Basic building blocks of signals Agenda for 01-03-2010

Basic building blocks of signals Basic building blocks of signals

Agenda for 01-03-2010

• Signals derived from unit step function

C ti ti (CT) i l

• Continuous-time (CT) impulse

• Some basic CT signal functions

• Pretest results

• Matlab Tutorial-1

• Matlab Tutorial-1

The CT Signum Function The CT Signum Function

1 , t  0

 

sgn

 

t ^{ 0 , t  0} 

1 , t  0



 

  2 u

 

t ^1

Precise Graph Commonly-Used Graph

h i f i i i di i f

The signum function, in a sense, returns an indication of the sign of its argument.

The CT Unit Ramp Function e C U t a p u ct o

ramp

 

t ^{ t , t  0} 0 t  0







  u

 

 ^d



t ^{ t u}

 

^t

p

 

0 , t  0

 

 



  

Continuous time impulse Continuous-time impulse

• Generation of impulse

• Impulse as a sampler Impulse as a sampler

Introduction to the CT Impulse Introduction to the CT Impulse

D fi f ti 

 

^t  1

a , t  a 2





Define a function, _a

 

^t 

0 , t  a 2





Let g(

t

) be finite and continuous at

t

= 0.

Introduction to the CT Impulse p

a

The area under the product of the two functions is A  1

a g

 

t ^dt

a a



2

a 2

As the width of approaches zero, _a

 

t lim0 A  g 0

 

^lim

0

1

a dt

a



2 ^{ g 0}

 

^lim₀ ¹_a

 

^{a  g 0}

 

a0 a0 a

^a 2



_{a0 a}

The CT unit impulse is implicitly defined by g 0

 

^{ }





 

t ^g

 

^{t dt}

e C u pu se s p c y de ed by



CT Unit Step and CT Unit Impulse CT Unit Step and CT Unit Impulse

As

a

approaches zero, g(t) approaches a CT unit step and g 

 

t As

a

approaches zero, g(t) approaches a CT unit step and approaches a CT unit impulse

g

 

The CT unit step is the integral of the CT unit impulse and the CT unit impulse is the

generalized derivative

of the CT unit step

Graphical Representation of Graphical Representation of

the CT Impulse

The CT impulse is not a function in the ordinary sense because its value at the time of occurrence is not defined It is represented value at the time of occurrence is not defined. It is represented a vertical arrow. Its strength is either written beside it or is represented by its length.p y g

MATLAB function: dirac(t) = d/dt (heaviside(t))

Properties of the CT Impulse Properties of the CT Impulse

The sampling property

g

 

t 



^t  t₀



^dt





^{ g}

 

^t0

p g p p y

 



  

The sampling property “extracts” the value of a function at i

The scaling property a point.





a t



 t₀

 

^{ 1}

a 



t t₀



a

This property makes the impulse different from ordinary functions

functions.

The CT Unit Comb The CT Unit Comb

The CT unit comb is defined by The CT unit comb is defined by

comb

 

t 



^ 



t  n



^{n an integer}

comb

 

t  



t n



n



 ^{, n an integer}

The comb is a sum of uniformly-spaced impulses.

Some basic continuous-time signal functions

• Rectangular pulse

• Triangular pulse Triangular pulse

• Sinc function

• Dirichlet functions

The CT Unit Rectangle Function The CT Unit Rectangle Function

1 , t  1

 



rect

 

t 

, t

2 1

2 , t  1

 2





 







Rectpuls(t)

 

2 2

0 , t  1 2

















Rectpuls(t,w)

 2

 

The product signal, g(

t

)rect(

t

), can be thought of as the signal, g(

t

),

“turned on” at timeturned on at time,

t t

= -1/2 and “turned back off” at time t = 1/2 1/2 and turned back off at time, t 1/2.

The CT Unit Triangle Function The CT Unit Triangle Function

tri

 

t  1 t , t  1 0 , t 1









0 , t 1

 

tripuls(t) tripuls(t,2)

tripuls(t) tripuls(t,w)

Th t i l d fi d thi i l t d t th it t l

tripuls(t,w,s) The triangle, defined this way, is related to the unit rectangle through an operation called

convolution

to be introduced in Chapter 3

Chapter 3.

The CT Unit Sinc Function The CT Unit Sinc Function

 

^sin

 

^t

The unit sinc

sinc

 

t  sin

 

^t

t function is related to

the unit rectangle

f ti th h th

function through the

Fourier transform

, to be introduced in

d



i

  

to be introduced in Chapter 5.

lim

t0 sinc

 

t  lim

t0

dt



sin

 

t



d

 

^{t  limt0}

 ^cos

 

^t

 ^{ 1} dt

 

^t

The CT Dirichlet Function The CT Dirichlet Function

drcl

 

t, N ^{ sin}



^Nt



N sin

 

t N sin

 

t

F dd

N

h Di i hl f i i d i f i

For odd

N

, the Dirichlet function is a repeated sinc function.

Combinations of CT Functions

Ease of questions

Ease of questions

Success in answering questions

Success in answering questions

Election and team roles Election and team roles

0351128 ل لا ا لاف ز ز لا

Team 1

0351128 يولبلا دعاس حلاف زيزعلادبع

Team 1

0453026

يثراحلا دمح يزوف

Team 2

0516835 يكلاملا ةيطع ﷲدبع ريمس

Team 3

0611133 هبيش يماس ميھاربا

Team 1

0611133

هبيش يماس ميھاربا

0611234

ةيطع دمحم نيساي

Team 2

T 3

0611620

يوامربز نب قازرلادبع نميأ

Team 3

0612966

يقشاشن دمحأ ديمحلادبع_ي

Team 1

0620879 هضيھ دمحم رصان دمحم

Team 2

0704754 د أ ن ن د

Team 3

0704754

دمحأ نينسح دمحم

Team 3

Using m-files in MATLAB Using m files in MATLAB

Open a blank m-file window in MATLAB and type in:

%Generation of step function clear

tmn=-2; dt=0.1; tmx=5;

tmn 2; dt 0.1; tmx 5;

t=tmn:dt:tmx;

t1=0;t2=1; %starting points

B1=1;B2=1.5*B1; %Magnitudes; ; g

u1=B1*stepfun(t,t1);% Generation of 1st step function u2=B2*heaviside(t-t2);%Generation of 2nd step function plot(t,u1,t,u2),axis([tmn tmx -B1 2*B1]),xlabel('Time (s)'),...

ylabel('u(t)'),title('Step function'),grid

•Try to interpret every statement and special character

•Save the file under file name step_function and run it!

•Try to link the features in the figure (plot) to the statements in the file

•Change “dt” to 0.5 and 0.01 and observe its effects

Duties for Wednesday 3/03/2010 Duties for Wednesday 3/03/2010

• Study 2.5 – 2.7 (pages 43 - 63) from Roberts

• Prepare active-learning exercises ALE 4-7 G t d f MATLAB t t i l

• Get ready for MATLAB tutorial