Overview:
Continuous functions
Periodic functions; fundamental period and frequency
Complex exponential function
Linear combination of periodic functions
Energy and Power functions
Reflection, translation, scaling, stretch, compression
Unit step function
Rectangular impulse function
Signum function
Ramp function
Sampling function
Unit impulse function (Dirac delta function)
Properties of Unit impulse function
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 A Continuous Function: f(t) is said to be continuous at t0 if for
every there exists  such that
| f(t) - f(t0 ) | < whenever | t - t0 | < 
t0     t0  and lim
t0     t0  . If
 Another definition: Let lim
0
0
f t 0   = f t 0   = f t 0  then f is continuous at t0 .
If a function is continuous at every point in its domain, it is called
a continuous function.
2
 “Continuous-time function f(t) over [a,b]” --- The function f(t)
need not be continuous. The domain on which it is defined is
continuous.
 If x(t) satisfies x(t) = x(t + nT) where n = 1, 2, 3, … then x(t) is
periodic of period T.
Example:
(i) x(t) = A sin(t) is periodic of fundamental period 2

(ii) t = A sin(t) is periodic of period 2Here is called the
frequency of the sinusoidal function.
 Consider a collection of complex exponential functions defined
by  k (t) = e jkw t where j  1 , k = 1, 2, 3, …., and w 0 is some
real number.
0
Question: Is  k (t) periodic? If yes, what is the period?
3
 Let us assume that x(t) and y(t) are periodic with fundamental
periods T1 and T2 respectively.
Question: Is z(t) = a x(t) + b y(t) periodic ( a and b are some
real numbers)?
“ The linear combination of two periodic functions is periodic if
the ratio of their fundamental periods is a rational number”
4
 Energy and power signals:
Let, x(t) = voltage across a resistance R.
i(t) = current produced
then i(t) = x(t) / R.
The instantaneous power = R i2 ( = x2 / R) and the energy
expended during the incremental time interval dt is ( x2 / R dt ).
Let us assume R = 1 ohm. Then the energy over time interval of
length 2L is given by
L
2
 L x dt .
L
2

2
Total Energy = lim
 L x dt   x dt
L 
Average power P = lim
L 
1 L 2
 L x dt
2L
5
 Consider the function x(t).
-x(t)
x(-t)
x(t) + c
x(t+c)
x(ct)
x(ct)
c x(t)
Reflection in t-axis
Reflection in x-axis
Translation along x-axis by c units; positive c  up
Translation along t-axis by c units; positive c  left
c < 1  Stretch by a factor of c along t axis;
c > 1  compression by a factor of c along t axis;
Stretch/compression by a factor of c along x-axis
Example:
(1) Sketch the graph of x(t) = sin(t)
(2) Sketch the graph of x(t) = 2 sin(/4 –t) + 1/2
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Some special functions
1 t  0
0 t  0
 Unit Step function: u(t) = 
Sketch u(t-a), u(t+a) and [u(t+a) – u(t-a)]
1 t  

2
 Rectangular pulse function: rect(t/) = 
0 t  

2
Example: Sketch rect(t/2a) and compare this with [u(t+a) – u(t-a)]
7
 Signum function:
 1 t  0

Sgn(t) = 0 t  0
1 t  0

t t  0
0 t  0
 Ramp function: r(t) = 
 Sampling function: Sa(t) = sin(t) / t
Sinc function: Sinc(t) = sin(t) / t
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 Unit Impulse function: (t)  “Dirac delta function”
By technical definition of a function, the Dirac delta function
is not a function. (t) has following properties:
(1) (0)  infinity
(2) (t) = 0, t not equal to 0.

(3)    (t)dt  1

(4)
(t) = ( - t)
Example:
0

Consider P1(t) = 1


t
t

2

2
It can be shown that lim
P1(t) = (t)
 0
Note that P1(t) can be rewritten as
0 t  
1 
2
P1(t) =  

 1 t 

2

t 
 rect  
 

1
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 Any function x(t) can be approximated by infinite sum of
weighted rectangular pulse functions of varying heights.
Suppose we know x(t) at points {234k
t  /2
.
t  / 2
1
  0
 t 
rect    
Recall,
1
t  k 
  
   0
t  k   / 2
t  k   / 2

rect 

t  k  x(k) k   / 2   t  k   / 2 
x(k) rect 
  
   
0 t  k   / 2 or t  k   / 2 


t  k 

x(t)   x(k) rect 
k  
  


1  t  k 
 
x(t)   x(k)  rect 
k  

    
For infinitesimally small let us denote  by d and k by 
The right hand side reduces to (recall the Riemann Sum definition
of an integral)


 x( ) lim
 0


 x(t)   x( )  (t   )d
Example: Evaluate

 e
 1
1  t   

d  
rect 
x( )  (t   )d .

    

 x(t)   x( )  (  t)d
 
sin   (  2)d
4 
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 Even and Odd functions:
x(t) is even if x(- t) = x(t) for all t.
x(t) is odd if x(-t) = - x(t) for all t.
Example 1:
x(t) = cos(t) and x(t) = t2 + 4 t4 are even functions. Verify.
Example 2:
x(t) = sin(t) and x(t) = 2t + 3t3 are odd functions. Verify.
Example 3:
x(t) = (t – 2)2 is neither odd nor even. Verify.
 Any arbitrary function x(t) can be written as sum of two
function xe(t) and xo(t) where xe(t) is an even function and xo(t)
is an odd function.
Let x(t) be an arbitrary function. Let us assume that there exists an
even function xe(t) and an odd function xo(t) such that
x(t) = xe(t) + xo(t)
then x(-t) = xe(-t) + xo(-t) = xe(t) - xo(t)
By solving these two equations we get
xe(t) = 1/2 [x(t) + x(-t)] and xo(t) = 1/2 [x(t) – x(-t)]
Exercise: Show that x(t) = (t – 1)2 + sin (t) is neither even nor odd.
Find an even function xe(t) and an odd function xo(t) such that
x(t) = xe(t) + xo(t)
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