​​​​  ​​ ​​ ​​ ​​​​ The Properties of DFT

• Linearity  ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​​​

• Time shifting

• Frequency shifting​​

• Conjugation

• Duality

• Convolution

• Parseval’s Theorom​​

Property :01

Linearity:​​

Prove: ​​ DFT{αx1(n) +​​ βx2(n)}​​ =​​ αX1[K] +​​ βX2[K]

Code:

n = 1:10;

x1 = sin(n);

x2 = cos(n);

w = -pi+2*pi/N:2*pi/N:pi;

x = x1+x2;

N = length(z);

N1 = length(x1);

N2 = length(x2);

f1 = fft(x1,N1);

f2 = fft(x2,N2);

f = fft(x,N);

subplot(2,1,1)

stem(w,f);

f3 = f1+f2;

subplot(2,1,2);

stem(w,f3);

OUTPUT:

Property :02

Time shifting:

Prove: ​​ DFT{x(n-l)} = X(K)e^(-j(2*pi/N)kl

Code:

n = 1:10;

x = sin(n);

t = -pi+2*pi/N:2*pi/N:pi;

k = +2;

[y n1]=sigshift1(x,n,k);

N = length(y);

f = fft(y,N);

subplot(2,1,1)

stem(n1,f);

​​

N1 = length(x);

f1 = fft(x,N1);

w = 2*pi/N;

f2 = f1.*exp(-1i*w*10*k);

subplot(2,1,2)

stem(n,f2)

OUTPUT:

Property :03

Frequency shifting:

Prove: ​​ DFT{x(n)e^(j(2*pi/N)nl = X(K-l)

Code:

n = 1:5;

x = sin(n);

l = -2;

w = 2*pi/N;

x1 = x.*exp(-1i*w*l*5);

N = length(x1);

f =​​ fft(x1,N);

subplot(2,1,1)

stem(f);

​​

N1 = length(x);

f1 = fft(x,N1);

[y n1]=sigshift1(f1,n,1);

subplot(2,1,2)

stem(y);

OUTPUT:

Property :04

Conjugation:

Prove: ​​ DFT[x*(n)] = X*(-K)

Code:

n = 1:10;

x = sin(n);

m = conj(x);

N = length(m);

f = fft(m,N)

subplot(2,1,1)

stem(n,f)

​​

N1 = length(x);

f1 = fft(x,N1);

[y n1] = sigfold(f1,n);

z = conj(y)

subplot(2,1,2)

stem(n1,z)

OUTPUT:

Property :05

Duality:

Prove: DFT[Nx(-n)] = X(K)

Code:

n = 1:10;

x = sin(n);

Ni = -1;

w = -pi+2*pi/N:2*pi/N:pi;

[y n1] =​​ sigfold(x,n);

y1 = y.*Ni;

N1 = length(y1);

f = fft(y1,N1);

subplot(2,1,1);

stem(n1,f);

​​

N = length(x);

f1 = fft(x,N);

subplot(2,1,2);

stem(n,f1);

OUTPUT:

Property :06

Convolution:

Prove: DFT{x1(n)*x2(n)} = X1(K)X2(K)

Code:

n1 = 1:10;

u = sin(n1);

n2 =​​ 1:10;

v = cos(n2);

x = conv(u,v);

N = length(x);

f = fft(x,N);

w = 2*pi/N:2*pi/N:2*pi;

subplot(2,1,1);

stem(f);

​​

N1 = length(u);

f1 = fft(u,N1);

N2 = length(v);

f2 = fft(v,N2);

y = f1.*f2;

N3 = length(y);

subplot(2,1,2);

stem(y);

OUTPUT: