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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119
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6answers
70k views

What does frequency domain denote in case of images?

I was just learning about the frequency domain in images. I can understand the frequency spectrum in case of waves. It denotes what frequencies are present in a wave. If we draw the frequency ...
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3answers
69k views

Why is the FFT “mirrored”?

If you do an FFT plot of a simple signal, like: t = 0:0.01:1 ; N = max(size(t)); x = 1 + sin( 2*pi*t ) ; y = abs( fft( x ) ) ; stem( N*t, y ) 1Hz sinusoid + DC ...
5
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1answer
5k views

What happens when N increases in N-point DFT [duplicate]

I am curious about DFT, and I wrote a simple MATLAB code to test what happens when $N$ increases. I took a rectangular signal with length $L=15$, an then found th DFT of 16, 32 and 64 points. I looked ...
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4answers
6k 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 ...
11
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2answers
19k views

What is normalized frequency

I am working on DSP and finding difficulty to understand the term Normalized frequency often used with DFT & DTFT. What is normalized frequency in DSP ? and how it is different from analog ...
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2answers
207 views

DFT exercise in the book Understanding digital signal processing 3 Ed

I am trying to solve exercises from the book Understanding digital signal processing 3 Ed - Richard Lyons. I will repeat the question as it is in the book: 3.3 We want to calculate an N-point DFT ...
2
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3answers
4k views

How to calculate resolution of DFT with Hamming/Hann window?

The frequency resolution of a DFT with a rectangular window of size $N$ is given by $f_s/N$. However, when using other window functions like a Hamming or Hanning window the resolution gets worse. How ...
6
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1answer
2k views

Zero Padding of FFT

There are many question related to the zero padding a time domain signal to get more frequency bins after performing Fourier transform. As I understand this process is equivalent to trigonometric ...
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2answers
80 views

Does zero-padding distort the spectrum?

It's said to "sample the DTFT", revealing what "DFT fails to see". And I fail to see how this sampling isn't distortion. The "spectrum" aims to provide the sinusoidal ...
0
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1answer
565 views

Using fourier coefficients to reconstruct data in matlab

When doing a discrete fourier transform on some data using matlab's fft function, its output is a set of fourier coefficients but I was wondering how do I go about converting these into an and bn so I ...
12
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3answers
2k views

Why do I have frequency leakage in DFT after zero padding if frequency resolution is fine?

Let's consider this example: Fs=1000; Ns=500; t=0:1/Fs:(Ns-1)*1/Fs; f1=10; f2=400; x=5+5*sin(2*pi*f1*t)+2*sin(2*pi*f2*t); X=fft(x); In this scenario, frequency ...
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9answers
13k views

Is there an algorithm for finding a frequency without DFT or FFT?

I was looking in the Android app store for a guitar tuner. I found a tuner app that claimed it was faster than other apps. It claimed it could find the frequency without using the DFT (I wish I still ...
10
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1answer
12k views

What is the meaning of the DFT? [duplicate]

Possible Duplicate: Real Discrete Fourier Transform What is the most lucid, intuitive explanation for the various FTs - CFT, DFT, DTFT and the Fourier Series? Discrete-time Fourier transform ...
7
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3answers
25k views

Number of DFT (FFT) Points Required for a Specific Frequency Resolution for an Oversampled Signal

I have a bandpass signal centered at 2 MHz and bandwidth of 50 kHz (the signal frequency varies from 2 MHz - 25 kHz to 2 MHz + 25 kHz). This signal is being sampled at 10 MHz. I want a frequency ...
4
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3answers
494 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 ...
3
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2answers
778 views

Symmetry and periodicity of frequency-shifted discrete Fourier transform

The unitary discrete Fourier transform (DFT) of a sequence of numbers $x_n$ to $X_k,$ with integer $0 \le n < N$ and $0 \le k < N,$ can be defined as: $$X_k = \frac{1}{\sqrt{N}} \sum_{n=0}^{N-1} ...
4
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1answer
6k views

Non Uniform FFT / DFT with FFTW?

Can FFTW perform the FFT on data which is not uniform in time? I can't seem to find a straight-forward answer to this question anywhere online.
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1answer
85 views

PyWavelets CWT: normalization? Vs Scipy?

Related. The equation being implemented normalizes by sqrt(1 / scale): $$ C_{a, b} = \frac{1}{\sqrt{a}} \sum_k s(k)\left( \int_{-\infty}^{k+1} \overline{\psi \left(\...
0
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3answers
189 views

DFT coefficients meaning?

What "are" they? What's a sensible way to interpret the coefficients (and what isn't)? To pose specifics: DFT coefficients describe the frequencies present in a signal They describe the ...
80
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4answers
65k views

What is the difference between a Fourier transform and a cosine transform?

In speech recognition, the front end generally does signal processing to allow feature extraction from the audio stream. A discrete Fourier transform (DFT) is applied twice in this process. The first ...
12
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7answers
12k views

Why Does the DFT Assume the Transformed Signal Is Periodic?

In many signal processing books, it is claimed that the DFT assumes the transformed signal to be periodic (and that this is the reason why spectral leakage for example may occur). Now, if you look at ...
15
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3answers
17k views

When to use the DTFT vs the DFT (and their inverses) in analysis?

In many of my readings, whenever some author speaks about working in the frequency (transform) domain (of a digital signal), they often times take the DFT, or the DTFT, (and of course their ...
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2answers
5k views

Frequency-domain zero padding - special treatment of X[N/2]

Suppose we wish to interpolate a periodic signal with an even number of samples (e.g. N=8) by zero-padding in the frequency domain. Let the DFT X=[A,B,C,D,E,F,G,H] ...
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2answers
2k views

Could a DCT be used for an audio magnitude spectrum rather than DFT?

From what I understand, the DCT has half the bin size as a DFT of the same size N. The DFT also includes phase information, but often this is not needed when only the magnitude spectrum is desired. ...
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2answers
1k views

Different way to separate a particular frequency from a signal

Let's say I have a signal which is composed of 2Hz,10Hz,17Hz,19Hz and 25Hz discrete time sinusoidal waves and I need to allow only 17Hz component to pass. As far as I know, the standard way to ...
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2answers
2k views

Interpolation of magnitude of discrete Fourier transform (DFT)

For example for peak frequency finding, it seems valid to use band-limited interpolation methods on the complex DFT bins, or separately on their real and imaginary parts and to calculate the ...
3
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1answer
1k 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
52 views

PyWavelets CWT implementation

I seek to understand PyWavelets' implementation of the Continuous Wavelet Transform, and how it compares to the more 'basic' version I've coded and provided here. In particular: How is integrated ...
0
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1answer
2k views

Specific Frequency Resolution

I have an audio signal SampleRate Fs: 44100 Hz TotalSamples: 94144 samples Duration t: 2.1348 s The frequency resolution is given by ...
5
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2answers
155 views

Amplitude after Fourier transform

How to obtain the correct amplitude after the numerical Fourier transform of a signal? Example: consider an exponential decaying wave $y(x)=e^{-x}\sin(100\pi x)$ with Fourier transform $y_f(x_f)$ ...
3
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2answers
414 views

Should the phase coefficients of the DFT of a real, even input signal all be zero?

The phase coefficients of a real, even input signals should all apparently be $0$ or a multiple of $\pi$. That's a property of the DFT I've learned about in the Audio Signal processing course on ...
18
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3answers
8k views

Why are magnitudes normalised during synthesis (IDFT), not analysis (DFT)?

In most examples and FFT code that I've seen, the output (frequency magnitudes) of the forward DFT operation is scaled by N -- i.e. instead of giving you the magnitude of each frequency bin, it gives ...
12
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1answer
10k views

Real discrete Fourier transform

I am trying to understand the real DFT and the DFT and why the distinction exists. From what I know so far the DFT uses $e^{i2\pi kn/N}$ for basis vectors and gives the representation $$x[n]=\sum_{k=...
13
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2answers
13k views

Why do we say that “zero-padding doesn't really increase frequency resolution”

Here is a sinusoid of frequency f = 236.4 Hz (it is 10 milliseconds long; it has N=441 points at sampling rate ...
12
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3answers
4k views

STFT and DWT (Wavelets)

STFT can be successfully used on sound data (with a .wav soundfile for example) in order to do some frequency-domain modifications (example : noise removal). With ...
7
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1answer
2k views

Failed to implement Goertzel algorithm in Python

After some questioning on stackoverflow, I tried to implement a Goertzel algorithm in Python. But it doesn't work : https://gist.github.com/4128537 ...
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2answers
7k views

How can I compute a log-spaced power spectrum?

I would like to compute a power spectrum in which the frequencies are logarithmically spaced. In Welch's method there is a trade-off between the frequency resolution of the resulting power spectrum ...
3
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2answers
2k views

Frequency Analysis (DFT / FFT) of a Signal Without a Constant Sampling Frequency (Non Uniform Sampling in Time Domain)

I'm a stack exchange user for some time and now I'm registering to ask a simple question (I think!). I have a vibration signal with an amplitude and time (sampling frequency not constant) in a $...
6
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2answers
10k views

Frequency resolution and timestep in DFT

I am learning DFT and I am not sure I understand it correctly. I am just going to write out the process of DFT and I am asking you to tell me if I am thinking the right way. Suppose we have a signal ...
4
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1answer
6k views

The necessity of null subcarrier in OFDM?

In looking to LTE specifications, with the subcarrier spacing $\Delta f = 15 \textrm{kHz}$, for bandwidth $10 \textrm{MHz}$, fft size $N_{fft} = 1024$ that needs a sampling rate $F_s \geq 1024 \times ...
5
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3answers
956 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 the ...
3
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2answers
8k views

Hermitian symmetry in OFDM systems

I am trying to understand the usage of Hermitian symmetry in OFDM systems and have a couple of questions regarding this. What is the reason of using the Hermitian symmetry in OFDM? How can we arrange ...
3
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2answers
678 views

A system that perfoms Fourier Transform operation - is it an LTI system?

If a system takes input as the time domain signal and outputs the frequency domain signal, is such a system an LTI system? For if the input time domain signal can be represented as a linear ...
2
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2answers
123 views

Effect of changing sample rate, window duration and zero padding on DTFT and DFT

Let $T$ be the window duration, $N$ be the DFT size, $F_s$ be the sample rate, and $F_{max}$ be the frequency of the highest bin. In the context of image below: halving the $F_s$ (keeping $T$ ...
4
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1answer
8k views

Difference Between Linear Convolution and Circular Convolution

If I understood correctly (and this page should confirm: http://www.cs.ioc.ee/~khoros2/linear/convolution-teo/front-page.html) if I convolve linearly (the usual point-to-point multiplication and ...
4
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2answers
411 views

please give the reason why every notation for DFT is valid?

$X$ represents sample in frequency domain and $x$ represents samples in time domain. NOTATION 1 $ X[k] = \sum\limits_{n=0}^{N-1} x[n] \ e^{-j \frac{2\pi}{N} n k} $ $ x[n] = \frac{1}{N} \sum\limits_{...
3
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2answers
2k views

Link between DFS, DFT, DTFT

My understanding of DFT is as follows For a signal $x[n]$ of finite-length, the DFT is DFS of the periodic extension, $\tilde{x}[n]$, of that signal $x[n]$ and also another way to view DFT is that it’...
2
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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 ...
4
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1answer
1k views

Recursive version of DFT as presented in Cooley-Tukey paper

The seminal paper of Cooley and Tukey provides an iterative method for computing the DFT for a sequence of length $N$. Specifically, they mention a method which utilizes the fact that $N$ can be ...
4
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1answer
448 views

Discrete Fourier Transform a dummy approach

I'm trying to understand how DFT works exactly. However, when experimenting around, I compared both Matlab generated FFT result with a dummy approach result and I get similar result. However, the ...