Questions tagged [fourier-transform]

The Fourier transform is a mathematical operation that decomposes a function into its constituent frequencies, known as a frequency spectrum.

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About Discrete Fourier Transform vs. Discrete Fourier Series

I am new to the field of signal processing. I am wondering what is the difference between DFS(Fourier Series) vs. DFT(Fourier Transform). For common applications, usually we get a segment(length <...
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3 answers
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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 ...
1 vote
3 answers
814 views

Does Zero Padding Distort the Spectrum of a Signal?

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 ...
140 votes
8 answers
169k views

Why is the Fourier transform so important?

Everyone discusses the Fourier transform when discussing signal processing. Why is it so important to signal processing and what does it tell us about the signal? Does it only apply to digital signal ...
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38 votes
6 answers
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Difference between discrete time fourier transform and discrete fourier transform

I have read many articles about DTFT and DFT but am not able to discern the difference between the two except for a few visible things like DTFT goes till infinity while DFT is only till N-1. Can ...
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36 votes
7 answers
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What is the most lucid, intuitive explanation for the various FTs - CFT, DFT, DTFT and the Fourier Series?

Even after having studied these for quite sometime, I tend to forget (if I'm out of touch for a while) how they are related to each other and what each stands for (since they have such similar ...
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24 votes
1 answer
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Is the Laplace transform redundant?

The Laplace transform is a generalization of the Fourier transform since the Fourier transform is the Laplace transform for $s = j\omega$ (i.e. $s$ is a pure imaginary number = zero real part of $s$). ...
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9 answers
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Where is the flaw in this derivation of the DTFT of the unit step sequence $u[n]$?

This question is related to this other question of mine where I ask for derivations of the discrete-time Fourier transform (DTFT) of the unit step sequence $u[n]$. During my search for derivations I ...
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6 votes
1 answer
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Applying Image Filtering (Circular Convolution) in Frequency Domain

To filter an image we can: Use a 3x3, 5x5, 7x7, etc. filter, that is convolve the image and the filter in the space domain. Use a FFT on both the image and the filter, multiply them together in the ...
2 votes
1 answer
828 views

Power Spectrum: Definition

I am new to the study of time series. Recently I have asked a question about the covariance of real and imaginary part of a real(in time domain) stochastic time series and I have received an answer ...
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3 answers
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What effect does a delay in the time domain have in the frequency domain?

If I have a signal that is time limited, say a sinusoid that only lasts for $T$ seconds, and I take the FFT of that signal, I see the frequency response. In the example this would be a spike at the ...
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5 answers
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Difference between DC component and zero frequency component of signal

We know that Fourier Transform of a signal exists if it is absolutely integrable and it exists for periodic signals if impulse functions are allowed. If we consider the fourier transform of $\text{...
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2 answers
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One integral inverse CWT

MATLAB's icwt docs state inversion to be done by a single integral: $$ f(t) = 2 \Re e\left\{ \frac{1}{C_{\psi, \delta}} \int_0^\infty \left< f(t), \psi(t) \right> \frac{da}{a} \tag{1} \...
6 votes
2 answers
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Image Processing and applicability of 2D Fourier Transform

As a newbie in the world of signal processing, I am having a hard time in appreciating image 2-D fourier transforms. I am fully able to appreciate the concept of 1-D Fourier transform. Essentially, ...
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24 votes
3 answers
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Tips for improving pitch detection

I'm working on a simple web app that allows the user to tune his/her guitar. I'm a real beginner in signal processing, so please don't judge me too harshly if my question is inappropriate. So, I ...
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16 votes
1 answer
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Fourier transform 4 times = original function (from Bracewell book)

I was glancing through "The Fourier Transform & Its Applications" by Ronald N. Bracewell, which is a good intro book on Fourier Transforms. In it, he says that if you take the Fourier ...
10 votes
1 answer
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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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11 votes
4 answers
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What's the difference between the Gabor-Morlet wavelet transform and the constant-Q transform?

At a glance, the constant-Q fourier transform and the complex Gabor-Morlet wavelet transform seem the same. Both are time-frequency representations, based on constant-Q filters, windowed sinusoids, ...
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13 votes
2 answers
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Discrete-time Fourier Transform of the unit step sequence $u[n]$

From text books we know that the DTFT of $u[n]$ is given by $$U(\omega)=\pi\delta(\omega)+\frac{1}{1-e^{-j\omega}},\qquad -\pi\le\omega <\pi\tag{1}$$ However, I haven't seen a DSP textbook that ...
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6 votes
1 answer
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Replicate MATLAB's `conv2()` in Frequency Domain

When conv2d is on same mode, the image needs no padding, because the result is the same size as the image. When ...
5 votes
2 answers
820 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 $y_f(x_f)$ ...
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3 answers
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When can we write Heisenberg uncertainty Principle as a equality?

We know that Heisenberg uncertainty Principle states that $$\Delta f \Delta t \geq \frac{1}{4 \pi}.$$ But (in many case for Morlet wavelet) I have seen that they changed the inequality to an equality....
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25 votes
1 answer
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How to circularly shift a signal by a fraction of a sample?

The shift theorem says: Multiplying $x_n$ by a linear phase $e^{\frac{2\pi i}{N}n m}$ for some integer m corresponds to a circular shift of the output $X_k$: $X_k$ is replaced by $X_{k-m}$, where ...
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1 answer
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How do I convert a real baseband signal to a complex baseband signal?

I have radio telescope observations that have resulted in two real-valued signals (corresponding to the right- and left-handed circular polarizations). The signals are sampled at rate $2B$, and ...
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Image Reconstruction:Phase vs. Magnitude

Figure 1.(c) shows the Test image reconstructed from MAGNITUDE spectrum only. We can say that the intensity values of LOW frequency pixels are comparatively more than HIGH frequency pixels. Figure 1.(...
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What is the difference between $X(j\omega)$ and $X(\omega)$ notation?

What is the difference between $X(j\omega)$ and $X(\omega)$ notation? What is the meaning of $j\omega$? Does it represent frequency, and if yes, what is the meaning of an imaginary frequency?
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How to Zero Pad in Order to Perform Filtering in the Fourier (Frequency) Domain?

Consider an $M\times N$ image $f$ and an $G \times K$ filter $h$. Given that convolution in the spatial domain corresponds to multiplication in the Fourier domain, then we can perform a convolution of ...
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2 votes
2 answers
5k views

Relation between the DTFT and the spectrum of a sampled signal

In the $\rm DTFT$ (Discrete Time Fourier Transform) the spectrum is periodic with a period of $2\pi$ . A continuous signal when sampled has a spectrum which is a repeated version of its original ...
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Aliasing after downsampling [duplicate]

Let me start with time domain representation of the original signal \begin{equation} x_n=\sum_{k=0}^{2N-1}X_ke^{j\frac{2\pi nk}{2N}} \end{equation} where $2N$ is number of time/frequency samples ...
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50 votes
3 answers
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What is the sparse Fourier transform?

MIT has been making a bit of noise lately about a new algorithm that is touted as a faster Fourier transform that works on particular kinds of signals, for instance: "Faster fourier transform named ...
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6 votes
1 answer
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$2\pi$ periodicity of discrete-time Fourier transform

In my signals and systems course, we have learned that the discrete-time Fourier transform is $2\pi$ periodic, but the continuous-time Fourier transform is not periodic in general. For reference, we ...
11 votes
2 answers
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Effect of windowing on noise

I understand that truncating a signal in time 'smears' the frequency response depending on the window chosen. In general, the shorter the signal duration, the more 'flattened' the frequency response, ...
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4 answers
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Why do we have to rearrange a vector and shift the zero point to the first index, in preparation for an FFT?

I am trying to learn how to implement the FFT as a way to approximate the continuous-time Fourier transform, and as a "nice easy example" I have chosen to test it with a simple Gaussian pulse in the ...
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3 votes
1 answer
605 views

why do we use $X(e^{j\omega})$ instead of $X(j\omega) $ in Discrete Time FT

I am studying DT-FT. But I cannot figure out why we use $X(e^{j\omega})$ instead of $ X(j\omega) $ in DT FT Thanks in advance..
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1 answer
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What are the units of my data after an FFT?

Magnetometer measures the derivative of the magnetic field, or dB/dt, with an output in microvolts (mV). The Sampling rate is 128 Hz, so if we collect data for 2 minutes, $2 \times 60 \times 128=...
12 votes
3 answers
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Slow Down Music Playing While Maintaining Frequency

Playing a piece of music audio at a slower speed would lower its pitch (frequency). Is there a tool and theory to slow down the song playing while keep the frequency the same? I suppose one can do ...
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1 answer
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What exactly is a complex envelope?

I have seen this be mentioned a couple of times in some books I read, so I want to make sure. Is the complex envelope simply the summation of the real and quadrature components of a signal, whereby ...
4 votes
1 answer
946 views

3D wiggle plot for an analytic signal: Heyser corkscrew/spiral

Just reading The Analytic Impulse, A. Duncan, 1988, I met the name "Heyser corkscrew" for the first time in my DSP life, for a 3D display of a cisoid or complex exponential $e^{i\omega }$ (often ...
4 votes
3 answers
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How condition for existence of Fourier transform is valid?

The condition for Discrete time Fourier transform to exist for function $f(n)$ is given as $$\sum_{-\infty}^\infty |f(n)| < \infty.$$ In case of continuous Fourier transform the difference is ...
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8 votes
2 answers
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Kernel Convolution in Frequency Domain - Cyclic Padding

I don't know whether this is the right place to post this, but I suppose it is. I know that frequency multiplication = circular convolution in time space for discrete signals (vectors). I also know ...
0 votes
1 answer
182 views

Confusion in CT Fourier Transform Proof

I am confused trying to understand the Proof of Fourier Transform from Oppenheim book Signals and Systems. I am pasting the equations directly from the book: $$\widetilde{x}(t)=\sum_{k=-\infty}^{+\...
0 votes
2 answers
593 views

Discreteness and periodicity in Fourier transform

Why discreteness in time / frequency domain dictates periodicity in the other frequency / time domian? For example the DTFT is perodic in frequency? Why it doesn't contain all the frequencies? Why ...
18 votes
8 answers
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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 ...
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Intuitive explanation of cross-correlation in frequency domain

According to the cross-correlation theorem : the cross-correlation between two signals is equal to the product of fourier transform of one signal multiplied by complex conjugate of fourier transform ...
16 votes
2 answers
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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 ...
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13 votes
4 answers
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Why is a negative exponent present in Fourier and Laplace transform?

could anyone explain why there is a need of negative exponent in fourier and laplace transform.I looked through the web but I couldn't get anything.Does anything happen if a positive exponent is ...
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8 votes
1 answer
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STFT: why overlapping the window?

For STFT, we impose window of certain size onto the original signal, then we perform fft on each window. The uncertanty about frequency and time is determined by the width of the window, however, I ...
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6 votes
5 answers
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What is an Intuitive Explanation of the Phase of a Signal

I understand that the meaning of the phase response of a system is simply how much the system delays a frequency component. However, I do not find an intuitive explanation for the phase of a signal. ...
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15 votes
3 answers
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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 ...
18 votes
3 answers
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FFT with asymmetric windowing?

Common non-rectangular window functions all seem to be symmetric. Is there ever a case when one would want to use a non-symmetric window function before an FFT? (Say if the data on one side of the ...
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