Questions tagged [dtft]

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Mathematical advantages of the ZT, DTFT and DT?

I apologize if this question is too general to answer concretely, but I was hoping more to perhaps be pointed towards some resources that could help more extensively. Essentially, I have a Discrete-...
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3answers
3k views

Why DTFT coefficients are periodic and why continuous Fourier transform coefficients are not periodic?

As I understand, when the input signal is discrete in time and we want to find the coefficients of Fourier transform, DTFT is used and the coefficients in frequency domain are periodic, but I can't ...
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1answer
664 views

Tricks for plotting the magnitude of a DTFT?

To preface, this isn't a homework question but rather a self-study question to help me to understand the basics of finding the DTFT and magnitude of the DTFT based on a discrete time signal sampled ...
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1answer
507 views

DTFT of $x[-n-1]$

How can I determine the DTFT of $x[-n-1]$? I searched for DTFT problems and checked several references but I couldn't find a similar case. My background is a little lacking, so excuse me if it's too ...
3
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1answer
369 views

Why is this DFT of a real symmetric signal resulting in complex valued coefficients?

I am trying to understand exactly how sampling the DTFT to get the DFT works. The signal I'm trying to analyze is $x(n)$ seen below. $$x(n) = \delta(n\pm2) + 2\delta(n\pm1) + 3\delta(n)$$ Taking the ...
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2answers
581 views

Frequency Response with Delta Function?

I am trying to find frequency response and magnitude of the frequency response of the following system impulse response: $$h[n] = 2\delta [n] + 2\delta [n-1]$$ I understand, that through the DTFT: $$...
3
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1answer
483 views

2D Fourier Transform of Rotated Discrete Domain Signal

Assume we know that the Fourier transform of a signal $x(n_1,n_2)$ is $\mathcal{F}(x(n_1,n_2))=X(\omega_1,\omega_2)$. What is the Fourier transform of the signal after being transformed by a rotation ...
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2answers
344 views

Discreteness and periodicity in Fourier transform

Why discreteness in time / frequency domain dictates periodicity in the other frequency / time domian? For example the DTFT is perodic in frequency? Why it doesn't contain all the frequencies? Why ...
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1answer
440 views

Finding the deterministic autocorrelation function (ACF) from its power spectrum

The power spectrum of a stationary discrete-time random signal is $$\Phi_{xx}(e^{j\omega})=\begin{cases} 1 & |\omega|<\pi/2 \\ 0 & \pi/2 <|\omega| \le\pi \end{cases} $$ (a) ...
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1answer
1k views

DTFT and Inverse DTFT Homework Problem

I'm trying to solve this signals homework problem: So for part a, since multiplication in the time domain is convolution in the frequency domain, I just used a DTFT table, found the DTFT for $\left(\...
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2answers
2k views

Multiplication property DTFT

I was truing to solve an example of DTFT which is following multiplication property. The problem is $$ a^n \sin(\omega_0 n) u[n]$$ we know that the definition of DTFT is $$ X(j \omega) = \sum _ {n=-\...
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2answers
2k views

Difference between Fourier Transform and DFT? - Example

I have read many excellent answers to similar questions, but never one this specific. Here is another way to ask it. Why is the modulation transfer function (MTF) of $\textrm{rect}(x/5) = \textrm{...
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1answer
74 views

Find the Fourier transform of $g(k)$ from $G(z)$ for frequency $=1/2$

$$G(z)=\displaystyle \frac{\frac{1}{z}}{1+\frac{5}{6z}+\frac{1}{6}z^{-2}}$$ I found: $$g(k)=\displaystyle \left(\frac{-1}{3}\right)^k - \left(\frac{-1}{2}\right)^k$$ I don't understand how I can ...
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4answers
3k views

What does the exponential term in the Fourier transform mean?

We know that Fourier transform $F(\omega)$ of function $f(t)$ is summation from $-\infty$ to $+\infty$ product of $f(t)$ and $e^{-j \omega t}$: $$ F(\omega) = \int\limits_{-\infty}^{+\infty} f(t) \ e^...
2
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1answer
284 views

What is the interpretation of the discrete-time spectrum?

The CTFT of an analog signal is a representation of that analog signal in terms of the frequency parameter of sinusoidal (cosine specifically) functions whose weighted sum make up that signal. The ...
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2answers
427 views

Why this DTFT plot from CTFT? [closed]

I have a sample test with an answer but don't understand how they got to the answer: $x(t)$ has info only between $2 < |\omega| < 4$ $X^F(\omega) = 0$ for other frequencies. All ...
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2answers
49 views

DTFT inconsistency

Let $x[n] = u[n] - u[n-4]$ (a discrete pulse of length 4), and $X(\omega)$ is its DTFT. Let $x_1[n] = x[n]*x[n]$. I expect DTFT of $x_1[n]$ to be same as $X(\omega)$, because $x_1[n]$ has the same ...
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1answer
37 views

Given a discrete time signal, what is the sequence of possible frequencies I can get from DTFT?

I know that when I have a discrete time signal, let's say: The definition of the DTFT is given by: Now, my question is regarding Omega(n). I know the frequencies will be discret because we can't ...
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1answer
153 views

why $-$ sign in DTFT pair for constant

In discrete time Fourier transform, The DTFT of constant 1 is $$\sum_{l=-\infty}^{+\infty} \delta(\omega-2\pi l) $$. I have confusion that why there is $-$ sign, why it can't be $$\sum_{l=-\infty}^{+\...
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0answers
106 views

Confusion in understanding the Proof of DTFT

While understanding the proof of DTFT from Signals and Systems by Oppenheim, I have confusion in understanding few steps. $$ x'[n]=\sum_ {k=<N>} a_ke^{jk(2\pi/N)n}$$ $$ a_k= \frac{1}{N} \sum_ {...
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1answer
261 views

Finding values of DTFT without explicitly computing

My attempt : a) Summation of all values? b)c)d) Failed e) Parserval's theorem
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3answers
5k views

How condition for existence of Fourier transform is valid?

The condition for Discrete time Fourier transform to exist for function $f(n)$ is given as $$\sum_{-\infty}^\infty |f(n)| < \infty.$$ In case of continuous Fourier transform the difference is ...
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1answer
174 views

Inverse DTFT Problem

Having trouble finding the inverse DTFT of $\ X(\ e^{j \omega}) = \frac{3 - \frac{1}{4} e^{-j\omega}}{1 - \frac{1}{4} e^{-2j\omega}} $ Given the IDFT of $Xe^{j \omega}$ as : $x(n) = \frac{1}{2\pi} ...