Introduction to Digital Signal and System Analysis - Tutorsindia

Introduction to Digital Signal and System Analysis
Discrete Fourier Transform
6. Even and odd signals
From the DFT definition
N
= − 1
2πkn
2πkn
X [ k]
x[
n]
cos
− j sin
n= 0 N N
2πkn
2πkn
where cos is an even function, and sin is an odd function.
N
N
Let
Real( X[
k])
=
Imag
( X[
k]
)
=
N −1
n=
0
N −1
n=
0
2πkn
x[
n]cos
N
2πkn
x[
n]sin
N
When x[n] is real signal,
a) if x[n] is an even function,
Im(X[k]) =0 (6.9)
b) if x[n] is an even function,
Re(X[k]) =0 (6.10)
This property can be used to simplify and save the calculation.
7. Conjugation
N
1
N
1
If x[n] is real, ∑ − X [0] = x[
n]
and ∑ − n
X [ N / 2] = ( −1)
x[
n]
are real coefficients, and the other N-2 are complex
coefficients.
n=
0
n=
0
X [ −k]
=
X * [ k]
or
X [ N − k]
=
X * [ k]
(6.11)
X [ −k]
= X [ k]
,
X [ N − k]
= X [ k]
(6.12)
Only X[0], X[N/2] and X(k), k=1,2,N/2-1 are needed to represent the whole X[k] (k=0,1,2,…,N-1). i.e. there are a total
of 2 real and N/2-1 complex coefficients. It can also be proved
92
Download free ebooks at bookboon.com