Basic Operation on Signals
Continuous-Time Signals
• The signal is the actual physical
phenomenon that carries information, and
the function is a mathematical description
of the signal.
Complex Exponentials & Sinusoids
• Signals can be expressed in sinusoid or complex exponential.
g(t) = A cos (2Пt/To+θ)
= A cos (2Пfot+ θ)
= A cos (ωot+ θ)
g(t) = Ae(σo+jωo)t
= Aeσot[cos (ωot) +j sin (ωot)]
sinusoids
complex exponentials
Where A is the amplitude of a sinusoid or complex exponential, To is
the real fundamental period of sinusoid, fo is real fundamental cyclic
frequency of sinusoid, ωo is the real fundamental radian frequency of
sinusoid, t is time and σo is a real damping rate.
• In signals and systems, sinusoids are expressed in
either of two ways :
a. cyclic frequency f form - A cos (2Пfot+ θ)
b. radian frequency ω form - A cos (ωot+ θ)
• Sinusoids and exponentials are important in signal
and system analysis because they arise naturally in
the solutions of the differential equations.
Singularity functions and related
functions
• In the consideration of singularity functions,
we will extend, modify, and/or generalized
some basic mathematical concepts and
operation to allow us to efficiently analyze
real signals and systems.
The Unit Step Function
1 , t  0

u  t   1/ 2 , t  0
0 , t  0

Precise Graph
Commonly-Used Graph
The Signum Function
 1 , t  0


sgn  t    0 , t  0   2 u  t   1
1 , t  0 


Precise Graph
Commonly-Used Graph
The signum function, is closely related to the unit-step
function.
The Unit Ramp Function
t , t  0  t
ramp  t   
   u    d   t u t 
0 , t  0  
•The unit ramp function is the integral of the unit step function.
•It is called the unit ramp function because for positive t, its
slope is one amplitude unit per time.
The Rectangular Pulse Function
1/ a , t  a / 2
Rectangular pulse,  a  t   
, t  a/2
0
The Unit Step and Unit Impulse Function
As a approaches zero, g  t  approaches a unit
step and g  t  approaches a unit impulse
The unit step is the integral of the unit impulse and
the unit impulse is the generalized derivative of the
unit step
Graphical Representation of the
Impulse
The area under an impulse is called its strength or weight. It is
represented graphically by a vertical arrow. Its strength is either
written beside it or is represented by its length. An impulse with a
strength of one is called a unit impulse.
Properties of the Impulse
The Equivalence Property
The Sampling Property

 g  t   t  t  dt  g  t 
0
0

The sampling property “extracts” the value of a function at
a point.
The Scaling Property
  a  t  t0   
1
  t  t0 
a
This property illustrates that the impulse is different from
ordinary mathematical functions.
The Unit Periodic Impulse
The unit periodic impulse/impulse train is defined by
T  t  

  t  nT 
, n an integer
n
The periodic impulse is a sum of infinitely many uniformlyspaced impulses.
The Unit Rectangle Function
 1 , t  1/ 2 


rect  t   1/ 2 , t  1/ 2   u  t  1/ 2   u  t  1/ 2 


0
,
t

1/
2


Precise graph
Commonly-used graph
The signal “turned on” at time t = -1/2 and “turned back off” at
time t = +1/2.
The Unit Triangle Function
1  t , t  1
tri  t   

, t  1
0
The unit triangle is related to the unit rectangle through an
operation called convolution. It is called a unit triangle because
its height and area are both one (but its base width is not).
The Unit Sinc Function
The unit sinc function is related to the unit rectangle function
through the Fourier transform.
sinc  t  
sin  t 
t
The Dirichlet Function
drcl  t , N  
sin  Nt 
N sin  t 
The Dirichlet function is the sum of infinitely many
uniformly-spaced sinc functions.
Combinations of Functions
• Sometime a single mathematical function may
completely describe a signal (ex: a sinusoid).
• But often one function is not enough for an
accurate description.
• Therefore, combination of function is needed to
allow versatility in the mathematical
representation of arbitrary signals.
• The combination can be sums, differences,
products and/or quotients of functions.
Shifting and Scaling Functions
Let a function be defined graphically by
and let g  t   0 , t  5
1. Amplitude Scaling, gt   Agt 
1. Amplitude Scaling, gt   Agt 
(cont…)
2. Time shifting, t  t  t0
Shifting the function to the right or left by t0
3. Time scaling, t  t / a
Expands the function horizontally by a factor of |a|
3. Time scaling, t  t / a
(cont…)
If a < 0, the function is also time inverted. The time inversion
means flipping the curve 1800 with the g axis as the rotation axis
of the flip.
 t  t0 
g
t

A
g


4. Multiple transformations  
a


Amplitude scaling, time scaling and time shifting can be applied
simultaneously.
A multiple transformation can be done in steps
 t  t0 
 t  t t t0
g  t  
 A g  t  
 A g    A g 

a
a
 


amplitude
scaling, A
t t / a
The order of the changes is important. For example, if we
exchange the order of the time-scaling and time-shifting
operations, we get:
 t  t0 
t

t t t0
t t / a
g  t  
 A g  t  
 A g  t  t0  
 A g   t0   A g 

a

 a 
amplitude
scaling, A
tt
Multiple transformations, gt   Ag  0 
 a 
A sequence of amplitude scaling , time scaling and time shifting
Differentiation and Integration
• Integration and differentiation are common
signal processing operations in real systems.
• The derivative of a function at any time t is
its slope at the time.
• The integral of a function at any time t is
accumulated area under the function up to
that time.
Differentiation
Integration
Even and Odd CT Functions
Even Functions
Odd Functions
gt   gt 
gt    gt 
Even and Odd Parts of Functions
The even part of a function is g e  t  
The odd part of a function is g o  t  
g  t   g  t 
2
g  t   g  t 
2
A function whose even part is zero is odd and a function
whose odd part is zero is even.
Combination of even and odd
function
Function type
Sum
Difference
Product
Quotient
Both even
Even
Even
Even
Even
Both odd
Odd
Odd
Even
Even
Neither
Odd
Odd
Even and odd Neither
Products of Even and Odd Functions
Two Even Functions
Cont…
An Even Function and an Odd Function
Cont…
An Even Function and an Odd Function
Cont…
Two Odd Functions
Function type and the types of
derivatives and integrals
Function type
Derivative
Integral
Even
Odd
Odd + constant
Odd
Even
Even
Integrals of Even and Odd Functions
a
a
a
0
 g  t  dt  2 g  t  dt
a
 g  t  dt  0
a
Signal Energy and Power
All physical activity is mediated by a transfer of energy.
No real physical system can respond to an excitation unless it has
energy.
Signal energy of a signal is defined as the area under the square
of the magnitude of the signal.
The signal energy of a signal x(t) is

Ex 
 x t 
2
dt

The units of signal energy depends on the unit of the signal.
If the signal unit is volt (V), the energy of that signal is expressed
in V2.s.
Signal Energy and Power
Some signals have infinite signal energy. In that case
it is more convenient to deal with average signal power.
The average signal power of a signal x(t) is
T /2
2
1
Px  lim
x  t  dt
T  T 
T / 2
For a periodic signal x(t) the average signal power is
2
1
Px   x  t  dt
T T
where T is any period of the signal.
Signal Energy and Power
A signal with finite signal energy is
called an energy signal.
A signal with infinite signal energy and
finite average signal power is called a
power signal.
Basic Operation on Signals
Discrete-Time Signals
Sampling a Continuous-Time Signal
to Create a Discrete-Time Signal
• Sampling is the acquisition of the values of a
continuous-time signal at discrete points in time
• x(t) is a continuous-time signal, x[n] is a discretetime signal
x  n   x  nTs  where Ts is the time between samples
Complex Exponentials and
Sinusoids
• DT signals can be defined in a manner analogous to their continuoustime counter part
g[n] = A cos (2Пn/No+θ)
= A cos (2ПFon+ θ)
sinusoids
= A cos (Ωon+ θ)
g[n] = Aeβn
= Azn
complex exponentials
Where A is the real constant (amplitude), θ is a real phase shifting
radians, No is a real number and Fo and Ωo are related to No through
1/N0 = Fo = Ωo/2 П, where n is the previously defined discrete time.
DT Sinusoids
• There are some important differences
between CT and DT sinusoids.
• If we create a DT sinusoid by sampling CT
sinusoid, the period of the DT sinusoid may
not be readily apparent and in fact the DT
sinusoid may not even be periodic.
DT Sinusoids
4 discrete-time sinusoids
DT Sinusoids
An Aperiodic Sinusoid
A discrete time sinusoids is not necessarily periodic
DT Sinusoids
Two DT sinusoids whose analytical expressions look different,
g1 n   Acos 2 F01n   
and
g 2  n   A cos  2 F02 n   
may actually be the same. If
F02  F01  m, where m is an integer
then (because n is discrete time and therefore an integer),
A cos  2 F01n     A cos  2 F02 n   
(Example on next slide)
Sinusoids
The dash line are the CT function. The CT function are obviously
different but the DT function are not.
The Impulse Function
1 , n  0
  n  
0 , n  0
The discrete-time unit impulse (also known as the “Kronecker
delta function”) is a function in the ordinary sense (in contrast
with the continuous-time unit impulse). It has a sampling property,

 A n  n  x n  A x n 
n
0
0
but no scaling property. That is,
  n     an  for any non-zero, finite integer a.
The Unit Sequence Function
1 , n  0
u  n  
0 , n  0
The Unit Ramp Function
n
n , n  0 
ramp  n  
   u  m  1
0 , n  0  m
The Rectangle Function
1 , n  N w 
rect N w  n   
 , N w  0 , N w an integer
0 , n  N w 
The Periodic Impulse Function
 N  n 

  n  mN 
m
Scaling and Shifting Functions
Let g[n] be graphically defined by
gn  0 , n 15
Scaling and Shifting Functions
1. Amplitude scaling
Amplitude scaling for discrete time function is exactly the
same as it is for continuous time function
2.
Time shifting
n  n  n0 , n0 an integer
3. Time compression, n  Kn
K an integer > 1
4. Time expansion
n  n / K, K 1
For all n such that n / K is an integer, g  n / K  is defined.
For all n such that n / K is not an integer, g  n / K  is not defined.
Differencing and accumulation
• The operation on discrete-time signal that is
analogous to the derivative is difference.
• The discrete-time counterpart of integration
is accumulation (or summation).
Even and Odd Functions
g  n  g  n
ge  n 
g  n  g  n
2
g  n   g  n
go  n 
g  n  g  n
2
Combination of even and odd
function
Function type
Sum
Difference
Product
Quotient
Both even
Even
Even
Even
Even
Both odd
Odd
Odd
Even
Even
Even or odd
Odd
Odd
Even and odd Even or Odd
Products of Even and Odd
Functions
Two Even Functions
Cont…
An Even Function and an Odd Function
Cont…
Two Odd Functions
Accumulation of Even and Odd
Functions
N
N
n N
n1
 g  n   g  0   2 g  n 
N
 g  n  0
n N
Signal Energy and Power
The signal energy of a signal x[n] is
Ex 

 x  n
n
2
Signal Energy and Power
Some signals have infinite signal energy. In that case
It is usually more convenient to deal with average signal
power. The average signal power of a signal x[n] is
2
1 N 1
Px  lim
x  n

N  2 N
n N
For a periodic signal x[n] the average signal power is
2
1
Px 
x  n

N n N
 The notation  n N means the sum over any set of

 consecutive n 's exactly N in length.





Signal Energy and Power
A signal with finite signal energy is
called an energy signal.
A signal with infinite signal energy and
finite average signal power is called a
power signal.