Transcript lecture7

Continuous-time & Discrete-time
Systems.
• Physical Systems are interconnection of
– components, devices, or subsystems.
• System can be viewed as a process in which
– input signals are transformed by the system or
– cause the system to response in some way,
– resulting in other signals as outputs.
Continuous-time & Discrete-time
Systems.
x(t)
y(t)
y(t)
x(t) Continuous-time
system
x[n]
Discrete-time
System
y[n]
x[n]
y[n]
Examples Of Systems
i(t)
vs(t)  vc(t)
,
R
Relationsh ip of current and voltage for a capasitor : dv (t)
i(t)  C c , and substituti ng this into the above equation : dt
We have the differenti al equation describing the relationsh ip
From Ohms' s Law : - i(t) 
between th e input v s (t ) and the output v c (t ) : 
dvc(t) 1
1

vc(t) 
vs(t)
dt
RC
RC
Example of Mechanical System
 v (t )
f(t)
net force f(t) -  v(t)  mass * accelerati on of car
dv(t )
 f(t) -  v(t)  m *
,
dt
dv(t ) 
1
i.e.
 v(t )  f (t ).
dt
m
m
Generally 1st order differenti al equation : dy (t )
 ay (t )  bx(t ).
dt
Example of Discrete-time System
Simple Model for Monthly Bank Balance
y[n]=present current balance.
x[n]=net deposit(deposits-withdrawals).
Accrue 1% interest on monthly past balance.
y[n]=1.01y[n-1]+x[n].
or y[n]-1.01y[n-1]=x[n].
Digital Simulation of Differential Equation
Through Difference Equation.
dv(t ) 
1
 v(t )  f (t ).
dt
m
m
dv(t) v[n ] - v[n - 1]
By first backward difference :

dt

The differenti al equation can be expresses as : v[n ] - v[n - 1] 
1
 v[n]  f [n].

m
m
1 
v[n  1] 1
v[n](  ) 
 f [n].
 m

m
m

v[n] 
v[n  1] 
f [n],
m  
m  
Letting, v[n ]  v[n] and f[n]  f[n ].
m

v[n] 
v[n  1] 
f [n],
m  
m  
Interconnections of Systems
Series or Cascade Form
Input
System1
System2
Output
Parallel Form
Input
System 1
+
System 2
Output
Interconnections of Systems
Series - Parallel Form
System 1
System 2
+
Input
System 4
Output
System 3
Feedback Form
Input
System 1
+
System 2
Output
Interconnections of Systems
Simple Electrical Circuit
i1 (t )
i2 (t )
V(t)
Feedback Block Diagram Form of Circuit
Input
i (t )
i1 (t )
Capasitor
1 t
v(t )   i1 ( )d
C 
+
+i2 (t )
Resistor
v (t )
i2 (t ) 
R
Output
v (t )
Basic System Properties
•
•
•
•
•
•
Systems with and without memory.
Invertibility and Inverse Systems.
Causality.
Stability
Time Invariance
Linearity.
Systems without memory.
• System is memoryless if its output at any
one time depends only on the input at the
same time.
• E.g. y[n]=(2x[n]-x2[n])2
• A resistor is memoryless because y(t)=Rx(t)
• So too an identity system is memoryless
because y(t)=x(t), y[n]=x[n].
Systems with memory.
• System with memory depicts its output at
any one time that is dependent not only on
the present input but also past(future) values
of input and output.
• E.g. accumulator/summer
y[n] 
n
 x[k ],
k  
Delay, y[n]  x[n - 1].
1 t
A capacitor is a memory analog device, y(t)   x( )d ,
C -
Invertible System
• Systems whereby distinct inputs lead to
distinct outputs
• As such an inverse system exits that, when
cascaded with the original system, yields an
output w[n] equal to the input x[n] to the
first system.
x[n]
System
y[n]
Inverse
System
w[n]=x[n]
Examples of an invertible
continuous-time system.
1. y(t)=2x(t) for which the inverse system is
w(t)=y(t)/2.
2. y[n] 
n
n 1
k  
k  
 x[k ], y[n]   x[k ]  x[n],
y[n]  y[n  1]  x[n], x[n]  y[n]  y[n  1].
 the inverse system is : w[n]  x[n]  y[n]  y[n  1].
Two Examples of Inverse System
x(t)
y(t)=2x(t)
y(t)
w(t)=x(t)
w(t)=y(t)/2
w[n]=x[n]
x[n]
y[n] 
n
y[n] w[n]  y[n]  y[n  1].
x
[
k
]

k  
Examples of Noninvertible
Systems
• y[n]=0.
• The output is always zero for any value of
input x[n].
• y(t)=x2(t).
• The sign for the input x(t) cannot be
determined for a certain value of output y(t)
• I.e. for both cases the values of the output
is not distinct for distinct values of input,
Causality
• A system is causal because its output
depends only on present and past values of
the input
• Such a system does not anticipate future
values of input.
• y[n]=x[n]-x[n+1] and y(t)=x(t+1) are noncausal systems.
Stability
• A stable system is one in which small inputs
lead to response that do not diverge.
y(t)
stable pendulum
unstable pendulum
Time Invariance
• System is time invariant if the behavior
and characteristics of the system are fixed
over time.
• E.g. the RC circuitry where the values of
the parameter of the components R and C
do not changed with time I.e. constant.
• A system is time invariant if a time shift in
the input signal results in an identical time
shift in the output signal.
Linearity
• The system is linear if it possesses the
superposition property.
Let y1 (t ) be the response of a continuous - time
system to an input x 1 (t ),
and let y 2 (t ) be the output correspond ing
to the input x 2 (t ).
System is linear if : 1) The response to x1 (t )  x2 (t )
is y1 (t )  y2 (t ).
2)The response to ax1 (t ) is ay1 (t ), where " a"
is any complex constant.
Combining the two property for
linearity.
ax1 (t )  bx2 (t )  ay1 (t )  by2 (t )
ax1[n]  bx2 [n]  ay1[n]  by2 [n].
Example 1.12. Causality???
Is the system given by y[n]  x[-n] causal or not?
If n  0 , e.g n  4, y[4]  x[-4].
This says that at n  4, the output y[n]
depend on past value of x[n].
However n  0, e.g n  -3, y[-3]  x[-(-3)],
y[-3]  x[3], i.e. the output y[n] depends
on future value of input x[ 3].
System is not causal.