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

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How is Linear Canonical Transform a generalization of Fractional Fourier Transform?

I have studied that Fourier transform changes the domain of a signal from time to frequency, and in that way it is a 90 degree shift. When it comes to Fractional Fourier Transform a generalization of ...
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41 views

Hilbert transform pair proof

I am looking for the proof that the Hilbert transform of $\displaystyle\frac{\sin(at)}{at}$ is given by $$\frac{\sin^2(at/2)}{at/2}.$$ How do we prove this? This is a $\operatorname{sinc}(at)$ ...
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31 views

What does it mean for the DFT phase to be relative to a cosine wave?

The following paragraph from Understanding Digital Signal Processing got me puzzled: The answer is: The DFT phase at the frequency $mf_s/N$ is relative to a cosine wave at that same frequency of ...
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27 views

Deriving the DFT magnitude of $A\cos(2\pi nk/N)$

Given that $$x(n) = A\cos(2\pi nk/N),$$ the $N$-point DFT of $x(n)$ can be expressed as follows—the derivation can be found in here: $$X(m) = \color{red}{\frac{A}{2}\sum_{n=1}^{N-1}e^{-j2\pi ...
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30 views

Where does $\frac{N}{2}$ came from in approximating an N-point DFT?

I've came across the author saying that ... for a real cosine input having k cycles in the N-point input time sequence, the amplitude response of an N-point DFT bin in terms of the bin index m is ...
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44 views

Why is N-point DFT approximated by the sinc function?

While looking into DFT leakage, I've came across the author saying that "..., the amplitude response of an N-point DFT bin in terms of the bin index m is approximated by the sinc function." ...
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15 views

Should S-Matrix components for scattering response in a Polarimetric Radar be Real or Complex?

I am trying to implement some polarimetric decomposition algorithms on my MIMO polarimetric radar. At first, I will try the Pauli Decomposition algo. But instead of doing signal processing on the ...
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39 views

Why does Fourier zero-fill interpolation pad the middle frequencies?

In this: http://dspguru.com/dsp/howtos/how-to-interpolate-in-time-domain-by-zero-padding-in-frequency-domain the author says that we can interpolate a signal by zero-padding the middle frequencies in ...
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60 views

what is nyquist rate of $h(t)\cdot h(t)$ and $h(t)\circledast h(t)$

Let's say we have $h_c(t)$ as a continuous-time signal with bandwidth $B$ and we would like to sample it. To be able to reconstruct it correctly, the sampling rate must be greater than $2B$. Now ...
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18 views

The signal in MRI

I am trying to understand the signal formation in MRI and have a confusion. I understand that in the presence of the external magnetic field $B0$, the protons are precessing at the frequency given by ...
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19 views

How to analyze baby speech time series

I've been collecting speech data for my baby brother (who is now 6 months old) with the intention of doing computational analysis of the development of his speech patterns. I haven't much deep ...
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35 views

How to interpret output of matched filter with complex input?

I have implemented a matched filter based on the Fourier Transform approach. In the real numbers domain that means that I use as the coefficients of my filter (B) the inverted time-samples of the ...
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1answer
25 views

Bandwidth range for Fast Fourier vs principal component analysis?

I've read somewhere that the Fast Fourier is only applicable to those processes exhibiting bandwidth. Where as principal component analysis can be applied to a process exhibiting any finite ...
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22 views

Calculate 2D Windowed Sinc Kernel

I am implementing a Windowed Sinc Filter for Scaling images by the factor of 2. Therefore I need to calculate a Filter Kernel. I already know how to calculate the kernel including a Blackman Window ...
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2answers
95 views

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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20 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 cisoid. This representation is quite common for analytic signals, but ...
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1answer
33 views

Why Fourier transform and Stockwell-transform retain the absolute phase information of one signal?

Hello friends am studying the topic of signal processing and the Fourier transform and the s-transform and in most books as for example "Time-Frequency Signal Analysis and Processing. 2nd" of Boashash ...
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1answer
23 views

Autocorrelation of a noisy linear map

I am interested in calculating the autocorrelation function of a linear map with some noise (model given below) but am slightly confused in doing so. At first, I did not realize there were two ...
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35 views

In fourier space, how to apply transfer function with n frequencies to input data with m>>n frequencies

I have a transfer function in Fourier space with $N=2028$ frequencies $(\frac {0, 1}{(N\cdot dx)} \dots ) $ Where $dx = 0.1m$. I need to apply this transfer function to a signal with 20000 samples ...
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86 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 ...
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69 views

Why are the basis functions for DFT so?

When you get a DFT of a signal, you use the basis functions as: $e^{-j2\pi kn/N}$ Why is it so? Why don't we use the conjugate, $e^{j2\pi kn/N}$, or any other function?
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104 views

DFT of real sinusoids - why sum over -$N/2$ to $N/2-1$ as opposed to $0$ to $N-1$?

I'm going through a Coursera course on signal processing, and we're just introduced to DFTs. We are told that if you have a complex sinusoidal signal $x[n]$ where $n=0,1...\ N-1$, its DFT is given ...
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33 views

Convolution theorem for cross-correlation

Forgive me is this is an ill-posed question. Is there any such thing as a 'convolution theorem' for the cross-correlation. Namely, the convolution theorem states that: $$ x[n] * h[n] = ...
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28 views

How to find Galois Field $\textrm{GF}(p)$ from any $n$-point DFT

How to find a finite field $\textrm{GF}(p)$ (where $p$ is a prime number) (here I want to find the value of $p$ only)that is as small as possible and efficient computation can be performed for (let's ...
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41 views

Polar form of the Fourier transform of $\sin(t)$

I came across the Fourier transform of sin(t). It ends up being a purely imaginary (dirac delta) impulse pair. But when considering the frequency domain representation of a signal, we consider the ...
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7 views

A 2 input 1 output model with time delay

What assumptions should be taken into consideration if I need to build a 2 by 1 black box process model on Matlab where the inputs samples I have are sampled each minute to result eventually after 1 ...
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22 views

Approximation of Step Response from Data with delay

How can I approximate a Step Response curve from a measured data of a single input and a single output using Matlab while the delay between the input and the output is 30 minutes?
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1answer
53 views

How do I understand Fourier descriptors more visually and intuitively?

I read the book Image Processing, Vision and Machine Vision and find the concept Fourier descriptors hard to understand, although literally its derivation is somewhat reasonable. Can anyone give me a ...
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47 views

I have a pressure signal and want to do SPL analysis on it

Signal I have an acoustic signal from a Ffowcs Williams Hawkings CFD analysis and would like to convert it to the frequency domain and see the SPL and OASPL. I know I need to use fft() but I am ...
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113 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 period of $2\pi$ . A continuous signal when sampled has a spectrum which is a repeated version of its original ...
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59 views

$2\pi$ 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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44 views

What is the interpretation of the discrete-time spectrum?

The CTFT of an analog signal is a representation of that analog signal in terms of the frequency parameter of sinusoidal (cosine specifically) functions whose weighted sum make up that signal. The ...
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1answer
38 views

square wave frequency representation

New to this signals stuff and i'm confused about the frequency representation of the square wave. Correct me if i'm wrong, a periodic square wave is composed of odd harmonics sine waves which are ...
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61 views

Anomaly at bin centre frequency

I'm building an additive synth using a real IFFT. Spectrum size 512, Samplerate 44100. I've noticed that when I alter the fine tune of one my oscillators, I get a blip sonic artefact exactly at the ...
3
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1answer
61 views

Fourier transform of Image to identify sinusoidal sources of interference

I'm a statistics grad student, and I just started getting into Digital-Image-Processing (an analogy for processing super-large contingency tables). In the book "Digital Image Processing" by Gonzalez ...
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1answer
38 views

Which transformation in frequency domain equals a x-axis shift of a signal in time domain?

I have discrete Fourier transformation results from measurements. Looking at the signal from the time domain perspective, I want to shift the signal on the $x$-axis to the left or right. Which ...
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36 views

Discrete Fourier Transform for text analysis?

I would like to determine the number of text-blocks verifying a roughly similar pattern. I have the intuition that I could do it using some Discrete Fourier Transform (DFT)-like methodology. Example: ...
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1answer
55 views

Order Analysis Signal Processing

I'm currently working on vibration detection machine used to detect unbalance from a rotating shaft, rotating from 300-2500 Hz. I have both the tacho signal ( 1 PPR ) and the vibration from ...
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1answer
42 views

Fourier Transform of triangle function $x(t)=\Delta\left(\frac{t-1}{2}\right)$

Can you please tell me if my working is right for the Fourier Transform of this function: $$x(t)=\Delta\left(\frac{t-1}{2}\right)$$ My workings are: I have used the fourier transform standard ...
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2answers
59 views

Convolution & FT properties

I have been trying hardly to check if these equalities are true or false. However, I have not been able to conclude anything. Could you help me, please? $$y[2n]=h[n]\star x[2n] $$ ...
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1answer
43 views

Find $X_s(f)$ of a sampled continuous signal

I've been trying to find the transform of the following signal, but have not been successful, any help would be greatly appreciated: Find $X_s(f)$ of the following signal the "mathematical DAC" ...
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1answer
40 views

convert Fourier tranform of a signal to $\omega$ form?

Suppose we have this signal: $x(t)=5 \sin(2\pi 1000 t) \cos(2\pi 10000 t)$ I calculated the Fourier transform as below: ...
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48 views

DFT have independent variable in between 0 to $\frac{1}{2}$ no of samples

I was learning of Fourier Transform from this book 'Scientist and Engineer Guide to DSP'. Suppose we have a digital periodic signal and there are 128 of samples of it in the time domain i.e X[0] to ...
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1answer
48 views

Relation between Fourier Series & Fourier transform [duplicate]

So i was just revising some basic DSP concepts. Just wanted to verify this fact. Fourier series represents a periodic signal $\hat{x}(t)$ with period P as a countably infinite sum of sinusoids of ...
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2answers
62 views

What information does fourier transform carry? [duplicate]

As one starts learning signal processing, then comes inevitably the topic of Fourier Transforms. Unfortunately I have difficulties not in computing but in interpreting the results of the Fourier ...
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1answer
66 views

How does shifting a signal in the time domain affect its frequency domain?

Suppose the signal is shifted by dt (signal 'starts' later, say after 1s instead of 0s), does that correspond to a positive or a negative phase shift df in the frequency domain? There are certainly ...
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51 views

Zero-order interpolation problem

Let $x_c(t)=\cos(\omega_0t)$. This signal is sampled with $\omega_s$, which is greater than the Nyquist rate. It is then interpolated with a zero-order interpolator. The signal obtained is ...
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117 views

Problem designing a specific filter

I have the next problem. $H_{c1}(j\omega )$ is the ideal antialising filter and $H_{c2}(j\omega )$ is a real one. I'm asked to design $H(e^{j\Omega })$ so that $y[n]$ in the second diagram (the one ...
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86 views

Is it possible exctract sinusoids from non periodic signal?

Digital signal UT1-UTC is not periodic but is including many sinusoids (periodic elements in IERS nomenclature) that are not multiples of some fundamental. For example tidal sinusoids are not ...
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94 views

Why this DTFT plot from CTFT? [closed]

I have a sample test with an answer but don't understand how they got to the answer: $x(t)$ has info only between $2 < |\omega| < 4$ $X^F(\omega) = 0$ for other frequencies. All ...