Digital Signal Processing Reference
In-Depth Information
A signal having finite power is called a
Power Signal
.
2.5
DISCRETE TIME SYSTEMS
2.5.1 LTI SYSTEMS
A processing system that receives an input sample sequence
x
]
in response is called a
Discrete Time System
. If we denote a discrete time system by the operator
DT S
,
we can then state this in symbolic form:
[
n
]
and produces an output sequence
y
[
n
y
[
n
]=
DT S
[
x
[
n
]
]
A number of common signal processes and/or equivalent structures, such as FIR and IIR filtering
constitute discrete time systems; they also possess two important properties, namely, 1) Time or Shift
Invariance, and 2) Linearity.
A discrete time system
DT S
is said to be
Shift Invariant
,
Time Invariant
,or
Stationary
if,
assuming that the input sequence
x
[
n
]
produces the output sequence
y
[
n
]
, a shifted version of the input
sequence,
x
[
n
−
s
]
produces the output sequence
y
[
n
−
s
]
, for any shift of time
s
. Stated symbolically,
this would be
DT S
[
x
[
n
−
s
]
]
=
y
[
n
−
s
]
A discrete time system
DT S
that generates the output sequences
y
1
[
n
]
and
y
2
[
n
]
in response,
respectively, to the input sequences
x
1
[
n
]
and
x
2
[
n
]
is said to be
Linear
if
DT S
[
ax
1
[
n
]+
bx
2
[
n
]
]
=
ay
1
[
n
]+
by
2
[
n
]
where
a
and
b
are constants. This is called the
Principle of Superposition
.
A system that is both shift or time invariant and linear will produce the same output sequence
y
[
n
]
in response to the sequence
x
[
n
]
regardless of any shift in time of
n
samples. Such systems are referred
to as
Linear, Time Invariant (LTI)
systems.
Example 2.1.
Demonstrate linearity and time invariance for the system below using MathScript.
y
[
n
]=
2
x
[
n
]
can be scaled by the constant
A
. The
code below generates a cosine of frequency
F
, scaled in amplitude by
A
as
x
[
n
]
We begin with code to compute
y
[
n
]
= 2
x
[
n
]
where
x
[
n
]
, computes
y
[
n
]
, and then
plots
x
. You can change the scaling constant
A
and note the linear change in the output, i.e.,
if the input signal is scaled by
A
, so is the output signal (comparison of the results from running the two
example calls given in the script above will demonstrate the scaling property).
[
n
]
and
y
[
n
]
