# Tag Info

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• 4,920
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### What does the exponential term in the Fourier transform mean?

It's a complex exponential that rotates forever on the complex plane unit circle: $$e^{-j\omega t} = \cos(\omega t) - j \sin(\omega t).$$ You can think of Fourier transform as calculating correlation ...
• 12.6k
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• 82.9k
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### How condition for existence of Fourier transform is valid?

As mentioned in Batman's answer, the condition of the sequence being absolutely summable is only sufficient but not necessary. The Fourier transform can be extended to $\ell_2$ sequences, i.e. ...
• 82.9k

### What does the exponential term in the Fourier transform mean?

If you don't like thinking about imaginary numbers, complex numbers and functions, you can alternatively think of the complex exponential in the FT as just shorthand for mashing together both a ...
• 34.5k

### Link between DFS, DFT, DTFT

Yes your understanding is basically correct. The 1st paragraph (2 lines) expresses the fundamental relation between the DFS and the DFT of a finite-length sequence $x[n]$ while the 2nd paragraph tries ...
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• 82.9k
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• 82.9k
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### why $-$ sign in DTFT pair for constant

It can also have a $+$ sign, there's no difference. Write down a part of the sum (around index $l=0$) and try to see that in both cases you're summing the same terms, just in a different order. More ...
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### Discrete-time Fourier Transform of the unit step sequence $u[n]$

I'll provide two relatively simple proofs that do not require any knowledge of distribution theory. For a proof that computes the DTFT by a limit process using results from distribution theory, see ...
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### system function $H(\omega)$ relationship to odd and even components of h[n]

The DTFT relationships $$x_{even}[n]=\frac12\left(x[n]+x^*[-n]\right)\Longleftrightarrow\textrm{Re}\left\{X(e^{j\omega})\right\}$$ and x_{odd}[n]=\frac12\left(x[n]-x^*[-n]\right)\...
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### Aliasing and DTFT of a real signal

We are analyzing a real signal with the DTFT. Since we are using a limited number of samples it's like we are transforming a finite signal. No. The DTFT takes an infinite discrete time signal as an ...
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### Why DFT is used for approximating CTFT when you can approximate CTFT-integral itself?

Assuming that future vistors won't take the time to read all the comments, I'd like to give a very simple and straightforward interpretation of the discrete Fourier transform (DFT) as an approximation ...
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### Difference between Fourier Transform and DFT? - Example

The difference is between $\mbox{sinc}$ and the periodic version obtained using the DFT. See this answer for a comparison. It strikes me that asinc = ratio of two sincs. The $\mbox{asinc}$ ...
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DTFT is the Z-transform at the unit circle. So if $z=re^{j\omega}$ then for DTFT $r = 1$. i.e If you have the Z-transform of a signal then plug-in $e^{j\omega}$ for every $z$