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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1answer
276 views

What Is the Point of Doing the Zero Padding? [duplicate]

What are the advantages and disadvantages of doing Zero-padding, in particular the case of speech signals?
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211 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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115 views

Is there a scientific name for this technique?

The following algorithm takes a set of signals, and return a complex number: For a given set of signals - $\{s_1(n),s_2(n),...,s_M(n)\}$, compute the DFT of each of them - $\{S_1(k),S_2(k),...,S_M(k)\...
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198 views

Whitening signal vs. whitening its DFT

A whitening transformation (PCA) is simply a rotation into a space in which variables become uncorrelated. Because a DFT is a transformation into a coordinate space of orthogonal frequency components,...
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Understanding LPC for Formant Estimation

I went through the Matlab tutorial on Formant Estimation using LPC Coefficients. Though I vaguely understand the details, it's not entirely clear why we need to do this. From http://person2.sol.lu.se/...
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Expression for discrete fourier transform of linear ramp

I am trying to compute a single coefficient of the DFT of a linearly ramping sequence, $x[n]=an$ where $a$ is a constant that changes from sequence to seqeunce. I have looked at loads of DFT transform ...
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170 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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2k views

Difference between 2D DFT's and 1D DFT's of Linearized Matrices

I have recently left the safe and easy MATLAB environment and begun to use CUDA-C/C++ for image processing. Since CUDA doesn't allow 2D arrays to be passed into kernels I am now used to linearizing ...
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414 views

Exercise Related to Frequency Resolution and SNR

I'm studying a book about DSP and trying to make exercises. Here is one I'm interested in: A scientist acquires 65,536 samples from an experiment at a sampling rate of 1 MHz. He knows that the signal ...
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1answer
620 views

Sliding DFT: I think my results are accurate, but I can't buy a swap-swap IDFT

I've got a sliding DFT implementation that appears to be working (judging from an output plot). I would like to be able to invert this implementation using the standard tricks of swapping the real and ...
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Does chirp have constant magnitude frequency response?

Pg. 223 claims so, yet my results via DFT differ: Is the textbook wrong? My attempted explanations: (code) DFT vs DTFT: "frequency response" is computed via latter. Still, DFT should ...
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45 views

Cross Domain Equivalent to Nyquist Sampling Theorem?

In attempting to answer this question by @Oliver here: What characterizies 'causality' for a finite FFT? I have considered the minimum requirement to avoid time domain aliasing in the Discrete ...
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447 views

2D DFT in image processing in python

I try to compute 2D DFT in a greyscale image with this formula: I write the code bellow with python ...
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71 views

Leakage in power spectral density estimation

What does it mean for a power spectral density exhibiting leakes from one frequency bin to other. I am reading a book which states the following: "Often the time series we use as input for our ...
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2answers
172 views

What is the DFT of a pure cosine wave cos(θ)

I want to find a DFT of a pure cosine wave cos(θ) sampled at N equally spaced points on the interval $[0, 2\pi)$ so for our cosine wave, I put my $x$ like this $x=cos(\phi)$ then I just put it in ...
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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 $...
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323 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
383 views

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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201 views

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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58 views

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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1answer
311 views

Limitation on the shift theorem of DFT due to frequency resolution?

The shift theorem states that shifting a sine wave in time domain by t is equivalent to multiplying the corresponding DFT coefficient of the signal by a complex exponential e^(-jwt). Described by the ...
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1answer
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Expressing 2N point DFT in terms of N point DFT

I have a problem with expressing odd samples of X2 in terms of X1. I understand that the resulting DFT will be more precise in terms of expressing the exact spectrum of signal x[n], due to more ...
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119 views

How to do DFT of this signal

I am trying to get $X[k]$ when $x[n]$ is equal to $$x[n] = \cos\left(\tfrac{\pi}{4}n-\tfrac{\pi}{4}\right)$$ I'm using this equation: $$X[k] = \sum_{n=0}^{N-1}x[n]e^{-j \frac{2 \pi}{N}kn}$$ but ...
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Computation of the Inverse DCT (IDCT) Using DCT Or IFFT

Is there a way to compute the inverse discrete cosine transform (type-2) by leveraging either a DCT, FFT, or IFFT algorithm? I have seen ways to compute the DCT using FFTs, and I've seen ways to ...
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1answer
196 views

Understand Inverse DFT (FFT)

I was reading this article to understand the fft https://jackschaedler.github.io/circles-sines-signals/dft_walkthrough.html I understand that the fft takes a block (256 or 512) of the signal, ...
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775 views

DFT shift theorem proof

Learning DSP on my own time. Can't figure out the proof for DFT shift theorem which states the following: Given, $x[n]$ to be a periodic with period $N$, $\text{DFT}\{x[n]\} = X[k]$, then $$ DFT\{x[n-...
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Finding the correct $x[n]$ from a given DFT $x[k]$ [closed]

I'm struggling to solve this question from an exam. The question gives you a set of IDFTs from a given DFT, only one of those is the correct. The first thing I did is notice that $2\sin(\pi k)$ will ...
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Circular Convolution of Length L of Sequences of Length Greater Than L

I'm trying to understand how may I obtain the circular convolution of length L when the sequences I'm trying to convolve are of length greater than L. For instance this Matlab code using sequences ...
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251 views

Can I test any frequency with DFT?

Ladies, Gentlemen, Please let me the subject question, for I see in DFT equations testing frequencies sub-multiples of sampling frequency. Regards.
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3k views

The Phase Information of the DFT

As known, we can get a 'phase spectrum' from the DFT of a input signal. Assume, we do a DFT on a given equal interval sampled 50Hz AC signal, the signal is a complete whole cycle, but the start sample ...
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1answer
298 views

Applying Convolution (In 2D) Theorem Swaps Quadrants

For educational purposes I implemented the DFT and inverse DFT for images using OpenCV. Applying the DFT to an image and taking the inverse DFT of the spectrum yields the original image, so this works....
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538 views

Color Artifacts in Fourier Transformed Image

I am working with images and Fourier transforms. I am trying to understand what might be causing some artifacts in my output image. I am starting with a 512x512, RGB image of Lenna. I FFT the image,...
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Why would you use DCT as oppose to DFT for detecting oscillation in a signal?

For my project, I needed to detect the primary oscillation frequency in a signal data. As I googled around, it turns out that DCT is the best tool to find this (http://www.mech.pku.edu.cn/robot/...
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252 views

How to prove that Dilution of a time domain signal results in duplication in Frequncy domain?

While reading a description on FFT from this book, I found the following statement: ...diluting the time domain with zeros corresponds to a duplication of the frequency spectrum. How can I prove ...
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1answer
28 views

Proof that module of FT of 2 independent signals is sum of modules

I found on these posts (PSD subtraction and PSD of a sum of two stationary real signals) what I expected : that, just like the variance of the sum of 2 independent signals is the sum of the variances ...
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1answer
62 views

Amplitude estimation of sinusoid in known spiky spectral noise

What is the "best" way to estimate amplitude of a known-frequency sinusoid in the presence of known spiky spectral noise (i.e. noise comprising a few spectral peaks at known frequencies)? By "best", ...
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1answer
62 views

Goertzel derivation

Referring to this link, about Goertzel,I am confused about the final equation after N iterations Where does real = (q1 - q2 * cosine) and ...
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1answer
55 views

Evaluate DFT-downsampler-upsampler-IDFT block diagram

I am trying to solve the above question. I am not sure how to proceed. I know the formula for 64 point DFT of $x[n]$. $X[k]=\sum_{n=0}^{63} x[n] e^{-j2\pi nk/64}$ But how can I find $R[k]$ and $Y[k]...
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1answer
123 views

Representing stereo signal with complex numbers as input to DFT

Usually, when we have a stereo signal, we process each channel separately in order to extract the frequencies using Fourier transform. However, Fourier transform can also be applied to complex ...
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1answer
659 views

Relation between the DTFT and CTFT in sampling- sample period isn't as the impulse train period

I will quote an answer of Matt L. from this post (I didn't comment there because I can't) If you have a continuous-time signal $x(t)$, then the two signals you're talking about are $$\begin{...
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1answer
138 views

Which of DFT, DCT and DWT transforms is more robust to noise and geometric transformations in image processing [closed]

I am currently studying about image watermarking and I have been testing these 3 domains to embed the watermark at. My question is, how do these domains hold up to noise addition or geometric ...
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1answer
914 views

Circular vs Linear Convolution

When deriving DFT from DTFT,we sample the frequency domain with uniformly spaced samples,hence adding periodicity to time domain. But DFT requires a limited support: we take only 1 period. Does that ...
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1answer
537 views

How to perform a block-wise frequency down-shift?

I'm writing a Python function that shifts the frequency of an audio signal down to a specified range. I perform the downshift by multiplying the time series with a complex exponential, and then low-...
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1answer
83 views

FIR Filter design by non-equal samples

I'd like to design a Digital Differentiator with a pre-specified lowpass characteristic. I've tried Wiener Filters (using inverseFFT to transform from frequency to time-domain), Savitzky Golay and ...
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2answers
2k views

Upsampling DFT at frequency domain

Suppose we have a vector of length N of DFT, which every odd index (in frequency) equals 0: $$X[k] = 0\quad \forall k:\, k\mod2\equiv1$$ What does that mean for the series in the time domain?
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550 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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3answers
421 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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1answer
228 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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1answer
223 views

Performed the Danielson-Lanczos shuffling for FFT, but I don't know what to do next

I'm writing an audio analysis program and need to do some FFT to frames of the data. I've got some code (rather verbose so I'll leave out the details) that successfully performs the shuffling of the ...
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1answer
1k views

FIR filter design and implementation - sample rate and number of taps

I have been designing and using IIR filters to process audio for a while now but want to design some FIR filters for delay equalization. I have done some preliminary work using the following approach: ...

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