Using Laplace to solve differential and integrodifferential equations
Example: Solving for where and and
Example 2: Solving for where and and
Zeroinput and zerostate components of the response
Initial
condition
term
Input
terms
Using Laplace transfor
Example 2: Determine for and
g(
x()
x()g(t
t
c
t
Determine for and
Example 3: Determine c=
for and
x()
g(
g()x(t
t
c
t
Calculate c=
for and
Interconnected System
Parallel system
S1
S2
Cascade system
S1
S2
Cascade system
S1
S2
S2
S1
Cascade with an integ
Chapter 4: ContinuousTime System Analysis using Laplace Transform
The Laplace Transform
where is the complex frequency
The Inverse Laplace Transform
where
is a constant chosen to ensure the
convergence of the above integral.
The Laplace Transform is line
Chapter 3 TimeDomain Analysis of DiscreteTime Systems
A discrete time signal or
is the sampling time
Energy and power of a discrete signal
Energy
Power
Useful signal operations
Shifting
Time Reversal
Useful signal operations
Decimation (Downsampling)
I
Find the inverse Laplace Transform
Find the inverse Laplace Transform
Find the inverse Laplace Transform
Find the inverse Laplace Transform
Properties of Laplace Transform
Time shifting
for
Example: Find the Laplace Transform
Example: Find the inverse Lap
Chapter 5: DiscreteTime System Analysis using zTransform
The zTransform
where is a complex variable
The Inverse zTransform
indicates an integration in counterclockwise direction
around a closed path in the complex plane.
Im
Re
ZTransform is linear
Un
Chapter 2 Timedomain analysis of continuoustime systems
A linear, timeinvariant, continuous time (LTIC) system can be described by the
following set of differential equation:
Using , the above equation can be written as
We can also further rewritten it
Zero state response and the transfer function H(s)
Remember that at the end of Chapter 2, a LTIC system is expressed as
Q(D)y(t) P(D)x(t)
where
Q(D) DN a1DN1 . aN1D aN
P(D) bNM DM bNM1DM1 . bN1D bN
We let and also , and derive the transfer function
This r
Calculate for input and
Frequency Response of the Ideal Delay, Ideal Differentiator and Ideal Integrator
(a) Ideal Delay
(b) Ideal Differentiator
(c) Ideal Integrator
Bode Plot
decibel (db) =
Log scale
In log scale, the magnitude and angles of transfer fu
Analysis of Active Circuit
+

+
+

+

+
+

+

+

+

+

Find the transfer function
+


+
+

+

+
+
+



System Realization
The transfer function of an arbitrary LTIC system is
There are two main ways to realize such a system:
(1) Direct Form
DiscreteTime Exponential
For a discrete signal , it can be written as
where
plane
increasing
decreasing
(oscillation)
Exponentially increasing
Exponentially decreasing
Exponentially
increasing
Exponentially
decreasing
plane
increasing
decreasing
(osc
Signal size
Signal Energy (for finite signal)
if is real
if is complex
Signal Power (for periodic signal)
for is real
for is complex
Root Mean Square (RMS) Energy
Root Mean Square (RMS) Power
Example: Calculate the energy and the RMS energy of the signal
The Laplace Transform
where is the complex frequency
The Inverse Laplace Transform
where
is a constant chosen to ensure the
convergence of the above integral.
Example 2: For a signal , find the Laplace transform and its ROC.
Both of them has the same Lapl
The unit impulse (delta) function (t)
for
Properties of the delta function
1)
2)
3)
4)
5)
Examples:
The Exponential
exponential function with
est complex frequency s
where
Complex frequency
The conjugate of
Also,
est
est
t
est
t
est
t
t
Complex frequency
Chapter 2 Timedomain analysis of continuoustime systems
A linear, timeinvariant, continuous time (LTIC) system can be described by the
following set of differential equation:
Using , the above equation can be written as
We can also further rewritten it
Frequency shifting
Example: From , find the Laplace Transform
TimeDifferentiation Property
Find the Laplace transform using the timedifferentiation and timeshifting properties
TimeIntegral Property
Scaling Property
Proof is deferred to Chapter 7
Time
Associative property
Shift property
If
Convolution with an impulse
The width property
If x1(t) and x2(t) are finite and has a with of T1 and T2
x2(t)
x1(t)
T1
t
T2
The duration of is T1+T2
T1+T2
t
t
Causality and zerostate response
Generally,
If the syst
Example: Use OpAmp to realization a circuit of the following transfer function.
Control System Analysis
The transfer function of the system is
Assume the motor transfer function
Step input
The tracking system response to a step input
Ramp input
The track
Unit impulse response
Before we discuss how to get the zerostate response y s(t), we need to
discuss unit impulse response h(t).
(t)
Q(D),
P(D)
If M = N,
characteristic modes
h(t)
To find the impulse response , we can use simplified impulse matching met
The Unit Impulse Response
When
with initial conditions
Using iterative approach to solve for
Example:
with
can be found in closed form by assuming
where
Example: Determine the unit impulse response for the equation
ZeroState Response with convolution
An
If
If
unit step response
Also
h(t)
h(t)
The transfer function H(s)
h(t)
The transfer function H(s)
h(t)
The Transfer Function H(s)
The Transfer Function H(s) can be considered as
For the linear, timeinvariant, continuous time (LTIC) system
dN y(t)
dN1 y(
Chapter 1 Signals and Systems
What is a signal?
What is a system?
Systems
f
Input functions (signals)
Output functions
What is a Linear systems?
there is a system representing by a function , and there is a bunch of input signal ,
If
to , and
If the s
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