Digital Signal Processing Reference
In-Depth Information
N
−
1
C
=
x
[
n
]
(
cos
[
2
πnk/N
]+
j
sin
[
2
πnk/N
]
)
n
=
0
or, using the complex exponential notation
N
−
1
x
[
n
]
e
j
2
πnk/N
C
=
n
=
0
The real part of
C
contains the correlation of
x
[
n
]
with cos
[
2
πnk/N
]
, and the imaginary part contains
the correlation with sin
[
2
πnk/N
]
.
In m-code, the expression
exp (x)
means
e
raised to the
x
power and hence if
x
is imaginary, i.e., an amplitude
A
multiplied by
√
−
1,we
get
exp
(jA)
=
cos
(A)
+
j
sin
(A)
Example C.8.
Compute the correlation (as defined above) of cos(2
π
(0:1:3)/4) with the complex expo-
nential exp(j2
π
(0:1:3)/4).
The call
sum( cos (2*pi*(0:1:3)/4).*exp(j*2*pi*(0:1:3)/4))
produces the answer2+j0.
C.8 USES FOR SIGNAL PROCESSING
• Complex numbers, and in particular, the complex exponential, can be used to both generate and
represent sinusoids, both real and complex.
• Complex numbers (and the complex exponential) make it possible to understand and work with
sinusoids in ways which are by no means obvious using only real arithmetic. Complex numbers are
indispensable for the study of certain topics, such as the complex DFT, the
z
-Transform, and the
Laplace Transform, all of which are discussed in this topic.
