signal and system

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Lecture 1
Signals and Systems
Introduction
This course is about signals and their processing by systems. It involves:
 Modeling of signals by mathematical functions
 Modeling of systems by mathematical equations
 Solution of the equations when excited by the functions
What are Signals?
A Signal is something that represents information. It is a function representing a
physical quantity or variable. Mathematically, a signal is represented as a function of
an independent variable t. Usually t represents time. Thus, a signal is denoted by x(t).
Our world is full of signals, both natural and man-made.
Examples are:
 Variation in air pressure when we speak
 Voltage or current waveform in a circuit
 Stock prices of Intel
What are systems?
A system is a generator of signals or it is a transformer of signals
What is Signal Processing?
Signal processing involves enhancing, extracting, storing and transmitting useful
information. Electrical signals are best suited for such manipulations. It is common to
convert signals to electrical form for processing. Two conceptual schemes for signal
processing are shown below.

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Why study Signals and Systems?
Signals and systems are fundamental to all of engineering!
Steps involved in engineering are:
 Model system: Involves writing a mathematical description of input and output
signals
 Analyze system: Study of the various signals associated with the system
 Design system: Requires deciding on a suitable system architecture, as well as
finding suitable system parameters
 Implement system/Test system: Check system, and the input-output signals, to
see that the performance is satisfactory.
Classification of Signals
A. Continuous-Time and Discrete-Time Signals:
A signal x(t) is a continuous-time signal if t is a continuous variable. If t is a discrete
variable, that is, x(t) is defined at discrete times, then x(t) is a discrete-time signal.
Since a discrete-time signal is defined at discrete times, a discrete-time signal is often
identified as a sequence of numbers, denoted by {xn} or x[n], where n=integer.
Illustrations of a continuous-time signal x(t) and of a discrete-time signal x[n] are
shown in Fig. 1-1.
Fig. 1-1 Graphical representation of (a) continuous-time and (b) discrete-time signals.

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A discrete-time signal x[n] may be obtained by sampling a continuous-time signal x(t).
Many physical systems operate in continuous time.
 Mass and spring
 Leaky tank
Digital computations are done in discrete time
 State machines: given the current input and current state, what is the next output
and next state.
B. Analog and Digital Signals:
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 x(t) is called an
analog signal. If a discrete-time signal x[n] can take on only a finite number of
distinct values, then we call this signal a digital signal.
C. Real and Complex Signals:
A signal x(t) is a real signal if its value is a real number, and a signal x(t) is a complex
signal if its value is a complex number. A general complex signal x(t) a function of the
form
x (t) = x1(t) + jx2(t)
where x1(t) and x2(t) are real signals and √ .
D. Deterministic and Random Signals:
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.
E. Even and Odd Signals:
A signal x ( t ) or x[n] is referred to as an even signal if
x(-t) = x(t)
x[-n] = x[n]
A signal x (t) or x[n] is referred to as an odd signal if
x(-t) = - x(t)
x[-n] = - x[n]

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Examples of even and odd signals are shown in Fig. 1-2.
Fig. 1-2 Examples of even signals (a and b ) and odd signals (c and d)
F. Periodic and Nonperiodic Signals:
A continuous-time signal x (t) is said to be periodic with period T if
x(t +T) = x (t) for all t
An example of such a signal is given in Fig. 1-3(a). From Fig. 1-3(a) it follows that
x(t +mT) = x (t) for all t and any integer m.
The fundamental period T0, of x (t) is the smallest positive value of T.

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Fig. 1-3 Examples of periodic signals.
Any continuous-time signal which is not periodic is called a nonperiodic (or aperiodic)
signal.
Periodic discrete-time signals are defined analogously. A sequence (discrete-time
signal) x[n] is periodic with period N if there is a positive integer N for which
x[n +N] =x[n] for all n
An example of such a sequence is given in Fig. 1-3(b). From Fig. 1-3(b) it follows that
x[n+mN] =x[n] for all n and any integer m.
The fundamental period N0 of x[n] is the smallest positive integer N.
Any sequence which is not periodic is called a nonperiodic (or aperiodic) sequence.
G. Energy and Power Signals:
Consider e(t) to be the voltage across a resistor R producing a current i(t). The
instantaneous power p(t) per ohm is defined as
2( ) ( )
( ) ( )
e t i t
p t i t
R
 
Since power is the rate of energy, the total energy expended over the time interval
t1tt2 is :
2 2
1 1
2
( ) ( )
t t
t t
E p t dt i t dt  
and the average power over this interval is:
2 2
1 1
2
2 1 2 1
1 1
( ) ( )
( ) ( )
t t
avg t t
P p t dt i t dt
t t t t
 
  

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If t1 = -T and t2 = T then,
2
( )
T
T
E i t dt

  and
21
( )
2
T
avg T
P i t dt
T 
 
If i(t) is a continuous-time signal, the total energy E and average power P on a per-
ohm basis are
2 2
lim ( ) ( )
T
T
T
E i t dt i t dt joules


 
  
and
21
lim ( )
2
T
avg
T
T
P i t dt watts
T

 
For an arbitrary continuous-time signal x(t), the total energy normalized to unit
resistance is defined as
2 2
lim ( ) ( )
T
T
T
E x t dt x t dt joules


 
  
and the average power normalized to unit resistance is defined as
21
lim ( )
2
T
avg
T
T
P x t dt watts
T

 
Based on the above definition, the following classes of signals are defined:
1. x(t) is an energy signal if and only if 0<E< (having finite energy), so that P=0.
2. x(t) is a power signal if and only if 0<P<, thus implying that E=.
3. Signals that satisfy neither property are therefore neither energy nor power
signals.
An energy signal has zero average power, whereas a power signal has infinite energy.
Thus the periodic signals are classified as power signals.
In discrete time
2 2
lim [ ] [ ]
N
N
n N n
E x n x n


 
  
and
21
lim [ ]
(2 1)
N
avg
N
n N
P x n
N





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Basic Continuous-Time Signals
A. The Unit Step Function:
The unit step function u(t) is defined as:
1 0
( )
0 0
t
u t
t

 

which is shown in Fig. 1-4(a).
Similarly, the shifted unit step function u(t - to) is defined as:
1
( )
0
o
o
o
t t
u t t
t t

  

which is shown in Fig. 1-4(b).
Fig. 1-4 (a) Unit step function; (b) shifted unit step function.
B. The Unit Impulse Function:
The unit impulse function (t), also known as the Dirac delta function (or simply Delta
function), defined as the conventional function having unity area over an infinitesimal
time interval as shown in Fig. 1-5
Fig. 1-5: The unit impulse function (t)

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and possesses the following properties:
0
1. ( )
0 0
t
t
t

 
 

2. ( ) 1t dt



3. ( ) ( ) (0)t t dt  



where (t) is any regular function continuous at t=0.
Similarly, the delayed delta function (t - to) is defined by –
( ) ( ) ( )o ot t t dt t  


 
where (t) is any regular function continuous at t = to.
For convenience, (t) and (t- to) are depicted graphically as shown in Fig. 1-6.
Fig. 1-6 ( a ) Unit impulse function; (b) shifted unit impulse function.
Some additional properties of (t) are
1
4. ( ) ( )
5. ( ) ( )
6. ( ) ( ) (0) ( )
7. ( ) ( ) ( ) ( )o o o
at t
a
t t
x t t x t
x t t t x t t t
 
 
 
 

 

  
if x(t) is continuous at t = to.
Thus any continuous-time signal x(t) can be expressed as
( ) ( ) ( )x t x t d   


 

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C. Unit Ramp Function:
0
( ) ( ) ( )
0 0
t t
r t x t u t
t

  

( )b
t
( )r t
1
1
0
( )a
t
( )x t
0t
1
0t0
Fig 1-7: (a) unit ramp function (b) unit pulse function
D. Unit Pulse Function: ( ) ( ) ( ) ( )o ox t t u t t u t t    
E. Complex Exponential Signals:
Let s = +j be a complex number. We define x(t) as
( )
( ) (cos sin )st j t t
x t e e e t j t  
 
   
Real exponential function: ( ) t
x t e

Fig. 1-8 Continuous-time real exponential signals. (a)  > 0; (b)  < 0.

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Sinusoidal Function: x(t) = A cos(ot+)
Fig. 1-9 Continuous-time sinusoidal signal.
Waveform for ( ) cost
ox t e t

Fig. 1-10 (a) Exponentially increasing sinusoidal signal; (b) exponentially decreasing sinusoidal
signal.