useful to fractionally decompose <math>\frac{Y(z)}{z}</math> before multiplying that quantity by ''z'' to generate a form of ''Y(z)'' which has terms with easily computable inverse Z-transforms.
| |
| | |
| ==See also==
| |
| * [[Advanced Z-transform]]
| |
| * [[Bilinear transform]]
| |
| * [[Difference equation]] (recurrence relation)
| |
| * [[Convolution#Discrete_convolution|Discrete convolution]]
| |
| * [[Discrete-time Fourier transform]]
| |
| * [[Finite impulse response]]
| |
| * [[Formal power series]]
| |
| * [[Laplace transform]]
| |
| * [[Laurent series]]
| |
| * [[Probability-generating function]]
| |
| * [[Star transform]]
| |
| * [[Zeta function regularization]]
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| | |
| ==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==
| |
| * {{springer|title=Z-transform|id=p/z130010}}
| |
| * [http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/LaplaceZTable/LaplaceZFuncTable.html Z-Transform table of some common Laplace transforms]
| |
| * [http://mathworld.wolfram.com/Z-Transform.html Mathworld's entry on the Z-transform]
| |
| * [http://www.dsprelated.com/comp.dsp/keyword/Z_Transform.php Z-Transform threads in Comp.DSP]
| |
| * [http://math.fullerton.edu/mathews/c2003/ZTransformIntroMod.html Z-Transform Module by John H. Mathews]
| |
| * [http://www.youtube.com/watch?v=4PV6ikgBShw A graphic of the relationship between Laplace transform s-plane to Z-plane of the Z transform]
| |
| | |
| {{DSP}}
| |
| | |
| [[Category:Transforms]]
| |