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Questions tagged [dtft]

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0answers
11 views

IFFT, negative frequencies and noncausal impulse

When using the IFFT to compute an approximation to the IDTFT, frequency range is 0 to 2$\pi$ versus $-\pi$ to $\pi$. This produces a different impulse sequence(causal I think). How do you calculate ...
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1answer
33 views

Matlab FFT not producing symmetric spectrum

I am plotting a FFT of a sampled RC pulse but my resulting spectrum isn't symmetric - it's offset. ...
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1answer
29 views

Calculating DTFT

When calculating DTFT of (1/2)^n u[n]. We evaluate the sum as follows: Please correct statements and answer questions below: 1) So to go from STEP 1 to STEP 2, the limits of the series are changed ...
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1answer
64 views

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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2answers
84 views

Proof that first difference filter amplifies noise

I'm a bit befuddled by noise's effect on derivative filters. I've always 'known' that straightforward first difference derivative filters of discrete signals amplifies noise, but I'm struggling to ...
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2answers
68 views

Support of the convolution of two rectangular signals

I'm trying to convolve two rectangular signals in the frequency domain $$H_1(\omega) = u[\omega +.2\pi] - u[\omega -.2\pi]$$ and $$H_2(\omega) = u[\omega +.1\pi] - u[\omega -.1\pi]$$ My result is a ...
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1answer
42 views

DTFT of sawtooth wave through DTFT of rect signal

In a course i'm currently taking, the lecturer computed DTFT for the following signal: $$r[n] = \begin{cases} 1& 0 \le n \le N\\ 0& \mbox{otherwise} \end{cases} $$ For $N = 32$ i pictured $\...
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1answer
32 views

system function $H(\omega)$ relationship to odd and even components of h[n]

What qualities of $h[n]$ are necessary for: $$ H(e^{j\omega}) = DTFT\{h_{even}[n]\} + j\ DTFT\{h_{odd}[n]\} $$ Do all real / causal h[n] have the property that: $$ H(e^{j\omega}) = DTFT\{h_{even}[n]...
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1answer
79 views

Is possible reach the DFT if I have the DTFT?

My teacher told me that DFT is DTFT sampled, i.e.: $$X[k] = X(e^{j \omega})\Bigg|_{\omega = \frac{2\pi k}{N}}$$ But, if I have the sine $$ x[n] = \sin(\omega_0 n) $$ the DTFT is: $$X(e^{j \...
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1answer
33 views

Conversion between DTFT in radians/sample to DTFT in cycles/sample

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 ...
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1answer
105 views

Relationship between the IDFT of a sampled DTFT and its discrete-time domain signal

Suppose we are given an input signal s[m,n] with DTFT $S(\omega_1, \omega_2)$. We sample it at $\omega_1 = \frac{2 \pi k}{256}$ and $\omega_2 = \frac{2 \pi l}{256}$ to get a 256 point DFT S[k,l]. ...
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3answers
89 views

Maximum Magnitude Deviation between DFT and DTFT

Let $x[n]$ be a finite-length discrete-time signal with length $N$. The continuous DTFT $X(\omega)$ is then $$ X(\omega) = \sum_{n = 0}^{N-1} x[n] e^{-j \omega n}. $$ The length-$N$ DFT of $x[n]$ is $...
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1answer
59 views

Using the given identities, find the inverse DTFT

Using the given identities, $ a^nu[n]$ <===> $\frac{1}{(1-ae^{-jw})}$ $\delta[n-k]$ <===> $e^{-jwk}$ Find the inverse DTFT of, $ H(e^{jw}) = B \frac{e^{-jw}}{(1-ae^{-jw})}$ my attempt: $ ...
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1answer
142 views

Bridging CTFT and DTFT for a cosine

I'm trying to understand how I can start from the CTFT of a signal and end up with a DTFT. For example if I take a basic example: $$ x(t) = cos(\omega_x \cdot t) = \frac{1}{2} \cdot \left( e^{j\...
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1answer
198 views

How can I show DTFT result in MATLAB? [closed]

I wanna show DTFT result and convolution result in f-domain are same. Additionally, I wanna show sampled function and inverse fourier transform result are same too. How can I show this? If I use ...
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0answers
9 views

Draw 2-D DTFTs from 2-D LSI system with impulse response

Consider the 2-D LSI system with impulse response h[n1,n2] and input x[n1,n2]. The 2-D DTFTs associated with these signals are H(ω1,ω2) and X(ω1,ω2) respectively. How do I figure out and sketch what X(...
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1answer
159 views

The concept of normalized frequency

This question has already been asked and answered, but the motivation behind the use of normalized frequency units still evades me. The Discrete Time Fourier Transform $$X(\tilde{ \omega }) = \sum_{n=...
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0answers
169 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 ...
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0answers
22 views

How to calculate the DTFT of ${a^{|n|}}$for |a| < 1?

I know how to calculate the DTFT of ${a^{n}}$ for |a| < 1. But how to proceed to find the DTFT of ${a^{|n|}}$ for $|a| < 1$?
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1answer
141 views

DTFT of $(-1)^n \cdot \mathrm{sinc}()$

I'm trying to find the DTFT of $$(-1)^n \cdot \frac{\sin (\pi n/2)}{\pi n}$$ I know the DTFT of $\frac{\sin \pi n/2}{\pi n}$ = a box function of amplitude 1, cutoff $\pi/2$. And I know that ...
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1answer
63 views

Question regarding DTFT of a complex signal

I have been doing DTFT practice problems for my DSP course, and I encountered this problem in the textbook that completely stumped me. The question asks to find the DTFT of the shown signal and to ...
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1answer
204 views

Time scaling of discrete-time sequences and the DTFT

In the second edition of Signals and Systems by Alan Oppenheim, he discusses the DTFT of a "time-expanded" sequence that is effectively a slowed down version of the original sequence and can be ...
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3answers
368 views

Periodicity of the discrete-time Fourier Transform

The DTFT of a sequence $x[n]$ can be written as $$X(e^{j\omega}) = \sum_{n = -\infty}^{\infty} x[n] e^{-j\omega n}.$$ Is the smallest (fundamental) period in frequency of the DTFT always $2\pi$? Or ...
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1answer
53 views

IDTFT of $\sum_{k=-\infty}^{+\infty}(u(\Omega+\pi)+u(\Omega+\frac{\pi}{4})-u(\Omega-\frac{\pi}{4})-u(\Omega-\pi))\star \delta(\Omega-2k\pi)$

Compute the IDTFT of the following signal: $$X(\Omega)=\sum_{k=-\infty}^{+\infty}\left(u(\Omega+\pi)+u\left(\Omega+\frac{\pi}{4}\right)-u\left(\Omega-\frac{\pi}{4}\right)-u(\Omega-\pi)\right)\...
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1answer
122 views

Inverse DTFT of $H_1(\Omega)=\begin{cases} 10,& \frac{\pi}{3} \leq |\Omega| < \pi\\ 0,& 0 \leq |\Omega| < \frac{\pi}{3}\\ \end{cases}$

What is the inverse DTFT of the $2\pi$-periodic extension of following function: $$H_1(\Omega)=\begin{cases} 10,& \text{for } \frac{\pi}{3} \leq |\Omega| < \pi\\ 0,& \text{for } 0 \leq ...
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9answers
407 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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1answer
42 views

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]=\...
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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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1answer
169 views

Does the DTFT of $\frac{u[n-1]}{n}$ exist?

I have started learning DSP on my own and I have this doubt. I have done some googling but haven't found an answer. I hope that someone here would give the answer. It will be of great help.
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1answer
48 views

How is this was derived? DTFT for symmetric pulse

do not understand how the middle expression was derived from above expression, after all summation\multiplication i get: $$\frac{e^{-jwN} + e^{jwN} - e^{-jw(N+1)} + e^{jw(N+1)}}{2 - e^{-jw} - e^{jw}...
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2answers
222 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 ...
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2answers
78 views

Difference in Interpretation: $ω$ (rads/s) vs. $ω$ (rads) and $X(ω)$ vs. $X(e^{jω})$

The fourier transform of a continuous time signal $x(t)$ is $X(ω)$ where the unit of $ω$ is radians/second. And for a discrete signal $x(n)$, the DTFT is $X(e^{jω})$ where the unit of $ω$ is radians. ...
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1answer
564 views

Circular vs Linear Convolution

When deriving DFT from DTFT,we sample the frequency domain with uniformly spaced samples,hence adding periodicity to time domain. But DFT requires a limited support: we take only 1 period. Does that ...
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2answers
224 views

How is a continuous spectrum for the DTFT possible?

So we that a complex sinusoid of the form $e^{j\omega_0n}$ is periodic over $N=2\pi/\omega_0$ only if $\omega_0$ is a rational multiple of $\pi$, otherwise the exponential is not periodic. (see EDIT!) ...
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0answers
74 views

DTFT of $ f[k] = 3^k u(-k-1)$

Find the Discrete-time Fourier transform of $ f[k] = 3^k u(-k-1)$ (then sketch it and find its magnitude & angle). It doesn't fit any templates on the Fourier table, and I don't see how one ...
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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
174 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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1answer
3k views

DTFT of any finite sequence in matlab

I think freqz in a MATLAB toolbox, is the way to obtain DTFT of sequence. freqz can calculate frequency response of: H(z)=(Num)/...
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2answers
389 views

Calculating the inverse DTFT of a signal

There is a signal $y[n]$ with a differentiable DTFT $Y(e^{i\omega})$. How do I find the inverse DTFT of $i\frac{dY(e^{i\omega})}{dw}$ in terms of $y[n]$ (where of course $i = \sqrt{-1}$)?
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1answer
121 views

Does $H(-z)$ produce aliasing? [closed]

Given $H(z)$ is the z-transform of a signal, I know that $H(-z)$ results in shifting of frequencies in DTFT by $\pi$ or $-\pi$. Does it produce aliasing ? How ?
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2answers
466 views

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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1answer
241 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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1answer
179 views

What additional information do we get from z-transform that we don't get from DTFT? [duplicate]

As an engineer analyzing a system (whether it be a circuit or an audio sample), you should know when to apply the analysis tools you've been given--such as Discrete Time Fourier Transform and Z-...
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1answer
233 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
264 views

The DTFT of $\{1,1\}$ is $1+e^{-j\omega}$ but what is the DTFT of $\{1,-1\}$?

So I know that the DTFT of $\{1,1\}$ is equivalent to $1+e^{j\omega }$. But what is the DTFT of $\{1,-1\}$ equivalent to? Is it equivalent to $1-e^{j\omega }$?
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1answer
887 views

How to find the DTFT of unit step with $(-1)^n$ scalar multiplied?

What are the DTFTs of the following two signals? $$x[n] = e^{j \pi n}\left\{u[n] - u[n-8]\right\}\quad\text{and}\quad h[n] = (-1)^n\left\{u[n] - u[n-4]\right\}$$ I am trying to find $X(\omega)$ and $...
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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
605 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
490 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 ...
0
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2answers
548 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: $$...