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Questions tagged [dft]

The Discrete Fourier Transform (DFT) is a mapping between a finite set of discrete points in a (primal) domain (time, space) and the dual frequency domain. DFT requires an input sequence which is discrete, such as a sampling from an analogue audio signal.

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How to express STFT and ISTFT as a 1d convolution and 1d deconvolution in tensorflow/keras

I'm trying to implement this paper in tensorflow and keras. At the end of section 3 it says. ...
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DFT equivalent circular convolution weight matrix with a symmetric filter of length 2K+1

$\DeclareMathOperator{\diag}{diag}$In a research paper, I read that: For optimization, the $n\times n$ weight matrix of DFT can be equivalent to circular convolution with a symmetric filter of length ...
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How to select the sign of the square root of each element of a DFT in obtaining the square root of a polynomial?

I want to find the square root of a polynomial by the following process: Compute the N-element DFT of its coefficients, maybe padded with zeros. Compute the complex square root of each of the N ...
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STFT Spectrum Analyzer: power drops over 15 kHz with songs, not with pure signals

I'm developing an audio file spectrum analyzer for a University Project. My main goal is to have an application that plots the Db Spectrum of a 16 Bit WAV PCM audio file (at this time only mono files) ...
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Division of sines as sinc

$\DeclareMathOperator\sinc{sinc}$I've come across a text that states that $\frac{\sin(\frac{3\pi}{10}n)}{\sin(\frac{\pi}{10}n)}, n=0\ldots9$ is a 'weighted' $\sinc$ function. Thus, its Fourier ...
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Resolution of Discrete Fourier Transform is 1/T - Mathematical proof?

In many articles I see that the frequency resolution of the Discrete Fourier Transform (DFT) equals Fs/N where Fs is the sampling rate and N is the total number of samples. Fs/N is equivalent to 1/T ...
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Cepstrum Calculation of Rational Function H(z)

I am trying to solve my first problems at cepstrum calculation. I want to calculate the complex cepstrum $\hat{h}[n]$ of a signal $h[n]$ with Z-Transform: $$H(z)=\frac{(1-0.5z^{-1})(1+4z^{-2})}{(1-0....
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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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Calculating N for DFT

We want to compute an N-point DFT of a one-second duration compact disc (CD) audio signal x[n], whose sample rate is $f_s = 44.1Khz$ with a DFT sampling of 1 Hz. (a) What is the number of necessary x[...
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How Could I extract the signal in the below Image

This is my first post so sorry if I make any mistake. The entire signal length in the screen shot below is 10 second. I ran a first order low pass filter (digital) at 10Hz cut-off frequency, and I ...
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Circular Convolution Matrix of $ {H}^{H} {H} $

We all know that Discrete Fourier Transform (DFT) corresponds to circular (not linear) convolution. That is to say, if $x(n),h(n)$ and $y(n)$ is the original signal, the filter and output signal in ...
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dft of sampled sine using python

I'm trying to write a python script to perform a 100-point DFT over a finite length sample of a sinewave at 1/8 the sampling frequency. I'm curious why my DFT magnitude plot has two spikes in it when ...
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PSD subtraction

I am interested seeing the difference between two power spectral densities (PSD) as a noise reduction exercise The blue line is the psd of my signal, and the the orange line is the psd of the ...
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Interpreting results of multiplying signal and wavelets in the frequency domain

I have followed a lot of stuff on mike cohens website and have managed to make a filterbank of Morlet wavelets and convolve with an incoming block of samples. This of course is a computationally heave ...
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Phase differences between DFT and FFT (in MatLab)

I have implemented a (direct) DFT in MatLab (following script) and compared it to the built-in FFT routine. The magnitude response seems to be identical (excluding some possibly round-off errors), but ...
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39 views

DTFT frequency range

$$X(e^{j\omega}) = \sum_{n=-\infty}^\infty x[n] e^{-j\omega n} $$ The frequency term $\omega$ in DTFT is normalized as $\omega = \frac{\Omega}{f_\mathrm{s}}$ $\Omega= 2 \pi f$ is the angular ...
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Equivalence of the Power Spectral Density definitions

I am trying to show the equivalence of the following Power Spectral Density definitions in Matlab: Definition 1: $$ P(\omega) = \sum_{k=-\infty}^{\infty} r(k)e^{-j\omega k} $$ Definition 2: $$ P(\...
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DFT of 2d real signal and Hermitian symmetry

Knowing that DFT of n-values real signal in 1d consists of n/2+1 different values where the second half of the spectrum is complex conjugate of the first one (Hermitian symmetry). However in the ...
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Criteria for Inverse STFT

So far I have studied the STFT and how it works. Following this question: Inverse Short Time Fourier Transform algorithm described in words , I got a grasp of how it works and how to implement it. ...
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Wavelet Transform Implementation Method

I have been watching the wonderful youtube videos here: https://www.youtube.com/watch?v=4TTpwIZrUAo&list=PLn0OLiymPak2G__qvavn3T8k7R8ssKxVr&index=3 detailing morlet wavelets and fit ...
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What should my reference value be when converting FFT bin amplitudes to dB?

I want to transform my FFT output values into a dB scale, but I'm struggling to determine the function I should run each bin amplitude through. My understanding of the decibel scale is that a value ...
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Processing using Matlab

I'm new to Matlab.As a beginner, I want to work on few signals which are used in day today life and extract useful information through it. Can someone please help me from where I could start or share ...
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Amplitude Estimating Using a Windowed DFT

Let's say we want to estimate the amplitude A of a mono-frequent signal using a windowed DFT. The frequency of the signal is unknown, and the frequency resolution of the DFT is limited, thus it cannot ...
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Significance of modular arithmetic in DFT?

In what ways does modular arithmetic plays a part in DFT? Why is it a so integral part of DFT?
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Help me understand the stages involved in filtering a signal using Discrete Fourier Transform

I have a series of discrete values measured from a sensor. I want to filter the frequencies coming from this sequence of values. Then, if I understood the process correctly this is what I do: I ...
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49 views

How can I find the value of Inverse DFT

I have dt signal $x[n]={[6.29, 8.11,-7.46,8.26,2.64,-8.04,-4.43,0.93,-9.29]}$ And I need to give the function value of: 1) sum of $x[n] = \sum\limits_{k=0}^{N-1} X[k] \, e^{-j 3 \pi k/5}$ from k=0 ...
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Is there a name for the result of summing the bins of an FFT?

Is there a name for the result of summing the bins of a DFT? I don't mean to sum the squares of the bins, but to simply add the magnitude of the frequency bins together to get a single result. Is ...
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37 views

Finding n Amplitudes by DFT, what is correct normalization

Please forgive me if this has already been asked. Let us assume an example with $x(t) = \sum_{i=1}^N A_i \sin(2\pi f_i t) $ under a given sampling frequency $f_s$, frequencies $\omega_i$ and ...
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In case of Complex DFT spectrum, why the x axis range from N/2 to N point mean a negative frequency?

When we transform a complex signal into frequency signal by using a complex DFT, The range from N/2 to N point on the X axis of the spectrum mean a negative frequency... But i cant understand why ...
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71 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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DFT of a series of RC exponentials

Context: I'm trying use matlab to apply a single-pole filter to a time-domain ramp waveform that is generated by a sequence of time-shifted "RC steps" that are added together. The time domain voltage ...
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31 views

Frequency Resolution Problem

Im trying to tackle the following problem while still not having a firm idea on what "frequency resolution" means : Suppose we sample a continuous time signal with sampling period Ts = 1/2000, and ...
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DFT sample point k < N has negative frequency

(From: Schaum's DSP outline, 2nd edition, page 254, problem 6.35) A signal $x_a(t)$ that is bandlimited to 10 kHz is sampled with a sampling frequency of $f_s = 20$ kHz. The DFT of N=1000 samples of ...
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Why is this recursive DFT algorithm not equivalent to this iterative method?

Edit 1/30 - Taking @Fat32's edits into account, it seems like there is still an issue with the scale of the frequency axis. While version 2 correctly identifies the response at 1 HZ, version 1 seems ...
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DFT zero-padding of signals starting before n=0

If a signal starts before n=0, what part of the signal should be used to compute DFT after zero-padding? For example, x(n) = {1, 2, 3, 4, 5}, where x(-2) = 1 and x(0) = 3. If this signal is zero-...
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56 views

time downsampling vs. frequency downsampling [closed]

$x[n]_M$ is a finite length sequence of length M. if: $$ y = x[nN]_M \tag{1} $$ is called downsampling in the time-domain. then what do you call the process of converting going from a M-point ...
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32 views

DFT product of sinusoids

(From Shaums DSP outline, 2nd edition, page 248, problem 6.21) Book says, evaluate the Sum: $$ S = \sum^{N-1}_{n=0} \Bigg( x_1[n] \ x^{*}_2[n] \Bigg) $$ when: $$ \begin{aligned} x_1[n] = \cos\left(...
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Estimators for improved spectral subtraction of noise

Real zero-mean Gaussian white noise, independent of a clean signal $x$ and of known variance is added to $x$ producing a noisy signal $y.$ Discrete Fourier transform (DFT) $Y$ of the noisy signal is ...
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80 views

downsampling DFT with aliasing

(Schuam's DSP Outline, 2nd edition, problem 6.11(c), page 241). Is there a DFT down-sampling property that looks something like this: Given $x[\![n]\!]_M$ we want to downsample from M to N to obtain ...
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DFT conjugate of $X^*[k]$, how to prove its formula in terms of $x^*[n]$?

Trying to prove that: $$ X^{*}[k] = \sum^{N-1}_{n=0} x^{*}\left((N-n)\right)_N\ W_N^{nk} $$ Where: $$((x))_N \text{ = x modulus N}$$ $$W_{N}^{nk} = e^{-j\ 2\pi / N}$$ So I start out with ...
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81 views

Why do DFT frequency buckets need to be divided by sample period?

edit: This is my first post in this community. I'm sure that my downvoter had a good reason to do so, but could someone please comment and tell me how I can better format my question? -- I have a ...
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33 views

Calculating phase of DFT sinewave?

I have been attempting to make a basic, slow, DFT in Matlab and have noticed peculiar behavior that I don't understand. I have been trying to plot the phase of a 100Hz sinewave captured at 250kHz ...
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Voice classification

I'm working to prepare research article for my project. While preparing for it, I've gone through the topics like Gaussian mixture model and Fourier transform for voice classification problems. I've ...
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153 views

What's the difference between “Discrete Fourier Series” and “Discrete Fourier Transform”? [duplicate]

I look at the equations for DFS (Discrete Fourier Series) and DFT (Discrete Fourier transform) and the only difference I notice is that one has a squiggle above the letter and the other doesn't. The ...
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Amplitude and Phase spectrum of a signal [closed]

Please help me to solve this question.
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40 views

Performing inverse DFT after taking conjugate of the result of DFT

The following is a question I got in my school assignment. Pick an image and follow the operations Multiply image by (−1)x+y. Compute the DFT. Take the complex conjugate of the ...
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performing FFT on Voltage measurments to get Z(f)

I have a csv file containing measurments a system's step response when the step is current and the output is the voltage. output looks something like this and the it's evenly sampled provided I know ...
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1answer
200 views

Performing DFT twice on an image. Why am I getting an inverted image? [duplicate]

I was asked to perform DFT on an image twice as a part of my school assignment. Why am I getting a blurry inverted image when I perform DFT on an image twice? Sorry, I'm new to image processing and ...
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1answer
66 views

Autocorrelation sequence in terms of Fourier transform of the underlying signal

Let $x(n)$ be a sequence of length $N$, which is zero outside the interval $(0,N-1)$. Let $X(k), k=0,1,\cdots,N-1$ be the FFT coefficients of $x(n)$, that is, $X(k)=\sum_{n=0}^{N-1}x(n) \exp\left( -\...
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Compute the two-dimensional DFT

Compute the two-dimensional DFT [4x4] for the following 4x4 image $ \begin{matrix} 0.5 & 0.5 & 0.5 & 0.5 \\ 0.5 & 0.5 & 0.5 & 0.5\\ 0.5 & 0.5 & 0.5 & 0.5\\ ...