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The Properties of the DTFT 69
8. Multiplication: This is a dual of the convolution property.
F [x 1 (n)·x 2 (n)] = F [x 1 (n)] ∗○ F[x 2 (n)] △ = 1
2π
∫ π
−π
X 1 (e jθ )X 2 (e j(ω−θ) )dθ
(3.12)
This convolution-like operation is called a periodic convolution and
hence denoted by ∗○. It is discussed (in its discrete form) in
Chapter 5.
9. Energy: The energy of the sequence x(n) can be written as
E x =
=
∞∑
|x(n)| 2 = 1
2π
−∞
∫ π
0
|X(e jω )| 2
dω
π
∫ π
−π
|X(e jω )| 2 dω (3.13)
(for real sequences using even symmetry)
This is also known as Parseval’s theorem. From (3.13) the energy density
spectrum of x(n) isdefined as
Φ x (ω) = △ |X(ejω )| 2
π
Then the energy of x(n) inthe [ω 1 ,ω 2 ] band is given by
∫ω 2
ω 1
Φ x (ω)dω, 0 ≤ ω 1 > x1 = rand(1,11); x2 = rand(1,11); n = 0:10;
>> alpha = 2; beta = 3; k = 0:500; w = (pi/500)*k;
>> X1 = x1 * (exp(-j*pi/500)).^(n’*k); % DTFT of x1
>> X2 = x2 * (exp(-j*pi/500)).^(n’*k); % DTFT of x2
>> x = alpha*x1 + beta*x2; % Linear combination of x1 & x2
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