The Z-transform converts a discrete time-domain signal, which is a sequence of real or complex numbers, into a complex frequency-domain representation.

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Differentials - Differences: Non causality in the system

I'm still learning DSP and referring to Oppenheim video lectures. In that lectures, differential difference equation is obtained for IIR filter design, in Lecture 14. $$\mathcal{L}[\frac{\mathrm ...
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Closed form of $\mathcal Z$-transform : decomposition signal $x(n)$

The text of my exercise ask : Determine the closed form of the $\mathcal{Z}$-transform for this $x(n)$ $$ x(n) = \begin{cases} |n-N| & \text{if 0<$n$<2N} \\ 0 & \text{elsewhere} ...
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28 views

IDFT of H(z) sampled in N values

If a have a causal IIR filter described by $H(z)$ and I sample it in $N$ equispaced values around the unit circle, I get a DFT of $N$ points. That DFT corresponds to $h[n]$ truncated in $n=N-1$ or to ...
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32 views

Analysis of a LTI system using DFT

Consider an LTI system $$H(z)=1-\frac{1}{2}z^{-1}+\frac{3}{4}z^{-2}$$ Let $x[n]=(\frac{1}{3})^n\cdot u[n]$ be the input signal. It is desired to determine the output for $n=0,1...,N_a$. To ...
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63 views

Downsampling and Then Upsampling

Given this system: I need to show the $\mathcal Z$-transform of $y[n]$ as a function of the $\mathcal Z$-transform of $x[n]$. Now I know that for downsampling alone: $$Y(z) = ...
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Sampling H(z) to get DFT

Suppose that I have a $H(z)$ and I sample it to get a DFT of 15 values. Let's call this DFT $H_{1}[k]$. Then, suppose I antitransform $H(z)$ and grab the first 10 values of the sequence, and then I ...
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44 views

Problem designing a specific FIR filter

Consider an LTI system whose impulse response is $$h[n]=\frac{1}{2^n}u[n]+\frac{1}{3^n}u[n]$$ The input signal to this system is $x[n]$ and is null for $n<0$ but may or may not be null for $n=0$. ...
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43 views

Identifying the magnitude and impulse response from pole zero plot quickly

I have an exam next week and it's verty certain that a task of this kind will be there. Are there some good tips how to match the right pole zero plot to the right responses? No proof is needed in ...
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1answer
31 views

Is this system LTI?

Assuming the system $h[n]$ is LTI (and has an associated $H(z)$ transform), is the whole system below LTI? I found the impulse response of the system and I got that it is $$h_{0}[n]=\alpha ...
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39 views

ROC of the product of two Z-Transforms

Suppose I have an LTI system $$H(z)=\frac{z}{(z-2)(z-\frac{1}{2})}$$ and I want to know its response to the step function $u[n]$. The LTI system $H(z)$ has three possible ROCs: $$|z|<\frac{1}{2}$$ ...
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Relationship between z-transform and DFT

I'm studying for a Signals Processing exam and came across an exercise that I'm finding pretty difficult to solve. It says: Asume there is a signal $x[n]$ of length $N$. Its ...
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62 views

Z-transformation

Hi everybody i'm a student. Yesterday i had a test about my Engineering subject about signal processing and there was this problem: You have the sequence $x(n) = N+1 - |n|$. With $|n|\leq N.$ ...
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75 views

DFT/FFT Transfer function

I want play and record a sine sweep. When i have both signals the recorded one and the send one i can create a Transferfunction. That is what i know so far. $$ H_0 = \frac{OUT}{IN} = \frac{Y}{X} $$ ...
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56 views

Z-transform of x[a -n]…where a is int

i try to calculate the $\mathcal z$-transform of $x[a-n]$ (where $n$ is my variable) i can't find any transform. the best suited transform is $x[-n] \longleftrightarrow X(z^{-1})$ i took the sum ...
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30 views

doubt of intersection of ROC Z Transform

dear friends of StackExchange. I have a doubt of the intersection of two ROC. I have H(Z), X(Z) and and i have to determine: $$ \begin{align} Y(Z)= H(Z)X(Z)\end{align}$$ $$ \displaystyle $$ $ ...
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67 views

Is the below filter linear phase

$$h(t) = \frac{1}{1 + t^2}$$ and is it IIR or FIR filter. I tried finding the Laplace transform of this filter to get the data flow diagram with 5 taps and T=2s, however, I am unable to solve this. ...
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48 views

Inverse Z-transform with complex conjugate poles

I was computing an inverse z-transform here, and I am facing some problems. So, the z-transform is: $$ X(z) = \frac{2+3z^{-1}}{1 - z^{-1} + 0.81z^{-2}} , |z| > 0.9 $$ I found the following ...
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58 views

Question about z transform

After studying z transform from different books and literature on internet I want to ask few which makes me confuse. a) From the Discrete Time Fourier Transform we have drive equation for z ...
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108 views

What does 'z' in Z-transform represent ? Is it frequency or something else?

my question is about the Z- transform. My first question is what the title says. What does 'Z' in Z-transform represent ? Say in Fourier transform, 'w' (omega) represents frequency ? From Fourier ...
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31 views

What is the right way to calculate the inverse Z-transform of $zX(z^{-1})$

say the signal $x(n)$ has the z transform $X(z)$ and there is signal $x_1(n)$ that $X_1(z)=zX(z^{-1})$ I tried 2 different approach to get the relationship between $x(n)$ and $x_1(n)$ and the ...
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67 views

Z-Transform of $x(n) = 3^n$

First of all, thank you all for your answers. I know the z transform for $$ x(n)=3^n \space ; \space n\geqslant 3 $$ or rather $$ x(n)= 3^n u(n-3) ...
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27 views

Phase response for conjugate zeros

If a second order system has 2 poles/zeros that are conjugate symmetric, how does this affect the phase response? I know that if there are 4 zeros/poles that are conjugate reciprocals, then it is a ...
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30 views

the ROC of a Z-transform for shifted signal

I have got two different answers for the ROC of the signal. In that PIC, I have solved it in 2 methods, but I'm getting different answer. Which one is correct? Also please explain how to find the ...
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70 views

The right way to approach z transform?

I am a student learning dsp. I like the subject. I could understand the discrete time signals. When I move into z transform. I could not understand it. Z transform is the mapping from discrete ...
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What does z-transform imply?

As z tranform is the transformation of discrete time signals into complex frequency domian. What do you get out of complex Stuff. As wikipedia calls it complex frequency domain. Why do you need it ? ...
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38 views

System Stability: Can we derive stability of a discrete system (Frequency domain, Z-transform) by applying analogous methods?

So given some analogue system function in the complex s-domain. Can we perform a stability analysis in the $s$-domain, before actually transfer it into the $z$-domain? So in other words analysis in ...
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22 views

Non-causal z transform in MATLAB?

In MATLAB, a causal z transfer function can be specified with filt(). Is it possible to specify a transfer function with acausal terms (positive exponent on z)?
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53 views

Z-transform of an FIR filter

QUESTION Compute the Z-transform of $y[n] = x[n] + 2x[n-1]$. and find the poles and zeros. I just bombed an interview where I couldn't do this (because I have no grounding in fundamentals and have ...
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70 views

Inverse z transform - Pair of complex conjugate poles

How can I perform the inverse z-transform on the following $H(z)$ to be able to calculate a real-valued impulse response $h[n]$? $$ H(z)=\frac{z^2}{z^2+0.8\sqrt{2}z+0.64} $$ My idea was to find an ...
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81 views

pole/zero locations for real and imaginary signal

In Z-Transform, For a real signal, $x(n)$ =$x^*(n)$ . Taking Z-transform on both sides, $X(z)$=$X^*(z^*)$ , which gives certain pole/zero condition similarly for a purely imaginary signal ...
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43 views

Find the fourier transform of g(k) from G(Z) for frequency=1/2

$$G(z)=\displaystyle \frac{\frac{1}{z}}{1+\frac{5}{6z}+\frac{1}{6}z^{-2}}$$ I found: $$g(k)=\displaystyle \left(\frac{-1}{3}\right)^k - \left(\frac{-1}{2}\right)^k$$ I don't understand how can i ...
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64 views

Finding fourier transform of a discrete signal from the z transform

Is it possible to find the fourier transform of a discrete signal if you know the $\mathcal{Z}$-transform of it?
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57 views

What is the ROC for this discrete signal:

$$ x(k)=4[u(k-2)-u(k)*δ(k-3)]$$ I found that the $\mathcal{Z}$ transform of the signal is $X(z)=4/(z^2)$. What would the ROC be?
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How can the order of a transfer function be derived from its equivalent state space representation?

Suppose I have a discrete state space model: $\begin{eqnarray*} \theta[k+1] &=& A \theta[k] + B u[k]\\ y[k] &=& C \theta[k] \end{eqnarray*}$ I know that the equivalent transfer ...
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42 views

What is the inverse Z transform of this:

$X(z) = \displaystyle \frac{1}{z}{\left(1-\frac{z^2}{4}\right)\left(1+\frac{1}{z}\right)\left(1-z\right)}$ Using partial fractions expansion i came up to this: $\displaystyle \frac{1}{X(z)} = ...
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41 views

Bilateral z-tranform of exponential

We all know that $a^nu(n)$ has unilateral $Z$-transform. But what is the $Z$-transform of $a^n$? (bilateral) When i tried to solve, i got answer as 'zero'. But bilateral Laplace transform of $e^t$ ...
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52 views

FIR filter design for response inside unit circle

I would like to design an FIR filter such that its Z-transform has a certain profile in certain regions. For example, if I'd like to have an FIR filter that nulls the decaying exponential ...
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34 views

Discrepancy when calculating LTI system output using inverse z-Transform

I'm given a difference equation, $y[n]-0.4y[n-1]=x[n]$, and asked to find the natural response $y_n[n]$, forced response $y_f[n]$ and complete response $y[n]$ if $x[n]=4 (0.25)^nu[n]$ and $y[0]=0$. ...
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25 views

transform function with non-linearity

I'm a newbie to Signal Processing - my apologies if this question is too obvious (I'm a financial trader trying to use DSP techniques). For a linear filter: $y[n] = (1-p) x[n]+p y[n-1]$ we can the ...
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Evaluating the inverse Z transform on the unit circle

I am trying to understand the math. The inverse z-transform is given by: $x[n] = \displaystyle\frac{1}{j2\pi} \int_cX(z)z^{n-1}dz$ where $\int_c$ is a contour integral. The inverse Fourier ...
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117 views

Determining which Filter from a Z-Plane Plots?

How do i determine which FIR filter (LP, HP, BS, BP) it is from looking at it's z-plane plot?
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47 views

For linear IIR digital filter, what happens for negative frequencies?

By negative frequency, I refer to Fourier transform. Often, the frequency response of a digital filter is only displayed for positive frequencies. For a linear IIR digital filter, what happens for ...
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Z-plane/S-plane to time domain?

I'm working through the IIR chapter in Dick Lyons' Understanding DSP, and there's something I'm having a hard time wrapping my head around. He'll draw poles and zeros and then show the associated time ...
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53 views

Why Z-transform is considered as separate transform?

The mathematical formula of the Laplace and Z transforms are same with just one difference. I.e. in the first we use $t$ for continuous-time signal and in the latter uses $n$ for discrete-time ...
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What is the mathematical interpretation of using direct inversion around a single pole/zero

Let $F[z]=N^2\frac{z(z-(3+j\sqrt{7})/2)(z-(3-j\sqrt{7})/2)}{(z-1)^3}$ This has 3 poles at $z=1$; one zero at $z=0$; and a conjugate pair of zeros at $z=\frac{3\pm{}j\sqrt{7}}{2}$ Assuming a contour ...
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Finding minimum phase version of 2D impulse response?

For a impulse response $\mathbf{h}=[h_0 \space h_1 \space … \space h_n]$, one can find a always find a minimum phase version of $\mathbf{h}$ by using an appropriate All-Pass system. For my problem, ...
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100 views

Can the Z-Transform be used to create smoothed 3D surfaces from point clouds?

According to Dr. Math the Z-transform can create closed-form solutions for 1D series defined by difference equations (e.g. the Fibonacci series). My 3D surface $z=LC(x,y)$ is defined by difference ...
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23 views

When sinusoidal input starts at n=0, why are transient response associated with z-transform poles of digital filter?

In http://www.eng.ucy.ac.cy/cpitris/courses/ECE623/presentations/DSP-LECT-10-11-12.pdf, it says that when sinusoidal input $X(z)$ starts at n=0 (with n<0 having zero input) and the input passes ...
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58 views

Transfer function, amplitude response and difference equation for a filter

I've found a paper with a filter described in terms of transfer function, amplitude response and difference equation: transfer function of the second-order low-pass filter: $$ H(z) = ...
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37 views

Inverse Z-transform mystic simplification

I have the following expression: $$X(z) = \frac{16}{15}\frac{1}{1-\frac14z^{-1}} - \frac{16}{15}\frac{1}{1-4z^{-1}}$$ According to my understanding this should become: $$x(n) = ...