Questions tagged [z-transform]

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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Digtial FIR Impulse response & transfer function

I am currently working the figure through below. as it is an FIR Filter i have worked out using convolution that the output is 4,2,4,6,0,0. i am trying to obtain the 'z' domain transfer function of ...
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How do I represent any signal in the form x[n]u[n] and x[n]u[-n-1]?

For eg-: If I have the signal x[n]={1,a,a²...}I can represent it as a^n * u[n]. Similarly if I havethe sequence as x[-1]=-a^-1 x[-2]=-a^-2 Then I can represent it as x[n]=-a^n*u[-n-1].
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DSP: newbie not understanding z transform/complex sinusoidal frequency and phase

Im reading Will Pirkles Designing audio effect plugins book and I'm not sure if I'm understanding the z transform correctly. I got up to differential equations in college, but haven't done math in ...
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How to find the difference equation directly from Direct Form II signal flow graph

I am trying to solve for the difference equation of the following signal flow graph: I am aware that Direct Form II can be converted to Direct Form I, which finding the difference equation directly ...
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44 views

Polar Fourier Transform and its similarity to the Z-Transform

The DFT is given by $X_{\mathcal{F}}(k) = \sum_{n=0}^{N-1} x_n e^{-j 2 \pi k n / N}$ Knowing that complex numbers can also be represented using a polar form $A \angle\theta$, I was looking for a ...
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Inverse z-transform. Where is mistake?

I've already wrote about that trouble (link here), but I don't understand where I've made a mistake. Full description of the task is as follows: Z-transform of sequence {x(k)} describe by the ...
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119 views

Potential issues arising from too stable discretization

When numerically simulating a system, usually some kind of discretization is necessary, obtained by some kind of z-transform, such as, for instance, the bilinear transform $s\mapsto \frac{2}{\triangle ...
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105 views

Having trouble interpreting z-transform description of a predictor from a codec paper

I've been looking at the opus paper (https://arxiv.org/pdf/1602.04845.pdf); in particular, in section 4.1, they describe the predictor for the current band energy based on energy from both the current ...
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how to calculate $H_{lp} = z(1- H_{hp}) $ , given coefficients for $H_{hp}$?

Given a high pass transfer fn of the form $H_{hp}=a_{1}*z^0 +a_{2}*z^{-1} + ... a_{n}*z^{-n}$ Is it possible to calculate a causal low pass filter using $H_{lp} = z*(1-H_{hp})$ ? attempting $H_{...
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Why does Simulink generate this code for a PID controller?

For the Simulink PID Controller model The Simulink generated code (rewrite for better understanding) is: ...
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30 views

Mixed - Discrete and Continuous system Laplace domain stability - Effect of Sampler and DAC

I have a system whose the plant transfer function is continuous and the compensation is discrete. I have an ADC which allows to measure the output of the system and a DAC which allows to control the ...
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80 views

Z-transform of not quite an upsampler

I know the z-transform of an upsampler is: $$ y[n] = \begin{cases} x(n/L) &n=0,\pm L, \pm 2L, ...\\ 0&otherwise \end{cases} \longrightarrow Y(z)= X(z^{L}) $$ if $x[n]_L$ is defined to zero ...
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Identify out signal from in signal and impulse response

Hi there I am studying a course in signal processing and systems. I was given an excecise which I am having a great deal of trouble solving. I am allowed to solved using matlab, but for top marks I ...
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factoring poles / zeros: off by constant gain compared with textbook

(From Schaum's DSP outline, 2nd edition, problem 5.32) Book says factor it and extract H(z) from the factored product: $$ H(z)H(z^{-1})= \frac{ \frac{5}{4} - \frac{1}{2}z - \frac{1}{2}z^{-1} }{ \...
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Zeros and poles from transfer function

I have a transfer function $$ H(z) = \frac{Y(z)}{X(z)} = 1 - 0.5z^{-1} \text{.}$$ I'm interested in zeros and poles. I know I need to adjust the function to $$ H(z) = \frac{\prod_i(z-n_i)}{\prod_i(z-...
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Difference equation to FIR filter coefficients

I have a difference equation $$y[n] = y[n-1] + 0.5x[n] - 0.5x[n-2] \text{.}$$ According to this answer, the filter should have finite impulse response. I just can't figure out how to get the FIR ...
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Calculating the output of a pole eigen signal in a difference equation

Let an IAR system be defined by the following difference equation: $$y[n]-\frac{1}{4} y[n-2]=x[n]+3x[n-1]$$ and an input signal $x[n]=(-0.5)^n$. The transfer function is defined as $H^z(z)=\frac{1+3z^{...
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$Z$-transform of a multilinear function/ consecutive multplication of $k$ signals $y_1(n), \ldots, y_k(n)$

How should one go about calculating the $Z$-transform of a signal that is the multiplication of $k$ signals (i.e. a multilinear function with regards to signals $y_1(n) \ldots y_k(n)$ ? Namely, $\...
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Summations in Z-Transform

I'm currently working on a problem that involves a Z-Transform. Basically, the essence of the problem is that if: \begin{equation} H\left(z\right) = \sum_{n=0}^{N-1}h\left(n\right)z^{-n} \end{equation}...
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inverse z transform performed on 6th order IIR filter

we are told to find coefficients and impulse response of IIR filter of order of 6. There are 6 zeros and 6 poles in the design. Pole and zero pairs are conjugate and poles are within the unit circle ...
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If we take Z-transform of a signal & find its ROC. How to use this ROC? There are ∞ signals in ROC, suppose we choose any signal what to do with it?

If we take Z-transform of a signal & find its ROC. How to use this ROC? There are infinite signals in ROC, suppose we choose any signal from ROC, what to do with it?
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Suppose a signal is growing exponentially. We take its Z transform & find its ROC. So what are we supposed to do practically with the signals in ROC?

Suppose a signal is growing exponentially. We take its Z transform & find its ROC. So what are we supposed to do practically? Are we supposed to manipulate our O/P by multiplying any signal from ...
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Impulse response of a second order LTI

I have a set of measurement which I want to model with 2nd-order difference equations (first order eqs don't model well enough). The equation is $$y[n] = \alpha_1 y[n-1] + \alpha_2 y[n-2] + \beta_0 ...
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First order low pass filter

I am trying to better understand the first-order low pass filter: Summary: Per wikipedia, a first order low pass filter yields the following in discrete time: $$ \frac{Y(s)}{U(s)}= \frac{\omega_{c}}...
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I tried two approaches and gained the different conclusions of judging the stability of the transfer function of the system

We want to judge whether the system is stable or not. Given the below transfer function. $$ H\left( z \right) =\frac{\left( 1+2 z^{-1} \right) }{\left( 2+z^{-1} \right) } $$ $$ H\left( z \right) ...
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Filter odd or even harmonics with notch or inverse notch filter

Hi i had the following question. I have a signal containing a 200Hz sine wave and it's odd and even harmonics (no other frequencys or disturbing signals are contained). What i'm looking for is a kind ...
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Let a LTI system be causal and stable with the transfer function being… show that

if the system is an IIR LTI causal and stable one, and the transfer function is \[H(z)=\sum_{n=0}^{\infty}h[n]z{^{n}}= \frac{G}{1 -\sum_{k=1}^{p}a_kz{^{-k}}}\] show that the cepstrum of this system ...
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How do I find the ROC of a system if it has no poles

The output of a system of discrete time $y[n]$ is corellated with the input $x[n]$ through the equation $y[n]$. $$y[n] = \frac 13\big(x[n-1]+x[n]+x[n+1]\big)$$ It then asks me to find the system ...
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185 views

Where to start with DSP?

I have a DSP exam coming up this summer and I got an abysmal mark on an assignment earlier this year. Obviously, the lecture slides are not enough for me to understand DSP, so I am wondering where can ...
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211 views

transfer function and 'causal' signal - evaluate transfer function or use z-transform of input?

From my studying difference equations and transfer functions, I understand that when a complex exponential input $x[n]=z_1^n$ is applied to an LTI system with transfer function $H(z)$, determining the ...
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How do I convert this simple Laplace equation to Z-domain?

A basic model of coupled strings (eg. piano) is provided here as: The principle is that it has two identical string simulations each formed by a delay line and LPF. The outputs of these are summed at ...
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Power Spectral Density of a Filter

I need to calculate the output power spectral density of the following digital filter My calculations are as follows: $y\left(n\right)\:=\:x\left(n-1\right)+d\left[x\left(n-1\right)+x\left(n\right)\...
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Design discrete controller for zero steady state error

I have the following system where $$G(s)=\frac{0.5}{s+1}+\frac{5}{s+10}$$ How can I design the C(z) controller so that the steady state error for a step input r(t)=1(t) is zero? I know that this ...
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Determine whether the system is a FIR or IIR by looking the transfer function

I have the following system: $$ y[n]=\frac{1}{3}(x[n+1]+x[n]+x[n-1]) $$ After the Z-Transform we get $$ \frac{y[z]}{x[z]}=\frac{z^2+z+1}{3z} $$ which is of course the transfer function of the system. ...
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K-Path sampling overall transfer function

I'm trying to derive the overall Z-transform for the K-path sampling system shown below. I numbered some of the equations that I'll reference. I don't think I understand going from equation 3 to ...
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Transfer Function Block Diagram Confirmation

Can someone confirm if this is the correct block diagram for the following transfer function? The original equation provided was: y[n+1] = y[n] + 0.01x[n] Which I rearranged into H(z) = Y(z)/X(z) = ...
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LCCDE in simple words?

What is LCCDE?I only know its abbreviation/full form :linear constant-coefficient difference equation I know that in s domain we have differential equations and in z domain we have difference ...
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Frequency warping when integrators are replaced with backward-euler and forward-euler integration

Resonant controllers are used in the power industry. The transfer function is $G_{res}(s) = \frac{K_i s}{s^2 +2\omega_c s+ \omega_o^2}$ The "textbook" discrete implementation is depicted in ...
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Z-transform of an affine function

What is the transfer function of the system described by the following affine input ($x$)-output ($y$) relationship: $$ y[n] = \alpha x[n] + \beta. $$ Using the Z-transform we find: $$ Y[z] = \alpha X[...
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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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How to find H(z) from just zeros and poles

I have a system with a DC gain of 8, poles at z = +- j/2 and zeroes at e^+-j5, I need to find the H(z). I have tried this but not sure if it is right. $$ H(z) = G_o * z^{-1} \frac{(z-z_0)(z-z_1)}{(z-...
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Intuitive interpretation of Laplace transform

So I am getting to grasps with Fourier transforms. Intuitively now I definately understand what it does and will soon follow some classes on the mathematics (so the actual subject). But then I go on ...
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Inverting a sampled system

I'm doing some self-study for an upcoming exam and came across the following question: My first idea to solve it was using the bilinear transform to get some approximation of $H(Z)$ (or just using ...
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What is the difference between natural response and zero input response?

I am new to DSP and was going through different responses of a system subjected to an input. My understanding of zero input response is: it is the response/output of the system when the input signal ...
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370 views

Octave - freqz strange results

I wrote octave code to find transmittance based on zeros and poles: ...
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Names of system functions in frequency domain

I was just trying to refresh my systems theory known from long ago, and I realized that I had forgetten the name of the basic functions. Specifically, what are the names of these functions ...
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Deconvolution and Polynomial division

I found this comment inside MATLAB's deconv.m function Deconvolution and polynomial division are the same operations as a digital filter's impulse response $B(z)/A(z)$. What does this statement ...
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What is the transfer function of this block diagram

I and my friend have different answer for this block diagram Mine: $$y[n] = -\frac23x[n] + x[n-1] -\frac12[n-2]$$ Hence $$H(z) = \frac{1}{-\frac23 + z^{-1} -\frac12z^{-2}}$$ My friend: $$q[n]=x[n]-...
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329 views

Justification of bilinear transform

I would like to understand the "justification" for the bilinear transform. The basic idea as I understand it is that by integration rule of Laplace transform we have for continuous $y(t)$: $$\mathcal{...
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40 views

Filter impulse response order calculation

I want to generate the following periodic sequence: h[n]={1,2,3,4,0,0,0,0,1,2,3,4,0,0,0,0,1,2...}. I am planning to generate this as an impulse response of a filter with an order no more than 8. I ...

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