Page 337 - Schaum's Outline of Theory and Problems of Signals and Systems

FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS [CHAP. 6

Then

Interchanging the order of summation and integration, we get
1 ( w
X ) = ( 0 1 x,[n] e'j("-@"
n= -00

Hence,

6.27. Verify the properties (6.62), (6.63~1, and (6.63b); that is, if x[n] is real and

x[n] =x,[n] +xo[n] ++X(fl) =A(R) + jB(R) (6.140)
where x,[n] and xo[n] are the even and odd components of x[n], respectively, then
X(-R) =X*(R)
x,[n] ++ Re{X(R)} = A(i2)

x,[n] ++ j Im{X(i2)} = jB(fl)
If x[n] is real, then x*[n] =x[n], and by Eq. (6.45) we have
x*[n] -X*( -R)

from which we get
x(n) =x*( -n) or x( -0) =x*(n)

Next, using Eq. (6.46) and Eqs. (1.2) and (1.3), we have
X[ -n] =x,[n] - x,[n] c-, X( -0) = X*(R) =A(R) - jB(R) (6.141)
Adding (subtracting) Eq. (6.141) to (from) Eq. (6.1401, we obtain
x,[n] -A(R) = Re(X(R)}
x,[n] H jB( 0) = j Im{ X(R))

6.28. Show that

Let
44 ++X(R>
Now, note that
s[n] = u[n] - u[n - 11
Taking the Fourier transform of both sides of the above expression and by Eqs. (6.36) and
(6.431, we have