Questions tagged [dtft]

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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(jw) = 2 at w=0 and 0 at w = -10000pi and 10000pi. Looks like a triangle. I am told to sketch X(e^jw) the frequency response of a ...
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1 vote
2 answers
78 views

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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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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1 answer
51 views

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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1 answer
77 views

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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1 answer
28 views

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 answer
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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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1 answer
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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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2 votes
0 answers
117 views

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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1 vote
1 answer
67 views

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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3 votes
1 answer
90 views

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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1 answer
138 views

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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1 answer
73 views

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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477 views

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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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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1 vote
1 answer
64 views

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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1 answer
123 views

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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1 vote
1 answer
65 views

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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2 votes
5 answers
376 views

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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144 views

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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4 votes
1 answer
294 views

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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0 answers
125 views

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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1 vote
3 answers
759 views

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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1 vote
0 answers
45 views

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 answers
95 views

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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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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1 answer
131 views

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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1 answer
72 views

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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0 votes
1 answer
115 views

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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1 answer
37 views

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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0 votes
1 answer
109 views

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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1 vote
1 answer
69 views

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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2 votes
1 answer
85 views

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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1 vote
1 answer
98 views

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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1 vote
2 answers
158 views

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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2 answers
1k views

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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1 vote
1 answer
71 views

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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0 answers
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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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0 answers
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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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1 answer
231 views

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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How do you change an instance of dsp.DigitalDownConverter object in MATLAB to work with filters other than the ones it is originally defined with?

If you look at this website: https://www.mathworks.com/help/dsp/ref/dsp.digitaldownconverter-system-object.html you will see an example (with code) that attempts to up convert and down convert a ...
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1 answer
564 views

Deriving expression for the DTFT of a rectangular window

Looking at the picture above, how did the author get from point A) to B)? My Approach: Multiply A) by $e^{j\omega/2}/e^{j\omega/2}$. Now I am stuck with simplying the numerator.
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3 answers
116 views

DFT of the same signal with different values of N

Let $x[n]$ be a discrete signal of 2 samples. We know that its DFT with N=4 is $X[k]=[0, 1+j, 2, 1-j]$. Without calculating $x[n]$, how can we know the DFT with N=2? I have tried to use the relation ...
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1 answer
144 views

CTFT to DTFT why can't you always just substitute $\Omega = \omega/T_S$

This is something I've always wondered about in DSP class, but just accept as a fact because I never really fully understand why this is the case: Given CTFT: $$X_s(j\Omega) = 6000 \pi \sum \limits_{...
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2 votes
2 answers
528 views

Effect of changing sample rate, window duration and zero padding on DTFT and DFT

Let $T$ be the window duration, $N$ be the DFT size, $F_s$ be the sample rate, and $F_{max}$ be the frequency of the highest bin. In the context of image below: halving the $F_s$ (keeping $T$ ...
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2 answers
1k views

Ideal high pass filter for discrete signal

there. I currently get stuck on a question. I was asking to find an inverse discrete-time Fourier transform for the ideal high pass filter. Here is the question It is getting more confused after I ...
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160 views

Characterizing a non-LTI system

How should we characterize a non-LTI system? For example we have: $y[n]=x[3n]+x[2n]+x[n]$ which is clearly not LTI. Also, the impulse response will be $h[n]=3\delta[n]$ and if we take the DTFT of this ...
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1 vote
2 answers
588 views

Getting the DTFT from the DFT samples

How would you get the DTFT from the DFT samples? How will the DFT indexes map to the discrete frequency and what kind of an interpolation would be required?
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359 views

What is the formula for the frequency spectrum?

A signal $f[n]$ is given, the corresponding DTFT as $F(e^{j\omega})$ and a plot of the frequency spectrum $f(t)$. Unfortunately I can't find a formula for the frequency spectrum in my documents. When ...
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