Questions tagged [dtft]
The dtft tag has no usage guidance.
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Discrete time Fourier transform of an exponential decaying sigal [duplicate]
I have a fundamental question about the discrete-time Fourier transform. I used two methods but got two results.
Background knowledge
The discrete time signal is given by:
$$x[ n ] = x[ {n{T_s}} ] = x(...
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How to approach signal reconstruction with sampling frequency not equal to reconstruction
I'm trying to figure out what kind of equations i can use in a situation like this:
I know the relation between $x(t)$ and $x[n]$ is :
but then $x[n]$ is multiplied by a pulse train, with a ...
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1
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206
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Gaussian filter: Plotting DTFT and DFT (by hand) from the continuous-time impulsive response
I am trying to make an algorithm that plots out the Discrete-Time Fourier Transform (DTFT) and the Discrete Fourier Transform (DFT) of the Gaussian filter. The impulsive response and its transfer ...
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Intuitive or physical explanation of DTFT$\{1\}=2\pi\delta(\omega)$
I am trying to understand the fact that
"The DTFT of 1 (an infinite discrete sequence of unit impulses from from $-\infty$ to $+\infty$) is $2\pi\delta(\omega)$"
in an intuitive or physical ...
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Computing an infinite sum of time-shifted sequence
Given a discrete-time domain signal $x[n]$ defined as
$$x[n] = \begin{cases}1 & 0 \leq n \leq L-1 \\
0 & \textrm{otherwise}\end{cases} $$
we are tasked with computing $$\sum_{k = -\infty}^{\...
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56
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Is Hann-windowing applicable when calculating a DTFT?
I have read that people often use a zero-padded DFT with Hann-windowing to get the amplitude+phase information at one frequency (where the Hann window is used to reduce the effect of a small/finite ...
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1
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45
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DTFT Pair Transformation of unit step [duplicate]
I am not seeing a direct pair of DTFT transform of the unit step.
5
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189
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Convolution theorem for inverse DTFT
in trying to understand the convolution theorem for DTFT, I'm faced with the following problem which I can't get my head around.
First, let me state the convolution theorem for the DTFT as follows:
\...
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3
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Difference in having even number and odd number of samples in DFT?
In the DFT we sample one period of the spectrum in the frequency domain. What is the difference between having an odd or an even number of samples?
We know that DFT is just a sampled version of the ...
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2
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163
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Easy (?) DTFT calculation
I'm asked to compute the DTFT of the following signal but i'm quite stuck
$$
\begin{cases}
(-1)^{\frac{n}{2} + 1} & \text{ if } n \text{ is even} \\
0 & \text{ if } n \text{ is odd}
\end{...
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Given the Fourier Transform of a continuous signal how can I sketch the sampled signals discrete time fourier transform
I am given the frequency response for a continuous time signal $X(j\omega)$ = 2 at $\omega=0$ and 0 at $\omega = -10000\pi$ and $10000 \pi$. Looks like a triangle. I am told to sketch $X(e^{jw})$ ...
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Multiplication term $ \frac{ 1}{T_s} $ in sampling theorem
\begin{equation}
X(\Omega) = \frac{ 1}{T_s} \sum ^{\infty}_{k=-\infty} X_a\left \lbrace \frac{\Omega /( 2 \pi) - k}{T_s}\right \rbrace
\end{equation}
What is the purpose of multiplying sampled ...
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1
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381
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Intuition of odd and even complex conjugate symmetry definition of DFT/DTFT so that $X(e^{j w})=X_{e}\left(e^{j w }\right)+X_{o}\left(e^{j w}\right)$
I have been reading through my courses DSP slides and came across something which was not really taught in detail. You can look up here for reference, it is stated almost identical.
Given the ...
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Ft and DTFT of negative frequency
I have a question that might sound silly but if I have a real and even signal x(t) can I define the FT and DTFT of the negative frequency if I can show:
$$X(-\omega) = \int_{-\infty}^{\infty} x(-t)e^{...
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436
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Zero padding affects the DTFT?
I wanted to understand better how zero padding affects a signal:
Which is just N ones. where $ N > 0$ is an Integer
$$ X[n] = 1, 1, 1, ... 1 $$
Zero padding it gives:
$$ X[n] = 1, 1, 1, ... 1, 0, 0,...
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Understanding graphs of DTFT with time shift of$~y\left[n\right]=x\left[n-2\right]~$
$$x\left[n\right]:=\text{discrete time signal}\tag{1}$$
The following plot is DTFT of$~x\left[n\right]~$
What I know so far are as below.
$$x\left[n\right]=\frac{1}{2\pi}\int_{0}^{2\pi}X\left(\exp\...
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1
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Is there a simple way to express this DTFT in polar form?
Consider the discrete-time system
$$
H(z) = a_0 + a_1 z^{-1} + a_2 z^{-2}
$$
To compute the DTFT, let $z = e^{j\omega}$ such that
$$
H(e^{j\omega}) = e^{-j\omega} \left(a_0 e^{j\omega} + a_1 + a_2e^{-...
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I'm having problems simplifying this discrete-time fourier tranform
I have this problem, and I can't get to the solution.
$$X(e^{j\omega}) = \sum_{n=-\infty}^{\infty} {(0.6)^{|n|}[u(n + 10) − u(n − 11)]}e^{-j\omega n}$$
The solution is
$$X(e^{j\omega}) = \frac{0.64 − ...
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DTFT and Eigenvalues in frequency domain
Consider an LTI system with impulse response $h[k]$. Does the frequency response $H(e^{j\Omega})$ equal the eigenvalue corresponding to an eigensignal of frequency $\Omega$?
So if I convolve an ...
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Moving average frequency response over an image
I'm studying image denoising by linear filtering with cross-correlation, in particular with a moving-average kernel (K x K kernel of all equal elements which sum is 1).
For clarity, I'd like to refer ...
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Where did we get the DC term of the Accumulator from DTFT?
Define $y[n]:=\displaystyle\sum_{m=-\infty}^{n}x[m]$. The DTFT is found as follows:
\begin{align*}
y[n]&=\sum_{m=-\infty}^{n}x[m]\\ \\
&=\sum_{m=-\infty}^{n-1}x[m]+x[n]\\ \\
&=y[n-1]+x[n]\\...
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395
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Zero padding DFT intuition
I'm trying to grasp some intuition about why zero-padding the time domain sequence $x[n]$ interpolates the frequency domain bins of the $DFT\{x[n]\} = X[k]$ and how does this relate to the $DTFT$ of $...
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143
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Proof of DTFT equal to DFT when signal is periodic?
I was using the Wikipedia page on the discrete time Fourier transform to understand the connection between DFT and DTFT. The following is claimed in the article - I was wondering if anyone had a proof ...
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2
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DTFT of sine wave using freqz
As mentioned in the title, is it possible to use freqz to find the DTFT of a sine wave? I am confused about what the 'a' and 'b' vectors would look like, since there are only impulses in the numerator....
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38
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DTFT Pairs confusion
When I am in the DT Fourier Domain, and I want to come back to the time domain, which pair do I use? Asking because both pairs have the exact same "form" in the Fourier domain, and that is ...
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98
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Recovering DTFT from Z-transform
The relationship between the Z-transform and DTFT can be expressed like:
$$ H(e^{j \omega}) = H(z)|_{z = e^{j \omega}}$$
Graphically, evaluating the Z-transform on the unit circle is shown as sweeping ...
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Why do the DTFT and FFT give me completely different results for magnitude at a specific frequency?
I am trying to write a program to compute the magnitude and phase of a specific, non-integer frequency component (i.e. given a sampled finite signal of length $N$, I want to know the magnitude and ...
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131
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Evaluate expressions without computing DTFT
Let $X(\omega)$ be the DTFT of the sequence $x[n]$ given by:
$$
x[n] = \{4, 2, -1, 5, -3, 1, -2, 4, 2\},\quad\text{for}\quad n \in [-6, 2]
$$
I do want to compute
$X(0)$
$X(\pi)$
$\displaystyle\int_{-...
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How is the DTFT of a periodic, sampled signal linked to the DFT?
I am trying to understand the connection between FT, DTFT and ultimately the DFT. But I am failing to link the DTFT to the DFT.
This is how far I am getting: Say I have a function $f(t)$, and its ...
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Discrete-time Fourier transform of $a^{|n|} u[n]$
I have a problem calculating the DTFT of this pair:
Could anyone tell me why the DTFT for $a^{|n|} u[n]$ is different from $a^{n} u[n]$'s?
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Why DFT is used for approximating CTFT when you can approximate CTFT-integral itself?
I was using MATLAB for approximating FTs. Why DFT is used if we can approximate the transform-integration using summation.
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Can different Discrete-Time-Fourier-Series(DTFS) coefficients have the same discrete sequence in the time domain?
Please, check the following discrete periodic sequence when the period $N=2$.
$x[k]=\exp(j\frac{2\pi}{N}k), N=\text{period}$
$..., x[0]= 1, x[1]= -1, x[2]= 1, x[3]= -1, ... , N=2$
According to my ...
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Does Zero Padding Distort the Spectrum of a Signal?
It's said to "sample the DTFT", revealing what "DFT fails to see". And I fail to see how this sampling isn't distortion.
The "spectrum" aims to provide the sinusoidal ...
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Deriving Fourier Transform of Time-Windowed Discrete Signal
I'm trying to derive the Fourier Transform of a finite-length discrete signal to show the effect of windowing,e.g. spectral leakage and resolution, but I can't seem to arrive at the same answer.
Just ...
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2
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Inverse DTFT of phase shifted complex exponential
I have been working on this problem for a few days now and I think this is the closest I have gotten.
I am getting an Answer of zero and I would like to know if that is correct and if someone could ...
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88
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find time domain sequences using DTFT definition (NOT IDTFT)
The sequence is X(e^jw) = 3 + 2cos(w) + 4cos(2w), and the problem asks to use the definition of the DTFT to find the corresponding sequence.
I have tried using the IDTFT and integrating, but I could ...
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How to find minimum length of a FIR symmetric filter if I am given DTFT
I am practising for upcoming exams and came across this question.
Let $h[n]$ be an FIR filter such that $h[n] = 0$ when $|n| > M$ and $h[n] = h[−n]$. A plot of $H(e^{j\omega})$ (DTFT of $h[n]$) is ...
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Inverse discrete time Fourier transform with differentiation
Consider a signal x[n] and its DTFT $X(e^{jω})$ . Assume $X(e^{jω})$ is differentiable. Compute the inverse DTFT of
$j\frac{dX(e^{jω})}{d\omega}$
You should write your answer in terms of $x[n]$ and ...
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186
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How to calculate DTFT of cosine function divided by n
I'm having a hard time to calculate the next function, and I don't really know Matlab good enough to calculate it there.
Help would be appreciated:
$$h[n]=\frac{A_1 \cos[\theta_1(n-N/2)]}{n-N/2}$$
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Evaluate phase of $X(\omega)$ without computing $X(\omega)$
$$x(n) = \{ -1, 0, 1, 2, 1, 0, 1, 2, 1, 0, -1 \}$$
Let $X(\omega)$ be the DTFT of $x(n)$. I need to find the phase of $X (\omega)$ without computing $X(\omega)$. I notice that $x(n)$ can be a type I ...
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1
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207
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Finding causal impulse response given the imaginary part of the frequency response
I understand that I would need to calculate inverse Discrete Time Fourier Transform (iDTFT) to find $h(n)$. Since $h(n)$ is real, iDTFT of the imaginary part of $H(e^{j\omega})$ gives the odd part of ...
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Expressing DTFTs in terms of one another with similar time domain signals
I cam across a question in my DSP book asking this:
1). Express $X_2(e^{j\omega})$ in terms of $X_1(e^j\omega)$ without explicitly computing $X_1(e^{j\omega})$. ($X_1(e^j\omega)$ represents the DTFT ...
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Is my solution to the DTFT of a delta function correct?
I am newer to signal processing math and just figured out something cool (hopefully). I was trying to see how the DTFT of the delta function is 1, because thats what my book says. I could only find ...
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DTFT and a Downsampled Sinc Function
I found the answers to this question and this question to be extremely helpful in understanding the derivation of the downsampling or decimation property of the DTFT. Thank you! I am now struggling ...
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219
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Question on N point DTFT - Fourier transform
I have been trying to use the logic that both X and Y should have same Z transform, but according to the definition, Y is not anti causal.
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Continuous frequency vs discrete frequency? [closed]
I have understood idea of discrete time and continuous time but I am feeling difficult to comprehend this idea in regard to frequency.
As for example the DFT output is discrete and the DTFT output ...
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129
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Evaluating discrete spectral density at only a few frequencies
I'm trying to obtain the spectral density at three particular frequencies for a computational chemistry problem that I'm working on (if you are curious, it has to do with the estimation of Nuclear ...
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$2\pi$ Periodicity is not working for me for Fourier of Discrete Time Signal
please help me find the error in the following counter example.
Consider we take sinus with period of $2\pi$. We sample it many time, and more than 3. We make convolution with rectangle of height 1 ...
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A DT sequence y[n] is constructed from another DT sequence x[n]
A DT sequence $y[n]$ is constructed from another DT sequence $x[n]$ according to the
formula $y[n]=x[nN]$, where $N$ is a constant positive integer greater than one. (This process is usually called ...
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Why is there a negative in front of the phase response equation for this complex exponential?
first time on here!
I'm working through "Digital Signal Processing using MATLAB" by Vinay and Proakis. Good book.
I am stuck on this example tho.
Shouldn't the imaginary part in the denominator (...
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