Questions tagged [dtft]

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1answer
47 views

Why is DTFT of $e^{jn\omega_0}$ an impulse train?

update : After asking the question, I figured out that DTFT result is an impulse train. Now my question evolved to, how it is derived in this way? Using the DTFT formula seems not to be working, ...
3
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1answer
407 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: $$\begin{aligned} x(t) &= \cos(\omega_x \cdot t) = \frac{1}{2} \...
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1answer
117 views

Using the given identities, find the inverse DTFT

Using the given identities, $$ a^nu[n] \Longleftrightarrow \frac{1}{(1-ae^{-jw})}$$ and $$\delta[n-k]\Longleftrightarrow e^{-jwk}$$ Find the inverse DTFT of, $$ H(e^{jw}) = B\cdot\frac{e^{-jw}}{(...
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1answer
60 views

About the proof of an equality related to the DFT [sampling the DTFT to obtain the DFT]

This wiki page about the DTFT says that the DFT can be obtained from the DTFT by sampling the latter in one cycle at $N$ points: When the DTFT is continuous, a common practice is to compute an ...
3
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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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1answer
46 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: $$f(t_n),\;t_n=\dfrac{n}{F_s}$$ The definition of the DTFT is given by: $$F_n(\omega)=\sum_{n=0}^{N-1}f(t_n)\cdot e^{-i\omega_nt_n}$$ Now,...
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1answer
64 views

relation between DFT to CTFT

The signal $$x(t)\;\;\;\;0\leq t\leq 0.2s $$ We know that the CTFT of $x(t)=0$ when $|w|>2*\pi*10^4$ We sample $x(t)$ in sample space of $$T=25\mu s$$ or $$F_s=1/T=40000Hz$$and we get a series ...
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1answer
56 views

Finite sequence input to DTFT

i'm studying the practical utility of Fourier transforms and i have some questions. I hope to receive answers in layman terms. 1) Does the DTFT take only infinite input sequences? 2) If i apply the ...
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1answer
37 views

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. As I remember, the FT of a finite signal has an infinite ...
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1answer
24 views

Linearity and time-shifting of $\mathcal{F}\{0.8^n\cos(0.1πn)u[n]\}$

To preface, this is not a homework related question but purely for self-study purposes. Hi there, I try to calculate $\mathcal{F}\{0.8^n\cos(0.1πn)u[n]\}$ by using the properties of Discrete time ...
3
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1answer
1k views

Calculate the Inverse DTFT of the DTFT Derivative in Terms of $ x \left[ n \right] $

Consider the signal $ x \left[ n \right] $ and its DTFT transform $ X \left( {e}^{j \omega} \right) $. Assume $ X \left( {e}^{j \omega} \right) $ is differentiable. What is the Inverse DTFT of: $$ j ...
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1answer
102 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
76 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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2answers
51 views

What is the meaning of the DTFT of the unit impulse sequence?

In an exercice, I'm asked to draw the $X_{imp}(\omega)$ Discrete-Time Fourier Transform (DTFT) of the $x_{imp}(n)$ unit impulse sequence defined as: $$ x_{imp}(n) = \begin{cases} 1 & \text{if } ...
2
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2answers
31 views

Invertibility of Time-Dependent Fourier Transform

I am reading Oppenheim & Schafer's (O&S) Discrete Time Signal Processing (2nd or 3rd edition, does not matter) and I find hard to understand a technicality behind the Time-Dependent Fourier ...
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1answer
35 views

DTFT of window function applied to input signal

$$x[n] = cos(\omega_1n) + cos(\omega_2n)$$ $w[n] = 1/N$ for $0 \leq n < N, 0$ for everything else Find the DTFT of $y[n]=x[n]w[n]$ expressed by the DTFT of $w[n]$, $W(\omega)$ I was thinking ...
2
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1answer
87 views

Discrete Time Fourier Transform (DTFT) cross correlation property

I came across this property of the Discrete Time Fourier Transform (DTFT) and I am having a tough time proving it. In general, consider two real signals $x[n] \: \& \: y[n]$. If $$ x[n] \...
2
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1answer
67 views

Frequency estimation of circularly shifted single tone signal

I have a discrete signal $y[n] = <e^{j ~ 2 \pi f ~ n}>_J + ~w[n]$ with $n \in [0, N[$ and $w[n]$ AWGN, $<x[n]>_K$ denotes the signal $x[n]$ circularly shifted by $K$ samples. Let's define $...
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1answer
249 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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1answer
177 views

Find the period of a signal with the DTFT plot

I have an exercise and I'm struggling to resolve it. Here it is : My problem is about the DTFT. I've always been taught that we use DTFT for infinite-lenght signal that are not periodic (if the ...
0
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1answer
79 views

Continuous-time vs Discrete-time Fourier transform

case 1) to calculate the Fourier transform of discrete-time signal(sampled signal) we use Discrete-time Fourier transform. but my question is: case 2) if I consider that discrete-time signal as ...
1
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1answer
67 views

Method for determining probe angle by analyzing skewed sine wave

I've got a fun problem and would be curious to get feedback on how some of you would go about solving this. Imagine I have a probe and am scanning the surface of some material. This material surface ...
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0answers
41 views

z-transform and DTFT properties

I actually do not understand what to do with the third property of the impulse response g[n] and how it has to be determined. Thanks in advance!
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3answers
564 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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9answers
578 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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2answers
95 views

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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0answers
38 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 ...
3
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3answers
580 views

Question about z transform

After studying z transform from different books and literature on internet I want to ask few which makes me confuse. a) From the Discrete Time Fourier Transform we have drive equation for z ...
3
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2answers
303 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 ...
2
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1answer
314 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 ...
3
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2answers
116 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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4answers
1k 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 ...
1
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1answer
81 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
37 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
110 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 \...
2
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3answers
162 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
38 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 ...
0
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1answer
291 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
329 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 ...
2
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2answers
3k views

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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1answer
448 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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1answer
277 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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2answers
3k 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 ...
1
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1answer
66 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 ...
2
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1answer
659 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 ...
3
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3answers
702 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 ...
2
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1answer
848 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 ...
2
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1answer
217 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 ...
1
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1answer
64 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)\...
0
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1answer
43 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]=\...