Digital Signal Processing Reference
In-Depth Information
p
(
t
)
x
s
(
t
)
x
(
t
)
X
x
(
t
)
t
0
p(t)
δ
(t)
δ
(t−1)
t
−T
s
T
s
2T
s
0
x
s
(
t
)
t
−T
s
T
s
2T
s
0
x(n)
o
o
o
o
o
o
o
x(2)
o
o
o
o
n
−1
0
1
2
Fig. 2.1
The sampling process in the time domain
impulse train p
ð
t
Þ¼
P
n
¼1
d
ð
t
nT
s
Þ
(See Fig. (
2.1
)). The resulting sampled
signal in the continuous-time domain is x
s
(t), which corresponds to a train of delta
functions weighted by the values of x(t) at integer multiples of T
s
, i.e., weighted by
x(nT
s
). The discrete-time representation of the same sampled signal is x(n), which
is the sequence of actual values of x(t) at the discrete-time instants t = nT
s
. (Note
that x(n) is actually x(nT
s
), although, T
s
is normally omitted from the notation for
simplicity). In practical DSP, the discrete-time representation (x(n)) is more
commonly used than the continuous-time representation of sampled signals (x
s
(t)).
2.2.1.1 Definitions for Some Important Discrete-Time Signals
Formal definitions for two important discrete-time signals are presented below.
The Discrete-Time Unit Pulse
The discrete-time counterpart for the continuous-time unit impulse (delta function)
is the discrete-time unit pulse. It is defined as:
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