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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2k views

Spectral Analysis of a Time Series with Missing Data Points

I use a PC to record time series of some physical property. The problem is that, for some reason, I did not record the time series as a whole, rather I record first segment, then second, third, etc. ...
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127 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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1answer
53 views

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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3answers
207 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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6answers
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Definition of the DFT / FFT Bin Size

When reconstructing the frequency domain for an FFT, what is the most self-consistent way to do this -- i.e. how is it best to define the bin width, $\Delta f$? For example, previously I thought it a ...
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2answers
99 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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2answers
142 views

Fourier Like Spectral Analysis with Uneven Intervals and Redesigned DFT Matrix

I intended to use a discrete Fourier transform (DFT) on a time series sampled at uneven intervals. What I did was to calculate a DFT matrix where the elements are the values at the uneven locations ...
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2answers
1k views

Sinc Interpolation Using DFT (FFT)

Lets say I want to double the number of points in an array f. I had the idea to do this: F=fft(f);N=length(f); FF=[F(1:N/2) zeros(1,N) F(N/2+1:N)]; f=ifft(FF); ...
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1answer
38 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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How do I calculate peak amplitude of the signal components after zero padding and FFT?

I am learning about DFT and trying to apply it to some audio processing. I am new to DSP but experienced in programming and have some background in math and physics. The FFT algorithm I use (lomontFFT)...
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25 views

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
134 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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1answer
387 views

Comparing arithmetic complexity of FFT radix-2 and convolution

Let's assume we have a discrete linear time invariant system and we have a real signal $x[n]$ with length N=50 as input for the system. The impulse response $h[n]$ of the system is considered to be ...
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2answers
76 views

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

I have the following MATLAB code: ...
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2answers
94 views

Non-zero DFT components where zero is expected?

I am implementing DFT in Octave. Here's my code: ...
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2answers
39 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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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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536 views

Absolute convergence of periodic sinc interpolation

An $N$-periodic complex discrete-time sequence $[x_0, \dots, x_{N-1}]$ can be resampled to an $M$-periodic sequence $[y_0, \dots, y_{M-1}]$ with $M>N$, using sinc interpolation: $$\begin{align}y_m ...
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1answer
35 views

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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2answers
184 views

Why Zero Padding in the Center of the DFT Interpolates / Upsamples the Signal (Sinc Interpolation / DFT Interpolation / Periodic Interpolation)

I'm experimenting with the Inverse Discrete Fourier Transform. Starting from the two-cycles continuous $x(t)$ signal below: I have the discrete signal $x(n) = \{ 1, 0, -1, 0, 1, 0, -1, 0 \}$ leading ...
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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
35 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
39 views

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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5answers
198 views

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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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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72 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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1answer
82 views

DFT and integer valued basis functions'sf requencies

In Matlab the function W=dftmtx(N) gives the DFT matrix of size N. Each row is computed for an integer frequency k. $W_{k,n} = e^{-i2\pi kn/N}$, k-th frequency, ...
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1answer
35 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
35 views

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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1answer
169 views

Spatial spectrum of EEG data: non-uniform DFT?

I want to compute the spatial spectrum of EEG data collected using a non-uniformly sampled grid of sensors (as in the figure below). One way to do this would be to interpolate the data on a ...
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2answers
5k views

Z-transform of a downsampler

In this paper or multirate filtering, the author establishes the following mathematical relationship. Let $y_D$ be the output of a downsampler such that $$y_D[n] = x[Mn]$$ where $M$ is the ...
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1answer
125 views

resilient codes to phase shift

I need to define an encoding to beacon IDs. The codes are going to be periodic but in the receiver side I can have phase shifts so I' wondering the best way to decode: - 8 bits code groups that ...
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2answers
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The difference between DFT and DFS

In the literature, I've found that DFS and DFT are one and the same. If they are one and the same why to use two different names for them? If there is really a difference what is it and what is the ...
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1answer
18 views

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
1k views

harmonics of a signal and FFT

Consider that we have a discrete signal of finite length. How can we find the amplitude and phase corresponding to different harmonics of this signal in Matlab? Thanks.
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Correct magnitude spectra of a cosine DFT?

I've just started my course on DSP and haven't laid my hands on MATLAB yet. I was wondering if the plot of the magnitude spectra was correct for the below shown $x(n)$:
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91 views

Why is it assumed that $x[n]$ is limited from $0$ to $N-1$ while evaluating DFT?

I am a total beginner in this topic of DFT. I get that the series must be finite for DFT calculation. But everywhere we are assuming that this series must be limited from $0$ to $N-1$. How to evaluate ...
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1answer
32 views

Find signal power in a band by integrating the PSD in a frequency band (and zeroing the out-of-band DFT bins)

I know that directly zero DFT bins outside a frequency band has the side effect of introducing ringing, as this post says Why is it a bad idea to filter by zeroing out FFT bins?. But what about ...
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2answers
149 views

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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1answer
628 views

Mapping the Value of a Sample in a 2D DFT to Cycles/Pixel

If I have an image and its 2-D DFT of that image, what is the mapping between the value of the DFT at (u,v), and the frequency in the spatial domain in the x and y components, in cycles/pixel? I want ...
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3answers
641 views

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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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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2answers
275 views

Prove that, in DFT, Upsampling in time domain is equal to replication in frequency domain

Given: $x[n]$ is an $N$-point sequence whose DFT is $X[k]$ $$x[n]\xrightarrow{\mathcal{DFT}} X[k]$$ then, Prove that: DFT of the same sequence after insertion of $(M-1)$ zeroes between successive ...
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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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5answers
8k views

How to get Fourier coefficients to draw any shape using DFT?

I'm teaching myself about Fourier Series and the DFT and trying to draw a stylised $\pi$ symbol by fourier epicycles as detailed by Mathologer on youtube (from 18:39 onwards), and the excellent ...
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1answer
7k views

How to calculate the power of a discrete signal? / Clarification on PSD estimates

I want to calculate the channel power $ P_\mathrm{x}$ of a given discrete and complex signal $x[n]$ (with a length of N) in a given bandwidth $B$. I'm aware, that I could probably apply a sharp ...
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1answer
511 views

How to Combine 8 N/8 FFT's into one N FFT

I need to make in FPGA (using Verilog) an FFT. Input data is N=8192 points at 1 GSPS. However, the FPGA operates at 125 MHz, therefore the data is split into 8 channels (each one at 125 MHz). This ...
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1answer
52 views

Bin sizes for non-uniform discrete Fourier transforms

For a non-uniform discrete Fourier transforms, do the specified frequencies – i.e., $f_k$ in – refer to the midpoint of the bin or the lower bound? I read the answer here, but that stated that ...
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3answers
80 views

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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