Digital Signal Processing Reference
In-Depth Information
where
ξ
[
n
]
is a “selector” sequence defined as
N
1 r
n
multiple of
N
0 rwie
ξ
[
n
]=
N
l
g
r
,
y
i
d
.
,
©
,
L
s
The question now is to find an expression for such a sequence; to this end,
let us recall a very early result about the orthogonality of the roots of unity
(see Equation (4.4)), which we can rewrite as follows:
N
for
n
multiple of
N
0 t r ie
N
−
1
W
kn
=
(11.8)
N
k
=
0
e
−j
2
N
. Clearly, we can define our desired selector
where, as per usual,
W
N
=
sequence as
N
−
1
1
N
W
kn
ξ
[
n
]=
N
N
k
=
0
and we can therefore write
∞
N
−
1
1
N
W
k
N
x
[
n
]
z
−n
X
a
(
z
)=
n
=
−∞
k
=
0
N
−
1
∞
]
W
N
z
−
1
n
1
N
=
x
[
n
n
=
−∞
k
=
0
N
−
1
1
N
W
N
z
=
X
(
)
(11.9)
k
=
0
so that finally:
N
−
1
1
N
X
W
N
z
1
/
N
X
ND
(
z
)=
(11.10)
k
=
0
The Fourier transform of the downsampled signal is obtained by evalu-
ating
X
ND
(
z
)
on the unit circle; explicitly, we have
N
−
1
X
e
j
(
N
−
)
1
N
2
N
k
e
j
ω
)=
X
ND
(
(11.11)
k
=
0
The resulting spectrum is, therefore, the scaled sum of
N
superimposed
copies of the original spectrum
X
e
j
ω
)
(
; each copy is shifted in frequency
by a multiple of 2
N
and the result is stretched by a factor of
N
.Weare,in
many ways, in a situation similar to that of equation (9.33) where sampling
π/
