Questions tagged [dtft]
The dtft tag has no usage guidance.
2
votes
1answer
54 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 ...
1
vote
2answers
35 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 ...
3
votes
2answers
63 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 ...
1
vote
1answer
35 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 $\...
0
votes
1answer
31 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:
$$
H(e^{j\omega}) = DTFT\{h_{even}[n]...
5
votes
1answer
68 views
Is possible reach the DFT if I have the DTFT?
My teacher told me that DFT is DTFT sampled, i.e.:
$$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 \...
0
votes
1answer
27 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 ...
1
vote
1answer
67 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]. ...
2
votes
3answers
77 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
$...
0
votes
1answer
42 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:
$ ...
2
votes
1answer
101 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:
$$
x(t) = cos(\omega_x \cdot t) = \frac{1}{2} \cdot \left( e^{j\...
0
votes
1answer
162 views
How can I show DTFT result in MATLAB? [closed]
I wanna show DTFT result and convolution result in f-domain are same.
Additionally, I wanna show sampled function and inverse fourier transform result are same too.
How can I show this? If I use ...
0
votes
0answers
9 views
Draw 2-D DTFTs from 2-D LSI system with impulse response
Consider the 2-D LSI system with impulse response h[n1,n2] and input x[n1,n2]. The 2-D DTFTs associated with these signals are H(ω1,ω2) and X(ω1,ω2) respectively. How do I figure out and sketch what X(...
-1
votes
1answer
113 views
The concept of normalized frequency
This question has already been asked and answered, but the motivation behind the use of normalized frequency units still evades me.
The Discrete Time Fourier Transform $$X(\tilde{ \omega }) = \sum_{n=...
0
votes
0answers
131 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 ...
0
votes
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 $|a| < 1$?
1
vote
1answer
116 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
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 ...
1
vote
1answer
145 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 ...
2
votes
3answers
310 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 ...
1
vote
1answer
47 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:
$$X(\Omega)=\sum_{k=-\infty}^{+\infty}\left(u(\Omega+\pi)+u\left(\Omega+\frac{\pi}{4}\right)-u\left(\Omega-\frac{\pi}{4}\right)-u(\Omega-\pi)\right)\...
2
votes
1answer
82 views
Inverse DTFT of $H_1(\Omega)=\begin{cases} 10,& \frac{\pi}{3} \leq |\Omega| < \pi\\ 0,& 0 \leq |\Omega| < \frac{\pi}{3}\\ \end{cases}$
What is the inverse DTFT of the $2\pi$-periodic extension of following function:
$$H_1(\Omega)=\begin{cases}
10,& \text{for } \frac{\pi}{3} \leq |\Omega| < \pi\\
0,& \text{for } 0 \leq ...
11
votes
9answers
382 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 ...
0
votes
1answer
39 views
DTFT fourier transform (modified property)
I know there are 3 properties of DTFT that help with my problem
$$ a^{n}u[n]=\frac{1}{1-ae^{-jΩ}} $$
$$ (n+1)a^{n}u[n]=\left(\frac{1}{1-ae^{-jΩ}}\right)^{2} $$
$$ \frac{(n+r-1)!}{n!(r-1)!}a^{n}u[n]=\...
9
votes
2answers
774 views
Discrete-time Fourier Transform of the unit step sequence $u[n]$
From text books we know that the DTFT of $u[n]$ is given by
$$U(\omega)=\pi\delta(\omega)+\frac{1}{1-e^{-j\omega}},\qquad -\pi\le\omega <\pi\tag{1}$$
However, I haven't seen a DSP textbook that ...
1
vote
1answer
137 views
Does the DTFT of $\frac{u[n-1]}{n}$ exist?
I have started learning DSP on my own and I have this doubt. I have done some googling but haven't found an answer. I hope that someone here would give the answer. It will be of great help.
0
votes
1answer
35 views
How is this was derived? DTFT for symmetric pulse
do not understand how the middle expression was derived from above expression,
after all summation\multiplication i get:
$$\frac{e^{-jwN} + e^{jwN} - e^{-jw(N+1)} + e^{jw(N+1)}}{2 - e^{-jw} - e^{jw}...
3
votes
2answers
210 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 ...
1
vote
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. ...
0
votes
0answers
124 views
Finding the output sequence given non-causal input and impulse response
Given the impulse response $$h(n)=0.9^{|n|}$$ and the input signal $$x(n)=3\cos\left(\frac{\pi n}{2}+\frac{\pi}{3}\right)$$ find the output signal.
I found the frequency response to be
$$H(e^{j\...
0
votes
0answers
33 views
DTFT problem.Need answer
Can any one suggest the method in which above problem can be solved.If the answer is know please guide me to solve this
2
votes
1answer
529 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 ...
1
vote
2answers
183 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!)
...
0
votes
0answers
71 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 ...
2
votes
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’...
0
votes
1answer
165 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
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)/...
1
vote
2answers
367 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}$)?
0
votes
0answers
182 views
Energy of output signal
so I got stuck at the following problem:
I've got an input signal $x[n]$ that goes through an ideal LPF, and an output signal $a[n]$. The input signal has the following DTFT:$$X(e^{j\omega})=0.1+0....
-1
votes
1answer
113 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 ?
0
votes
2answers
443 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
1answer
223 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 ...
-2
votes
1answer
157 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-...
0
votes
1answer
222 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-...
2
votes
1answer
224 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 $1-e^{j\omega }$?
1
vote
1answer
768 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 $...
0
votes
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 ...
0
votes
1answer
541 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 ...
-1
votes
1answer
470 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
votes
2answers
513 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:
$$...
