Fourier series, Fourier transform, short-time Fourier transform formula

Fourier series, Fourier transform and short-time Fourier transform are commonly used tools in signal processing, which can help us analyze the spectral structure and periodic characteristics of signals. The following is a detailed introduction to these three concepts:

Fourier series
The Fourier series is a representation that decomposes a periodic signal into a set of sine and cosine functions. It is based on Fourier's theorem that any periodic signal can be expressed as the sum of a series of sine and cosine functions. These sine and cosine functions are called fundamental frequencies, and they have different frequencies and amplitudes that describe the spectral structure and periodicity of periodic signals.

The formula for the Fourier series is:

$$f(t)=\frac{a_0}{2}+\sum _{n=1}^{\infty }[a_{n}cos(n\omega t)+b_{n}sin(n\omega t)]$$

where $f\left(t\right)$ is a periodic signal,$\omega$ is the fundamental frequency,$a_0$、$a_n$and $b_n$are coefficients and can be computed by computing the projection of the signal onto the fundamental frequency. Fourier series is very useful in the periodic analysis of signals, which can help us understand the periodic characteristics and frequency components of signals.

Fourier Transform
The Fourier Transform is a representation that decomposes a non-periodic signal into a set of sine and cosine functions. It is based on Fourier's theorem that any signal can be expressed as the sum of a series of sine and cosine functions. These sine and cosine functions are called fundamental frequencies, and they have different frequencies and amplitudes that describe the spectral structure and frequency characteristics of the signal.

The formula for the Fourier transform is:

$$F\left(\omega \right)={\int }_{-\infty }^{\infty }f(t)e^{-i\omega t}dt$$

where $f\left(t\right)$ is a non-periodic signal,$\omega$ is the frequency,$F\left(\omega \right)$ is the signal at frequencyω \omegaThe projection on$\omega$$e^{-i\omega t}$ is a complex exponential function. Fourier transform is very useful in the frequency domain analysis of signals, which can help us understand the frequency components and spectral structure of signals.

Short-time Fourier transform
The short-time Fourier transform is a method for spectral analysis of non-stationary signals. It divides the signal into time slices and performs a Fourier transform on each time slice. In this way, the representation of each time segment in the frequency domain can be obtained, so as to better understand the frequency characteristics and spectral structure of the signal.

The formula for the short-time Fourier transform is:

$$STFT(t,oh)={\int }_{-\infty }^{\infty }f\left(\tau \right)w(\tau -t)e^{-i\omega \tau }d\tau$$

where $f\left(\tau \right)$ is a non-stationary signal,$w(t-t)$ is the window function,$STFT(t,\omega )$ is the signal at time$t$ and frequencyω \omegaprojection on$\omega \; .$The short-time Fourier transform is very useful in the time-frequency analysis of signals, which can help us understand the time-frequency characteristics and spectral structure of signals.

In general, Fourier series and Fourier transform are methods for spectral analysis of periodic and non-periodic signals, while short-time Fourier transform is a method for spectral analysis of non-stationary signals. These tools are very useful in signal processing, which can help us understand the time domain and frequency domain characteristics of the signal, so as to better analyze and process signal data.