Download Chapter 2 One Dimensional Continuous Time System

Chapter 2
One Dimensional Continuous
Time System
2.2 One Dimensional Continuous
Time System
Definition 1: An one dimensional analog system
or continuous time system can be defined as a
mapping function T which maps a real-valued
analog signal f(t) to another real-valued g(t) such
that
g ( t )  T [ f ( t )]
input
f(t)
system
T
output
T[f(t)]
Definition 2: The mapping T is linear if it satisfies
the following equations
T[af1(t) + bf2(t)] = aT[f1(t)] + bT[f2(t)] (additivity)
T[af(t)] = aT[f(t)]
(homogeneity)
for any a , b  R .
If a system is not linear it is called a nonlinear
system.
Example 1: A multiplier system is defined as
T [ f ( t )]  af ( t ).
The multiplier is an amplifier that converts a very
small audio signal into a large audio to drive the
speaker. The multiplier is a transformer that
convert a low voltage sinusoidal wave into a high
voltage sinusoidal wave or vice versa.
Example 2: A differentiator is defined as
df (t )
T[ f (t )] 
dt
Example 3: An integrator is defined as
t
T [ f ( t )]   f ( a ) da
0
Example 4: A delay system is defined as
T [ f ( t )]  f ( t  D )
where D > 0 is a time delay.
Example 5: The following square wave
 1, if    t  0;
f (t )  
 1, if 0  t   ;
with the period 2 can be approximated by
N
2
f N (t )  
(1  ( 1) n ) sin( nt )
n 1 n
fN(t) is the superposition (linear combination)
of sine waves. It is easy to know that
f1 (t ) 
4

sin( t )
4
1
f 3 (t )  (sin( t )  sin( 3t ))

3
4
1
1
f5 (t )  (sin( t )  sin( 3t )  sin( 5t ))

3
5
4
1
1
1
f 7 (t )  (sin( t )  sin( 3t )  sin( 5t )  sin( 7t ))

3
5
7
Figure2.1: (a) N = 1 (b) N=3 (c) N = 5
(d) N = 10 (e) N=50 (f) N = 100
Figure 2.1 shows f1(t), f3(t), f5(t), f10(t), f50(t) and
f100(t). As N increase There are overshoot and
undershoot also increases at t = 0. the following
MATLAB program show the plot of f100(t)
<Matlab Program>
%
% f(wc) = -1, -pi < wc < 0
%
= 1 , 0 <= wc < pi
wc = 0.5 * pi;
ws = 0.01* pi;
N = pi /ws;
Nc = wc/ws;
f = zeros(1, 2*N-1);
f(1:N-1) = -1 * ones(1,N-1);
f(N:2*N-1) = ones(1, N);
Nop = 100;
w = (-pi + ws) : ws : (pi -ws);
s = zeros(size(w) );
for i = 1:1:Nop
s = s + 2/pi * (1 - (-1)^i )/i * sin(i*w);
end;
plot(w, f, w,s);
Example 6: Figure 2.2(a) shows
f(t) = 1 + 0.5cos(200t) + 0.25 cos(1000t)).
It is easy to know that
df ( t )
 100 sin( 200t )  250 sin(1000t ).
dt
Figure 2.2(b) shows the analog signal
df ( t )
.
dt
Obviously the differentiation system attenuates
low frequency signal and magnifies high
frequency. It is a high pass system.
(a)
(b)
(c)
Figure 2.2: (a) f(t) = 1 + 0.5 cos(200t) + 0.25cos(1000t)
df ( t )
dt
(b)
(c)

t

f (a )da
Also, if f(t) is processed by an integration system
then
t

1
1
f ( a ) da  t 
sin( 200t ) 
sin(1000t ).
400
4000
t
Figure 2.2(c) show the analog signal
 f (a )da.
The integration system attenuates high frequency
and magnifies low frequency. it is a low pass
system.
Example 7: A square system is defined as
T [ f ( t )]  f ( t )
2
Example 8: A exponential system is defined as
T [ f ( t )]  e
f (t )
Example 9: A natural logarithm system is
defined as
T [ f (t )]  log( f (t ))
Definition 3: The system T is linear timeinvariant if
T [ f ( t  a )]  g ( t  a ).
Example 10: The system g(t)  g(t  D) = f(t)
is a linear time invariant system.
<Proof:>
Let h’(t) = T(d(t a), a  0, is the output when
d (t  a) Thus,
h  ( t )  h  ( t  D )  d ( t  a ).
Assume that the system is linear time
invariant.It is known that the output of
the system is the impulse response h(t)
when the input is d(t). At time t  a if the
input is f(t  a) = d(t  a) then the
output is
h(t  a )  h(t  a  D)  d (t  a )
From the above two equations we can easily
obtain
h (t )  h(t  a )  h (t  D)  h(t  a  D)
The equality holds if
h  ( t )  h ( t  a )  T (d ( t  a )).
Therefore, the system is time invariant.
Example 11: The system g(t)  tg(t  D) = f(t)
is a time varying system
<Proof:>
Let h’(t) be the output when the input d ( t  a ),
a  0. applies to the system. Thus,
h  ( t )  th  ( t  D )  d ( t  a ).
At time t  a we obtain
h ( t  a )  ( t  a ) h ( t  a  D )  d ( t  a ).
Assume that the system is linear time invariant.
It is known that the output of the system is the
impulse response h(t) when the input is d(t  a).
Therefore,
h ( t  a )  th ( t  a  D )  d ( t  a ).
From the two previous equations we can
easily obtain
ah ( t  a  D )  0,
which implies
h ( t  a  D )  0,
Thus,
h ( t  a )  d ( t  a ).
If the system is time invariant
h ( t  a  D )  d ( t  a  D ),
which is contraction to Equation h(t  a  D) = 0,
Therefore, h  ( t )  h ( t  a ) and the impulse
response of the system is time varying.
Theorem 1: If h(t) is the impulse response of
a linear invariant system and f(t) is the input of
the system then the output is

g (t )   h(t  a ) f (a )da.


f(t)
h(t)
g (t )   h(t  a ) f (a )da.

Definition 4: A system T is called causal
if its present output does not depend on its
future input
Definition 5: A system T is call BIBO stable if
its input and output is bounded.
That is, if
then
| f (t ) | 
| T [ f (t )] | 
Theorem 2: If the impulse response h(t)
of the system is absolute integrable
the system is BIBO stable.
<Proof:>
Assume that the input f(t) is bounded. There
exist a real number M1 such that
| f (t ) | M1
Since h(t) is absolute integrable there exists a
real number M2 such that


| h( a )| da  M 2
Then, the absolute output is
t
| g ( t )| |  h ( a ) f ( t  a ) da |
0
t
  | h( a )|| f ( t  a )| da
0
t
 M 1  | h ( a )| da
0
 M1 M 2
The output g(t) is bounded.
Example 12: A multiplier system is defined as
g ( t )  kf ( t )
where k is a constant and is called the gain of the
system.The system is an all pass filter.
Example 13: A time delay system is defined as
g ( t )  f ( t  T ),
where T is called the delay time.
It can be easily
that the system is linear time invariant. The system
is casual since the impulse response h(t) = d(tT)
= 0 for t < 0. The output depends on its past input
but does not depends on the future input.
Note that the system
g (t )  f (t  T )
is linear time invariant. However, it is not causal
because the current output g(t) depends on future
input f(t + T). In fact, its impulse response
h(t )  d (t  T )  0
for t < 0.
Example 14: A differentiation system is defined
as
df (t )
g (t ) 
dt
It is a high pass filter.
If f(t) = e jw t then g(t) =jwe jw t. For small w, |g(t)|
is very small while for large w, |g(t)| is very large.
It is a high pass filer.
Example 15: An integration system is defined as
t
g ( t )   f ( a ) da
0
It is a low pass filter.
If
j wt
f (t )  e , g (t ) 
1
jw
jw
( e t  1).
For small w, |g(t)|
is very large while for large w, |g(t)| is very small.
It is a low pass filter. The integration filter is
linear time invariant.
Example 16: A fall-wave rectifier is
defined as
g ( t )  abs( f ( t ))
Example 17: A half-wave rectifier is defined as
 f (t ),
g (t )  
 0,
if
f ( t )  0;
otherwise.
Both the full wave rectifier and the half wave
rectifier are nonlinear system.
Definition 6: the modulation of a signal f(t) is
defined as
g ( t )  m( t ) f ( t ),
where m(t) is called the modulating signal.
2.3 Linear Differential Equations
Definition 7: an ordinary differential equation is
an equation that has derivatives with respect to an
independent variable only. If the equations has the
derivatives with respect to at least two variables it
is called an partial differential equation.
Definition 8: The ordinary differential equation
d N g (t )
d N 1 g ( t )
dg ( t )
a N (t )
 a N 1 ( t )
... a1 ( t )
 a0 ( t ) g ( t )  f ( t )
N
N 1
dt
dt
dt
is a linear differential equation of order N. If a
differential equation can not be written as above
equation it is nonlinear.
Each Coefficient ak(t) depends on on the variable t.
If f ( t )  0 the
equation is said to be non- homogeneous; otherwise it it said to be homogeneous.
If the initial conditions
n
d g (0)
n
dt
 ck ,
0n N
are given the differential equation is called the
initial-value problem.
In practical systems,
is usually assumed to be
constant so that the system is
LTI.
ak (t )
Example 18: The following equations
dg ( t )
2
t
 g (t )  e
dt
dg ( t )
3t
 sin( t ) g ( t )  cos( t )
dt
are ordinary differential equations.
Example 19: The following equations
u ( x , y )
v ( x , y )
2
5 0
x
y
 u( x, y )  v( x, y ) u( x, y ) v( x, y )
2

2
2
2
 x
 y
y
y
2
2
are partial differential equations
Example 21: The following ordinary differential
equations
dg ( t )
3t
 sin( t ) g ( t )  e t
dt
2
d g (t )
 g (t )  0
2
dt
d 2 g (t )
dg ( t )
3
 2 g (t )  2
2
dt
dt
d 5 g (t )
dg ( t )
3
 2 g ( t )  sin( t )
5
dt
dt
are linear.
Example 22: The following ordinary differential
equations
dg ( t )
2
t
 g (t )  e
dt
2
dg ( t ) d g ( t )

g
(
t
)

0
2
dt
dt
are nonlinear.
i (t )
+
R
-
VR (t )  Ri (t )
Voltage across the resistor
i (t )
i (t )
+
L
-
di (t )
VL ( t ) L
dt
Voltage across the inductor
i (t )
i (t )
+
C
-
1 t
VC (t )   i (t )dt
C 0
Voltage across the capacitor
i (t )
The Kirchhoff’s Current Law
The sum of the currents at a node
in a circuit is zero.
i1 (t )
i3 (t )
i2 (t )
i4 (t )
i1 (t )  i2 (t )  i3 (t )  0
i2 (t )  i4 (t )  i5 (t )  0
i5 ( t )
The Kirchhoff’s Voltage Law
The sum of the voltages around
a loop in a circuit is zero.
v2 (t )
+
-
+
+
v1 (t )
v3 (t )
-
+
-
v4 (t )
- v1(t ) + v2 (t ) + v3 (t ) + v4 (t ) = 0
Example 23: A R-L series circuit shown in
Figure 2.5
R
i(t)
+
V

Figure 2.5: The R-L series circuit.
L
The current i(t) that satisfies
di (t )
L
 Ri (t )  V (t ), i (0)  0
dt
where V(t) is the input voltage.
It is a first order differential equation.
Example 24: Figure 2.6(a) show a circuit where
R and C in series.
i(t)
R
+
V

Figure 2.6(a): The R-C series circuit.
C
The current flows in the circuit
is i(t). The voltage across the capacitor is
t
1
i (t )dt
C 0

and the voltage across the resistor is Ri(t). Thus,
1 t
V  Ri (t )   i (t )dt
C 0
The equation can be also written as
dq (t ) q (t )
VR

dt
C
where q(t) is the charge across the capacitor.
Example 25: A series R-L-C circuit shown in
Figure 2.7
R
i(t)
V(t)
L
q(t)
C
Figure 2.7: A R-L-C series circuit.
The input voltage V(t) satisfies
d 2 q (t )
dq ( t ) 1
L
R
 q ( t )  V ( t ), q ( 0)  q0 , i ( 0)  0
dt
dt
C
where q(t) denotes the charge in the capacitor
and i(t) denotes the current of the circuit. It is a
linear second differential equation.
The R, L, and C values in the
circuit can change with time.
However, they are assumed to be
constant for analysis.