Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

1
vote
2answers
738 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 ...
1
vote
1answer
37 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 ...
0
votes
1answer
194 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 $$ x(n)= 3^n u(n-3) $$$$\begin{align}X(z)&=\sum_{n=-\infty}^{\...
0
votes
1answer
74 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 ...
0
votes
2answers
96 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 ROC ...
0
votes
3answers
256 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 ...
-2
votes
1answer
120 views

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 ? ...
0
votes
1answer
75 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 ...
0
votes
1answer
2k 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 ...
1
vote
0answers
364 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 ...
0
votes
2answers
1k 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 ...
0
votes
1answer
70 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 I can ...
2
votes
2answers
103 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?
0
votes
2answers
1k 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?
4
votes
1answer
148 views

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{align} \theta[k+1] &= A \theta[k] + B u[k]\\ y[k] &= C \theta[k] \end{align} I know that the equivalent transfer function can be found by ...
1
vote
1answer
58 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)} = \frac{-...
2
votes
1answer
105 views

Bilateral $\mathcal Z$-transform of exponential

We all know that $a^nu(n)$ has unilateral $\mathcal Z$-transform. But what is the $\mathcal Z$-transform of $a^n$? (bilateral) When i tried to solve, i got answer as 'zero'. But bilateral Laplace ...
0
votes
1answer
128 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 $\left(\...
0
votes
2answers
55 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 $y[0]=0$. ...
0
votes
2answers
30 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 ...
1
vote
1answer
94 views

Evaluating the inverse $\mathcal Z$-transform on the unit circle

I am trying to understand the math. The inverse $\mathcal Z$-transform is given by: $$x[n] = \displaystyle\frac{1}{j2\pi} \int_cX(z)z^{n-1}dz$$ where $\displaystyle \int_c$ is a contour integral. ...
3
votes
2answers
1k 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?
0
votes
1answer
119 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 ...
0
votes
1answer
77 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 signal....
0
votes
1answer
39 views

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 $...
3
votes
1answer
199 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 ...
0
votes
1answer
30 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 ...
1
vote
1answer
179 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) = \frac{(1-z^{-6})...
0
votes
1answer
89 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) = \frac{16}{15}\left(\...
0
votes
1answer
107 views

what is the name of this curve

I was drawing a high pass filter response ,in a polar coordinates, the function is (z+1)/z and then it is 2+2*cos(w) the plot is ...
-1
votes
2answers
63 views

Can poles of z-transform transfer function be zero to eliminate transient response?

I believe that answer to "Can poles of z-transform transfer function be zero to eliminate transient response?" is no, but I am not sure why it's no.
0
votes
1answer
124 views

transient response of simple digitalized RC low-pass filter

In http://en.wikipedia.org/wiki/Bilinear_transform#Example, digital version of simple RC low-pass filter is presented: $$\frac{1 + z^{-1}}{(1 + 2RC / T) + (1 - 2RC / T) z^{-1}}$$ where $T$ is ...
0
votes
1answer
143 views

Checking convolution property II of z transform

I have two sequences x and y of lengths, say 5 and 10. I multiply them in time domain element by element. I get another sequence. Now this as per the convolution theorem should be equal to the ...
0
votes
1answer
861 views

Inverse z transform - contour integration

Here is my task: Find inverse z transform of $X(z)=\frac{1}{2-3z}$, if $|z|>\frac{2}{3}$ I need to find it using definition formula, $x(n)=\frac{1}{2\pi j}\oint_{C}^{ } X(z)z^{n-1}dz$. How can I ...
2
votes
1answer
7k views

What is the significance of Z-transform?

As we have in Laplace transform that the roots decide the stability of the system i.e. if the roots are complex and lie in the left side of the plane you get a sinusoidal response with decreasing ...
0
votes
1answer
101 views

Visualising a Z-transformed Transfer Function?

For designing any analog filter and various other outputs of filter we use laplace transform,I can visualise a laplace transform like for ex. s[X(s)] can be ...
1
vote
1answer
74 views

Z Transform problem

I have a class exercise of an inverse Z transform and I have some trouble. I will render an example to make my point. Let's asume the Z transform pairs: $$a^n \cdot u[n] \Leftrightarrow \frac{1}{1-az^{...
2
votes
3answers
272 views

Can I study continuous time Fourier Transform and treat the rest as special cases

Say I learned the theoretical result of continuous time Fourier transform. And I want to extends some results(say "convolution rule") to Lapace transform, Z transform, DTFT, DFT, Fourier sequence ...
0
votes
1answer
58 views

just getting started in Signal Processing - easy question

I am reading Cycle Analytics for Traders by John Ehlers and need help. in the first section "Transfer Response" he refers to: so if I had $4$ prices say $8+8+8+8$ then average would be $8$ and ...
1
vote
3answers
35k views

How to compute magnitude and phase response from transfer function in Z-domain?

I have a transfer function $$H(z)=\frac{1+1.2z^{-1}+0.8z^{z^-2}}{1-0.9z^{-1}}$$ from which I'm supposed to sketch the magnitude and phase response. I know that you can transform $z=e^{j\omega}$ to get ...
0
votes
1answer
295 views

Can someone explain complex mapping as appears in Ogata's textbook

Refer to diagram above, in Ogata's text on discrete time control, he showed that you can map a curve in the S plane, namely curve 1,2,3,4,5 onto a circle in the Z plane through the complex mapping $e^{...
0
votes
1answer
2k views

How to prove this theorem about the Z transform and final value theorem?

Claim: If $\lim_{k\rightarrow\infty} x[k]$ exists and is finite then $X(z)$, the Z-transform of $x[k]$, has no poles in the region $|z|>1$ and at most 1 pole at $z = 1$. Attempt: \begin{...
3
votes
2answers
512 views

How to intuitively understand the state space formulation of discrete time system?

The SS formulation of DT system is given by $$x[(k+1)T] = Ax(kT) + Bu(kT)$$ $$y(kT) = Cx(kT) + Du(kT)$$ Note: T is the sampling period and often omited Can someone explain to me why the state ...
1
vote
1answer
69 views

Claim: Given sampling time T, the hold operator is approximated at low frequency by a time delay of T/2

Can someone verify this statement? The hold operator is assumed to be a zero order hold Then the laplace transform of this hold operator has a well known form $(1-e^{-sT})/s$ Let w approach 0, we ...
4
votes
1answer
2k views

Do Causal Discrete-time systems have proper transfer functions?

In the case of continuous-time systems, if the system is causal, its Laplace transfer function is strictly proper (the degree of the numerator is less than the degree of the denominator). Is this ...
0
votes
2answers
852 views

Eigenfunction of LTI causal system Z-transform

I am studying for my DSP final and I came across this question from the Oppenhiem and Schafer book 3rd edition. The question says 3.18 A casual LTI system has the system function $$H(z)=\dfrac{...
0
votes
2answers
146 views

2-sided Regions of convergence for Z transforms

Given a z transform with one pole can you have a 2 sided Region of convergence or does 1 pole limit it to being only left or right sided? I know when you have two poles the 2 sided scenario is when a "...
0
votes
1answer
1k views

How to show that y[n] = x[n] * h[n] turns into the Y(z) = X(z).H(z)?

I'm trying to show that $y[n]=x[n]*h[n]$ turns into $Y(z) = X(z)H(z)$ in Z-domain by first applying convolution then by taking the inverse Z-transform of the $Y(z)$, stating that it's the same ...
3
votes
1answer
222 views

DSP interview question: use of the identity in development of a significant transform

I'm preparing interview and found this question. But I don't really understand what is the question. Does it ask about Fourier transform or Z transform? How the simple identity $$xy=\frac{1}{2}x^2 + ...
0
votes
1answer
465 views

Difference equation and Impulse response of an IIR filter - deconvolution

Given the input to the IIR filter $$ x[n]=\delta [n]+0.25\delta[n-2205], \quad \text{where}\quad \delta[n]= \begin{cases}1, &n=0\\0, & n\neq 0\end{cases} $$ What I want to do is to design an ...