Purpose
1. Master the characteristics of DFT conjugate symmetry of real sequences;
2. Learn the method of constructing frequency-domain sequences by applying the conjugate symmetry of real sequence DFT to ensure that the time-domain sequences are real numbers;
Experimental principle
1. Conjugate symmetry of DFT
in:
2. Conjugate Symmetry of DFT of Finite Length Real Sequences
Set
as a finite-length real sequence of length N, it 
is a circular conjugate symmetric sequence: 
. This symmetry can be expressed as: 
.
Experimental content
1. Try to use the conjugate symmetry of DFT to design two efficient algorithms, so that an N-point DFT can be calculated to obtain the N-point DFT of two real sequences. Assume:
(1) Algorithm 1: Let 
, calculate its 16-point discrete Fourier transform (hint: when N=16, 
.
It is verified whether the obtained result is correct by calculating the sums 
separately 
and 
by summing the IDFT method.
clc,clear,close all
N=16;
n=0:1:N-1;
x1=cos(pi/4*n);
x2=sin(pi/8*n);
xn=x1+x2;
Xk=fft(xn,16);
X1=real(Xk);
X2=imag(Xk);
x11=ifft(X1,16);
x22=ifft(1i*X2,16);
subplot(611)
stem(n,X1)
xlabel('k');
ylabel('real(X1)');
subplot(612)
stem(n,X2)
xlabel('k');
ylabel('real(X2)');
subplot(613)
stem(n,x1);
xlabel('n');
ylabel('x1');
subplot(614)
stem(n,x11);
xlabel('n');
ylabel('x1恢复');
subplot(615)
stem(n,x2);
xlabel('n');
ylabel('x2');
subplot(616)
stem(n,x22);
xlabel('n');
ylabel('x2恢复');
Experimental conclusion 1-1 : The relationship between 
, 
and 
?
answer:
(2) Algorithm 2: Let 
, repeat (1).
clc,clear,close all
N=16;
n=0:1:N-1;
k=0:1:N-1;
x1=cos(pi/4*n);
x2=sin(pi/8*n);
xn=x1+1i*x2;
Xk=fft(xn,16);
Xkx=conj(Xk);%取共轭
X1=1/2*(Xk+[Xkx(1),fliplr(Xkx(2:16))]);
X2=1/2*(Xk-[Xkx(1),fliplr(Xkx(2:16))]);
x11=ifft(X1,16);
x22=ifft(X2,16)*(-1i);
subplot(611)
stem(n,real(X1))
xlabel('k');
ylabel('real(X1)');
subplot(612)
stem(n,real(X2))
xlabel('k');
ylabel('real(X2)');
subplot(613)
stem(n,x1);
xlabel('n');
ylabel('x1');
subplot(614)
stem(n,x11);
xlabel('n');
ylabel('x1恢复');
subplot(615)
stem(n,x2);
xlabel('n');
ylabel('x2');
subplot(616)
stem(n,x22);
xlabel('n');
ylabel('x2恢复');
Experimental conclusion 1-2 : The relationship between 
, 
and 
?
answer:
2. Conjugate Symmetry of DFT of Finite Length Real Sequences
From the conjugate symmetry of the DFT of the finite-length real sequence, it can be known that the sequence with conjugate symmetry in the frequency domain is used as IDFT
The latter is a real sequence, and the sending of real numbers can greatly simplify the sending device. OFDM uses this feature to ensure that the sequence sent to the channel is a real sequence.
Program the following as required:
Let XK_in be a frequency domain complex number sequence, XK_in=[1+j,-3-j,-3+3*j,-1-3*j];
Try to use the conjugate symmetry formula of the DFT of the real sequence to 
convert the frequency domain sequence
XK_in is expanded into a conjugate symmetric form Xk to ensure that the corresponding time domain sequence xn =ifft(Xk,16) is a real number sequence.
(1) Find the frequency domain sequence Xk; and give the real and imaginary parts of Xk;
clc,clear,close all
format compact
N=16;
n=0:1:N-1;
k=0:1:7;
XK_in=[1+1i,-3-1i,-3+3*1i,-1-3*1i];
XKf=conj(fliplr(XK_in));
Xk=[0,XK_in,0,0,0,0,0,0,0,XKf];
subplot(211)
stem(n,real(Xk));
xlabel('k');
ylabel('real(Xk)');
subplot(212)
stem(n,imag(Xk));
xlabel('k');
ylabel('imag(Xk)');
Experimental conclusion 2-1: Explain the characteristics of the real and imaginary parts of Xk;
Answer: The real part is symmetrical about N/2 even, and the imaginary part is symmetrical about N/2 odd.
2）求xn =ifft(Xk,16)；
clc,clear,close all
format compact
N=16;
n=0:1:N-1;
k=0:1:7;
XK_in=[1+1i,-3-1i,-3+3*1i,-1-3*1i];
XKf=conj(fliplr(XK_in));
Xk=[0,XK_in,0,0,0,0,0,0,0,XKf];
xn =ifft(Xk,16)
subplot(211)
stem(n,real(xn));
xlabel('n');
ylabel('real(xn)');
subplot(212)
stem(n,imag(xn));
xlabel('n');
ylabel('imag(xn)');
Experimental conclusion 2-2: Whether xn is a sequence of real numbers can be explained by the graph of real and imaginary parts of xn.
Answer: As can be seen from the above figure, the imaginary part of xn is always 0, which is a sequence of real numbers.
experimental thinking
1. For the sequence x(n), how to get the N-point DFT by calculating the N/2-point DFT?
answer:
(1) For the sequence x(n), the radix 2FFT algorithm can be used to divide the parity sequence of x(n) to obtain the N/2-point DFT, thereby obtaining the N-point DFT.
(2) In particular, if x(n) is a real sequence, then 
when N=even, it is only necessary to calculate the front N/2+1 points of X(k), and when N=odd, only X is calculated. (N+1)/2 points in front of (k), thus calculating N-point DFT.