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realistic sampling - where am I wrong?

I’m given a signal $x(t)$, it's convolved with $h(t)$ and sampled at rate T=1. The result is called $\tilde{x}[n]$. For $$h(t) = \begin{cases} 1 & -0.5<t\le 0.5 \\ 0 & \text{else} \end{...
Piratemetaldrinkingcrew's user avatar
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The following four facts are given about a real-valued DT signal x[n] with discrete tine Fourier transform X(ejw) [closed]

I found this question while studying for a signals and systems course. It makes sense to me if they define the imaginary parts of the DTFT, but defining the real parts makes it much trickier. Does ...
sto's user avatar
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Need help with DTFT transform (sampling and reconstruction)

Given the continuous time signal $$ x \left( t \right)= 2 \cos \left( 100 \pi t \right) + 3 \sin \left( 250 \pi t \right) \tag{1} $$ The signal is sampled in point sampling with sampling interval $T_{...
EEngineer's user avatar
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1 answer
84 views

Downsampling of signal

I have two signals, for example, $x_1[n]$ and $x_2[n]$, which are the same for every $n$ except for $n_0$ and $n_1$. They are both bounded within a frequency range of $\pi/3$. I want to reconstruct $...
Moran Poco's user avatar
1 vote
3 answers
177 views

Phase of a Complex Exponential

It is known that the discrete-time Fourier transform (DTFT) of a complex exponential $$ x[n] = e^{j\omega_0 n} $$ is $$ X(e^{j\omega}) = 2\pi \sum_{k = -\infty}^{\infty} \delta(\omega - \omega_0 + 2\...
Sami Al-Dalahmah's user avatar
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40 views

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(...
Lynn's user avatar
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28 views

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 ...
Processor48's user avatar
2 votes
1 answer
277 views

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 ...
Juliana Xavier's user avatar
1 vote
1 answer
107 views

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 ...
Madavan Viswanathan's user avatar
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1 answer
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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}^{\...
MaxFrost's user avatar
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1 answer
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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 ...
Chillpadde's user avatar
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1 answer
47 views

DTFT Pair Transformation of unit step [duplicate]

I am not seeing a direct pair of DTFT transform of the unit step.
discretephysicscs's user avatar
5 votes
1 answer
208 views

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: \...
Tim Mak's user avatar
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3 answers
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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 ...
Carl's user avatar
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2 answers
186 views

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{...
Bozu's user avatar
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1 answer
280 views

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 $X(e^{jw})$ ...
Caleb Burke's user avatar
1 vote
2 answers
189 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 ...
abhilash's user avatar
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1 answer
478 views

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 ...
OuttaSpaceTime's user avatar
1 vote
1 answer
106 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^{...
makala's user avatar
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1 vote
1 answer
543 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,...
Edward Josef's user avatar
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1 answer
53 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\...
electrical apprentice's user avatar
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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^{-...
mhdadk's user avatar
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1 answer
95 views

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 − ...
Derteck pt's user avatar
2 votes
0 answers
242 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 ...
Eigenbrödler's user avatar
1 vote
1 answer
121 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 ...
LuxGiammi's user avatar
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3 votes
1 answer
159 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]\\...
SPARSE's user avatar
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1 vote
1 answer
454 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 $...
user3921's user avatar
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1 vote
1 answer
157 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 ...
Merry's user avatar
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3 votes
2 answers
1k 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....
Pranav Krishnan's user avatar
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1 answer
38 views

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 ...
Minato Namikaze's user avatar
1 vote
1 answer
112 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 ...
Halleff's user avatar
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1 vote
1 answer
194 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 ...
Tommy Wolfheart's user avatar
1 vote
1 answer
139 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_{-...
Anonymous's user avatar
2 votes
5 answers
762 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 ...
geo's user avatar
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0 votes
1 answer
787 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?
tin tan's user avatar
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5 votes
1 answer
536 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.
Lelouch Yagami's user avatar
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0 answers
155 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 ...
kappy super's user avatar
2 votes
3 answers
2k 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 ...
OverLordGoldDragon's user avatar
1 vote
0 answers
49 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 ...
Bawb's user avatar
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0 votes
2 answers
176 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 ...
Dom's user avatar
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0 answers
105 views

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 ...
Dom's user avatar
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0 votes
1 answer
260 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 ...
Tiklu's user avatar
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0 votes
1 answer
91 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 ...
amn_suryansh's user avatar
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1 answer
191 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: $$h[n]=\frac{A_1 \cos⁡[\theta_1(n-N/2)]}{n-N/2}$$
Jhon Margalit's user avatar
0 votes
1 answer
46 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 ...
my_knee_Hertz's user avatar
0 votes
1 answer
225 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 ...
Ruhi's user avatar
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1 vote
1 answer
80 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 ...
Dom's user avatar
  • 61
2 votes
1 answer
147 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 ...
Dom's user avatar
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1 vote
1 answer
196 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 ...
Steve J.'s user avatar
1 vote
2 answers
233 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.
Ruhi's user avatar
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