bmt415-chapter 1 2

8/10/2019 BMT415-Chapter 1 2

1/68

Basic Operation on Signals

Continuous-Time Signals

8/10/2019 BMT415-Chapter 1 2

2/68

The signal is the actual physical

phenomenon that carries information, and

the function is a mathematical descriptionof the signal.

8/10/2019 BMT415-Chapter 1 2

3/68

Complex Exponentials & Sinusoids

Signals can be expressed in sinusoid or complex exponential.

g(t) = A cos (2t/To+)

= A cos (2fot+ )

= A cos (

ot+)

g(t) = Ae(o+jo)t

= Aeot[cos (ot) +j sin (ot)]

Where A is the amplitudeof a sinusoid or complex exponential, Tois

the real fundamental periodof sinusoid,fois real fundamental cyclicfrequencyof sinusoid, o is the real fundamental radian frequencyofsinusoid, t is timeand ois a real damping rate.

sinusoids

complex exponentials

8/10/2019 BMT415-Chapter 1 2

4/68

In signals and systems, sinusoids are expressed in

either of two ways :

a. cyclic frequencyfform - A cos (2fot+ )

b. radian frequency form - A cos (ot+ )

Sinusoids and exponentials are important in signal

and system analysis because they arise naturally inthe solutions of the differential equations.

8/10/2019 BMT415-Chapter 1 2

5/68

Singularity functions and related

functions In the consideration of singularity functions,

we will extend, modify, and/or generalized

some basic mathematical concepts andoperation to allow us to efficiently analyze

real signals and systems.

8/10/2019 BMT415-Chapter 1 2

6/68

The Unit Step Function

1 , 0

u 1/ 2 , 0

0 , 0

t

t t

t

Precise Graph Commonly-Used Graph

8/10/2019 BMT415-Chapter 1 2

7/68

The Signum Function

1 , 0

sgn 0 , 0 2u 1

1 , 0

t

t t t

t

Precise Graph Commonly-Used Graph

The signum function, is closely related to the unit-step

function.

8/10/2019 BMT415-Chapter 1 2

8/68

The Unit Ramp Function

, 0

ramp u u0 , 0

tt t

t d t t t

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.

8/10/2019 BMT415-Chapter 1 2

9/68

The Rectangular Pulse Function

Rectangular pulse, 1/ , / 2

0 , / 2a

a t at

t a

8/10/2019 BMT415-Chapter 1 2

10/68

The Unit Step and Unit Impulse Function

As approaches zero, g approaches a unit

step andg approaches a unit impulse

a t

t

The unit step is the integral of the unit impulseand

the unit impulse is the generalized derivativeof the

unit step

Functions that approach unit step and unit impulse

8/10/2019 BMT415-Chapter 1 2

11/68

Graphical Representation of the

ImpulseThe area under an impulse is called its strengthor 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.

8/10/2019 BMT415-Chapter 1 2

12/68

Properties of the Impulse

0 0g gt t t dt t

The Sampling Property

0 01

a t t t t a

The Scaling Property

The sampling property extracts the value of a function at

a point.

This property illustrates that the impulse is different from

ordinary mathematical functions.

The Equivalence Property

8/10/2019 BMT415-Chapter 1 2

13/68

The Unit Periodic Impulse

The unit periodic impulse/impulse train is defined by

, an integer Tn

t t nT n

The periodic impulse is a sum of infinitelymany uniformly-

spaced impulses.

8/10/2019 BMT415-Chapter 1 2

14/68

The Unit Rectangle Function

1 , 1/ 2

rect 1/ 2 , 1/ 2 u 1/ 2 u 1/ 2

0 , 1/ 2

t

t t t t

t

The signal turned on at time t= -1/2 and turned back off at

time t= +1/2.

Precise graph Commonly-used graph

8/10/2019 BMT415-Chapter 1 2

15/68

The Unit Triangle Function

1 , 1

tri 0 , 1

t t

tt

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).

8/10/2019 BMT415-Chapter 1 2

16/68

8/10/2019 BMT415-Chapter 1 2

17/68

The Dirichlet Function

sindrcl ,

sin

Ntt N

N t

The Dirichlet function is the sum of infinitely many

uniformly-spaced sincfunctions.

8/10/2019 BMT415-Chapter 1 2

18/68

Combinations of Functions

Sometime a single mathematical function maycompletely describe a signal (ex: a sinusoid).

But often one function is not enough for anaccurate description.

Therefore, combination of function is needed toallow versatility in the mathematical

representation of arbitrary signals. The combination can be sums, differences,

products and/or quotients of functions.

8/10/2019 BMT415-Chapter 1 2

19/68

Shifting and Scaling Functions

Let a function be defined graphically by

and let g 0 , 5t t

8/10/2019 BMT415-Chapter 1 2

20/68

1. Amplitude Scaling, g t Ag t

8/10/2019 BMT415-Chapter 1 2

21/68

1. Amplitude Scaling,

(cont)

g t Ag t

8/10/2019 BMT415-Chapter 1 2

22/68

2. Time shifting,0t t t

Shifting the function to the right or left by t0

8/10/2019 BMT415-Chapter 1 2

23/68

3. Time scaling, /t t a

Expands the function horizontally by a factor of |a|

8/10/2019 BMT415-Chapter 1 2

24/68

3. Time scaling,

(cont)

/t t a

If a < 0, the function is also time inverted. The time inversion

means flipping the curve 1800with the g axis as the rotation axis

of the flip.

8/10/2019 BMT415-Chapter 1 2

25/68

0g g t t

t Aa

4. Multiple transformations

0

amplitude

scaling, / 0g g g gt t tA t t a t tt

t A t A Aa a

A multiple transformation can be done in steps

0amplitudescaling, / 0

0 0g g g g gt t tA t t a t tt

t A t A t t A t Aa 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:

Amplitude scaling, time scaling and time shifting can be applied

simultaneously.

8/10/2019 BMT415-Chapter 1 2

26/68

g t Ag t t0a

Multiple transformations,

A sequence of amplitude scaling , time scaling and time shifting

8/10/2019 BMT415-Chapter 1 2

27/68

Differentiation and Integration

Integration and differentiation are common

signal processing operations in real systems.

The derivative of a function at any time t isits slope at the time.

The integral of a function at any time tis

accumulated area under the function up tothat time.

8/10/2019 BMT415-Chapter 1 2

28/68

8/10/2019 BMT415-Chapter 1 2

29/68

Integration

8/10/2019 BMT415-Chapter 1 2

30/68

Even and Odd CT FunctionsEven Functions Odd Functions

g t gt g t gt

8/10/2019 BMT415-Chapter 1 2

31/68

Even and Odd Parts of Functions

g g

The of a function is g 2e

t t

t

even part

g g

The of a function is g2

o

t tt

odd part

A function whose even part is zero is oddand a functionwhose odd part is zero is even.

8/10/2019 BMT415-Chapter 1 2

32/68

Combination of even and odd

functionFunction type Sum Difference Product Quotient

Both even Even Even Even Even

Both odd Odd Odd Even Even

Even and odd Neither Neither Odd Odd

8/10/2019 BMT415-Chapter 1 2

33/68

Two Even Functions

Products of Even and Odd Functions

8/10/2019 BMT415-Chapter 1 2

34/68

Cont

An Even Function and an Odd Function

8/10/2019 BMT415-Chapter 1 2

35/68


Cont

8/10/2019 BMT415-Chapter 1 2

36/68

Two Odd Functions

Cont

8/10/2019 BMT415-Chapter 1 2

37/68

Function type and the types of

derivatives and integralsFunction type Derivative Integral

Even Odd Odd + constant

Odd Even Even

8/10/2019 BMT415-Chapter 1 2

38/68

Integrals of Even and Odd Functions

0

g 2 ga a

a

t dt t dt

g 0a

a

t dt

8/10/2019 BMT415-Chapter 1 2

39/68

Signal Energy and Power

2

x xE t dt

The signal energyof a signal x(t) is

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 squareof the magnitude of the signal.

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.

8/10/2019 BMT415-Chapter 1 2

40/68

Signal Energy and PowerSome signals have infinite signal energy. In that case

it is more convenient to deal with averagesignal power.

/ 22

x

/ 2

1lim x

T

TT

P t dtT

The average signal power of a signal x(t) is

For a periodic signal x(t) the average signal power is

2

x

1xTP t dtT

where Tis any period of the signal.

8/10/2019 BMT415-Chapter 1 2

41/68


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.

8/10/2019 BMT415-Chapter 1 2

42/68

Basic Operation on Signals

Discrete-Time Signals

8/10/2019 BMT415-Chapter 1 2

43/68

Sampling a Continuous-Time Signal

to Create a Discrete-Time Signal Samplingis 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 discrete-time signal

x x where is the time between sampless sn nT T

8/10/2019 BMT415-Chapter 1 2

44/68

Complex Exponentials and

Sinusoids DT signals can be defined in a manner analogous to their continuous-

time counter part

g[n] = A cos (2n/No+)

= A cos (2Fon+ )

= A cos (on+ )

g[n] = Aen

= Azn

Where A is the real constant (amplitude), is a real phase shiftingradians,N

o

is a real numberand Fo

and o

are related to No

through

1/N0 = Fo = o/2 , where n is the previously defined discrete time.

sinusoids

complex exponentials

8/10/2019 BMT415-Chapter 1 2

45/68

DT Sinusoids

There are some important differences

between CT and DT sinusoids.

If we create a DT sinusoid by sampling CTsinusoid, the period of the DT sinusoid may

not be readily apparent and in fact theDT

sinusoid may not even be periodic.

DT Si id

8/10/2019 BMT415-Chapter 1 2

46/68

DT Sinusoids4 discrete-time sinusoids

DT Si id

8/10/2019 BMT415-Chapter 1 2

47/68

DT SinusoidsAn Aperiodic Sinusoid

A discrete time sinusoids is not necessarily periodic

8/10/2019 BMT415-Chapter 1 2

48/68

DT Sinusoids

Two DT sinusoids whose analytical expressions look different,

g1 n Acos 2F01n 2 02g cos 2n A F n and

may actually be the same. If

02 01 , where is an integerF F m m

then (because nis discrete time and therefore an integer),

01 02cos 2 cos 2A F n A F n

(Example on next slide)

8/10/2019 BMT415-Chapter 1 2

49/68

Sinusoids

The dash line are the CT function. The CT function are obviously

different but the DT function are not.

Th I l F ti

8/10/2019 BMT415-Chapter 1 2

50/68

The Impulse Function

1 , 00 , 0

nnn

The discrete-time unit impulse (also known as the Kronecker

delta function) is a function in the ordinary sense (in contrastwith the continuous-time unit impulse). It has a sampling property,

0 0x xn

A n n n A n

but no scaling property. That is,

for any non-zero, finite integer .n an a

8/10/2019 BMT415-Chapter 1 2

51/68

The Unit Sequence Function

1 , 0

u0 , 0

nn

n

8/10/2019 BMT415-Chapter 1 2

52/68

The Unit Ramp Function

, 0

ramp u 10 , 0

n

m

n nn m

n

8/10/2019 BMT415-Chapter 1 2

53/68

The Rectangle Function

1 ,

rect , 0 , an integer 0 ,w

w

N w w

w

n Nn N N

n N

8/10/2019 BMT415-Chapter 1 2

54/68

The Periodic Impulse Function

Nm

n n mN

8/10/2019 BMT415-Chapter 1 2

55/68

Scaling and Shifting FunctionsLet g[n] be graphically defined by

g n 0 , n 15

8/10/2019 BMT415-Chapter 1 2

56/68

0 0 , an integern n n n Time shifting

Scaling and Shifting Functions

2.

1. Amplitude scaling

Amplitude scaling for discrete time function is exactly thesame as it is for continuous time function

8/10/2019 BMT415-Chapter 1 2

57/68

3. Time compression, n Kn

Kan integer > 1

8/10/2019 BMT415-Chapter 1 2

58/68

/ , 1n n K K Time expansion

For all such that / is an integer, g / is defined.

For all such that / is not an integer, g / is not defined.

n n K n K

n n K n K

4.

8/10/2019 BMT415-Chapter 1 2

59/68

Differencing and accumulation

The operation on discrete-time signal that is

analogous to the derivative is difference.

The discrete-time counterpart of integrationis accumulation (or summation).

8/10/2019 BMT415-Chapter 1 2

60/68

Even and Odd Functions

g g

g2

e

n nn

g gg

2o

n nn

g gn n g gn n

8/10/2019 BMT415-Chapter 1 2

61/68

Combination of even and odd

functionFunction type Sum Difference Product Quotient

Both even Even Even Even Even

Both odd Odd Odd Even Even

Even and odd Even or Odd Even or odd Odd Odd

P d t f E d Odd

8/10/2019 BMT415-Chapter 1 2

62/68

Products of Even and Odd

FunctionsTwo Even Functions

8/10/2019 BMT415-Chapter 1 2

63/68

Cont


8/10/2019 BMT415-Chapter 1 2

64/68

Cont

Two Odd Functions

Accumulation of Even and Odd

8/10/2019 BMT415-Chapter 1 2

65/68

Accumulation of Even and Odd

Functions

1

g g 0 2 gN N

n N n

n n

g 0N

n N

n

8/10/2019 BMT415-Chapter 1 2

66/68


The signal energyof a signal x[n] is

2

x xn

E n

Si l d

8/10/2019 BMT415-Chapter 1 2

67/68

Signal Energy and PowerSome signals have infinite signal energy. In that case

It is usually more convenient to deal with average signalpower. The average signal power of a signal x[n] is

1

2

x

1lim x

2

N

Nn N

P nN

2

x

1x

n N

P nN

For a periodic signal x[n] the average signal power is

The notation means the sum over any set of

consecutive 's exactly in length.

n N

n N

8/10/2019 BMT415-Chapter 1 2

68/68


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.

bmt415-chapter 1 2

Documents