What's the use of fftshift? Why does matlab need fftshift after FFT?

The article is only for personal understanding , Please correct the shortcomings

In use MATLAB do FFT When analyzing signals , Why do you sometimes need to fftshift, Sometimes you don't need , I've done it carefully today , Make a summary note .

First ,fftshift Usage of ：
                           Y = fftshift(X)
                           Y = fftshift(X,dim)
effect ： Swap the left and right halves of the row vector .
stay FFT The role of Li ： By moving the zero frequency component to the center of the array .

Why are you in FFT And then we use fftshift Well ？

I think this problem can be explained from the perspective of sampling law . Take baseband real signal as an example , We all know that the time domain is based on $f_s$ Sampling is in the frequency domain $f_s$ Translational continuation , Because the spectrum of the sampled signal will be in $f_s/2$ Produce aliasing , So the real signal sampling rate $f_s/2$ The bandwidth should be greater than or equal to , namely $f_s\ge 2B$, Here's the picture .
MATLAB Doing... On the sequence FFT When , It is equivalent to taking the spectrum $[0,f_s]$ Part of , Because the spectrum is based on $f_s$ Periodic extension , therefore $[f_s/2,f_s]$ Part of the spectrum and $[-f_s/2,0]$ Part of the same , If you want to see $[-f_s,f_s]$ part （ The red box part of the picture above ）, It needs to be done fftshift, Move the zero frequency component to the middle of the sequence , Here's an example .

close all; clear; clc;
fs = 300;            % The sampling rate is greater than 200
t = 0:1/fs:1;        % Define sampling points 
s = sin(2*pi*100*t)+cos(2*pi*40*t);  % The sampled signal sequence 

%--------- see [0,fs] The spectrum of ---------
F1 = fft(s);                %fft
f1 = linspace(0,fs,length(t));   % Spectrum horizontal axis 
subplot(211);plot(f1,abs(F1));xlabel('f');ylabel(' Range ');title(' see [0,fs]');

%--------- see [-fs/2,fs/2] The spectrum of ---------
F2 = fftshift(fft(s));                %fft
f2 = linspace(0,fs,length(t))-fs/2;   % Spectrum horizontal axis 
subplot(212);plot(f2,abs(F2));xlabel('f');ylabel(' Range ');title(' see [-fs/2,fs/2]');

give the result as follows ：

therefore ,fftshift To see $[-f_s/2,f_s/2]$ Operation required for partial spectrum , If you just look at $[0,f_s]$ The spectrum of , Not really .

The above is only the case of real sequences , However, we often use complex sequences in simulation experiments , Now let's talk about complex sequences .
Complex sequences have no negative frequencies , As long as the sampling rate is greater than the bandwidth , According to the above explanation, it is easy to understand this ,

The picture above shows , as long as $f_s\ge B$, The spectrum will not be aliased . So I won't go to see $[-f_s/2,f_s/2]$ The spectrum of , Will not use fftshift 了 .

close all; clear; clc;
fs = 300;            % The sampling rate is greater than 100
t = 0:1/fs:1;        % Define sampling points 
s = exp(1j*2*pi*100*t)+exp(1j*2*pi*40*t);  % The sampled signal sequence 

%--------- see [0,fs] The spectrum of ---------
F1 = fft(s);                %fft
f1 = linspace(0,fs,length(t));   % Spectrum horizontal axis 
plot(f1,abs(F1));xlabel('f');ylabel(' Range ');title(' see [0,fs]');

%--------- see [-fs/2,fs/2] The spectrum of ---------
% There is no negative frequency , It's meaningless here 
% F2 = fftshift(fft(s));                %fft
% f2 = linspace(0,fs,length(t))-fs/2;   % Spectrum horizontal axis 
% subplot(212);plot(f2,abs(F2));xlabel('f');ylabel(' Range ');title(' see [-fs/2,fs/2]');