The Discrete Fourier Transform (DFT) is a mapping between a finite set of discrete points in the frequency and time domains. DFT requires an input function which is discrete and non-zero, such as a sampling from an analogue audio signal.

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Mathematical justification for zero padding?

This question asks what's the point of zero padding. The accepted answer is certainly very insightful, but I don't understand a big chunk of it: Zero padding allows one to use a longer FFT, which ...
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29 views

DFT and periodicity

My problem is related to the periodicity of DFT. Having the following expression $$ Y_{k}=\sum_{n=0}^{2N-1}e^{-j\frac{2\pi mk}{2N}} $$ I can easly find that the upper function is $2N$ periodic. So ...
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21 views

Difference between PSD estimate and variance of DFT

In Bartlett's PSD estimate one averages over L segments of the squared DFT coefficients. From wikipedia I found this formula $$ \textrm{PSD}(k) = \frac{1}{L}\sum_{l=1}^{L} \frac{1}{M} \lvert X^{[l]}(...
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Butterworth filtering behaving unexpectedly -MATLAB

I'm new to signal processing but I'm trying to understand how the Butterworth filter works. To do so, I did the following: I assumed a sampling frequency of 100Hz and wanted to attentuate everything ...
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1answer
51 views

What can we understand when we increase the $N$-point DFT for a given signal?

I have an input sequence $$x(n)=\cos(0.48\pi n)+\cos(0.52\pi n).$$ I am determining its spectrum (amplitude of DFT values) based on the finite number of samples. For Example : Taking the first $10$ ...
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37 views

PSD looks to clean/continuous

I generated an signal with a base frequency of $50\textrm{ Hz}$ and its harmonics and quantization noise in it (as an array in MATLAB). It's sampled fast enough to make sure there is no aliasing etc. ...
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40 views

Integer and a half periods of sine wave in DFT window

Why does a pure sine wave starting and ending at 0 in the sample window of a DFT but containing 1.5 cycles (or periods) produce the same spectral leakage (i.e. not more) as a pure sine wave with ...
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The mathematical interpretation of DFT [closed]

We have the DFT(matrix form) $X = Wx$ ($W$ is the Fourier basis matrix, $x$ is the original signal in time domain, $X$ is in the frequency domain). In mathematics, $x$ represents the coordinates of $...
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59 views

Frequency resolution of DFT

I know that the frequency resolution in a DFT is given by sampling rate upon the total number of samples. $$\Delta f = \frac{f_s}{N}$$ But, what I want to know is where does this relation comes from? ...
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50 views

Proving DFT equation

How can I prove the equation $(3)$? I can't understand why there is a $2/N$ in $(3)$. Why he just get the second term in DFT? Consider a sinusoidal input signal of frequency $\omega$ given by ...
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Difference between Fourier Transform and DFT? - Example

I have read many excellent answers to similar questions, but never one this specific. Here is another way to ask it. Why is the modulation transfer function (MTF) of $\textrm{rect}(x/5) = \textrm{...
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61 views

DFT of a continuous quasi-periodic signal

If I have a continuous quasi-periodic signal and I want to do a DFT of signal then what could be appropriate length of continuous signal that I should consider for sampling? I want to do DFT of the ...
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25 views

what is the reason for the Change in DFT analysis after downsampling

firstly the original signal's dft analysis is this the signal is sammpled at 1000Hz the signal is then downsampled under different rates such as 25Hz this is the dft analysis of the signal ...
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1answer
30 views

Sub Band DFT response

With a DFT lets say I had a signal occupying 0Hz to 2048Hz and i am using 2048 bins. Can a DFT by design (say FFTW) only calculate the responses in a Sub Band like 768Hz to 1280Hz or -64Hz to -256Hz ...
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43 views

DFT analysis of a physical signal

what is the reason for the result that has been obtained from the DFT of the signal of these two images. The first image is a DFT analysis of a physical signal u which has a time history shown below ...
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1answer
46 views

Derive DFT of $x[2n]$ [closed]

Question is to find DFT, $G_7(k)$ of $g_7(n)$ = $x(2n)$. For purpose of illustration only, we can assume that $x(n)$ can be represented by envelope shown in corresponding figure. Shown in Figure ...
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How to convert a spatial frequency in a 2D-DFT into the units radians per pixel?

Let's say I have a 2D image, and I take the discrete Fourier Transform (via FFT) of that image. In the frequency domain, I get the following image: In this image, let's just assume all the spatial ...
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1answer
52 views

Computation of Only Even or Odd Frequency Bins of DFT

I have an algorithm where I am computing the FFT of a large signal. However, I desire only the even or odd terms of the DFT of the signal, but not both. Currently, I discard these undesired terms. Is ...
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1answer
49 views

Difference between fft(signal, nfft) and fft(signal)

I want to know the difference between these following commands (matlab) and why I get two different outputs? fft(signal); fft(signal, nfft); I found that ...
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1answer
30 views

Complex Conjugate Sinusoids in Forward DFT

I hope this isn't such a dumb question, but I'm finally getting to grips with the inner workings of the DFT. What I'm having trouble understanding is why the basis complex sinusoids in the "forward" ...
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34 views

What does it mean for the DFT phase to be relative to a cosine wave?

The following paragraph from Understanding Digital Signal Processing got me puzzled: The answer is: The DFT phase at the frequency $mf_s/N$ is relative to a cosine wave at that same frequency of ...
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1answer
29 views

Deriving the DFT magnitude of $A\cos(2\pi nk/N)$

Given that $$x(n) = A\cos(2\pi nk/N),$$ the $N$-point DFT of $x(n)$ can be expressed as follows—the derivation can be found in here: $$X(m) = \color{red}{\frac{A}{2}\sum_{n=1}^{N-1}e^{-j2\pi n(m-k)/N}}...
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31 views

Where does $\frac{N}{2}$ came from in approximating an N-point DFT?

I've came across the author saying that ... for a real cosine input having k cycles in the N-point input time sequence, the amplitude response of an N-point DFT bin in terms of the bin index m is ...
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48 views

Why is N-point DFT approximated by the sinc function?

While looking into DFT leakage, I've came across the author saying that "..., the amplitude response of an N-point DFT bin in terms of the bin index m is approximated by the sinc function." ...
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50 views

How to do the highest resolution FFT?

I wish to do the FFT of streaming audio data coming into a computer. I am just interested in frequencies from 80 Hz up to 1000 Hz but with a frequency resolution of 3 to 4 Hz and no more. Basically ...
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74 views

Performing DFT of streaming audio problem. Is there a limit?

I am trying to write software that will perform the discrete fourier transform of real time data coming from the microphone into the sound card on a computer. I am using Java with the javax.sound APIs....
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49 views

Possible to shift image pixel-wise with FFT?

I know we can shift an image entirely with FFT as http://stackoverflow.com/questions/25827916/matlab-shifting-an-image-using-fft However, is it possible to shift the image pixel-wise? I just found a ...
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91 views

DFT of discrete signals, why do we only analyze frequency bins equal to number of input samples?

If we have a signal $x[n]$ such that we have $N$ samples i.e. $n=0, \ldots, N-1$, then when we analyze the DFT $X[k]$ we only analyze for $k=0,\dots,N-1$ as well. Why is the range of $k$ tied to ...
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99 views

Why are the basis functions for DFT so?

When you get a DFT of a signal, you use the basis functions as: $e^{-j2\pi kn/N}$ Why is it so? Why don't we use the conjugate, $e^{j2\pi kn/N}$, or any other function?
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109 views

DFT of real sinusoids - why sum over -$N/2$ to $N/2-1$ as opposed to $0$ to $N-1$?

I'm going through a Coursera course on signal processing, and we're just introduced to DFTs. We are told that if you have a complex sinusoidal signal $x[n]$ where $n=0,1...\ N-1$, its DFT is given ...
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74 views

Effect of cyclic prefix and zero-padding in OFDM

What happens if zeros are added instead of cyclic prefix in OFDM system? How does it affect performance of system?
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29 views

How to get N point DFT using two 2N point DFT

If I have 2 8 point DFT, how will I compute 4 point DFT using this. And how can I compute 8 point DFT using 2 4 point DFT. Do we need to do zero padding for this?
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28 views

How to find Galois Field $\textrm{GF}(p)$ from any $n$-point DFT

How to find a finite field $\textrm{GF}(p)$ (where $p$ is a prime number) (here I want to find the value of $p$ only)that is as small as possible and efficient computation can be performed for (let's ...
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139 views

Extracting accurate phase and amplitude information from FFT with an arbitrary number of samples

This is a followup question to one I asked earlier based on the chat after the answer given by @hotpaw2, and cross-posted from stackoverflow since it was suggested it is more relevant to DSP. I have a ...
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1answer
73 views

What is the physical significance of the Fourier coefficients in the DFT of audio signals and how can they be best displayed in a spectrogram?

I am trying to display the spectrogram of an input audio signal (a newcomer to all things spectral here). The amplitudes of my audio signal can be assumed to be normalized between -1 and +1. Suppose ...
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2answers
38 views

non-uniform sampling on the unit circle - applications

If there is an N-point sequence, it has a Fourier Transform. DFT of this sequence is the sampling of the Fourier Transform at N equally spaced points on the unit circle. What happens if we take some ...
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68 views

What is plotted against the DFT of a windowed sampled signal?

I'm given a continuous-time analog signal $x_a(t) = \cos(2\pi f_1t)+\sin(2\pi f_2t)$, for some frequency $f_1, f_2$. I'm asked to sample $x_a(t)$ at $F_s=1024\textrm{ Hz}$, apply a 128-point Hamming ...
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39 views

Discrete Fourier Transform for text analysis?

I would like to determine the number of text-blocks verifying a roughly similar pattern. I have the intuition that I could do it using some Discrete Fourier Transform (DFT)-like methodology. Example: ...
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63 views

Would it be feasible to implement the Sliding DFT on a digital signal processor?

The Sliding DFT generates the spectrum for every input sample. Using the FFT, the spectrum is generated only after a certain number of samples (N) are obtained. It appears to me as though the cycles ...
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66 views

Number of samples needed to convert a analog->digital, frequency bound, signal with decent quality to the Fourier domain

Similiar question(s) have been asked before, but I didn't fully get the answer I was looking for, so I figured I'd ask again. My excuses if there's something really basic in signal processing I've ...
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DFT have independent variable in between 0 to $\frac{1}{2}$ no of samples

I was learning of Fourier Transform from this book 'Scientist and Engineer Guide to DSP'. Suppose we have a digital periodic signal and there are 128 of samples of it in the time domain i.e X[0] to X[...
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67 views

Transfer function of linear phase filter & its frequency domain representation

Text of my exercise request: Determine the transfer function $H(z)$ of a causal linear phase FIR filter with zeros at $z= \frac{1}{3}$ and $z=-2$. The value of the impulse response at $n=0$ equals ...
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1answer
28 views

Discrete signals in spectrum to Analog : DFT

An analog signal $x(t)$ was sampled for a period $T$ of $1 \text{ms}$. The resulting, discrete signal is denoted as $x[n]$. Now you are given the spectrum $X[k]$ of the sampled signal $x[n]$. Doubts ...
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1answer
34 views

IDFT of H(z) sampled in N values

If a have a causal IIR filter described by $H(z)$ and I sample it in $N$ equispaced values around the unit circle, I get a DFT of $N$ points. That DFT corresponds to $h[n]$ truncated in $n=N-1$ or to ...
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91 views

Time scaling in DFT?

Let $x[n]$ be a real signal whose length $N$ is even. Let $y[n]=x[2n]+jx[2n+1]$ a signal whose length is $M=\frac{N}{2}$. $X[k]$ for $k=0,...,N-1$ and $Y[k]$ for $k=0,...,M-1$ are the respective DFTs ...
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1answer
37 views

Analysis of a LTI system using DFT

Consider an LTI system $$H(z)=1-\frac{1}{2}z^{-1}+\frac{3}{4}z^{-2}$$ Let $x[n]=(\frac{1}{3})^n\cdot u[n]$ be the input signal. It is desired to determine the output for $n=0,1...,N_a$. To achieve ...
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40 views

DFT of a cosine/sine wave?

I'm seeing everywhere that the DFT of a sinusoid (i.e. a sine that has been windowed with a rectangular window) are just two deltas. What I don't understand is why the DFT is not the convolution of ...
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33 views

Applying Filter to Complex Data

I have data being streamed in the time domain, I would like to use a transfer function for a low pass filter in order to filter the data. I use DFT to get the data to the frequency domain, in the form ...
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1answer
72 views

Determine sequence's value from equally null circular convolution DFT

Can anyone help with this exercise from my Signal Processing exam? The text is: Given two sequences $x$ and $y$ whose supports are respectively $[0,1]$ and $[0,3]$ and their circular convolution'...
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89 views

How to realize DFT6 with 3 DFT2 blocks?

Is it possible to realize DFT of 6 samples by using 3 DFT2 blocks? For example if had 2 * N samples by one step of decimation in frequency or time is possible to reduce DFT(2N) to 2 * DFT(N). So if we ...