Questions tagged [dtft]

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11
votes
9answers
452 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 ...
9
votes
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 ...
5
votes
1answer
87 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 \...
3
votes
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 ...
3
votes
2answers
265 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
votes
1answer
375 views

Why is this DFT of a real symmetric signal resulting in complex valued coefficients?

I am trying to understand exactly how sampling the DTFT to get the DFT works. The signal I'm trying to analyze is $x(n)$ seen below. $$x(n) = \delta(n\pm2) + 2\delta(n\pm1) + 3\delta(n)$$ Taking the ...
3
votes
2answers
79 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 ...
3
votes
2answers
2k views

Difference between Fourier Transform and DFT? - Example

I have read many excellent answers to similar questions, but never one this specific. Here is another way to ask it. Why is the modulation transfer function (MTF) of $\textrm{rect}(x/5) = \textrm{...
3
votes
3answers
426 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
votes
1answer
500 views

2D Fourier Transform of Rotated Discrete Domain Signal

Assume we know that the Fourier transform of a signal $x(n_1,n_2)$ is $\mathcal{F}(x(n_1,n_2))=X(\omega_1,\omega_2)$. What is the Fourier transform of the signal after being transformed by a rotation ...
2
votes
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^...
2
votes
1answer
79 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 ...
2
votes
1answer
2k 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’...
2
votes
3answers
107 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 $...
2
votes
1answer
275 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 $1-e^{j\omega }$?
2
votes
1answer
183 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\...
2
votes
3answers
467 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
votes
1answer
159 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 ...
2
votes
3answers
181 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?
2
votes
1answer
653 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
votes
1answer
293 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 ...
2
votes
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=-\...
1
vote
1answer
206 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.
1
vote
4answers
880 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
vote
2answers
80 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. ...
1
vote
1answer
60 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 ...
1
vote
2answers
243 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!) ...
1
vote
2answers
2k 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 ...
1
vote
1answer
156 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 ...
1
vote
2answers
401 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 $i = \sqrt{-1}$)?
1
vote
2answers
137 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 ...
1
vote
1answer
57 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 $\...
1
vote
1answer
155 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]. ...
1
vote
1answer
64 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 ...
1
vote
1answer
296 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 ...
1
vote
1answer
54 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)\...
1
vote
1answer
1k 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 $...
1
vote
1answer
450 views

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) ...
1
vote
1answer
177 views

Inverse DTFT Problem

Having trouble finding the inverse DTFT of $\ X(\ e^{j \omega}) = \frac{3 - \frac{1}{4} e^{-j\omega}}{1 - \frac{1}{4} e^{-2j\omega}} $ Given the IDFT of $Xe^{j \omega}$ as : $x(n) = \frac{1}{2\pi} ...
1
vote
1answer
28 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 ...
1
vote
1answer
255 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-...
1
vote
1answer
1k views

DTFT and Inverse DTFT Homework Problem

I'm trying to solve this signals homework problem: So for part a, since multiplication in the time domain is convolution in the frequency domain, I just used a DTFT table, found the DTFT for $\left(\...
1
vote
1answer
262 views

Finding values of DTFT without explicitly computing

My attempt : a) Summation of all values? b)c)d) Failed e) Parserval's theorem
0
votes
2answers
597 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: $$...
0
votes
3answers
4k 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 ...
0
votes
1answer
34 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]...
0
votes
1answer
199 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 ...
0
votes
2answers
482 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 ...
0
votes
2answers
71 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 ...
0
votes
1answer
47 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 ...