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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149 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 ...
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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{-...
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117 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 ...
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159 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(\...
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61 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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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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1answer
117 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. ...
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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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139 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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106 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....
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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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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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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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262 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})...
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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(\...
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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 ...
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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.
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134 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 ...
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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 ...
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2k 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 ...
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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 ...
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105 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 ...
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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^{...
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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 ...
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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 ...
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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 ...
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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^{...
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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{...
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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 ...
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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 ...
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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 ...
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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{...
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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 "...
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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 ...
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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 + ...
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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 ...
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443 views

Pole/Zero existence at infinity

How can poles and zeros exist at infinity?Can anybody explain using a system function?
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Study Signal Processing

I'd Like to ask two questions : What is the difference between studying Signal processing (both Deterministic and statistical) in Department of Electrical Engineering versus Department of Mathematics ...
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2k views

Real and Imaginary Frequency Responses of a single complex pole

I'm filtering a real signal with a single complex pole with a complex coefficient (a_re [real part] and a_im [imaginary part]), I also have a gain coefficient but I'm gonna leave it out for the sake ...
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1answer
224 views

Z Transform - ROC values

I was reading a DSP book where they had a problem on Z Transform. Determine the z-transform and ROC for signal $\displaystyle x(n) = a^nu(n)$? The solution states that $$ \displaystyle X(z) = \...
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Are Z-transform time shifting and differentiation properties always compatible?

I'm currently studying the Z-transform, and I'm having issues in understanding the time shift and differentiation properties, to be precise: calculating a Z-transform explicitly, and obtaining it by ...
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Deriving finite impulse response for polylogarithm

As a part of my research i have to use the following z-transform in matlab 'filter' function so as to derive the convoluted signal from the original one. $$\frac{1}{{\rm Li}_{k}(z^{-1}e^{-b})}$$ ...
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Confusion Regarding Bi Linear Transform

I was reading my book where the z-transform of a signal is derived to be ${1-e^{-2bT}z^{-1}}$ . Then it goes on to say that by applying the bilinear transform we can get $$\frac{2(1+bT+(bT-1)z^{-1})}...
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969 views

Inverse Chirp Z Transform

I am working to understand and use the Chirp Z-Transform. I want to use the algorithm for simple signal processing on data sets that are not a power of two. I need to be able to inverse transform as ...
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Z-transform and binomial series

I am reading a paper on frequency warping and I need to do a little manipulation of the Z-transform. Can somebody help me on how can I go about deriving equations $(3)$ and $(4)$ from equations $(1)$ ...
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579 views

Does the ROC (Z-Transform) always start/end from/in a pole?

Can I assume that the R.O.C. from a Z-Transform will always start from or end in a pole or 0/infinity? Using an example: $$H(z)=\frac{(z+1)(z-1)(z+j)(z-j)}{z^4}\space,\space\space j=\sqrt{(-1)}$$ ...
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Z transform convergence

From the definition I know: $$|\sum_{n=-\infty}^{+\infty}x[n] \cdot z^{-n}| < \infty\space\space\space(1)$$ Is there any sequence of x[n] which can not be written as $$x[n] = y[n] \cdot u[an-b]\...
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Is it possible to calculate a z-transform for a filter calculated with Parks-McClellan?

I would like to know whether this can be done and if then how would this feat be acomplished in Matlab given a filter calculated with Remez? h = remez(...); If ...
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Connection with system analysis and laplace&Z transform

Laplace&Z transform is just frequency analysis of a system with a multiplication of decaying exponential. We analyse frequency with varying Laplace Exponential, which can be seen in the formula ...
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Generating a time series given a transfer function

I'm trying to work my way through a paper I found online, titled "Three Models of Wind-Gust Disturbances for the Analysis of Antenna Pointing Accuracy" by W. Gawronski, 2002. http://ipnpr.jpl.nasa....