Questions tagged [dtft]
The dtft tag has no usage guidance.
87
questions
0
votes
1answer
59 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 ...
1
vote
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 ...
0
votes
1answer
55 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 ...
0
votes
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 ...
1
vote
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 ...
1
vote
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, ...
2
votes
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:
$$
x_{imp}(n) = \begin{cases}
1 & \text{if } ...
2
votes
2answers
31 views
Invertibility of Time-Dependent Fourier Transform
I am reading Oppenheim & Schafer's (O&S) Discrete Time Signal Processing (2nd or 3rd edition, does not matter) and I find hard to understand a technicality behind the Time-Dependent Fourier ...
0
votes
1answer
35 views
DTFT of window function applied to input signal
$$x[n] = cos(\omega_1n) + cos(\omega_2n)$$
$w[n] = 1/N$ for $0 \leq n < N, 0$ for everything else
Find the DTFT of $y[n]=x[n]w[n]$ expressed by the DTFT of $w[n]$, $W(\omega)$
I was thinking ...
2
votes
1answer
83 views
Discrete Time Fourier Transform (DTFT) cross correlation property
I came across this property of the Discrete Time Fourier Transform (DTFT) and I am having a tough time proving it.
In general, consider two real signals $x[n] \: \& \: y[n]$. If
$$ x[n] \...
2
votes
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 $...
1
vote
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
votes
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 ...
0
votes
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!
1
vote
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 ...
0
votes
2answers
94 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
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 ...
0
votes
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.
...
0
votes
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 ...
2
votes
1answer
311 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
votes
2answers
302 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
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 ...
1
vote
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 $\...
0
votes
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:
$$
H(e^{j\omega}) = DTFT\{h_{even}[n]...
5
votes
1answer
110 views
Is possible reach the DFT if I have the DTFT?
My teacher told me that DFT is DTFT sampled, i.e.:
$$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 \...
0
votes
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 ...
1
vote
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]. ...
2
votes
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
$...
0
votes
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,
$$ H(e^{jw}) = B\cdot\frac{e^{-jw}}{(...
3
votes
1answer
403 views
Bridging CTFT and DTFT for a cosine
I'm trying to understand how I can start from the CTFT of a signal and end up with a DTFT.
For example if I take a basic example:
$$\begin{aligned}
x(t) &= \cos(\omega_x \cdot t) = \frac{1}{2} \...
0
votes
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 ...
3
votes
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:
$$ j ...
-1
votes
1answer
444 views
The concept of normalized frequency
This question has already been asked and answered, but the motivation behind the use of normalized frequency units still evades me.
The Discrete Time Fourier Transform $$X(\tilde{ \omega }) = \sum_{n=...
0
votes
0answers
328 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 ...
1
vote
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 ...
1
vote
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
votes
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
votes
3answers
699 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
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:
$$X(\Omega)=\sum_{k=-\infty}^{+\infty}\left(u(\Omega+\pi)+u\left(\Omega+\frac{\pi}{4}\right)-u\left(\Omega-\frac{\pi}{4}\right)-u(\Omega-\pi)\right)\...
2
votes
1answer
216 views
Inverse DTFT of $H_1(\Omega)=\begin{cases} 10,& \frac{\pi}{3} \leq |\Omega| < \pi\\ 0,& 0 \leq |\Omega| < \frac{\pi}{3}\\ \end{cases}$
What is the inverse DTFT of the $2\pi$-periodic extension of following function:
$$H_1(\Omega)=\begin{cases}
10,& \text{for } \frac{\pi}{3} \leq |\Omega| < \pi\\
0,& \text{for } 0 \leq ...
11
votes
9answers
576 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
43 views
DTFT fourier transform (modified property)
I know there are 3 properties of DTFT that help with my problem
$$ a^{n}u[n]=\frac{1}{1-ae^{-jΩ}} $$
$$ (n+1)a^{n}u[n]=\left(\frac{1}{1-ae^{-jΩ}}\right)^{2} $$
$$ \frac{(n+r-1)!}{n!(r-1)!}a^{n}u[n]=\...
10
votes
2answers
3k views
Discrete-time Fourier Transform of the unit step sequence $u[n]$
From text books we know that the DTFT of $u[n]$ is given by
$$U(\omega)=\pi\delta(\omega)+\frac{1}{1-e^{-j\omega}},\qquad -\pi\le\omega <\pi\tag{1}$$
However, I haven't seen a DSP textbook that ...
1
vote
1answer
314 views
Does the DTFT of $\frac{u[n-1]}{n}$ exist?
I have started learning DSP on my own and I have this doubt. I have done some googling but haven't found an answer. I hope that someone here would give the answer. It will be of great help.
0
votes
1answer
66 views
How is this was derived? DTFT for symmetric pulse
do not understand how the middle expression was derived from above expression,
after all summation\multiplication i get:
$$\frac{e^{-jwN} + e^{jwN} - e^{-jw(N+1)} + e^{jw(N+1)}}{2 - e^{-jw} - e^{jw}...
3
votes
2answers
337 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
81 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. ...
2
votes
1answer
847 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
260 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
90 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 ...
