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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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Matching 2 audio files

I'm doing research on how we can do matching between 2 audio files using Android to see if they have the same content or not. After doing lots of research, I have concluded that I must do 2 main ...
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baseband and passband modulation

I have a simulink model over 2.4GHz. In the transmitter part, the modulation is O-QPSK modulated baseband instead of passband modulation. In Matlab Help this is the reason of using baseband ...
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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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Fourier transform of a Fourier transform

I generate a Gaussian noise and then I filter it with a passband FIR Kaiser window filter. When I perform the Fourier transform of the output of the filter and plot its magnitude spectrum, it is ...
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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=...
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Fourier transform of cosine to the power of 3

How can I find the Fourier transform of $$ f(x) = ( \cos(x) )^3$$ I know that for $ g(x) = \cos(x) $ $$\mathcal F \Big\{ g(x) \Big\} = \mathcal F \Big\{ \cos(x) \Big\} = \pi \Big [ \delta(w-\pi / 2)...
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Signal Processing using Fourier Transform

How can I derive the fourier transform of ...
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Differentiation of sine in Fourier domain

The derivative of $\sin(\omega_o t)$ is $\cos(\omega_o t)$. The Fourier transform of $\sin(\omega_o t)$ is $\frac{\pi}{j}[\delta(\omega-\omega_o) - \delta(\omega+\omega_o)]$. Differentiation in the ...
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Conceptual question on FFT and chirp signal

If I take the FFT of a sinusoid I will get a plot whit all the energy of the signal concentrated at the sinusoid frequency. But what happens if I have a signal in which the frequency keeps changing?(...
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Why are Fourier analysis and transform only applicable for LTI systems?

Why are Fourier analysis and transform only applicable for LTI systems? What if the system is not LTI, won't Fourier analysis or transform be possible?
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Shannon-Nyquist theorem reconstruct 1Hz sine wave from 2 samples

Lets say I want to set the minimal sampling rate to reconstruct a 1Hz sine wave, according to the Nyquist-Shannon theorem that states that the maximum recoverable frequency is Fs/2 i.e. we must sample ...
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Does sampling in the frequency domain cause time-domain aliasing?

Let's say I have an impulse response $h[n]$. I analyze the power spectrum of that impulse response similar to fourier transformed $h[n]$ corresponding to roughly $H[f]$. Now I compare $H[f]$ with ...
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What is difference between outputs of Fourier transform and Fourier series of a periodic square waveform?

We can use Fourier transform of an aperiodic signal and Fourier series of periodic signal. But we can use Fourier transform formula for periodic function also. Now, let us consider a periodic square ...
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Average FFT Magnitude in bins

I am analyzing sounds of daily activities recorded by a smartphone. For example walking, getting up from bed, falling, running etc.. (one at the time). Let’s take one them. My goal is to convert from ...
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Fourier Transform negative amplitude meaning

I am reading this example http://www.thefouriertransform.com/pairs/truncatedCosine.php What does it mean to have some of the frequency components be negative in its amplitude ? I am not talking about ...
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How do I calculate the eigenvalues of a covariance matrix which contains harmonic functions

I have read that $ce^{-\frac{\phi_i-\phi_j}{\rho}}$ is a harmonic function of the form $e^{-in\phi_i}$, and therefore it's eigenvalues are one of the Fourier components of $(|\phi_i-\phi_j|)$. My ...
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Understanding where the constant $2/N$ comes from in Fourier transformation

I'm implementing Fourier transformation in my analysis and I wanted dig a bit deeper on the reasons why the absolute value of Fourier transformation is usually multiplied by the constant $2/N$ to get ...
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What proportion of a padded FFT should be actual values

For a given signal, I've been told that you can pad the vector with $0$s at the end to get a larger DFT, and as a result get more precision in frequency bins. What are the limits to this approach? ...
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What are the statistics of the discrete Fourier transform of white Gaussian noise?

Consider a white Gaussian noise signal $ x \left( t \right) $. If we sample this signal and compute the discrete Fourier transform, what are the statistics of the resulting Fourier amplitudes?
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Comparison between Fourier transform, short time Fourier transform and wavelets

What is the difference between the Fourier transform, short time Fourier transform and wavelets?
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Understanding the Gabor filter function

I need to implement a script for generating features from an input image by using the Gabor filter. I have no past experience of wavelets and I'm just learning Fourier analysis (I understand the basic ...
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Which transform most closely mimics the human auditory system?

The Fourier transform is commonly used for frequency analysis of sounds. However, it has some disadvantages when it comes to analyzing the human perception of sound. For example, its frequency bins ...
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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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How were windows originally conceived?

I am aware of the common types of windows, (Hamming, Hanning, Kaiser, Tukey, etc etc). However while many books describe them - almost none tell me just how exactly they were derived. What is so ...
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Extracting frequencies from FFT

I performed 512 point FFT on a signal. I got another set of 512 Numbers. I understand that those numbers represent amplitude of the various sine and cosine waves having different frequencies. If my ...
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Harmonic Product Spectrum limitations in pitch detection

I've made a pitch detection algorithm using HPS and I'm facing a problem. I'm a beginner with signal processing and this site helped me before, so I though I should ask. For higher pitches ( ...
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Why are there so many windowing functions?

Many windowing functions are listed here in the Mathematica documentation. I tried using a few to reduce leakage when computing a Discrete Fourier Transform. From what I could tell it made little ...
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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 ...
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How do I explain a complex exponential intuitively?

What is a complex exponential, explained intuitively? How do I explain to an adolescent a complex exponential function?
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Comparison between 2 images in Frequency domain

As you can see there are 2 images from the Tom and Jerry cartoon program. In the 1st image both Tom and Jerry are present. But in the 2nd one,only Tom is present.Now,we can clearly see this ...
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How do I optimize the window lengths in STFT?

I have many EEG signals and I want to analyze them using linear methods such as STFT (Short Time Fourier Transform). In STFT , How can I optimize the analysis window length, to reflect the frequency ...
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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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Phase Correlation - Poor Performance on Noisy/Blurred Images?

I have successfully tested 1D phase correlation algorithm to determine vertical shift between two synthetic images. When I moved to real images, however, it is not able to detect translation at all (...
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Coherent Sampling And The Distribution Of Quantization Noise

I have a question concerning the distribution of noise in the frequency domain in case of Coherent Sampling. I read up about Coherent Sampling and understood that in order for a frequency $f_{in}$ to ...
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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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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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Why is signum function used to calculate Fourier transform of unit step function

I read in a standard textbook that the Fourier transform of unit impulse function is calculated with the help of approximations and signum function as the integration of unit impulse does not converge....
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How do you measure “detail” of a signal?

I have an image and I would like to measure the amount of detail in it. Another way to look at it is to measure how blurry an image is. One way is to analyse the high frequency components in the ...
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Fourier Transform Identities

We know the below, $$ \mathscr{F}\big\{x(t)\big\}=X(f) \tag{1} $$ $$ \mathscr{F}\big\{x(-t)\big\}=X(-f) \tag{2} $$ $$ \mathscr{F}\big\{x^*(t)\big\}=X^*(-f) \tag{3} $$ Now, if for some signal $$ x(-...
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Conceptually, how does a Fourier transform differ from an autocorrelation?

I realize the two are derived using different algorithms, and the units are different, but from a conceptual standpoint of the information they provide how do they differ? I'm thinking here about the ...
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How to combine bins of my DFT

I have a time series and apply the FFT to get a spectrum. Let's assume that my sampling frequency and the length of the time sample are chosen such that I end up with a $\Delta f = 0.1$ Hz. As this ...
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Is interpolation of an audio signal to increase frequency resolution possible?

I apologize if some of what I ask is not entirely correct, I'm new to this field, but extremely interested. I have an Audio signal of sample rate 44.1 kHz that I want to segment into 30 frames, and ...
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What's the difference between using DFT, IDFT or DCT to calculate cepstrum of a power spectrum?

I've seen different equations that calculate cepstrum from power spectrum, but the equations are not consistent. Some people use Fourier transform, some use the inverse Fourier transform, and some use ...
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What happens with signal in frequency spectrum when it is time shifted in time spectrum?

I have some trouble to understand what is going on with signal in frequency spectrum when it is time shifted in time spectrum. I am hoping that somebody will help me to understand that. Thanks you ...
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How moving part pixel intensity values of video frames becomes dominant compared to stationary part intensities in reconstructed frames?

Hello everyone i want to do dynamic texture video sementation using the Fourier transform in MATLAB. I am applying 3-D fft on dynamic texture video frames (using matlab function 'fftn') and ...
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FIR filter design by the Fourier transform method

I am having some problems understanding how the Fourier transform method is used to determine the FIR filter. As far as i have understood, you start by using the ideal impulse reponse for the ...
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410 views

Covariance between real and imaginary parts of Fourier transform of a stationary time series

Since Fourier transform of a random stationary process in time (in the case of existence) is not necessarily real, my question is what is the relation between the covariance of real and imaginary ...
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length of window and overlap rate in STFT

I want to use STFT to analyze my signal and am wondering what are differences between two solutions: Use short windows (for ex. 256 samples window) Use longer windows (to get higher resolution in ...
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Repeated Fourier transform - what happens? [duplicate]

I have a Fourier transformable complex function that is a function of independent real variable a. Now I take the Fourier transform of it, giving me a complex function of real variable b. Now I ...
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Formulas of the Fourier transform family

It has annoyed me that there doesn't seem to be a source online where the complete complex Fourier transform family is presented with every variable defined. The lack of definitions can be a nuisance ...