Digital Signal Processing Reference
In-Depth Information
W
k
, k = 0, 1, 2, Á , 7,
Figure 12.8 shows a polar plot of
in the complex plane.
Three key observations can be made from this plot:
1.
Adjacent vectors in the sequence are separated by angles of
2p/N
radians
(N = 8).
2.
Each vector in the plot has an opposite of equal magnitude so that the sum
of the two must be zero, and we can extend this finding to state that
7
W
k
= 0.
a
k= 0
3.
The vectors are in conjugate pairs, so that each vector has a conjugate mate
with an angle of equal value, but opposite sign.
Plotted in Figure 12.8, we have the sequence of complex vectors
W
0
= 1; W
1
= 1e
-jp
>
4
; W
2
= 1e
-jp
>
2
; W
3
= 1e
-j3p
>
4
;
Á ; W
7
= 1e
-j7p
>
4
.
W
kn
= (W
k
)
n
, n = 0, 1, 2, Á , N - 1,
If we evaluate
we have the sequence of
vectors
( W
0
)
n
= 1; (W
1
)
n
= 1e
-jnp
>
4
; (W
2
)
n
= 1e
-jnp
>
2
;
( W
3
)
n
= 1e
-j3np
>
4
;
Á ; (W
7
)
n
= 1e
-j7np
>
4
.
For all of the vectors have a value of 1. For the vectors are as shown
in Figure 12.8. For the vectors will repeat the pattern of Figure 12.8. For
example, if we examine the set of vectors for
n = 0,
n = 1,
n 7 1,
n = 5,
we have
( W
0
)
5
= 1; (W
1
)
5
= 1e
-j5p
>
4
; (W
2
)
5
= 1e
-j5p
>
2
= 1e
-jp
>
2
;
(W
3
)
5
= 1e
-j15p
>
4
= 1e
-j7p
>
4
; (W
4
)
5
= 1e
-j20p
>
4
= 1e
-jp
;
( W
5
)
5
= 1e
-j25p
>
4
= 1e
-jp
>
4
; (W
6
)
5
= 1e
-j30p
>
4
= 1e
-j3p
>
2
; (W
7
)
5
= 1e
-j3p
>
4
.
If we carry this on for all integer values of
k, 0 … k … N - 1,
and
n, 0 … n …
N - 1,
we find that
N- 1
N- 1
W
0n
W
kn
= N and
a
= 0, k = 1, 2, Á , N - 1.
(12.35)
a
n= 0
n= 0
It will now be shown that the discrete Fourier transform is valid; that is, given
N
val-
ues of
x[n],
the forward transform in (12.30) results in
N
values,
X[k].
If, then, these
N
values,
X[k],
are substituted in the inverse transform in (12.30), the original
N
values of
x[n]
are obtained.
