# 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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### Frequency content of a noisy signal

To find the frequency content of a noisy signal (PSD), there are two methods below: #1 Take the fourier transform of its power signal (square the noisy signal) #2 Find the autocorrelation function of ...
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### Finding Discrete Fourier Transform (DFT) for different DFT size

$N$ is an even integer, $x[n]$ is a finite length signal over the interval $n \in [0,N-1]$, and $X[k]$ is the $N$-point DFT of $x[n]$. Analytically find the DFT of sequence below in terms of $X[k]$. ...
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### Relationship between fourier transform and fourier series

Let $$x(t) = A\sin(2 \pi f_0 t + \alpha)$$ its Fourier transform is given by $$X(\omega) = \frac{A \pi}{i}(e^{ia}\delta(\omega-2\pi f_0) - e^{-ia}\delta(w+2\pi f_0)).$$ the Fourier series complex ...
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### Is it useful to think of a Fourier Transform as writing out a signal in terms of a basis?

The (modified) trigonometric functions $\{0, \cos(kx), \sin(kx)\}$ serves as a basis for periodic function. I have also seen (but not rigorously) that the Fourier transform can also be seen as an ...
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### Fourier Transform and Music Analysis

I am a senior in high-school and am currently trying to conduct an exploration on Fourier Analysis, specifically using the Discrete Fourier Transform to analyze a chord played on my piano. Basically, ...
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### CFO Estimation in LoRa Chirp Signal (Preamble part)

I am trying to estimate CFO in LoRa chirp signal (preamble part). I have seen the discussion about CFO on this forum but it is mainly related to CFO estimation in OFDM. I want to estimate the CFO in a ...
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### Why is the DC component of discrete fourier transform not the same as the signal's arithmetic mean?

In this question we have a mathematical proof that the DC component of normalized discrete Fourier transform should be the same as the signal's arithmetic mean. However, in the following example I ...
Suppose I have some real-valued signal $x\mapsto f(x)$. The amplitude of its Fourier transform $\mathcal{F}[f]$ then looks like a peak around the DC-term, decaying as we move towards higher ...