Sure! The Fourier Transform is a mathematical tool used to decompose a function of time (or space) into its constituent frequencies, and is used extensively in signal processing, audio and image compression, and many other fields.

The continuous Fourier Transform (CFT) of a function f(x) is defined as:

F(ω)=∫−∞∞f(x)e−iωxdxF(\omega) = \int_{-\infty}^{\infty} f(x) e^{-i\omega x} dxF(ω)=∫−∞∞​f(x)e−iωxdx

where F(ω) is the Fourier Transform of f(x), and i is the imaginary unit. This formula expresses the function f(x) in terms of a continuous spectrum of frequencies ω, each weighted by a complex exponential factor. The CFT is an integral transform that converts a function of time (or space) into a function of frequency.

The inverse Fourier Transform is:

f(x)=12π∫−∞∞F(ω)eiωxdωf(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty} F(\omega) e^{i\omega x} d\omegaf(x)=2π1​∫−∞∞​F(ω)eiωxdω

which converts a function of frequency back into a function of time (or space).

For a discrete-time signal, the Fourier Transform is replaced by the Discrete Fourier Transform (DFT), which is defined as:

X[k]=∑n=0N−1x[n]e−i2πnk/NX[k] = \sum_{n=0}^{N-1} x[n] e^{-i2\pi nk/N}X[k]=∑n=0N−1​x[n]e−i2πnk/N

where x[n] is the discrete-time signal, and X[k] is the DFT of x[n]. The DFT computes a finite number of frequencies k, each weighted by a complex exponential factor.

The inverse DFT is:

x[n]=1N∑k=0N−1X[k]ei2πnk/Nx[n] = \frac{1}{N}\sum_{k=0}^{N-1} X[k] e^{i2\pi nk/N}x[n]=N1​∑k=0N−1​X[k]ei2πnk/N

which converts a signal in the frequency domain back into the time domain.

In summary, the Fourier Transform is a mathematical tool used to decompose a function of time (or space) into its constituent frequencies, and is used extensively in signal processing and many other fields. The continuous Fourier Transform is expressed as an integral, while the Discrete Fourier Transform is a sum over a finite number of frequencies.
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