Digital Signal Processing Reference
In-Depth Information
2.1.4
Elementary Operators
In this Section we present some elementary operations on sequences.
l
g
r
,
y
i
d
.
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,
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s
Shift.
Asequence
x
[
n
]
,shiftedbyaninteger
k
is simply:
y
[
n
]=
x
[
n
−
k
]
(2.12)
If
k
is positive, the signal is shifted “to the left”, meaning that the signal has
been
delayed
;if
k
is negative, the signal is shifted “to the right”, meaning
that the signal has been
advanced
. The delay operator can be indicated by
the following notation:
k
x
]
=
[
n
x
[
n
−
k
]
Scaling.
Asequence
x
[
n
]
scaled by a factor
α
∈
is
y
[
n
]=
α
x
[
n
]
(2.13)
If
is real, then the scaling represents a simple amplification or attenuation
of the signal (when
α
is complex, amplifi-
cation and attenuation are compounded with a phase shift.
α>
1and
α<
1, respectively). If
α
Sum.
The sum of two sequences
x
[
n
]
and
w
[
n
]
is their term-by-term sum:
y
[
n
]=
x
[
n
]+
w
[
n
]
(2.14)
Please note that sumand scaling are linear operators. Informally, thismeans
scaling and sum behave “intuitively”:
α
x
]
=
α
[
n
]+
w
[
n
x
[
n
]+
α
w
[
n
]
or
k
x
]
=
[
n
]+
w
[
n
x
[
n
−
k
]+
w
[
n
−
k
]
Product.
The product of two sequences
x
[
n
]
and
w
[
n
]
is their term-by-
term product
y
[
n
]=
x
[
n
]
w
[
n
]
(2.15)
Integration.
The discrete-time equivalent of integration is expressed by
the following running sum:
n
y
[
n
]=
x
[
k
]
(2.16)
=
−∞
k
