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.

Filter by
Sorted by
Tagged with
2
votes
2answers
5k views

why is the z transform transfer function 1/(z-1) called an integrator?

I am reading up on delta sigma modulators and there this term $\frac{1}{z-1}$ that appears repeatedly and is referred to as an "integrator". Why is this so ?
8
votes
1answer
16k views

Position of poles and Stability in $z$ domain

We know in Laplace Transform, if the poles lie on the left of $j\omega$ axis, we can say the system is stable. Similarly can we comment on the stability based on poles position in $\mathcal Z$-...
0
votes
2answers
39 views

Average image over $N$ frames with one frame buffer

Can you give some filter that is averaging an image over $N$ samples in a recursive way that only needs one frame buffer? $\frac{1}{N} \sum_{k=1}^{N} y_{k}(m,n)$ I can only imagine filters that need ...
0
votes
1answer
37 views

What is the Z-transform of $0.8^{n+2}u(n-1)$?

I have 2 signals. One is $x(n)=(-0.5)^nu(n)$ and the other one is $y(n)=0.8^{n+2}u(n-1)$. I know that for the first one it is $X(z)= 1/(1+0.5z^{-1})$, but what about the other one? I know $y(n)$ is ...
0
votes
1answer
48 views

Calculating IIR Filter gain at given frequency

Let's consider an IIR filter with transfer function: $H(z)$. Given the sampling frequency $F_s$ how can I calculate gain at say $F$ ? When I was dealing with analog systems when I wanted to calculate ...
1
vote
1answer
2k views

Downsampling: Mathematical derivation

The problem I am having is related to sample rate conversion and more precise to sample rate reduction. I have been working on the paper Interpolation and Decimation of Digital Signals Tutorial Review ...
2
votes
3answers
134 views

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 ...
1
vote
1answer
53 views

Can time-invariance be determined from a given a transfer function?

I've the following function. $$ G(z) = 2 + \frac{-1+5z^{-1}}{(1-0.5z^{-1})(1-z^{-1})}$$ Calculating it's inverse using $\mathcal Z$-Transform, I get the following function: $$g[n] = 2\delta[n] + 8u[n] ...
4
votes
1answer
468 views

Attempting to use the trapezoidal rule to form a difference equation representing a circuit

I have a differential equation that has been proven to be correct. The transfer function obtained by Laplace domain analysis and Matlab freqs match up and all is well. The problem is somewhere in ...
3
votes
1answer
76 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 ...
0
votes
1answer
35 views

Determine causality and stability from given filter structure

I have the diagram above. I found the transfer function below from it; The question asks me to find out if the system is causal and stable, but didn't it have to indicate whether it was left-sided or ...
0
votes
1answer
47 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 ...
0
votes
1answer
36 views

How to determine whether a filter is high/low or band pass from the Z transform?

How to solve questions of these kind? I have tried by replacing $z=re^{jw}$ and taking the limits from $0$ to $\infty$. But I am not sure what $e^{j\infty}$ is.
0
votes
3answers
63 views

Is there a simpler way to calculate the amplitude response of the following filter

I would like to calculate the amplitude response $|H(z)|$, $z=e^{j\omega}$, of the following filter: $$H(z)=\frac{\frac{b}{2}+z^{-2}}{2+bz^{-2}}$$ and I would like to avoid using Euler's formula and ...
4
votes
3answers
4k views

Is $y[k] = y[k-1] + x[k]$ an integrator?

It looks exactly like an integrator to me. Since $$y[k] = y[k-1]+x[k] = y[k-2] + x[k-1] +x[k] = \sum{x}$$ Applying the Z-transform gives \begin{align} Y(z) &= Y(z)\cdot z^{-1} + X(z)\\ \...
0
votes
1answer
37 views

Convolution that outputs a unit impulse

Im thinking whether any convolutional operation can output a unit impulse, an example to further explain: where a convolution between system $h[n]$ and unknown system $g[n]$ would output $\delta[n]$. ...
1
vote
3answers
64 views

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} }{ \...
-1
votes
1answer
29 views

Z-transform of a cosine without a unit step [duplicate]

What is the $\mathcal Z$-transform of a cosine without a unit step, i.e. $x[n] = \cos(\omega_0 n)$ and not $x[n] = \cos(\omega_0n)u[n]$?
2
votes
2answers
61 views

Z transform of finite signals

I was trying to solve the Z-transform for u[n] - u[n-N], where u[n] means discrete unit step function, and N is some finite integer. I solved this using 2 methods. ...
1
vote
2answers
279 views

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 ...
0
votes
1answer
19 views

Frequency response of filter <-> signal mix

This might be a weird question but here's the setup: I have a few biquads that filter a signal $x[n]$ and output the filtered signal $y[n]$. I can calculate the frequency response of those biquads ...
8
votes
3answers
13k views

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}}...
1
vote
1answer
37 views

Determine the Z-Transform for the following sequence: $ |n|(\frac{1}{2})^{|n|} $

Determine the Z-Transform for the following sequence: $$ |n|(\frac{1}{2})^{|n|} $$ I have tried to solve the above problem. However, the answer that I got is the negative of what is given in the ...
0
votes
1answer
28 views

Finding inverse Z Transform

Find the inverse Z transform: I have done the solution but my answer does not match with the one given in the textbook. What I may have done wrong?
0
votes
1answer
131 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 ...
2
votes
2answers
84 views

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 ...
0
votes
1answer
29 views

Unilateral Z transform

I tried to calculate the unilateral Z transform of x[n-2], is it right?
1
vote
2answers
1k views

z-Transform Methods: Definition vs. Integration Rule

The definition of the z-transform is defined as $z = e^{sT}$ where "s" is complex frequency for continuous-time systems and "T" is the sample period. Why are rules such as the forward rectangular ...
1
vote
2answers
67 views

On the stability and causality of a discrete system

On MIT's open course a simple exercise with two questions is given. On the first part, they question about the properties of the following discrete system: $$ y[n]=x[n]+0.5y[n−1]−2y[n−2] $$ The ...
0
votes
2answers
501 views

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 ...
0
votes
2answers
196 views

How can I plot the frequency response on a bode diagram with Fast Fourier Transform?

Assume that we have an unknow dynamical system and we only want to estimate its parameters. The system can be discribed as: Continous time: $$G(s) = \frac{3s + 5} {5s^2 + 3s + 2}$$ Discrete time ...
0
votes
1answer
17 views

How to find H(z) and H(k) from a given causal function

Consider the causal function, $y[k] = 2x[k] - 40x[k - 1] + 10y[k - 1]$ 􀀀 $16y[k - 2]$; where $y[k]$ is the output and $x[k]$ is the input. Assume that the system is initially at rest. Please someone ...
0
votes
2answers
218 views

Why do we care about the multiplicity of poles and zeros in rational Z-transforms?

In the DSP class I'm taking, a lot of the questions ask me to list the multiplicity of a pole or zero in a rational Z-transform. For example, the multiplicity of the zero at $z=1$ in :$\frac{(z-1)^2}{(...
0
votes
1answer
46 views

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) = ...
0
votes
1answer
98 views

How can one know the polynome equation based on the output of a system?

I came across this situation in my textbook: However I have no clue about how you can (starting from the stepresponse on the left), get the polynome equation on the right. Could someone please ...
0
votes
3answers
53 views

Z-Transform vs. Fourier Transform convergence

Are there signals for which the Fourier transform is known to exist (perhaps including singularities) and for which the z-transform does not converge?
1
vote
3answers
102 views

Find the $\mathcal Z$-transform of this function?

I need to find the $\mathcal Z$-transform of $x(n)=a^{-n} u(n)$. Assume, $a$ is a positive constant , but the power of $a$ is negative. Looking at the transform table, I found that $\mathcal Z$-...
0
votes
2answers
42 views

Impulse response of an LTI system given the input and output signals

I have been given the input and output signals of an LTI system as: $x[n] = (\frac{1}{2})^nu[n] + 2^nu[-n-1]$ $y[n] = 6(\frac{1}{2})^nu[n] - 6(\frac{3}{4})^nu[n]$ I have found the system function $...
0
votes
1answer
27 views

Do all systems in z domain are filters?

Normally a system transfer function is represented by H(z) I want to know wether all those system transfer functions are only representing a digital filter? Or their any other thing/entity in z ...
0
votes
2answers
49 views

Using ROC to find stability of system in specific example

I've started learning about finding the ROC from the transfer function, but I'm confused about an example. $$H(Z) = \frac{2Z + 1}{Z^2 + Z - \frac{5}{16}}$$ I understand the poles lie at $z = \frac{-...
1
vote
2answers
93 views

Discrete Filter $y[n] = \frac{1}{3} x[n] + \frac{1}{3} x[n-1] + \frac{1}{3} x[n-2]$

Consider the filter which equation can be represented by $y[n] = \frac{1}{3}x[n] + \frac{1}{3}x[n-1] + \frac{1}{3}x[n-2]$, in $x[n]$ and $y[n]$ are sequence of input and output of the system ...
2
votes
2answers
87 views

In z-transform, if z means “delay”, why do we talk about the value of z?

I've been watching MIT's signals course and trying to understand $z$-transform. The course introduces $z^{-1}$ as an operator that delays the signal by $1$ time unit (which works very well with the ...
2
votes
1answer
481 views

When to use Fourier, Laplace and Z transforms?

If we have an LTI system, with an input signal $x(t)$, impulse response $h(t)$ and output $y(t)$, I was under the assumption that if the input and impulse response were continuous in time, then you ...
2
votes
1answer
69 views

Factored form vs partial fraction form?

I have already understood partial fraction and here is link for my relevant DSP SE question Finding inverse z transform for two sided ROC? But now i want to know, is there any difference between ...
1
vote
3answers
100 views

Is $\cos(\pi \sqrt n)u[n]$ stable?

I took a z transform and got a double pole at $z=1$, but I don't know if that's correct. I'm lost because I don't know if $\cos(\theta)$ converges or diverges or what that means for $h[n]$ being ...
0
votes
3answers
108 views

Necessary Conditions for stability in z domain?

What are the necessary(must) conditions for stability in z domain? I am sure about one(ROC must include unit circle) Is there any other such condition which states that there shouldn't be any poles in ...
0
votes
2answers
69 views

Finding inverse z transform for two sided ROC?

I have a z transform $Y(z)=(z^2-z) /(z^2+1.3z+0.3)$ In this case i have two poles ,one at -0.3 and one at -1.0 I want to find inverse z transform of Y(z) Following is my matlab code: ...
1
vote
1answer
41 views

Z transform getting different answer for transform of rational function

Why am i getting different answers and which one is correct?? In this also
1
vote
1answer
253 views

Determing inverse Z-transform using impulse response?

In Matlab there is a command iztrans for finding inverse Z-transform. But how can we find inverse Z-transform using impulse response? The Matlab command ...
9
votes
3answers
7k views

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 ...

1
2 3 4 5
8