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

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Fourier transform properties. Spectral density function [on hold]

I have the following signal for which amplitude and phase spectrums have to be computed: This exercise also has a solution which begins by deriving the signal twice. Next, they say the spectral ...
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45 views

Why is level of power spectrum dependent on FFT resolution?

I created a sinusoidal wave in some noise, and plotted the power spectrum of the signal using two periodogram estimates (welch procedure). One estimate is 'high resolution' - i.e. it uses a longer ...
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39 views

How to obtain mean Power Spectrum from two audio channels?

I have a source of noise in a duct, on either side of the noise source I have a microphone, which register the pressure, so that I get two pressure signals: $p_1$ and $p_2$. I want to get the acoustic ...
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79 views

Fourier Transform of mirrored exponential signal

I have to determine the spectral function for the following signal: $$ x(t) = \begin{cases} A\,e^{\alpha t}, & \text{for $t<0$} \\ A\,e^{-\alpha t}, & \text{for $t>0, \; \alpha ...
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How to compute the Fourier Transform of this ramp-like signal?

I have the following signal: and this as the solution to the problem: $$\begin{align} X(j\omega) &= \int_{-\infty}^{+\infty}x(t)e^{-j\omega t}dt \\ &= \int_0^\tau \frac{E}{\tau}t\; ...
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25 views

Implementation of the constant Q transform + property questions

I'm reading up on fourier theory, especially the transforms. I implement the math as spectrograms in C++ to get a better understanding of what is going on. I've made an implementation of the short ...
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32 views

What are the differences between STFT in time and frequency domain?

For a signal f(t), we can do a short-time Fourier transform in time domain to get the time-frequency information. However, we can also to a "short-frequency inverse Fourier transform" in frequency ...
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29 views

Basics of waveform, how to define the zero point

Audio is a sequence of waveforms which have peaks, troughs, 0, just like below. How to define the x axis (y = 0) in the picture, how to choose the 0 point in the waveform since there are so many ...
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23 views

Signal Plus Weakly Stationary Noise

I was reading the book "Spectral Analysis of Time Series" By Herman Koopmans. On Page 55, he explains that a specific type of non-stationary signal which is the result of adding weakly stationary ...
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459 views

disadvantages of FFT, it can not extract enough frequencies without enough samples

Let's say sampling rate is $Fs = 44\mathtt{kHz}$, now I have $N = 2048$ samples, then I can get $N/2 + 1 = 1025$ frequencies. I'm confused by Matlab's FFT documentation that says the frequencies are ...
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49 views

How do I construct input to neural network from audio signals?

Input: Microphone recordings of digits from 0 to 9 from different speakers. Output: The digit from 0 to 9. I am doing this for fun. So first I will train my neural network using some samples and ...
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39 views

Unit Impulse funciton FT

How is FT of $ \delta $(t) equal to 1 ? Normal FT gives the result $\infty$. Can someone please explain? I did the normal integration and substituted the limits. Is it because $ \delta $(t) is a ...
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66 views

FFT Matlab - Meaning of Frequency Vector

I'm following a tutorial about the FFT. It's well explained but I don't understand the meaning of the frequency vector: ...
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98 views

Using Goertzel Algorithm in under-sampling

I plan to calculate a signal's phase using Goertzel Algorithm. I have 2 signals coming to microcontroller's ADC. Need to measure the phase difference between them. Signals are 15MHz sinusoids. Sample ...
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110 views

How to compute the main lobe width of generalized Hamming windows (i.e. Hann, Hamming, etc.)?

Is there a way to compute the main lobe width of windows of the generalized Hamming window family (i.e. Hann, Hamming, etc.)? By main lobe I mean the first zeros left and right of the center of the ...
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81 views

Power Spectrum: Defnition

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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36 views

UWB radar to find heartbeat

I have a UWB radar that Im playing with. The problem is that my algorithms at the moment can clearly, reliably and validly pick up 1-2mm movements of a pendulum i've set up. However, when I use it on ...
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143 views

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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159 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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51 views

how to get the phasor(frequency domain) representation of a digitalized signal?

I know that we can get the phasor representation(frequency domain) of a signal through Fourier transform, but the picture above gives a method that can get the phasor representation of a signal in a ...
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1answer
45 views

Discrete Fourier Transform and Opposite Convolution Theorem

I am reading the Wiki for DFT. There is a part for circular convolution theorem which sounds a bit odd saying: $$ \mathcal{F} \left \{ \mathbf{x\cdot y} \right \}_k \ \stackrel{\mathrm{def}}{=} ...
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33 views

How to get the quaternionic analytic signal of a image?

The 2D discrete image signal's quaternionic fourier transform can be described as F(u,v), and its 2D analytic signal often be described as: $$ F_q(u,v) = (1 + \mbox{sgn}(u))(1 + \mbox{sgn}(v)) F(u,v) ...
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39 views

Why does the periodic signal in time always give a discrete frequency spectrum

I would like to know, why does the periodic signal in time always give a discrete frequency spectrum in FT? I know the equations, but I simply dont understand why is it so. Thanks!
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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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101 views

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 ...
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115 views

Intuitive explanation of the Fourier Transform for some of the functions

Does anyone have a mechanism to understand intuitively (and automatically) why the Fourier Transform of certain functions have certain shapes (at least for some functions, not necessarily for all)? I ...
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61 views

Frequency spectrum of a sinc function

I am doing one example from my book as a preparation for exam. The assignment is: It is given that: $$\mathbb{rect}(t)=pf(t) \leftrightarrow PF(f)=2AT_0 \cdot \mathbb{sinc}(2\pi fT_0)$$ you ...
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Filtered backprojection implementatoin

I am wondering the detail of the filtered backprojection. Suppose I have a projection file: $p(s,\theta)$=[1 2 3 1 2] Then to get its filtered backprojection ...
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34 views

How are phase values able to capture motion from video?

I know that the phase spectrum contains most of the structural information about the image importance of phase. But I want to know more about importance of phase spectrum related to video signals. I ...
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41 views

Is it possible to decompose an image into basis images using MATLAB?

I have read that: Using Fourier decomposition any arbitrary image can be represented as summation of orthogonal basis images. I want to see the basis images for any image say Lena or ...
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125 views

Properties of Spectral Transformations - Allocation (decomposition into even and odd part)

I am trying to understand the Allocation property of Spectral Transformations. I can't. I know that every function can be separated into an even part and into an odd part. My problem is ...
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65 views

Inverse Chirp Z Transform

I am working to understand and use the Chirp Z-Transform. I want to use the algorithm for simple signal processing on data sets that are not a power of two. I need to be able to inverse transform as ...
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How do we compare between Spatial Domain and Frequency Domain?

I am doing my academic project in image processing. I have few doubts that i want to ask Which information do we actually get from "phase spectrum" and "magnitude spectrum" about an image? What are ...
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118 views

What is obtained from the cross correlation plot?

Let’s assume that we have two audio signals, x(t) and y(t) affected by the noise as shown below. And we would like to cross-correlate these two signals and the cross-correlation plot is shown as ...
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60 views

Inverse DTFT of $e^{-j\omega}$

I've been trying to solve the following problem, but I noticed that I get two different answers by means of two different techniques: Find the impulse response, $h(n)$, of the following system, ...
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168 views

What is the general feature of a time domain signal that gives a step like spectrum?

I'm trying to study a nonlinear system by sending a pulsed signal into the system and look at the response spectrum. The signal I send to the system is something like and the response spectrum from ...
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Development of Hilbert transform relationship

In the development of Hilbert transform relationships, Prof. Oppenheim has chosen \begin{equation} ...
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Noisy Signal auto/cross-correlation

I am trying to understand the basics of the cross/auto-correlation if the noisy signal $y(t)$ which is received from the channel at the receiver. Here at receiver we want to estimate the noise in the ...
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79 views

Magnitude of gradient in frequency domain

I'm learning some basics of image processing. Recently I've read about image filtering and two-dimensional Fourier transform, because I'm preparing for exam. And I have one question I don't know ...
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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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52 views

Fourier Transform Form: two sin components & a phase shift & a magnitude for only one term

This is an example from my text book of a continuous signal: $$x_{in}(t)=\sin \left( 2\pi \cdot 1000 \cdot t\right) + 0.5\sin \left( 2\pi \cdot 2000 \cdot t + \dfrac{3\pi}{4} \right) $$ So to perform ...
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Low Pass Filter in AM Demodulation

I am trying to recover my baseband signal that was AM modulated. I am using the filter command in MATLAB to perform the low pass filtering ...the code is given below: ...
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Is Gabor uncertainty a feature of the Fourier transform, or of nature?

By Gabor uncertainty, I mean the principle of uncertainty as applied to signals — with the result that you can't have arbitrary time and frequency localization. By way of background to my question, ...
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54 views

How to perform a Stockwell Transform on a 2D matrix

I was looking at performing a Stockwell transform on a matrix (or an image as it has two directions - the matrix I am interested in has a spatial dimension and a time dimension, but this should make ...
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72 views

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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Autocorrelation - cyclic vs acyclic

In terms of signal analysis (especially frequency decomposition) would it be bad to use cyclic autocorrelation (AC) instead of acyclic/linear AC? My use case: I'm trying to characterize events in a ...
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157 views

Why can't DFT be used when samples are not equally spaced in time?

I found the following comment here The DTFT can be used when the samples are not equally spaced in time, the DFT cannot My initial thought was that this had to do with periodicity of the basis ...
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84 views

Simple FFT filtering vs. e.g. butterworth filtering

I am currently working on functional connectivity analysis of EEG, and need to bandpass filter my data into different frequency bands (Delta, Theta, Alpha and Beta). An important thing is that the ...
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Why we need fourier transform of periodic signal although we have fourier series for periodic signal?

I was going through some of the basics of fourier series and fourier transform. And I came across one topic "Fourier transform of periodic signal". But I am not able to understnad why we need to go ...
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Discrete Fourier transform of complex time series

I am working through previous exam questions for a class I'm currently taking in signal theory, and one problem, although it looks easy, stumps me. PROBLEM Two real $N$-periodic time seires $f_n$ ...