Overview of the Fourier Transform

Knowledge Supplement:

1. Time domain and frequency domain:

1) Time domain (time domain)—the independent variable is time, that is, the horizontal axis is time, and the vertical axis is the change of signal. Its dynamic signal is a function that describes the value of the signal at different times.

2) Frequency domain (frequency domain)—the independent variable is frequency, that is, the horizontal axis is the frequency, and the vertical axis is the amplitude of the frequency signal, which is commonly referred to as a spectrogram.

3) The frequency domain and the time domain are themselves symmetrical, and there is a mutual constraint relationship: the continuous signal in the time domain corresponds to the aperiodic in the frequency domain, and the discrete corresponds to the periodic; the periodic in the time domain corresponds to the discrete in the frequency domain, and the aperiodic corresponds to continuous.

1. Fourier series (Fourier series, FS)

Continuous, periodic signal --> non-periodic, discrete

Any periodic function can be represented by an infinite series composed of sine and cosine functions (the sine and cosine functions are chosen as the basis functions because they are orthogonal), and later generations call the Fourier series a special trigonometric Series, according to Euler's formula, trigonometric functions can be transformed into exponential form, also known as Fourier series as an exponential series.

$f\left ( t \right )=C_{1}\varphi _{1}\left ( t \right )+C_{2}\varphi _{2}\left ( t \right )+...+C_{i}\varphi _{i}\left ( t \right )+...=\sum_{i=1}^{\infty }C_{i}\varphi _{i}\left ( t \right )$

The above formula is called the orthogonal expansion of the function, also known as the generalized Fourier series.

Remarks: $\varphi _{1}\left ( t \right ),\varphi _{2}\left ( t \right ),...,\varphi _{n}\left ( t \right )$ It is an orthogonal function.

Fourier series of periodic functions

The complete orthogonal functions over a period $\left ( t_{0},t_{0}+T \right ),T=2\pi /\Omega$ are:

1) Trigonometric functions: $\left \{ 1,cos\left ( n\Omega \pi \right ),sin\left ( n\Omega \pi \right ),n=1,2,... \right \}$

2) Set of imaginary exponential functions: $\left \{ e^{jn\Omega t},n=1,2,... \right \}$

Suppose the periodic function f(t), the period is T, the angular frequency (the fundamental frequency) $\Omega =2\pi /T$ , when it satisfies the Dirichlet condition, it can be expanded into a triangular Fourier series.

Complete set of orthogonal functions:

Dirichlet conditions:

1) In a period, the function is continuous or has only a limited number of discontinuity points of the first type (there are both limits around the discontinuity point);

2) In one cycle, the number of maximum and minimum values of the function should be limited;

3) Within a period, the function is absolutely integrable.

2. Discrete-time Fourier series (DFS/DTFS)

discrete, periodic --> periodic, discrete

Unlike continuous-time periodic signals, the Fourier series of discrete-time periodic signals is periodic. For a discrete-time periodic signal with a period of N, its frequency component spacing is 1/N, and the frequency components of the discrete-time periodic signal only need N values. This is the fundamental difference between Fourier series for continuous-time periodic signals and discrete-time periodic signals. (For details, see: DFS (Discrete Fourier Series)_Baidu Encyclopedia )

3. Fourier Transform (Fourier Transform, FT)

Continuous, aperiodic --> aperiodic, continuous

From the change of FS, the continuous-time non-periodic signal can be regarded as the period of the continuous-time periodic signal tends to infinity. At this time, the frequency discrete harmonic fundamental frequency of the Fourier series (FS) tends to be infinitely small, and the discrete frequency becomes continuous frequency. FS has become FT.

4. Discrete-time Fourier Transform (DTFT)

Discrete, aperiodic, infinitely long --> periodic, continuous

It transforms a function (discrete time signal) that takes discrete time nT (where T is the sampling interval) as a variable to a continuous frequency domain, that is, generates a continuous spectrum of this discrete time signal. It is worth noting that this spectrum is periodic .

5. Discrete Fourier Transform (DFT)

Discrete, aperiodic, finite length --> periodic, discrete

The discrete Fourier transform is to make a periodic expansion of the finite-length non-periodic series (into discrete, periodic, and infinite length), and then perform a discrete-time Fourier series to transform the sampling of the time-domain signal into a DTFT frequency domain sampling. Then intercept a periodic definition in the discrete Fourier series.

DTFT is for people, and DFT is for machines. Where are the limitations of the machine? Machines cannot express an infinitely long sequence, nor can they express continuous frequency domain features. For general discrete-time signals, it is really good to use DTFT directly, which is very convenient for us to analyze the frequency domain characteristics of the signal, but the problem is that this set of machines cannot be used.

6. Fast Fourier transform (fast Fourier transform, FFT)

Fast Fourier transform (fast Fourier transform), that is, the use of computers to calculate discrete Fourier transform (DFT) is a general term for efficient and fast calculation methods, referred to as FFT. The time complexity is $O\left ( nlogn \right )$ .

FFT decomposes the original N-point sequence into a series of short sequences in turn according to the odd, even, imaginary, and real characteristics of the discrete Fourier transform. By making full use of the symmetric and periodic properties of the exponential factor in the DFT calculation formula, the corresponding DFTs of these short sequences are obtained and combined appropriately, so as to achieve the purpose of deleting repeated calculations, reducing multiplication operations and simplifying the structure.

There are many FFT algorithms, and a brief introduction to the base 2 algorithm:

1) $x\left ( n \right )$ Decompose the signal into two sub-signals

Even sample point signal; $x\left ( 2n \right )$

Odd sample point signal: $x\left ( 2n+1 \right )$

2) Understand the two sum terms as two DFTs of length n/2

3) The specific calculation process of FFT

(For details, see: Learn Fast Fourier Transform (Fast Fourier Transform) in one hour - Zhihu )

7. Discrete Cosine Transform (DCT)

Since many signals to be processed are real signals, when using DFT, due to the conjugate symmetry of the Fourier transform of the real signal, there is half of the data redundancy in the frequency domain after DFT.

because:

Even function * Even function = Even function

odd function * odd function = odd function

When x[n] is a real even function:

$Im\left [ k \right ]=\sum_{n=0}^{N-1}x\left [ n \right ]sin\left ( kt \right )=0$

Then DFT can be abbreviated as:

$X\left [ k \right ]=\sum_{n=0}^{N-1}x\left [ n \right ]cos\left ( \frac{2\pi kn}{N}\right )$

But in reality, there are not so many real even signals, so we think they are created. The length of the signal is doubled to 2N, and in order to make the created signal symmetrical about 0, the entire extended signal is shifted to the right by 0.5 units, and the final DCT transformation formula is:

(For details, see: Detailed Discrete Cosine Transform (DCT) - Zhihu )