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37 votes
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“The Fourier transform cannot measure two phases at the same frequency.” Why not?

It's because the simultaneous presence of two sinusoidal signals with the same frequency and different phases is actualy equivalent to a single sinusoidal at the same frequency, but, with a new phase ...
Fat32's user avatar
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20 votes
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Why is the time domain low-pass filter the "sinc" shape?

It is a good way to understand the lowpass behavior of sinc function (as well as the convolution) through visualization. I've made some modification on this animated convolution project and here are ...
ZR Han's user avatar
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9 votes

About Fourier transform of periodic signal

We can figure out what's going on if we first understand a simple identity and then just compute the Fourier transform of the periodic function. A useful identity First let's prove that $$D(\omega - ...
DanielSank's user avatar
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8 votes

Integer multiples of fundamental period in sine wave

A $T$-periodic signal doesn't necessarily have a (sinusoidal) fundamental with period $T$. It never has one if the signal is also periodic with period $T/2$ or $T/3$ etc. (as in your example). But ...
Matt L.'s user avatar
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7 votes
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Why do we get different imaginary parts of a zero centered Gaussian for the the same number of data points N?

Answer : When $x_2 = [-1024:1:1023]$, then $x_2[n]$ satisfies the condition $x_2[n] = x_2[(N-n)\mod N]$. That is why when $x_2 = [-1024:1:1023]$, then the FFT is real and hence the imaginary part is $...
DSP Rookie's user avatar
  • 2,611
6 votes

For an LTI system, why does the Fourier transform of the impulse response give the frequency response?

The intuitive answer is that an impulse in time at t=0 contains all frequencies of equal magnitude, so applying an impulse to an LTI system is the same as applying all frequencies at once, thus the ...
Pat Eblen's user avatar
6 votes
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The Number of Sine and Cosine Waves in an $ N $ Point DFT

Here is another more formal way to look at it, starting with the definition of the DFT: $$ \begin{align} X(k) &= \sum_{n=0}^{N-1} x[n] \exp\left(-i\frac{2\pi}{N} k n\right), & k=0,\...
SleuthEye's user avatar
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6 votes
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Fast & accurate convolution algorithm (like FFT) for high dynamic range?

Disclaimer: I know this topic is older, but if one is looking for "fast accurate convolution high dynamic range" or similar this is one of the first of only a few decent results. I wanna ...
oli's user avatar
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6 votes
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Can FFT tells us existance of same frequencies with different phases?

I am thinking of two numbers. They add to the number 15. Tell me what the two numbers are.
robert bristow-johnson's user avatar
6 votes

Why is the time domain low-pass filter the "sinc" shape?

One way to think about it is the requirement of what a filter does, and what is the relation between the time domain and frequency domain plots of the signal or the filter. This also requires to know ...
Justme's user avatar
  • 2,303
6 votes

beam pattern are fourier transform of the beam weight, is it true ? if it's true how?

I'll work out some math that should hopefully capture the gist of the derivation of why the DFT can be used. The magnitude squared of the array factor is the beampattern for an array of isotropically ...
Baddioes's user avatar
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5 votes

Can use of Fourier transform be minimized completely with the help of Laplace and Z transform?

The short answer is yes, if you have the Laplace or Z-transform of a function you do not need the Fourier transform. This is because the CFT is a special case of the Laplace transform and the DTFT ...
Matt's user avatar
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5 votes
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"Fourier Transform can localize signals in frequency domain, but not in time domain." -- What does it mean in layman's terms?

In the Fourier transform, the basis functions are complex exponentials. These functions are perfectly localized in the frequency domain, i.e., they exist at one frequency, but they have no time ...
Matt L.'s user avatar
  • 90k
5 votes

About Fourier transform of periodic signal

1.The FT of a periodic signal is not one, but (potentially) infinite impulses. Assume an arbitrary periodic function $f_T(t)$ with period $T$ and consider its Fourier series representation in which $\...
msm's user avatar
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5 votes
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How to prove that the peak of the autocorrelation function is at zero lag?

The Cauchy Schwarz inequality states that: $$ \left|\int_{-\infty}^{\infty}g_1(t)g_2(t) dt\right|^2 \leq \int_{-\infty}^{\infty}|g_1(t)|^2 dt \int_{-\infty}^{\infty}|g_2(t)|^2 dt $$ I'm going to ...
David's user avatar
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5 votes

Fourier Transform of a signal using direct integration and properties

Your first solution using the properties of the Fourier transform is correct. Your second solution is wrong, because you forgot to include the unit step function. Your function $g(t)$ should be ...
Matt L.'s user avatar
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5 votes
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Prove the dirac delta contains all frequencies

some things I've read online show the Fourier transform should look more like a complex exponential. Don't believe everything you read online! A signal "contains all frequencies" is a vague ...
Dilip Sarwate's user avatar
5 votes
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struggling to understand why Fourier basis is orthogonal

Just use the formula for the geometric series (I use $l=h-k\neq mN$): $$\sum_{n=0}^{N-1}e^{-j\frac{2\pi}{N}nl}=\frac{1-e^{-j\frac{2\pi}{N}Nl}}{1-e^{-j\frac{2\pi}{N}l}}=\frac{1-e^{-j2\pi l}}{1-e^{-j\...
Matt L.'s user avatar
  • 90k
5 votes
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What is issue in square wave reconstruction?

Well this goes to show that Fourier series is just approximation that gets more and more correct when you add more harmonics. Take a look at this: $\dfrac{4\sin\theta}{\pi}$ is just first harmonic. ...
Healow's user avatar
  • 90
5 votes
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Characteristic function of a random Gaussian variable

In general, the characteristic function of a random variable is related to the fourier transform of the distribution as follows: $$\varphi_Z(-\omega) = \mathscr F \big\{ f_Z(z) \big\}$$ Why? ...
DSP Rookie's user avatar
  • 2,611
5 votes

Why do we get different imaginary parts of a zero centered Gaussian for the the same number of data points N?

You are simply seeing the effect of the time delay due to being offset by a half a sample (a delay in time is a linear phase in frequency). If you have an odd number of samples then you can implement ...
Dan Boschen's user avatar
  • 51.4k
5 votes

Amplitude after Fourier transform

Relating the DFT to the FT (CTFT) is a big issue. Let's start with the basic definitions without any domain specification using a $\frac{1}{N}$ normalization on the DFT. $$ FT(x(t))(f) = \int x(t) e^{...
Cedron Dawg's user avatar
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5 votes
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Fourier series of cycloid

The Fourier series of the cycloid can be expressed in terms of the Bessel functions of the first kind: $$J_n(x)=\frac{1}{\pi}\int_0^{\pi}\cos(nt-x\sin t)dt,\qquad n\in\mathbb{Z}\tag{1}$$ Using the ...
Matt L.'s user avatar
  • 90k
5 votes

Why is the time domain low-pass filter the "sinc" shape?

Perhaps one way to see the sinc is as a special moving average filter. As you noted, the lower the cutoff frequency (filtering out higher frequencies), the wider the sinc mainlobe. This corresponds to ...
Gillespie's user avatar
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4 votes
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Fourier transform of $ne^{-an}u[n]$

You're right, the first Fourier transform correspondence in your reference is wrong. It should be $$\mathcal{F}\{e^{-\alpha n}u[n]\}=\frac{1}{1-e^{-\alpha}e^{-j\omega}}\tag{1}$$ You just need to ...
Matt L.'s user avatar
  • 90k
4 votes
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Can use of Fourier transform be minimized completely with the help of Laplace and Z transform?

The answer to your last question is definitely 'no'. The point hotpaw2 makes in his answer is very relevant: the FFT is an efficient implementation of the DFT, and there are no equivalently efficient ...
Matt L.'s user avatar
  • 90k
4 votes

"Fourier Transform can localize signals in frequency domain, but not in time domain." -- What does it mean in layman's terms?

To localize here means: to find where the signal is mostly concentrated, and with what precision. This could be either in the time or the frequency domain. An answer could be: the signal's center of ...
Laurent Duval's user avatar
4 votes

Real world application of signal sparsity?

Sparsity concept is extensively being used in computer vision and image processing. The Idea is that natural image can be pretty sparse when it is transformed to different bases. this bases can be ...
DoronPor's user avatar
  • 139
4 votes

Fast & accurate convolution algorithm (like FFT) for high dynamic range?

Rather than scrapping the fast convolution algorithm, why not use an FFT with a higher dynamic range? An answer to this question shows how to use the Eigen FFT library with boost multiprecision.
Mark Borgerding's user avatar

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