Questions tagged [dtft]
The dtft tag has no usage guidance.
79
questions
2
votes
2answers
26 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
34 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
53 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
$$ x[n] \...
2
votes
1answer
66 views
Frequency estimation of circularly shifted single tone signal
I have a discrete signal $y[n] = <e^{j ~ 2 \pi f ~ n}>_J + ~w[n]$ with $n \in [0, N[$ and $w[n]$ AWGN, $<x[n]>_K$ denotes the signal $x[n]$ circularly shifted by $K$ samples. Let's define $...
1
vote
1answer
124 views
Find the period of a signal with the DTFT plot
I have an exercise and I'm struggling to resolve it. Here it is :
My problem is about the DTFT. I've always been taught that we use DTFT for infinite-lenght signal that are not periodic (if the ...
0
votes
1answer
59 views
Continuous-time vs Discrete-time Fourier transform
case 1) to calculate the Fourier transform of discrete-time signal(sampled signal) we use Discrete-time Fourier transform.
but my question is:
case 2) if I consider that discrete-time signal as ...
0
votes
0answers
41 views
z-transform and DTFT properties
I actually do not understand what to do with the third property of the impulse response g[n] and how it has to be determined.
Thanks in advance!
1
vote
1answer
65 views
Method for determining probe angle by analyzing skewed sine wave
I've got a fun problem and would be curious to get feedback on how some of you would go about solving this.
Imagine I have a probe and am scanning the surface of some material. This material surface ...
0
votes
2answers
84 views
What is the interpretation of Fourier Transform containing only imaginary part?
The FT of a unit step function is taken as:
$$
X(\omega) = \int_0^\infty e^{-j\omega t}dt = \frac{-1}{jw}e^{-j\omega t} \Biggr |_{0}^{\infty} = \frac{j}{\omega}
$$
The transform only has the ...
0
votes
0answers
31 views
IFFT, negative frequencies and noncausal impulse
When using the IFFT to compute an approximation to the IDTFT, frequency range is 0 to 2$\pi$ versus $-\pi$ to $\pi$. This produces a different impulse sequence(causal I think).
How do you calculate ...
0
votes
1answer
81 views
Matlab FFT not producing symmetric spectrum
I am plotting a FFT of a sampled RC pulse but my resulting spectrum isn't symmetric - it's offset.
...
0
votes
1answer
59 views
Calculating DTFT
When calculating DTFT of (1/2)^n u[n]. We evaluate the sum as follows:
Please correct statements and answer questions below:
1) So to go from STEP 1 to STEP 2, the limits of the series are changed ...
2
votes
1answer
227 views
Proving that the IDTFT is the inverse of the DTFT?
The DTFT is given by:
$$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$$
I have been ...
1
vote
2answers
235 views
Proof that first difference filter amplifies noise
I'm a bit befuddled by noise's effect on derivative filters.
I've always 'known' that straightforward first difference derivative filters of discrete signals amplifies noise, but I'm struggling to ...
3
votes
2answers
100 views
Support of the convolution of two rectangular signals
I'm trying to convolve two rectangular signals in the frequency domain $$H_1(\omega) = u[\omega +.2\pi] - u[\omega -.2\pi]$$ and $$H_2(\omega) = u[\omega +.1\pi] - u[\omega -.1\pi]$$
My result is a ...
1
vote
1answer
75 views
DTFT of sawtooth wave through DTFT of rect signal
In a course i'm currently taking, the lecturer computed DTFT for the following signal:
$$r[n] = \begin{cases}
1& 0 \le n \le N\\
0& \mbox{otherwise}
\end{cases}
$$
For $N = 32$ i pictured $\...
0
votes
1answer
36 views
system function $H(\omega)$ relationship to odd and even components of h[n]
What qualities of $h[n]$ are necessary for:
$$
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
99 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:
$$X(e^{j \...
0
votes
1answer
36 views
Conversion between DTFT in radians/sample to DTFT in cycles/sample
I have found that most commonly the DTFT is defined as:
$X(\omega) = \sum_{n=-\infty}^{\infty} x[n]e^{-j \omega n}$.
However the class I am taking frequently uses the DTFT expressed in "normalized ...
1
vote
1answer
232 views
Relationship between the IDFT of a sampled DTFT and its discrete-time domain signal
Suppose we are given an input signal s[m,n] with DTFT $S(\omega_1, \omega_2)$.
We sample it at $\omega_1 = \frac{2 \pi k}{256}$ and $\omega_2 = \frac{2 \pi l}{256}$ to get a 256 point DFT S[k,l]. ...
2
votes
3answers
138 views
Maximum Magnitude Deviation between DFT and DTFT
Let $x[n]$ be a finite-length discrete-time signal with length $N$. The continuous DTFT $X(\omega)$ is then
$$
X(\omega) = \sum_{n = 0}^{N-1} x[n] e^{-j \omega n}.
$$
The length-$N$ DFT of $x[n]$ is
$...
0
votes
1answer
107 views
Using the given identities, find the inverse DTFT
Using the given identities,
$ a^nu[n]$ <===> $\frac{1}{(1-ae^{-jw})}$
$\delta[n-k]$ <===> $e^{-jwk}$
Find the inverse DTFT of,
$ H(e^{jw}) = B \frac{e^{-jw}}{(1-ae^{-jw})}$
my attempt:
$ ...
3
votes
1answer
328 views
Bridging CTFT and DTFT for a cosine
I'm trying to understand how I can start from the CTFT of a signal and end up with a DTFT.
For example if I take a basic example:
$$
x(t) = cos(\omega_x \cdot t) = \frac{1}{2} \cdot \left( e^{j\...
0
votes
1answer
284 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 ...
-1
votes
1answer
348 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 $$X(\tilde{ \omega }) = \sum_{n=...
0
votes
0answers
297 views
frequency spectrum of a sampled signal, PSD and power discussion
Before I go into my question, I first want to review the basics of sampling a signal and at the same time I build the basics of my questions so that they make more sense. I know I have asked couple of ...
1
vote
1answer
259 views
DTFT of $(-1)^n \cdot \mathrm{sinc}()$
I'm trying to find the DTFT of $$(-1)^n \cdot \frac{\sin (\pi n/2)}{\pi n}$$ I know the DTFT of $\frac{\sin \pi n/2}{\pi n}$ = a box function of amplitude 1, cutoff $\pi/2$. And I know that ...
1
vote
1answer
66 views
Question regarding DTFT of a complex signal
I have been doing DTFT practice problems for my DSP course, and I encountered this problem in the textbook that completely stumped me.
The question asks to find the DTFT of the shown signal and to ...
2
votes
1answer
581 views
Time scaling of discrete-time sequences and the DTFT
In the second edition of Signals and Systems by Alan Oppenheim, he discusses the DTFT of a "time-expanded" sequence that is effectively a slowed down version of the original sequence and can be ...
3
votes
3answers
620 views
Periodicity of the discrete-time Fourier Transform
The DTFT of a sequence $x[n]$ can be written as $$X(e^{j\omega}) = \sum_{n = -\infty}^{\infty} x[n] e^{-j\omega n}.$$
Is the smallest (fundamental) period in frequency of the DTFT always $2\pi$? Or ...
1
vote
1answer
61 views
IDTFT of $\sum_{k=-\infty}^{+\infty}(u(\Omega+\pi)+u(\Omega+\frac{\pi}{4})-u(\Omega-\frac{\pi}{4})-u(\Omega-\pi))\star \delta(\Omega-2k\pi)$
Compute the IDTFT of the following signal:
$$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
203 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:
$$H_1(\Omega)=\begin{cases}
10,& \text{for } \frac{\pi}{3} \leq |\Omega| < \pi\\
0,& \text{for } 0 \leq ...
11
votes
9answers
538 views
Where is the flaw in this derivation of the DTFT of the unit step sequence $u[n]$?
This question is related to this other question of mine where I ask for derivations of the discrete-time Fourier transform (DTFT) of the unit step sequence $u[n]$. During my search for derivations I ...
0
votes
1answer
43 views
DTFT fourier transform (modified property)
I know there are 3 properties of DTFT that help with my problem
$$ 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
2k 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
296 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
63 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
313 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
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. ...
2
votes
1answer
797 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
251 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
89 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
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’...
0
votes
1answer
220 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
4k 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
429 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
votes
1answer
139 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
523 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
312 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
260 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-...
