Questions tagged [dtft]

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73 questions
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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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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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Proving that the IDTFT is the inverse of the DTFT?

The DTFT is given by: $$X(e^{j\omega}) = \sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n}$$ The IDTFT is given by: $$x[n]=\frac{1}{2\pi}\int_{0}^{2\pi}X(e^{j\omega})e^{j\omega n}d\omega$$ I have been ...
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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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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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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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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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Magnitude and phase of $-\delta[n]$?

I was reading this document and it shows the computation of the magnitude and phase of $h[n]=-\delta[n]$. We can get the DTFT as: $$H(e^{j\omega})= -1$$ So the magnitude will be $1$, and according ...
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What is the interpretation of Fourier Transform containing only imaginary part?

The FT of a unit step function is taken as: $$X(\omega) = \int_0^\infty e^{-j\omega t}dt = \frac{-1}{jw}e^{-j\omega t} \Biggr |_{0}^{\infty} = \frac{j}{\omega}$$ The transform only has the ...
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DTFT fourier transform (modified property)

I know there are 3 properties of DTFT that help with my problem $$a^{n}u[n]=\frac{1}{1-ae^{-jΩ}}$$ $$(n+1)a^{n}u[n]=\left(\frac{1}{1-ae^{-jΩ}}\right)^{2}$$  \frac{(n+r-1)!}{n!(r-1)!}a^{n}u[n]=\...
I have found that most commonly the DTFT is defined as: $X(\omega) = \sum_{n=-\infty}^{\infty} x[n]e^{-j \omega n}$. However the class I am taking frequently uses the DTFT expressed in "normalized ...