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13 votes

What is the most lucid, intuitive explanation for the various FTs - CFT, DFT, DTFT and the Fourier Series?

I just finished writing an expository note on this topic recently. I hope you would find it useful. The link between Fourier transform (FT), Fourier series (FS), Discrete-Time Fourier Transform (DTFT)...
Krasjet's user avatar
  • 131
10 votes

How to get Fourier coefficients to draw any shape using DFT?

I'm not understanding the comments. Of course you can do this. It is simply a matter of understanding what a DFT means, how to calculate DFT bin values, and how to interpret those bin values as ...
Cedron Dawg's user avatar
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10 votes

Is Fourier series a sampled version of Fourier transform?

There are 4 versions of Fourier transforms that are all close cousins. It's all due to the basic property that "sampling in one domain corresponds to periodicity in the other domain". If a ...
Hilmar's user avatar
  • 45.7k
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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9 votes
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What to do after this last step?

This sum appears quite often in DSP. \begin{align} \sum_{n=0}^{N-1} \exp(-j\alpha n) &\stackrel{(a)}{=} \frac{1- \exp(-j\alpha N)}{1 - \exp(-j\alpha )}\\ &= \frac{e^{-j\alpha N/2}(e^{+j\alpha ...
AlexTP's user avatar
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8 votes
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Fourier series - time shift and scaling

Consider a continuous-time periodic signal $x_1(t)$ whose fundamental period is $T_1$, fundamental radian frequency is $\omega_1 = \frac{2\pi}{T_1}$ and CTFS (continuous-time Fourier series) ...
Fat32's user avatar
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8 votes
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Scaling property of Fourier Transform

Be careful about the orders of the argument transforms: When you interpret the argument $3t-6$ as $3\cdot(t-2)$, you should first scale (compress) $x(t)$ by $3$ along $t$ axis, and then shift the ...
Fat32's user avatar
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7 votes
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Slow Down Music Playing While Maintaining Frequency

Yeah some of us can do it, you can speed up or slow down without affect the pitch, some guys call this applications of Time Stretch, there different ways to do it, you can do in frequency domain or ...
ederwander's user avatar
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7 votes

Trying to understand how to get this basic Fourier Series

This is indeed a gnarly integral to do by hand, so it may be easier to leverage some properties of the Fourier Transform applied to simpler signals. Your signal is a trapezoid and it can be created ...
Hilmar's user avatar
  • 45.7k
6 votes

DSP interview question: use of the identity in development of a significant transform

This is related to Chirp Z-transform (CZT) (refer to the Bluestein's algorithm). Using this identity, the CZT can be expressed in terms of a convolution. Hence, it can be efficiently implemented using ...
msm's user avatar
  • 4,295
6 votes

Why Fourier series and transform of a square wave are different?

The Fourier series expansion of a square wave is indeed the sum of sines with odd-integer multiplies of the fundamental frequency. So, responding to your comment, a 1 kHz square wave doest not include ...
Tendero's user avatar
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6 votes
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inner product zero?

Well, when $m=k$ the integral is: $$ \int_0^T e^{j(m-k)\Omega_0t} dt = \int_0^T e^{j \cdot 0 \cdot\Omega_0t} dt = \int_0^T dt = T $$ So as Juancho says in the comments, it's the same signal and ...
Peter K.'s user avatar
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6 votes

Is Fourier series a sampled version of Fourier transform?

Yes, for time-limited functions it is possible to obtain the Fourier series coefficients by sampling the Fourier transform. This is the dual case of the more common form of the sampling theorem, ...
Matt L.'s user avatar
  • 90.5k
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
  • 4,295
5 votes

Why are Fourier analysis and transform only applicable for LTI systems?

The family of Fourier Transforms are specificaly developed for analysing frequency contents of the signals for which there is no definition of linearity or time invariance. Hence we can define the ...
Fat32's user avatar
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5 votes
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Why do frequencies of analog signals range from $-\infty$ to $\infty$ while frequencies of digital signals are restricted to $[0,2\pi]$?

The digital frequency span of 0 to $2\pi$ is the normalized angular frequency given in units of radians per sample. For example, if we had a frequency tone that went $0.2\pi$ radians/sample, then it ...
Dan Boschen's user avatar
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5 votes

How to get Fourier coefficients to draw any shape using DFT?

Complex Fourier series of a piece-wise linear waveform tracing the desired shape Instead of using discrete Fourier transform (DFT) / fast Fourier transform (FFT), a more direct approach is to define ...
Olli Niemitalo's user avatar
5 votes
Accepted

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
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5 votes
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Why Does ASK Modulation Create Fourier Sidebands?

If you did a continuous on off keying of a 10101010... pattern, then you would see sidebands as described since this is simply an up-conversion of the Fourier Transform of a 50% duty cycle square wave ...
Dan Boschen's user avatar
  • 52.3k
5 votes

Trying to understand how to get this basic Fourier Series

You're on the right track, just keep a cool head and solve those integrals. You can make use of the function's symmetry by noting that for even and real-valued $f(t)$, the Fourier coefficients can be ...
Matt L.'s user avatar
  • 90.5k
5 votes
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How to know if a continuous function can be represented by a finite sum of sinusoids?

I think a good rule of thumb is this: "If it isn't already written as a finite sum of sinusoids, then it probably can't be written as a finite sum of sinusoids." Most functions are not a ...
Tanner Swett's user avatar
4 votes

Why are Fourier analysis and transform only applicable for LTI systems?

Why are Fourier Analysis & Transform only applicable for LTI systems? That's simply not true. Won't Fourier analysis or Transform be possible? They are. The question is just whether they are ...
Marcus Müller's user avatar
4 votes
Accepted

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

The common formulation of the forward DFT preserves energy (Parseval's theorem). This means that a longer constant magnitude sinewave input to a DFT, which has proportionally more energy, must be ...
hotpaw2's user avatar
  • 35.4k
4 votes

Slow Down Music Playing While Maintaining Frequency

The tool/theory you describe is really a large area of research in music technology, broadly called audio time-scale modification. A large component of this field is how you might prevent audible ...
Speedy's user avatar
  • 436
4 votes

How to get the Fourier series using Python's $\tt fft$

I am no expert in this topic, but have some useful examples to share. 1 component example To keep the i-eth Fourier component, you can zero the rest of the ...
Agustín's user avatar
  • 141
4 votes

Why does Hilbert filter distorts the shape of the signal?

Assuming that the width of the rectangular signal in your figure equals half the period, then the corresponding Fourier series can indeed be written as a weighted sum of sines (left-hand side signal) ...
Matt L.'s user avatar
  • 90.5k
4 votes
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Fourier Series Coefficients

You should use the synthesis equation of an impulse train with period $T$ (which is easy to derive): $$x(t)=\sum_{k=-\infty}^{\infty}\delta(t-kT)=\sum_{k=-\infty}^{\infty}\frac{1}{T}e^{jk\frac{2\pi}...
msm's user avatar
  • 4,295
4 votes

Fourier components of $\cos(2\pi f_1t)$

HINT: Going from your last equation, $$\frac{\sqrt{T}}{2}\bigg(\frac{e^{j2\pi (f_1T-n)}-1}{j2\pi (Tf_1-n)} + \frac{e^{-j2\pi (f_1T+n)}-1}{-j2\pi (Tf_1+n)}\bigg)$$ This can be simplified further down ...
Gilles's user avatar
  • 3,406
4 votes

Scaling property of Fourier Transform

Fat32's answer is correct and that's the way you should do it. Here I just want to add an explanation of your error, and a way to solve the problem correctly continuing from where you went wrong. The ...
Matt L.'s user avatar
  • 90.5k
4 votes
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What does it mean for a function to have frequencies?

I understand that "to have frequencies" can be misleading. This is a shorthand for "having non-zero energy or amplitude at two (specific) frequencies". Rephrased, for the cosine: when a cosine signal ...
Laurent Duval's user avatar

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