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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39 views

Proof of DTFT equal to DFT when signal is periodic?

I was using the Wikipedia page on the discrete time Fourier transform to understand the connection between DFT and DTFT. The following is claimed in the article - I was wondering if anyone had a proof ...
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Why does a longer observation time improve DFT resolution, but repeating a signal does not?

As was proven here: https://math.stackexchange.com/questions/228614/why-doesnt-repeating-a-signal-give-rise-to-a-finer-resolution-of-dft-fft repeating a certain sequence does not improve DFT frequency ...
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3answers
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Summation limits in DFT

Assume a discrete time signal $(x_n)$ is given. Some texts define the DFT as $$ X[k] = \sum_{n=-N}^N x_n\exp\left(\frac{-2\pi j k n}{N}\right) $$ while others define it as $$X[k] = \sum_{n=0}^{N-1} ...
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Does it matter if the scaling term is in the DFT versus the inverse DFT?

From Wikipedia, the equation for the 1D DFT is From a separate source, the equation for the 2D DFT is Notice how the 2D DFT definition features a scaling term while the first definition does not. Is ...
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1answer
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What is the effect of a larger window size on “temporal resolution”?

I guess my question boils down to what "temporal resolution" means? I'm taking a signal processing class right now and we're learning about DFT and windowing at the moment. We've learned ...
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Finding natural frequency from vibration analyzer data

I was asked to find the natural frequency from a waveshape obtained from vibration analyzer, and I performed an FFT and told that the peak represents the natural frequency. Now I am called as an ...
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47 views

Is FFT sub matrix degeneracy a problem in OFDM when there is noise in the frequency domain?

Say that the discrete Fourier transform (DFT) is used in OFDM. There are a number of degenerate (singular, non invertible) sub matrices of some DFT matrices. Does this result in any problems? One ...
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1answer
28 views

Amplitude in frequency domain of the same signal measured at different bandwidth (f_max - f_min) and different FFT lines

Fourier series in complex form Fourier series in trigonometry form Here, I'm measuring the same signal (an impulse force - or at least as close as possible to an impulse, represented by a knocking ...
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What is “filter periodization”?

A library defines periodize_filter_fourier, which is an equi-spaced averaging formulated by $$ v_f[k] = \sum_{i=0}^{\text{n_periods}-1} h_f[i\cdot N + k], $$ where $v_f$ is periodization of $h_f$, $N=\...
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Confusion about spectrograms: conceptual problem

The following code generates a spectrogram as shown in the image. The matrix s which is the spectrogram is of size 129x40 for a ...
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1answer
39 views

Finding the discrete time fourier coefficients to this problem

I'm trying to find the fourier series to this discrete time signal. $$x_1[n] =\begin{cases} +\frac72&\text{if }0\le n \le 4\\ -\frac72&\text{if }5\le n \le 9 \end{cases}$$ My approach: We see ...
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Represent DFT coefficients with respect to Continuous time-Fourier series coefficients

Does anyone know how to represent the Discrete Fourier transform (DFT) coefficient, $X[k]$, with respect to the Continuous time-Fourier series (CT-FT) coefficient, $X_k$? I come to the conclusion as $...
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Approximate using DFT phase shifting property

I have a discrete signal $x(n)$ having $N$ samples with DFT $X(n)$. Here $N$ is large say $N=600$. Let the samples of $x(n)$ be, $x(n) = \big[x(0), x(1), x(2), ..., x(N-1)\big]$. But if suppose ...
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1answer
64 views

Importance of Phase in FFT of an image

While processing digital images in the Fourier domain, mostly we exploit the amplitude and not the phase. This could be because the amplitude is much more structured and the amplitude spectrum reveals ...
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228 views

Problems computing the DFT of finite length sequence

I am having trouble finding the same answer as the solution manual for this sequence. The problem asks to compute the DFT of $$ x[n] = \begin{cases} 1 & \text{for even } n \in \{0\...
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1answer
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Why do the DTFT and FFT give me completely different results for magnitude at a specific frequency?

I am trying to write a program to compute the magnitude and phase of a specific, non-integer frequency component (i.e. given a sampled finite signal of length $N$, I want to know the magnitude and ...
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1answer
44 views

What is the correct length for obtaining a true linear convolution from DFT?

In the linear convolution of two equal length sequences M and N, the length of the output is length(A)+length(B)-1, and if we apply the DFT property of converting convolution into multiplication, the ...
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136 views

In the context of DFT, Where Does the Nyquist Frequency Sample Belong In a Double Sided Frequency Spectrum (Positive / Negative Side)?

If we have an even number of data points $N$, after DFT in MATLAB, the output has the order: $$(\text{DC}, f_1, f_2, \ldots, f_{N/2-1}, f_\text{Nyq}, -f_{N/2-1}, -f_{N/2-2}, \ldots, -f_1)$$ For real ...
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Properly stagger an FFT? How to handle output data (Have highs weigh as much as lows?)

I'm new to audio processing so I'm mostly running into problems with not being able to properly google something. Sorry. Basically I want to create something like an equalizer (just visually without ...
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1answer
179 views

The Proper Way to Do Sinc Downsampling (DFT Downsampling) for Uniformly Sampled Discrete Signals with Finite Number of Samples

Given a signal $ \left\{ x [ 0 ], x [ 1 ], ..., x [ N - 1 ] \right\} $ what would be the correct way to downsample it in the frequency domain (Sinc interpolation)?
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83 views

Difference between $\tt fft$ and $\tt dftmtx$ in MATLAB

I have the following MATLAB code: ...
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67 views

Normalization factor in the convolution theorem

Maybe it's a trivial question, but I couldn't find any explanation for this. According to the convolution theorem, in the continues case we add normalization factor, i.e. $$ \mathcal F\left\{h\star g\...
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2answers
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Non-zero DFT components where zero is expected?

I am implementing DFT in Octave. Here's my code: ...
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Derivative of cosine at Nyquist

is negative sine, or zero, which is trouble; the imaginary DFT basis (sine) is likewise zero. Is there a way to meaningfully define the derivative of cosine when sampled at $f_s/2$ (such that it's not ...
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Blackman window magnitude attenuation

I am trying to compute the fundamental phasor using sliding window DFT. I have employed a Blackman window in conjunction i.e $$ \sum_{k=0}^{L_{DFT}-1}x(k) w(k) e^{-j2\pi k/N} $$ where $x(k)$ is the ...
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1answer
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Where 2-D DFT is necessary in image processing applications instead of 2-D DCT?

As we know the 2-D DCT is a real-valued kernel and less computational complexity and it used in several image processing applications like image compression etc. 2-D DFT is a complex kernel and high ...
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Why the inverse discrete fourier transform of the Ricker pulse isn't the same as the Ricker pulse in time domain?

Question I'm trying to use Python's scipy library to compute the IDFT of the Ricker wavelet function and compare it with the analytical time-domain version of the same function. When I compare the ...
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1answer
38 views

A basic question regarding frequency analysis of an EEG signal

Assume that an EEG signal is sampled at $f_s = 300$ Hz then a 10000-point segment of it is selected, called $x[n]$. The corresponding 10000-point DFT is then computed and called $X[k]$. Assume further ...
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1answer
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DFT Signal DFT Length N , FFT

If We sample an Signal let say sine(2 * pi * f) with f=1Hz and a sampling Frequecy of Fs = 8Hz, is it right that the length of the data schoul be N = Fs/f or multiple of Fs/f like N= d*(Fs/f) with d=...
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How to get the phase/angle from an Alglib complex variable

It is easy to get the magnitude using alglib.abscomplex function (https://radfiz.org.ua/files/temp/Lab1_16/alglib-3.4.0.csharp/csharp/manual.csharp.html#gs_stdfunctions), but I would like to know, how ...
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5answers
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How is the DTFT of a periodic, sampled signal linked to the DFT?

I am trying to understand the connection between FT, DTFT and ultimately the DFT. But I am failing to link the DTFT to the DFT. This is how far I am getting: Say I have a function $f(t)$, and its ...
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1answer
39 views

Strange behavior from frequency to angular frequency in FFT

I have written a code to compute the Fast Fourier Transform of a simple complex exponential with frequency $f=50.0$, using scipy.fft. The code is written below: ...
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1answer
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Spectral leakage from mathematical point of view

Does anyone can explain or propose a reference that mathematically and quantitively investigates the effect of leakage on the magnitude and phase of a signal?
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Discrete Fourier transform of a finite length signal which saturates at non-zero value

I am performing spectral analysis of a finite length signal that saturates to a non-zero value. The signal ($s(t)$) can, practically, be write as $s(t) = f(t) \big(1-H(t-t_0)\big) $, where $t_0$ is ...
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2answers
134 views

How is wavelet time & frequency resolution computed?

Mallat gives analytic wavelet time & frequency widths/uncertainties as $$ \begin{align} \sigma_{ts}^2 &= \int_{-\infty}^{\infty} (t - u)^2 |\psi_{u, s}(t)|^2 dt = s^2 \sigma_t^...
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2answers
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Multiplying the imaginary part of DFT with a linear ramp to get a derivative

I am trying to understand the statement in a relatively old publication from 1970s, when Fourier transforms found applications in chemical analysis. The author quotes the derivative theorem citing ...
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How to successfully resolve multiple pulsation frequencies in the FFT?

I am dealing with signals that in both extreme cases can have: a) Non-overlapping monochromatic pulsation-like events at different points within a time series. b) Overlapping pulsations at multiple ...
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1answer
30 views

Find Magnitude of a DFT signal using a Blackman window

Hi I've been given a signal made of a series of cosines. I have taken the DFT of the signal using a rectangular window (blue), hamming window (red) and a Blackman window (black). I have identified two ...
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3answers
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Do symmetric discrete signals have zero phase?

I generated a Hanning window having an even symmetry: where for even-sampled case we either take "left" and "right" to include or exclude the center sample. I was surprised to ...
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3answers
229 views

Interpreting N in DFT as the Number of Points vs. Number of Intervals

The "N" is DFT is understood to be the number of data points in a given sequence or in other words the length of the sequence. We recently have had discussions here Indexing in DFT (from an ...
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120 views

Why Is the Total Time Equal to $ N \cdot {T}_{s} $ and Not $ \left( N - 1 \right) \cdot {T}_{s} $ In the Context of DFT?

In the definitions of the DFT DFT $$ X(j)=\sum_{k=0}^{N-1} x(k) \exp \left(-i 2 \pi\left(\frac{j}{N}\right) k\right) $$ Let us say, if we have $10$ points, $N=10$, each sampled at $0.2$ seconds, why ...
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1answer
45 views

Indexing in DFT (from an old paper)

There is a nice paper on explaining DFT from the 1960s in IEEE A guided tour of the fast Fourier transform. The author uses the following definitions of DFT DFT $$ X(j)=\sum_{k=0}^{N-1} x(k) \exp \...
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1answer
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How to remove the modulation before doing frequency offset estimation?

To use DFT/FFT (or maximum likelihood) method to estimate the frequency offset introduced by the channel, we need to remove the modulation on the received data samples in the front. If the unknown ...
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3answers
2k views

Applying DFT twice does not actually reverse an array. Instead, the first element stays in place while the rest of the array is reversed. Why?

I've heard a million times that applying DFT twice will result in a reversed array, but that is not what actually happens. Instead, the first element remains where it was, and the rest of the elements ...
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1answer
49 views

How to detect resonance with DFT

In the last paragraph of this article, the author gives an example of the DFT of a song and his window's resonant frequency. As he points out (and it is a bit intuitive), if he listens to this song, ...
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2answers
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Why should the last point be excluded when performing a least-squares fit of a periodic discrete time signal?

I fitted the function: f(t)=A_o+A_1 cos(wt)+B_1 sin(wt) to the following periodic discrete signal: ...
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67 views

Linear convolution in the DFT domain

Let's say I have 2 sequences a and b in the time domain. Both are length N. A and B are the DFT of a and b. If I do a circular convolution of A and B in freq domain (A o B), then the IDFT of the ...
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1answer
120 views

Why DFT is used for approximating CTFT when you can approximate CTFT-integral itself?

I was using MATLAB for approximating FTs. Why DFT is used if we can approximate the transform-integration using summation.
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1answer
53 views

Discrete Fourier style transform with non-sinusoidal/arbitrary waveform components

A discrete Fourier transform will produce frequency samples sufficient to recreate the original time sampled signal from sinusoidal waveforms. That works great as long as the original signal is ...
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1answer
41 views

Confusion regarding frequency of Discrete Time Periodic Signal

While Analyzing the DFT Plot for a signal with N samples, say we find a peak in the magnitude of the DFT plot at index $k$, this implies that our signal has a high amount of similarity with an ...

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