IST module 1

Introduction to System Theory
Vijaya Laxmi
Dept. of EEE
BIT, Mesra
1Vijaya Laxmi, Dept. of EEE, BIT, Mesra

Course Objectives
The course enables the students :
• To outline the fundamentals of common signals, systems, and recall
the list of electrical and non-electrical components.
• To summarize different transform methods, state-space techniques
and different stability conditions of linear time-invariant systemand different stability conditions of linear time-invariant system
using Routh-Hurwitz criteria.
• To analyze the transient and steady-state performance of first order
and second order systems when subjected to different signals and
also the absolute stability using above criterion.
• Actualize the electrical, mechanical, hydraulic and thermal systems
using differential equations, transfer functions and state variables.

Syllabus:EE 3201 INTRODUCTION TO SYSTEM THEORY
• MODULE – I: Introduction to Signals and Systems: Definition, Basis of classification,
Representation of common signals and their properties, System modeling. (4)
• MODULE – II: Analogous System: Introduction, D Alembert's Principle, Force-voltage
and force-current analogies, Electrical analogue of mechanical, Hydraulic and thermal
systems. (5)
• MODULE – III: Fourier Transform Method: Introduction, Fourier transform pair,
Amplitude spectrum and phase spectrum of signals, Sinusoidal transfer function. (3)
• MODULE – IV: Laplace Transform Method: Introduction, Laplace transform pair, Laplace
transformation of common functions, Gate function, Step function and impulse
function, Laplace theorems shifting, initial value, final value and convolution theorems,
Inverse Laplace transform by partial fraction expansion and convolution integralInverse Laplace transform by partial fraction expansion and convolution integral
method. (12)
• MODULE – V: System Analysis: System Analysis by Laplace Transform method, System
response. Natural, forced, transient and steady state responses. Transfer function and
characteristic equation, Superposition integral, Concept of poles and zeros, Nature of
system response from poles and zeros. (6)
• MODULE – VI: System Stability: Concept of stability, Types, Necessary and sufficient
conditions, Routh-Hurwitz stability criterion, Limitations and its applications to closed
loop systems. (4)
• MODULE – VII: State-Space Concept: Introduction, Definition: State, State variable, State
vector and state space, State space representation, Derivation of State model from
transfer function, Bush form and diagonal canonical form of state model, Non-
uniqueness of state model, Derivation of transfer function from state model, Transition
matrix and its properties, Solution of time invariant state equation. (6)3Vijaya Laxmi, Dept. of EEE, BIT, Mesra

Books
• Text Books:
1. Analysis of Linear Systems – D.K.Cheng. Narosa Publishing
House, Indian Student Edition, 8th Reprint, 1994
2. Control System Engineering – Nagrath & Gopal, New Age
International Pvt. Ltd. New Delhi, 2nd Edition
3. Control System – A. Anand Kumar3. Control System – A. Anand Kumar
• Reference Books:
1. Networks and Systems – D. Roy Choudhury, New Age
International Pvt. Ltd. New Delhi
2. Signal and Systems- Haykin
3. Signal and Systems- B.P. Lathi
4. Signals and Systems - Basu & Natarajan

Course Outcomes
After the completion of this course, students will be able to:
• Describe characteristics of different signals and systems
• Interpret the transform domains like Laplace equations,
concept of poles and zeros, Fourier equations and time-
domain state-space techniques
• Solve mathematical model of electrical and non-electrical
components with the knowledge of system science, related
mathematics of simple engineering problems and concept ofmathematics of simple engineering problems and concept of
analogous quantities
• Relate Laplace equations, Fourier equations and time-domain
state-space techniques to solve any given linear ordinary
differential equations and analyze transient and steady-state
performance of a first order and second order linear time-
invariant system using standard test signals
• Evaluate and formulate electrical or non-electrical linear time
invariant systems for desired transient behaviour, steady state
error and stability.

Module I
Introduction to Signals and SystemsIntroduction to Signals and Systems

Objectives: Module I
• To introduce introduce common signals and systems needed
in almost all electrical and other engineering fields and
scientific disciplines as well.
• The mathematical description and representation of signals
and systems and their classification.and systems and their classification.
• Define several basic properties of signals and systems.

Signal
• It is a function representing a physical quantity or
variable, and typically it contains information about
the behavior or nature of the phenomenon.
• Example: In RLC circuit the signal may be the voltageExample: In RLC circuit the signal may be the voltage
across the capacitor or current flowing through the
inductor or resistor.
• Mathematically, a signal may be represented as a
function of an independent variable ‘t’ and denoted
by x(t).

Classification of Signals
• Continuous-time and Discrete-time Signals
• Analog and Digital Signals
• Real and Complex Signals
• Deterministic and Random Signals• Deterministic and Random Signals
• Even and odd signals
• Periodic and Nonperiodic Signals

Continuous-time and Discrete-time Signals
• A signal x(t) is called continuous-time signal, if t is a
continuous variable.
• A signal x(t) is called discrete-time signal, if t is a discrete
variable, i.e., x(t) is defined at discrete times, often called
sequence of numbers and denoted by x[n], where n=integer.sequence of numbers and denoted by x[n], where n=integer.
• Discrete-time signal may be also obtained by sampling a
continuous-time signal x(t) such as
       
,...,...,,,
],...[],...1[],0[
,....,...,,,
210
210
n
n
xxxx
or
nxxx
or
txtxtxtx )(][.,. nn txnxxei 
For uniform sampling,
 sn nTxnxx  ][
Where Ts is the sampling interval

Continuous-time signal Discrete-time signal

Analog and Digital Signal
• If a continuous-time signal x(t) can take on any value
in the continuous interval (a,b), where a may be -∞
and b may be +∞, then the continuous-time signal is
called analog signal.
• If the discrete-time signal x[n] can take on only a
finite number of distinct values, then it is called a
digital signal.

Real and Complex Signal
• A signal x(t) is a real signal, if its value is a real
number, and signal x(t) is a complex signal if its value
is a complex number.

Deterministic and Random Signal
• Deterministic signals are those signals, whose values
are completely specified for any given time. Thus, a
deterministic signal can be modeled by a known
function of time t.
• Random signals are those signals that take random
values at any given time and must be characterized
statistically.

Even and Odd signals
• A signal x(t) or x[n] is referred to as an even signal if
   
   nxnx
txtx


• A signal x(t) or x[n] is referred to as an odd signal if
   
   nxnx
txtx



Even signalsEven signals
Odd Signals

Periodic and Nonperiodic signals
• A continuous-time signal x(t) is said to be periodic with
period T, if there is a positive nonzero value of T for
which
• Any continuous-time signal which is not periodic is called
a nonperiodic (aperiodic) signal.
    tallfortxTtx 
a nonperiodic (aperiodic) signal.
• A discrete-time signal x[n] is said to be periodic with
period N, if there is a positive nonzero value of N for
which
• Any discrete-time signal which is not periodic is called a
nonperiodic (aperiodic) signal.
    nallfornxNnx 

Periodic Signals

Basic continuous-time signals
• Unit step function
• Ramp function
• Unit impulse function
• Complex exponential signals• Complex exponential signals
• Real exponential signals
• Sinusoidal signals

Unit step function
• The unit step function u(t) is defined as
• It is discontinuous at t=0.






00
01
)(
t
t
tu
• It is discontinuous at t=0.
• The shifted unit step function u(t-t0) is defined as






0
0
0
0
1
)(
tt
tt
ttu

Gate function
• It can be written as sum of two step functions.
     tftftf      
   
   
      btUatUKtfhence
btKUtfand
atKUtfwhere
tftftf




,
,
2
1
21
     tbUatKUtf  .
It can also be represented as the product of two step functions as

Ramp function
• A ramp function is described as
 
  0;
0;0


ttKr
ttf
 
  atatKr
attfand


;
;0

Impulse function
• Consider a pulse function as
• The area of pulse =TX1/T=1.
• It can be represented mathematically as
      TtUtU
T
Ltt
T


1
0


Problem
• Express the half sine wave function using step
functions.

Solution
• The sine wave can be represented as
   
     
2/,0
2/,2/
0,0
0,
2
1
Tt
TtTtUtSinVtvand
t
tttUSinVtv
m
m






2/,0 Tt 
     
      
    2/
2/
21
TtUtUtSinV
TtUtSinttUSinV
tvtvtv
m
m






Problem
• Express the triangular wave using ramp functions.

Solution
• We have,
     
    TtTTtrT
TttrTtv


2/;22//2
2/0;/2
    TtfortvwhileTtforTtrBut  0,02/,
             TtrTTtrTtrTtv
Hence
 /22//4/2
,

Complex exponential signal
• The complex exponential signal is defined as
• An important property of a complex signal x(t) is that it is a
periodic signal for any value of w0.
• The fundamental period T0 of x(t) is given by
  tjSintCosetx tj
00
0


• The fundamental period T0 of x(t) is given by
0
0
2


T
   
 tjSintCoseeetx ttjst
00 
 
General Complex exponential signal
  t
etx 
 Real exponential signal

Exponentially increasing
sinusoidal signal
Exponentially decreasing
sinusoidal signal
Real Exponential signal for σ>0 and σ<0

Sinusoidal Signals
• A continuous-time sinusoidal signal can be expressed
as
     tACostx 0

Basic discrete-time signals
• Unit step sequence
• Unit impulse sequence
• Complex exponential sequence
• Real exponential sequence• Real exponential sequence
• Sinusoidal sequence

Unit step sequence
• The unit step sequence u[n] is defined as
• It is discontinuous at n=0.
 






00
01
n
n
nu
• It is discontinuous at n=0.
• The shifted unit step sequence u[n-k] is defined as
 






kn
kn
knu
0
1

Unit impulse sequence
• The unit impulse sequence δ[n], is defined as
 






01
00
n
n
n
• The shifted unit impulse sequence δ[n-k], is defined as
 






kn
kn
kn
1
0


Complex exponential sequence
• The complex exponential sequence is defined as
• In order for x[n] to be periodic with period N(>0), Ω0 must
satisfy the following condition:
  njSinnCosenx nj
00
0
 
egerpositivem
m
int,0


• Thus, the sequence in not periodic for any value of Ω0 . It is
periodic only if the above ratio is a rational number.
• In case of continuous-time complex exponentials, the signals
are all distinct for distinct values of w0, but this is not the case
for discrete-time complex exponentials
General Complex exponential signal
Real exponential signal
egerpositivem
N
m
int,
2
0



  n
Cnx 
101,10,1   and

Real exponential sequences
α>1 1>α>0
0>α>-1 α<-1

Sinusoidal Sequence
• A continuous-time sinusoidal sequence can be
expresses as
    nACosnx 0
Sinusoidal periodic Sequence Sinusoidal non-periodic Sequence

Energy and Power signals
• Consider v(t) as the voltage across a resistor R producing a
current i(t). The instantaneous power p(t) per ohm is defined
as
• Total energy E and average power P on a per ohm basis are
 ti
R
titv
tp 2)()(
)( 
• Total energy E and average power P on a per ohm basis are
 
  attswdtti
T
P
joulesdttiE
T
T
T 







2/
2/
2
2
1
lim

• For arbitrary cont-time signal x(t), the normalized energy
content E of x(t) is defined as
• The power of a periodic signal with period T0 is the average
 


 dttxE
2
• The power of a periodic signal with period T0 is the average
energy
• The normalized average power P of any signal x(t) is defined
as
 


2/
2/
21
lim
T
T
T
dttx
T
P
 
0
2
0 0
1
T
P x t dt
T
 
The limit may be zero also like exponential function etc.

• Similarly, for discrete-time signal x[n], the normalized energy
content E of x[n] is defined as
• The normalized average power P of x[n] is defined as




n
nxE
2
][
• The normalized average power P of x[n] is defined as

 

N
Nn
N
nx
N
P
2
][
12
1
lim

• X(t) or x[n] is said to be an energy signal (or sequence) if and
only if 0<E<∞ and so P=0.
• X(t) or x[n] is said to be an power signal (or sequence) if and
only if 0<P<∞ and thus E= ∞.
• Signals that satisfy neither property are referred to as neither• Signals that satisfy neither property are referred to as neither
energy signals nor power signals.
• A periodic signal is a power signal if its energy content per
period is finite and then the average power of this signal need
only be calculated over a period.

Problem
• Determine whether the following signals are energy
signals, power signals or neither.
 
at
tAtxb
atuetxa
cos)(.
0),()(.

 
 
 
nj
n
enxf
nunxe
nunxd
ttutxc
tAtxb
3
0
2][.
][][.
][5.0][.
)()(.
cos)(.




 

• A.
• B.
   




a
dtedttxE at
2
1
0
22
Energy signal
• B. The sinusoidal signal is periodic with T0=2π/w0.
 
   
















00
0
2
00
2
0
0
2/
2/
2
11
lim
,lim
1
lim,
TT
k
T
T
T
dttx
T
dttxk
kT
P
kTTastakenbetoittheallowing
dttx
T
PpowerAverage

Systems
• A system is a mathematical model of a physical
process that relates the input (or excitation) signal to
the output (or response) signal.
• A system is a collection of components which workA system is a collection of components which work
together to provide an output for a given input.

Classification of systems
• SISO and MIMO
• Continuous-time and Discrete-time systems
• Systems with memory and without memory
• Causal and Noncausal systems
• Linear and Nonlinear systems• Linear and Nonlinear systems
• Time-invariant and Time-varying systems
• Linear time-invariant systems
• Stable systems
• Feedback systems

SISO and MIMO system

Continuous-time and Discrete-time
systems
• If the input and output signals are Continuous-time
signals, then the system is called Continuous-time
systems.
• If the input and output signals are Discrete-time
sequences, then the system is called Discrete-time
systems.

Systems with memory and without
memory
• A system is said to be memoryless, if the output at any time
depends on only the input at that same time.
• Otherwise the system is said to have memory.
• Example: Resistor R with the input x(t) taken as the current
and voltage taken as the output y(t). The relation betweenand voltage taken as the output y(t). The relation between
them is
• The capacitor C with current as input x(t) and the voltage as
the output y(t)
• In discrete-time system, the system with memory is
)()( tRxty 


t
dx
C
ty )(
1
)(
   

n
k
kxny

Causal and Noncausal systems
• A system is called causal, if its output y(t) at an arbitrary
time t=t0 depends on only the input x(t) for t≤0, i.e., the
output of a causal system at the present time depends on
only the present and/or past values of the input, not on
its future values.its future values.
• Thus, in case of causal system, it is not possible to obtain
an output before an input is applied to the system.
• A system is said to be noncausal, if it is not causal.
• All memoryless systems are causal, but not vice versa.

Linear and Nonlinear systems
• The systems with follows the principles of
superposition and homogeneity are called linear
systems and the systems, which does not follow
them are called nonlinear systems.
• Principle of superposition (additivity): If for any• Principle of superposition (additivity): If for any
signals x1 and x2
• Principle of homogeneity (scaling): For any signals x
and y
• Hence, a system is called linear if
  2121
2211
yyxxTThen
yTxandyTx


  yxT  
  22112211 yyxxT  

Time-invariant and Time-varying
systems
• A system is called time-invariant, if a time shift (delay
or advance) in the input signal causes the same time
shift in the output signal. Thus, for a continuous-time
system, the system is time-invariant if
     . ofvaluerealanyfortytxT 

Linear time-invariant systems
• If a system is linear and also time-invariant, then it is
called linear time-invariant (LTI) system.

Stable systems
• A system is bounded-input/bounded-output
(BIBO) stable, if for any bounded input x
defined by , the corresponding
output y is also bounded defined by
1kx 
2ky output y is also bounded defined by
where k1 and k2 are finite real constants.
2ky 

Feedback systems
• In the feedback system, the output signal is fed back
and added to/subtracted from the input to the
system.

IST module 1

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