Chapter 2

One Dimensional Continuous Time System

2.2 One Dimensional ContinuousTime System

Definition 1: An one dimensional analog systemor continuous time system can be defined as amapping function T which maps a real-valuedanalog signal f(t) to another real-valued g(t) suchthat

g t T f t( ) [ ( )]

Tf(t) T[f(t)]

input system output

Definition 2: The mapping T is linear if it satisfiesthe 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 verysmall audio signal into a large audio to drive thespeaker. The multiplier is a transformer thatconvert a low voltage sinusoidal wave into a highvoltage sinusoidal wave or vice versa.

Example 2: A differentiator is defined as

T f tdf t

dt[ ( )]

( )

Example 3: An integrator is defined as

T f t f a dat

[ ( )] ( )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

f t( ),

,

1

1 t 0;if0 t ;if

with the period 2 can be approximated by

f tn

ntNn

n

N

( ) ( ( ) ) sin( ) 2

1 11

fN(t) is the superposition (linear combination)of sine waves. It is easy to know that

f t t14

( ) sin( )

))3sin(3

1)(sin(

4)(3 tttf

))5sin(5

1)3sin(

3

1)(sin(

4)(5 ttttf

))7sin(7

1)5sin(

5

1)3sin(

3

1)(sin(

4)(7 tttttf

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 andundershoot also increases at t = 0. the followingMATLAB 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(200t) + 0.25 cos(1000t)).

It is easy to know that

df t

dtt t

( )sin( ) sin( ). 100 200 250 1000

Figure 2.2(b) shows the analog signaldf t

dt( ) .

Obviously the differentiation system attenuateslow frequency signal and magnifies highfrequency. It is a high pass system.

Figure 2.2: (a) f(t) = 1 + 0.5 cos(200t) + 0.25cos(1000t)

(b)

(c)

tdaaf )(

dttdf )(

(a) (b) (c)

Figure 2.2(c) show the analog signal f a dat

( ) .

The integration system attenuates high frequencyand magnifies low frequency. it is a low passsystem.

f a da t t tt

( ) sin( ) sin( ).

1

400200

1

40001000

Also, if f(t) is processed by an integration system then

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 isdefined as

))(log()]([ tftfT

Definition 3: The system T is linear time-invariant 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((t a), a 0, is the output when(t a) Thus,

h t h t D t a( ) ( ) ( ).

h t a h t a 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 (t). At time t a if the input is f(t a) = (t a) then theoutput is

From the above two equations we can easilyobtain

h t h t a h t D h t a D( ) ( ) ( ) ( )

The equality holds if

h t h t a T 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 ( ),t a

applies to the system. Thus,

h t th t D t a( ) ( ) ( ).

At time t a we obtain

h t a t a h t a D t a( ) ( ) ( ) ( ).

a 0.

Assume that the system is linear time invariant.It is known that the output of the system is theimpulse response h(t) when the input is d(t a).Therefore,

h t a th t a 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 t a( ) ( ).

If the system is time invariant

h t a D t a D( ) ( ),

which is contraction to Equation h(t a D) = 0,Therefore, h t h t a( ) ( ) and the impulseresponse of the system is time varying.

Theorem 1: If h(t) is the impulse response ofa linear invariant system and f(t) is the input ofthe system then the output is

g t h t a f a da( ) ( ) ( ) .

h(t)f(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 ifits input and output is bounded.

That is, if|)(| tf

then |)]([| tfT

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. Thereexist a real number M1 such that

1|)(| Mtf

Since h(t) is absolute integrable there exists areal number M2 such that

| ( )|h a da M

2

Then, the absolute output is

| ( )| | ( ) ( ) |g t h a f t a dat

0 | ( )|| ( )|h a f t a da

t

0

M h a dat

1 0| ( )|

M M1 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 thesystem.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 easilythat the system is linear time invariant. The systemis casual since the impulse response h(t) = (tT)= 0 for t < 0. The output depends on its past inputbut does not depends on the future input.

is linear time invariant. However, it is not causalbecause the current output g(t) depends on futureinput f(t + T). In fact, its impulse response

0)()( Ttth for t < 0.

Note that the system

)()( Ttftg

Example 14: A differentiation system is definedas

g tdf t

dt( )

( )

It is a high pass filter.

If f(t) = e jt then g(t) =je jt. For small , |g(t)|is very small while for large , |g(t)| is very large.It is a high pass filer.

Example 15: An integration system is defined as

g t f a dat

( ) ( )0It is a low pass filter.

If f t e g t e tj tj

j( ) , ( ) ( ).

1 1

For small , |g(t)| is very large while for large , |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 f t( ) ( ( ))abs

Example 17: A half-wave rectifier is defined as

g tf t

( )( ),

,

0

f t( ) ;0if

otherwise.

Both the full wave rectifier and the half waverectifier are nonlinear system.

Definition 6: the modulation of a signal f(t) isdefined 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 isan equation that has derivatives with respect to anindependent variable only. If the equations has thederivatives with respect to at least two variables itis called an partial differential equation.

Definition 8: The ordinary differential equation

a td g t

dta t

d g t

dta t

dg t

dta t g t f tN

N

N N

N

N( )

( )( )

( )... ( )

( )( ) ( ) ( )

1

1

1 1 0

is a linear differential equation of order N. If adifferential equation can not be written as aboveequation it is nonlinear.

ak(t)f t( ) 0

equation is said to be non- homogeneous; other-wise it it said to be homogeneous. If the initial conditions

,)0(kdt

gd cn

n

0 n N

are given the differential equation is called the initial-value problem.

Each Coefficient depends on on the variable t. If

the

In practical systems,

)(tak is usually assumed to be constant so that the system is LTI.

Example 18: The following equations

dg t

dtg t et( )

( ) 2

3tdg t

dtt g t t

( )sin( ) ( ) cos( )

are ordinary differential equations.

Example 19: The following equations

u x y

x

v x y

y

( , ) ( , ) 2 5 0

y

yxv

y

yxu

y

yxv

x

yxu

),(

2),(),(

2),(

2

2

2

2

are partial differential equations

Example 21: The following ordinary differentialequations

3tdg t

dtt g t et( )

sin( ) ( )

d g t

dtg t

2

20

( )( )

d g t

dt

dg t

dtg t

2

23 2 2

( ) ( )( )

d g t

dt

dg t

dtg t t

5

53 2

( ) ( )( ) sin( )

are linear.

Example 22: The following ordinary differentialequations

dg t

dtg t et( )

( ) 2

dg t

dt

d g t

dtg t

( ) ( )( )

2

20

are nonlinear.

)()( tRitVR

)(ti R+ - )(ti

Voltage across the resistor

dt

tdiLtVL

)()(

)(ti L+ - )(ti

Voltage across the inductor

)(ti C+ - )(ti

t

C dttiC

tV0

)(1

)(

Voltage across the capacitor

The Kirchhoff’s Current Law

The sum of the currents at a node in a circuit is zero.

)(1 ti

)(3 ti

) (4t i ) (5t i

) (2t i0)()()( 321 tititi

0)()()( 542 tititi

The Kirchhoff’s Voltage Law

The sum of the voltages arounda loop in a circuit is zero.

+ -

+

-+-

+

-

)(1 tv

)(2 tv

)(3 tv

)(4 tv

)(1 tv

)(2 tv )(3 tv )(4 tv- + + + = 0

Example 23: A R-L series circuit shown inFigure 2.5

R

LV+

i(t)

Figure 2.5: The R-L series circuit.

The current i(t) that satisfies

Ldi t

dtRi t V t i

( )( ) ( ), ( ) 0 0

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.

R

CV+

i(t)

Figure 2.6(a): The R-C series circuit.

The current flows in the circuit

is i(t). The voltage across the capacitor is 1

0C

ti t dt( )

and the voltage across the resistor is Ri(t). Thus,

t

dttiC

tRiV0

)(1

)(

The equation can be also written as

V Rdq t

dt

q t

C

( ) ( )

where q(t) is the charge across the capacitor.

Example 25: A series R-L-C circuit shown inFigure 2.7

R

LV(t)

i(t)

Figure 2.7: A R-L-C series circuit.

Cq(t)

The input voltage V(t) satisfies

Ld q t

dtR

dq t

dt Cq t V t q q i

2

01

0 0 0( ) ( )

( ) ( ), ( ) , ( )

where q(t) denotes the charge in the capacitorand i(t) denotes the current of the circuit. It is alinear 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.