What is Z-transform: Definition and 39 Discussions

In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of 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 theory of time-scale calculus.

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  1. F

    Link between Z-transform and Taylor series expansion

    Hello, I am reading a course on signal processing involving the Z-transform, and I just read something that leaves me confused. Let ##F(z)## be the given Z-transform of a numerical function ##f[n]## (discrete amplitudes, discrete variable), which has a positive semi-finite support and finite...
  2. hugo_faurand

    Help with the Python package Scipy and the Z-transform please

    Hello everyone ! I am working on ultrasound scan and the processing of the signal received by the probe. I made the model I wanted and as I do not have an ultrasound scan machine I want to simulate the signal processing. I will do that with the Python package Scipy and the function...
  3. M

    Engineering Z-transform of this Logistic Difference Equation

    Hi, I am trying to work out how I could obtain an expression for a z-transform for the following expression: x_{n + 1} = r x_{n} \left( 1 - x_{n} \right) I am hoping to derive X(z) and then use the final value theorem to show agreement with numerically calculated steady state values. I...
  4. C

    Basic Z-Transform Transfer Function

    Homework Statement Hi, I'm asked to find the transfer function for the following digital filter described by a difference equation Homework EquationsThe Attempt at a Solution [/B] Usually if there's no constant term (0.7) here, I can just rearrange in the form of Y(z)/X(z) to give me H(z)...
  5. M

    Region of convergence Z-transform

    Hello everyone. Iam just learning the z-transform for discrete signals and I can't get my head around the Region of covergence (ROC). As far as I have understood describes the ROC if the z-transform excists or not ? But how to I actually calculate it? Is there any kind of formula? I all...
  6. kostoglotov

    Infinite series question with z-transform addendum

    Homework Statement Hello, I am currently doing some holiday pre-study for signals analysis coming up next semester. I'm mainly using MIT OCW 6.003 from 2011 with some other web resources (youtube, etc). The initial stuff is heavy on the old infinite series stuff, that seems often skimmed...
  7. Avatrin

    Digital signal processing and the z-transform

    Hi I am currently learning Digital Signal Processing from Applied Digital Signal Processing by Manolakis and Ingle. However, my background is in mathematics, so I like being shown why an assertion is true. So, in the chapter about the z-transform is where I start seeing this becoming an issue...
  8. T

    Complex integral for z-transform causality

    This relates to z-transform causality, but I'll try to phrase it as a complex analysis question. Suppose I have a function ##X(z)## whose poles are all inside the unit circle, and which has the property \lim_{|z|\to\infty} \frac{X(z)}{z} = 0 Is that sufficient to guarantee that \frac{1}{2\pi...
  9. MathematicalPhysicist

    Z-transform generalization

    Has there been any applications in engineering to the following generelization to Z-transform? ##X(z) = \sum_{n=-\infty}^\infty x[n] e^{-zn}## We get the ##z##-trasnform by using the substitution ##z \mapsto Log(z)##. Thanks in advance.
  10. N

    Z transform of a sequence

    Hi guys, I am trying to find the z-tranform of the following equation: x(n) =-2n u(-n-1) Using the Z-transform definition,summation and geometric series I am getting 1/(1-2z-1) But according to my lecturer the answer is suppose to be 0.5/(1-2z-1) I have tried simplification/factorization and...
  11. J

    How would you match up these Z-transforms?

    Homework Statement In Figure Q1b (on the next page), two plots display samples of the continuous-time signal uct(t) = sin(2πt) and two plots display samples of the continuous-time signal vct(t) = e −t sin(2πt). For each signal, the samples in the corresponding plots are obtained with two...
  12. A

    Inverse z-transform problem

    Homework Statement Homework Equations H(z) = Y(z)/X(z) The Attempt at a Solution I realized this wasn't in partial fraction form because the 1+z-1+0.5z-2 has non-real roots. I multiplied the 1st fraction part by z1 and the 2nd fraction by z2, then I combined them into one fraction and I...
  13. ElijahRockers

    Z-Transform of finite series.

    Homework Statement Let ## x_j = \begin{Bmatrix} {1, 0 \leq j \leq N-1} \\ {0, else} \\ \end{Bmatrix} ## Show that ##\hat{x}(\phi) = \frac{e^{-i\frac{N-1}{2}\phi}sin(\frac{N}{2}\phi)}{sin(\frac{1}{2}\phi)}##Homework Equations [/B] ##\hat{x}(\phi) := \sum_{j = -\infty}^{\infty} x_j...
  14. O

    Calculating Inverse z-Transform for X(z) = z/(z-0.2)^2(z+0.1)

    Homework Statement Find inverse z-transform of X(z) = \frac{z}{(z-0.2)^2(z+0.1)} Homework Equations The Attempt at a Solution : partial fraction My method :\frac{X(z)}{z} = \frac{1}{(z-0.2)^2(z+0.1)} \frac{X(z)}{z} = \frac{-100/9}{(z-0.2)} + \frac{10/3}{(z-0.2)^2} + \frac{100/9}{z+0.1} X(z) =...
  15. D

    Solve Inverse Z-Transform for H(z)=1/(1-2/3*z-1)

    Homework Statement H(z)=1/(1-2/3*z-1) h(n)=? Homework Equations I tried to use the residue method to solve this problem but it doesn't give me the good result and i am not sure if i don't know how to use the method or this problem can't be solved using this method. Also, its pretty...
  16. A

    Z-transform and even/odd signals.

    Homework Statement "Let x[n] be a real-valued DT signal. Using the Pole-Zero plot of X(z), argue as to why: a. |X(e^jω)| is even in ω. b. ∠X(e^jω) is odd in ω. Hint: if x[n] is real, what can we say about the poles and zeros of X(z)?Homework Equations None. The Attempt at a...
  17. P

    Inverse Z-transform question.

    Hello, Homework Statement I would like to find the inverse Z transform of the following: F(z)=1-1.25z-1+0.25z-2/[1-(5/6)z-1+(1/6)z-2] using (a) partial fractions, and (b) residue theorem I have obtained different results and hence would appreciate some insight on the discrepancy and how...
  18. E

    MATLAB Graphing 3D Z-Transform in MATLAB

    Does anyone know how I might go about graphing the Z-Transform in 3D? It want it to look something like the picture below.
  19. J

    ROC of z-transform (derivative)

    I have a question about the z-transform: In the (Oppenheim's Signals and Systems) book it is written: If the z-transform of x[n] is X(z) with ROC=R, then the z-transform of nx[n] is -zdX(z)/dz with the same ROC. I don't understand why the ROC remains unchanged. Some books say "it...
  20. C

    Z-transform of conjugated sequence ( a straightforward exercise)

    Z-transform of a conjugated sequence ("a straightforward" exercise) Homework Statement The conjugation property is expressed as x^*[n] \stackrel{Z}{\leftrightarrow} X^*(z^*) This property follows in a straightforward maner from the definition of the z-transform, the details of which are left...
  21. T

    Z-Transform, Cant find this transform

    Homework Statement I am attempting to perform a z-transformation on the terms 1/(z-3) and 1/(z-5) but I cannot find a sequence which fits this form in any of my tables. Is there a transform for these terms or do you just leave them as they are? Homework Equations The Attempt at a...
  22. Z

    Using an impulse response and an output to find an input using Z-transform

    Homework Statement An LTI digital system with impulse response h[n] = 2(1/4)^n u[n-1] produces an output y[n] = (-3)^n u[n-1]. Determine the corresponding input x[n]. Homework Equations I believe that Y[z] = X[z]H[z] The Attempt at a Solution So I started by finding the...
  23. T

    Convolution for Z-transform

    Greetings, We just had 1.5 classes about this topic (due to some unavoidable circumstances in our country...) so our teacher just scribbled and roughly explained them on the board...So after i got home i couldn't get some steps he did for convolution in z-transform: find inverse z-transform...
  24. Z

    Z-Transform & DTFT: Finite-Length Signal & Two Poles

    1. Homework Statement [/b] You are given the following pieces of information about a real, stable, discrete-time signal x and its DTFT X, which can be written in the form X(\omega)=A(\omega)e^{i\theta_x(\omega)} where A(\omega)=\pm|X(\omega|. a) x is a finite-length signal b) \hat{X} has...
  25. Z

    Z-Transform and DTFT Homework: Poles, Zeros, and Endpoints

    Homework Statement You are given the following pieces of information about a real, stable, discrete-time signal x and its DTFT X, which can be written in the form X(\omega)=A(\omega)e^{i\theta_x(\omega)} where A(\omega)=\pm|X(\omega|. a) x is a finite-length signal b) \hat{X} has exactly two...
  26. S

    Z-transform of a discrete convolution

    Hi, Suppose we have these two functions and their z-transforms are P(r,z)=\sum_{t=0}^{\infty}P(r,t)z^t and F(r,z)=\sum_{t=0}^{\infty}F(r,t)z^t. Now we are going to transform the following convolution of P and F: \sum_{t'\le{t}}F(r,t')P(0,t-t'). The result is said to be F(r,z)P(0,z). But I don't...
  27. E

    Partial Fraction Expansion (Inverse Z-Transform)

    Homework Statement H(z) = \frac{6-z^{-1}}{1+0.5z^{-1}} + \frac{2}{1-0.4z^{-1}} = k + \frac{A}{1+0.5z^{-1}} + \frac{2}{1-0.4z^{-1}} where A = (6-z^{-1}) is evaluated at z^{-1}=-2. Homework Equations Partial fraction expansion. The Attempt at a Solution Why is z^{-1} set equal to -2? I...
  28. B

    ROC in Sign function Z-transform

    Homework Statement Calculate the Z-transform of the function x[n] = u[n]-u[-n-1].Homework Equations X(z) = ZT\{x[n]\} = \sum_{n=-\infty}^{\infty}x[n]z^{-n} ZT\{u[n]\} = \displaystyle\frac{1}{1-z^{-1}}, ROC: |z| > 1. ZT\{-u[-n-1]\} = \displaystyle\frac{1}{1-z^{-1}}, ROC: |z| < 1. ZT\{x[n]\} =...
  29. C

    Confusion About Convolution z-Transform ROC

    Homework Statement I need to find a transfer function using the given functions which i have no problem with. But i don't know how to exactly define the Region Of Convergence of the resulting transfer function. I have h1[n] and h2[n] as to be convoluted to find the transfer function. Homework...
  30. M

    Can the Z-transform be used for both continuous and discrete functions?

    Hi, I'm currently studying some introductory discrete control theory and I've run into a problem with the Z-transform, although I could pose the same question regarding the Laplace transform. I know I'm completely off with this question but I've just stared myself blind to it. Here goes: I...
  31. J

    Finding the impulse response of a system without using Z-Transform

    Homework Statement Find the impulse response of the following: Assume the system is initially at rest. y[n] - (1/2) y[n-1] = x[n] + 2 x[n-1] + x[n-2] The Attempt at a Solution To find the impulse response y[n]: we know that y[n] = homogenous solution + particular solution so... lamda -...
  32. M

    Z-transform for a signal with an unknown sampling rate

    Homework Statement Given a ramp function x(t) = t*heaviside(t) with the known z-transform ( z / (z-1)^2 ). If the time-domain signal has been delayed by 2 seconds, then sampled with an unknown sampling rate (T). How could I get the z-transform (one-sided) for the discrete signal, for any values...
  33. J

    Finding Z-Transform of sin(Bn)

    Homework Statement I'm trying to take the (two-sided, aka defined for all n) Z-transform of x(n)=sin(Bn)Homework EquationsThe Attempt at a Solution What I tried to do was split x(n) into x(n)=sin(Bn)u(n)+sin(Bn)u(-n-1) The Z transform of the first part of x(n) was found from a table. The...
  34. S

    If the z-transform of x[n] is X(z)

    If the z-transform of x[n] is X(z), then what is the z-transform of x[n+1] in terms of X(z) ?
  35. L

    Compute the Z-transform of a^{-n} and step function

    Homework Statement Compute the Z-transform of x[n] = (0.5)^{-n} * u[n-1] And find the ROC (Region of Convergence) Homework Equations Z-Transform for discrete time x(z) = \sum_{n=0}^\infty x[n] * z^{-n} The Attempt at a Solution To solve this, I used the definition of...
  36. F

    I am having some trouble with this z-transform, and would like a

    I am having some trouble with this z-transform, and would like a little help. Question: Find the Z-transform, and sketch the pole/zero plot with the region of convergence. x[n] = \cos( \Omega_0 n) u[n] Where u[n] is the step function. "Answer" Directly from a table in my book, x[n]...
  37. B

    MATLAB Matlab z-transform Bode plot

    Hi, I try to visualize a sine wave, z-transformed in a Bode plot. The frequency shows up correctly, but the amplitude is way off - why? Thanks for your comments. Bmalp function zdemo format compact syms w Ts n z phi t Ts = 1 w = 0.1 phi = 0 vz =...
  38. electronic engineer

    Inverse Z-Transform of \frac {Z^2-Zr Cos W0} {Z^2-r^2 Sin^2 W0}

    I need to find the inverse z-transform for this function: \frac {Z^2-Zr Cos W0} {Z^2-r^2 Sin^2 W0}
  39. 7

    Complete search about Z-transform

    I was asked to prepare a complete search about Z-transform can anyone help me find files or resources about it.