Questions tagged [dtft]
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161 questions
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Computing the DFT of three sine waves with Aliasing
I was asked to compute the DFT of the following:
$$x(t) = \sin(2 \pi 1000 t) + \sin(2 \pi 3500 t) + \sin(2 \pi 19000 t)$$
Sampled at $f_s = 20,000 [Hz]$ for $N=256$ samples.
can you please look at my ...
0
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1
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39
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Discrete-Fourier transform of $$u[-n+2]$$
I'm Trying to Find the fourier transform in discrete time for $$u[-n+2]$$ .
My steps :
Time-Reversal Property : $$ u[(-n+2)] \{\omega\} = u[-(-n+2)] \{-\omega\} = u[n-2] \{-\omega\} $$
Time-Shifting ...
- 101
1
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1
answer
126
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Showing analytically that sampling exactly 1 period of a sinusoid yields a spectrum with no lobes in the DFT
The DTFT of a discrete sinusoid $f[k] = \sin(\omega_0 k)$ is
$$F(\Omega)=i\pi(\delta[\Omega-\omega_0] + \delta[\Omega+\omega_0]), \: \: \: \Omega \in [-\pi, \pi)$$
The DTFT of a rect function $w[k] = ...
- 480
1
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1
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100
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I want to understand the fundamental difference/connection between DFS, DFT and DTFT
I'm an EE student and I seem to miss some basic concept of my Signals course.
We have learned about all the different Fourier methods available, but I don't seem to find a difference/understand it.
As ...
- 11
3
votes
1
answer
155
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Z-transform of the Unit Step and DTFT
In class we showed the the z transform of the unit step only exists for |z|>1 but we also calculated the DTFT of the unit step. Does convergence on the unit circle imply the DTFT exists but not the ...
- 33
0
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1
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95
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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 $$h(t) = \begin{cases} 1 & -0.5<t\le 0.5 \\ 0 & \text{else} \end{...
0
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0
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69
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Need help with DTFT transform (sampling and reconstruction)
Given the continuous time signal
$$ x \left( t \right)= 2 \cos \left( 100 \pi t \right) + 3 \sin \left( 250 \pi t \right) \tag{1} $$
The signal is sampled in point sampling with sampling interval $T_{...
0
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1
answer
99
views
Downsampling of signal
I have two signals, for example, $x_1[n]$ and $x_2[n]$, which are the same for every $n$ except for $n_0$ and $n_1$. They are both bounded within a frequency range of $\pi/3$.
I want to reconstruct $...
1
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3
answers
289
views
Phase of a Complex Exponential
It is known that the discrete-time Fourier transform (DTFT) of a complex exponential
$$
x[n] = e^{j\omega_0 n}
$$
is
$$
X(e^{j\omega}) = 2\pi \sum_{k = -\infty}^{\infty} \delta(\omega - \omega_0 + 2\...
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0
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41
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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:
$$x[ n ] = x[ {n{T_s}} ] = x(...
- 1
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0
answers
29
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How to approach signal reconstruction with sampling frequency not equal to reconstruction
I'm trying to figure out what kind of equations i can use in a situation like this:
I know the relation between $x(t)$ and $x[n]$ is :
but then $x[n]$ is multiplied by a pulse train, with a ...
- 49
2
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1
answer
379
views
Gaussian filter: Plotting DTFT and DFT (by hand) from the continuous-time impulsive response
I am trying to make an algorithm that plots out the Discrete-Time Fourier Transform (DTFT) and the Discrete Fourier Transform (DFT) of the Gaussian filter. The impulsive response and its transfer ...
1
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1
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120
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Intuitive or physical explanation of DTFT$\{1\}=2\pi\delta(\omega)$
I am trying to understand the fact that
"The DTFT of 1 (an infinite discrete sequence of unit impulses from from $-\infty$ to $+\infty$) is $2\pi\delta(\omega)$"
in an intuitive or physical ...
0
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1
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54
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Computing an infinite sum of time-shifted sequence
Given a discrete-time domain signal $x[n]$ defined as
$$x[n] = \begin{cases}1 & 0 \leq n \leq L-1 \\
0 & \textrm{otherwise}\end{cases} $$
we are tasked with computing $$\sum_{k = -\infty}^{\...
- 403
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1
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71
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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 ...
0
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1
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56
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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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1
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232
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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:
\...
- 225
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3
answers
3k
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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 ...
- 480
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2
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212
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Easy (?) DTFT calculation
I'm asked to compute the DTFT of the following signal but i'm quite stuck
$$
\begin{cases}
(-1)^{\frac{n}{2} + 1} & \text{ if } n \text{ is even} \\
0 & \text{ if } n \text{ is odd}
\end{...
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1
vote
1
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331
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Given the Fourier Transform of a continuous signal how can I sketch the sampled signals discrete time fourier transform
I am given the frequency response for a continuous time signal $X(j\omega)$ = 2 at $\omega=0$ and 0 at $\omega = -10000\pi$ and $10000 \pi$. Looks like a triangle. I am told to sketch $X(e^{jw})$ ...
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1
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2
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206
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Multiplication term $ \frac{ 1}{T_s} $ in sampling theorem
\begin{equation}
X(\Omega) = \frac{ 1}{T_s} \sum ^{\infty}_{k=-\infty} X_a\left \lbrace \frac{\Omega /( 2 \pi) - k}{T_s}\right \rbrace
\end{equation}
What is the purpose of multiplying sampled ...
- 460
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1
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643
views
Intuition of odd and even complex conjugate symmetry definition of DFT/DTFT so that $X(e^{j w})=X_{e}\left(e^{j w }\right)+X_{o}\left(e^{j w}\right)$
I have been reading through my courses DSP slides and came across something which was not really taught in detail. You can look up here for reference, it is stated almost identical.
Given the ...
- 307
1
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1
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113
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Ft and DTFT of negative frequency
I have a question that might sound silly but if I have a real and even signal x(t) can I define the FT and DTFT of the negative frequency if I can show:
$$X(-\omega) = \int_{-\infty}^{\infty} x(-t)e^{...
- 65
1
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1
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636
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:
$$ X[n] = 1, 1, 1, ... 1, 0, 0,...
- 113
0
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1
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81
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Understanding graphs of DTFT with time shift of$~y\left[n\right]=x\left[n-2\right]~$
$$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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137
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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
$$
H(e^{j\omega}) = e^{-j\omega} \left(a_0 e^{j\omega} + a_1 + a_2e^{-...
- 360
-1
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1
answer
100
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I'm having problems simplifying this discrete-time fourier tranform
I have this problem, and I can't get to the solution.
$$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 − ...
- 13
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0
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266
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
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1
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130
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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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3
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1
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173
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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]\\...
- 119
1
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1
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506
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 $...
- 267
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1
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166
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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 ...
- 141
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2
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1k
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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....
0
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1
answer
42
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 ...
1
vote
1
answer
118
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 ...
- 349
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1
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206
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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 ...
- 173
1
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1
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159
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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_{-...
- 11
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5
answers
873
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 ...
- 184
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1
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906
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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5
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1
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627
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.
- 185
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0
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161
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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 ...
- 19
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4
answers
3k
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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 ...
- 9,232
1
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0
answers
50
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 ...
- 11
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2
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201
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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 ...
- 61
0
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0
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119
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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 ...
- 61
0
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1
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278
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 ...
- 15
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1
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99
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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 ...
0
votes
1
answer
192
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:
$$h[n]=\frac{A_1 \cos[\theta_1(n-N/2)]}{n-N/2}$$
- 103
0
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1
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47
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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 ...
0
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1
answer
243
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 ...
- 15