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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348 views

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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1answer
329 views

Frequency response of $\mathrm{sinc}[n]$

In this image the frequency response of a discrete time filter given as $h[n]$. Can someone explain how the magnitude of the frequency response is found ?
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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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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’...
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Conditions for symmetric and unimodal windows in both time and frequency domains

After a lecture on harmonic analysis and time/frequency methods, I reconsidered the Gaussian kernel, defined in continuous time. It is unimodal and symmetric, and its continuous Fourier transform is ...
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periodic spectrum?

I'm relatively new to signal processing, so please excuse me if this is a trivial question. Why is the spectrum of a frame of speech samples periodic? What is the meaning of a periodic spectrum? And ...
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1answer
285 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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2answers
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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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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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2answers
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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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1answer
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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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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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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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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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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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Help with obtaining the power spectral density of a simple continuous cosine (using both forms of the definition for PSD)

I am trying to go through a simple example to teach myself about Parseval's theorem and calculating power spectral density (PSD) in practice and would be very grateful if someone could check my ...
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1answer
195 views

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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356 views

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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2answers
597 views

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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1answer
319 views

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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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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Inverse Short Time Fourier Transform algorithm described in words

I'm trying to conceptually understand what is happening when the forward and inverse Short Time Fourier Transforms (STFT) are applied to a discrete time-domain signal. I've found the classic paper by ...
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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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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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1answer
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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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Why is the Fourier transform of a Dirac comb a Dirac comb?

This doesn't make sense to me, because the Heisenberg inequality states that $\Delta t\Delta \omega$ ~ 1. Therefore when you have something perfectly localized in time, you get something completely ...
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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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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 ...
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1answer
755 views

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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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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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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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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1answer
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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 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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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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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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2answers
927 views

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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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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About Fourier transform of periodic signal

In Fourier transform for periodic signal, I checked different books and I found a different explanation in each book. Let's take the explanation in Signals and Systems by Rajeshwari & Rao: The ...
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Is there any alternative basis for a fourier-like transform?

Fourier transform of a continuous signal is just the projection of the signal on the sinusoidal family for imaginary part and the same family with phase offseted by a quarter of period for the real ...
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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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1answer
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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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1answer
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How does Adobe After Effects generate its “audio spectrum” effect?

I'm trying to replicate the "audio spectrum" effect from Adobe After Effects. An example can be seen in this video: Obviously, it has to be some variant of a fourier transform, but I've tried taking ...