Questions tagged [dtft]

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1answer
29 views

Is my solution to the DTFT of a delta function correct?

I am newer to signal processing math and just figured out something cool (hopefully). I was trying to see how the DTFT of the delta function is 1, because thats what my book says. I could only find ...
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1answer
23 views

DTFT and a Downsampled Sinc Function

I found the answers to this question and this question to be extremely helpful in understanding the derivation of the downsampling or decimation property of the DTFT. Thank you! I am now struggling ...
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0answers
21 views

Find the inverse DTFT of the following signal: [closed]

Can someone please help me out here in solving this question.
2
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2answers
38 views

Question on N point DTFT - Fourier transform

I have been trying to use the logic that both X and Y should have same Z transform, but according to the definition, Y is not anti causal. So, I am clueless!
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2answers
45 views

Continuous frequency vs discrete frequency? [closed]

I have understood idea of discrete time and continuous time but I am feeling difficult to comprehend this idea in regard to frequency. As for example the DFT output is discrete and the DTFT output ...
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1answer
49 views

Evaluating discrete spectral density at only a few frequencies

I'm trying to obtain the spectral density at three particular frequencies for a computational chemistry problem that I'm working on (if you are curious, it has to do with the estimation of Nuclear ...
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0answers
27 views

$2\pi$ Periodicity is not working for me for Fourier of Discrete Time Signal

please help me find the error in the following counter example. Consider we take sinus with period of $2\pi$. We sample it many time, and more than 3. We make convolution with rectangle of height 1 ...
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0answers
43 views

A DT sequence y[n] is constructed from another DT sequence x[n]

A DT sequence $y[n]$ is constructed from another DT sequence $x[n]$ according to the formula $y[n]=x[nN]$, where $N$ is a constant positive integer greater than one. (This process is usually called ...
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1answer
38 views

Why is there a negative in front of the phase response equation for this complex exponential?

first time on here! I'm working through "Digital Signal Processing using MATLAB" by Vinay and Proakis. Good book. I am stuck on this example tho. Shouldn't the imaginary part in the denominator (...
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0answers
12 views

How do you change an instance of dsp.DigitalDownConverter object in MATLAB to work with filters other than the ones it is originally defined with?

If you look at this website: https://www.mathworks.com/help/dsp/ref/dsp.digitaldownconverter-system-object.html you will see an example (with code) that attempts to up convert and down convert a ...
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1answer
17 views

Deriving expression for the DTFT of a rectangular window

Looking at the picture above, how did the author get from point A) to B)? My Approach: Multiply A) by $e^{j\omega/2}/e^{j\omega/2}$. Now I am stuck with simplying the numerator.
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3answers
58 views

DFT of the same signal with different values of N

Let $x[n]$ be a discrete signal of 2 samples. We know that its DFT with N=4 is $X[k]=[0, 1+j, 2, 1-j]$. Without calculating $x[n]$, how can we know the DFT with N=2? I have tried to use the relation ...
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1answer
90 views

CTFT to DTFT why can't you always just substitute $\Omega = \omega/T_S$

This is something I've always wondered about in DSP class, but just accept as a fact because I never really fully understand why this is the case: Given CTFT: $$X_s(j\Omega) = 6000 \pi \sum \limits_{...
2
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2answers
81 views

Effect of changing sample rate, window duration and zero padding on DTFT and DFT

Let $T$ be the window duration, $N$ be the DFT size, $F_s$ be the sample rate, and $F_{max}$ be the frequency of the highest bin. In the context of image below: halving the $F_s$ (keeping $T$ ...
0
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1answer
67 views

Ideal high pass filter for discrete signal

there. I currently get stuck on a question. I was asking to find an inverse discrete-time Fourier transform for the ideal high pass filter. Here is the question It is getting more confused after I ...
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0answers
22 views

Characterizing a non-LTI system

How should we characterize a non-LTI system? For example we have: $y[n]=x[3n]+x[2n]+x[n]$ which is clearly not LTI. Also, the impulse response will be $h[n]=3\delta[n]$ and if we take the DTFT of this ...
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2answers
95 views

Getting the DTFT from the DFT samples

How would you get the DTFT from the DFT samples? How will the DFT indexes map to the discrete frequency and what kind of an interpolation would be required?
0
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1answer
46 views

What is the formula for the frequency spectrum?

A signal $f[n]$ is given, the corresponding DTFT as $F(e^{j\omega})$ and a plot of the frequency spectrum $f(t)$. Unfortunately I can't find a formula for the frequency spectrum in my documents. When ...
1
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1answer
111 views

Real-valued DTFT

Now this is a simple question, but I still ask it for clarification: I know that an even signal $$h[n] = h[-n]$$ results in a real-valued DTFT (we have proven that in class). Now my question is the ...
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1answer
35 views

Orthogonality of filter impulse response to its even shift

I meet this problem but still do not know how to solve it. Could you guy give me some guides? Upsampling by 2 ($U_2$) followed by filtering by $g$, with operator $G$ And given: $<g_n,g_{n-2k}>...
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1answer
24 views

DTFT of inverse of any function

In my book solution is given like this. But i am solving like this , am i doing wrong??
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1answer
47 views

From Orthogonality to DTFT

I have met this question, but cannot prove it using DTFT definition. Given: $g$ is a discrete sequence filter and: $$ g \in l^2(Z)$$ $$\langle g_n, g_{n-2k} \rangle = \delta_{k}$$ Prove: $$|G(e^{j \...
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1answer
79 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 ...
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1answer
80 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
80 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
61 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
27 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 ...
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1answer
60 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, ...
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2answers
143 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
36 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
46 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
131 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
74 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
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1answer
524 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
124 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 ...
2
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1answer
75 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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2answers
150 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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1answer
124 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
124 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
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1answer
514 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 ...
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2answers
434 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
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2answers
150 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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1answer
109 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
42 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
144 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 \...
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1answer
41 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 ...
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1answer
282 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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3answers
200 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
156 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}}{(...
4
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1answer
520 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} \...