Questions tagged [dtft]

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Relation between the DTFT and the spectrum of a sampled signal

In the $\rm DTFT$ (Discrete Time Fourier Transform) the spectrum is periodic with period of $2\pi$ . A continuous signal when sampled has a spectrum which is a repeated version of its original ...
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9answers
445 views

Where is the flaw in this derivation of the DTFT of the unit step sequence $u[n]$?

This question is related to this other question of mine where I ask for derivations of the discrete-time Fourier transform (DTFT) of the unit step sequence $u[n]$. During my search for derivations I ...
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4answers
873 views

Formulas of the Fourier transform family

It has annoyed me that there doesn't seem to be a source online where the complete complex Fourier transform family is presented with every variable defined. The lack of definitions can be a nuisance ...
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2answers
252 views

$|X(e^{jω})|^2$ - Power or Energy Density?

If $x(n)$ is an aperiodic signal and $X(e^{jω})$ its DTFT, then, what is $|X(e^{jω})|^2$? Power or Energy Spectral Density? My understanging of Fourier transforms so far tells me that its energy ...
3
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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
1k views

Link between DFS, DFT, DTFT

My understanding of DFT is as follows For a signal $x[n]$ of finite-length, the DFT is DFS of the periodic extension, $\tilde{x}[n]$, of that signal $x[n]$ and also another way to view DFT is that it’...
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1answer
251 views

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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1answer
275 views

DTFT reconstruction

I have a sampled DTFT. If we assume that there isn't aliasing in time domain, what is the best way to reconstruct DTFT from its equidistant samples? I though about Dirichlet interpolation. Do you know ...
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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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2answers
1k views

Discrete-time Fourier Transform of the unit step sequence $u[n]$

From text books we know that the DTFT of $u[n]$ is given by $$U(\omega)=\pi\delta(\omega)+\frac{1}{1-e^{-j\omega}},\qquad -\pi\le\omega <\pi\tag{1}$$ However, I haven't seen a DSP textbook that ...
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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=-\...
2
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3answers
173 views

Finding Fourier transform of a discrete signal from its Z-transform

Is it possible to find the Fourier transform of a discrete signal if you know its $\mathcal{Z}$-transform of?
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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^...
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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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1answer
191 views

FFT-like algorithm for fast DTFT computation? [duplicate]

Good morning! I'm coding up a project on a microcontroller to read in some analog audio (specifically, the sound of someone whistling: a near perfect sine wave) and determine which piano note tones ...
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0answers
213 views

frequency spectrum of a sampled signal, PSD and power discussion

Before I go into my question, I first want to review the basics of sampling a signal and at the same time I build the basics of my questions so that they make more sense. I know I have asked couple of ...