Basc Signals Type

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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 (2П f ot + θ)

= A cos (ω

ot +θ)

g(t) = Ae(σ o+jωo)t

= Aeσ ot [cos (ωot) +j sin (ωot)]

Where A is the amplitude of a sinusoid or complex exponential, To is

the real fundamental period of sinusoid, f o is real fundamental cyclicfrequency of sinusoid, ωo is the real fundamental radian frequency of sinusoid, t is time and σo is a real damping rate.

sinusoids

complex exponentials



• In signals and systems, sinusoids are expressed in

either of two ways :

a. cyclic frequency f form - A cos (2П f ot + θ)

b. radian frequency ω form - A cos (ωot + θ)

• Sinusoids and exponentials are important in signal

and system analysis because they arise naturally inthe 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 andoperation to allow us to efficiently analyze

real signals and systems.



The Unit Step Function

1 , 0

u 1/ 2 , 0

0 , 0

t

t t

t

Precise Graph Commonly-Used Graph



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.



The Unit Ramp Function

, 0

ramp u u0 , 0

t t 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.



The Rectangular Pulse Function

Rectangular pulse, 1/ , / 2

0 , / 2a

a t at

t a



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 impulse and

the unit impulse is the generalized derivative of the

unit step

Functions that approach unit step and unit impulse



Graphical Representation of the

ImpulseThe 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

0 0

g gt t t dt t

The Sampling Property

0 0

1a 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



The Unit Periodic Impulse

The unit periodic impulse/impulse train is defined by

, an integerT

n

t t nT n

The periodic impulse is a sum of infinitely many uniformly-

spaced impulses.



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



The Unit Triangle Function

1 , 1

tri 0 , 1

t t

t t

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

sin

sinct

t t

The unit sinc function is related to the unit rectangle function

through the Fourier transform.



The Dirichlet Function

sindrcl ,

sin

Nt t N

N t

The Dirichlet function is the sum of infinitely many

uniformly-spaced sinc functions.



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.



Shifting and Scaling Functions

Let a function be defined graphically by

and let g 0 , 5t t



1. Amplitude Scaling, g t Ag t



1. Amplitude Scaling,

(cont…)

g t Ag t



2. Time shifting, 0t t t

Shifting the function to the right or left by t 0



3. Time scaling, / t t a

Expands the function horizontally by a factor of |a|



3. Time scaling,

(cont…)

/ t t a

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.



0g gt t

t Aa

4. Multiple transformations

0

amplitude

scaling, / 0g g g gt t t A t t at t t

t A t A Aa a

A multiple transformation can be done in steps

0

amplitudescaling, / 0

0 0g g g g gt t t A t t a t t t t A t A t t A t A

a 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.



g t Agt t 0

a

Multiple transformations,

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 isits slope at the time.

• The integral of a function at any time t is

accumulated area under the function up tothat time.



Differentiation



Integration



Even and Odd CT FunctionsEven Functions Odd Functions

g t g t g t g t



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 t t

odd part

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



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



Two Even Functions

Products of Even and Odd Functions



Cont…

An Even Function and an Odd Function




Cont…



Two Odd Functions

Cont…



Function type and the types of

derivatives and integralsFunction type Derivative Integral

Even Odd Odd + constant

Odd Even Even



Integrals of Even and Odd Functions

0

g 2 g

a a

a

t dt t dt

g 0

a

a

t dt



Signal Energy and Power

2

x x E t dt

The signal energy of 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.



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

it is more convenient to deal with average signal power.

/ 22

x

/ 2

1lim x

T

T T

P t dt T

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

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

2

x

1xT P t dt

T where T is any period of the signal.




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 discrete-time signal

x x where is the time between sampless s

n nT T



Complex Exponentials and

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

time counter part

g[n] = A cos (2Пn/No+θ)

= A cos (2ПFon+ θ)

= A cos (Ωon+ θ)

g[n] = Ae β n

= Azn

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

o

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.

sinusoids

complex exponentials



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 the DT

sinusoid may not even be periodic.

DT Si id



DT Sinusoids4 discrete-time sinusoids

DT Si id



DT SinusoidsAn Aperiodic Sinusoid

A discrete time sinusoids is not necessarily periodic



DT Sinusoids

Two DT sinusoids whose analytical expressions look different,

g1 n Acos 2 F 01n 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 n is discrete time and therefore an integer),

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

Th I l F ti



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 x

n

A n n n A n

but no scaling property. That is,

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



The Unit Sequence Function

1 , 0

u0 , 0

nn

n



The Unit Ramp Function

, 0

ramp u 10 , 0

n

m

n nn m

n



The Rectangle Function

1 ,

rect , 0 , an integer0 ,w

w

N w w

w

n N n N N

n N



The Periodic Impulse Function

N

m

n n mN



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

g n 0 , n 15



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



3. Time compression, n Kn

K an integer > 1



/ , 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.



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).



Even and Odd Functions

g g

g2

e

n nn

g gg

2o

n nn

g gn n g gn n



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



Products of Even and Odd

FunctionsTwo Even Functions



Cont…




Cont…

Two Odd Functions

Accumulation of Even and Odd



Accumulation of Even and Odd

Functions

1

g g 0 2 g N N

n N n

n n

g 0 N

n N

n




The signal energy of a signal x[n] is

2

xx

n

E n

Si l d



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

N n N

P n N

2

x

1x

n N

P n N

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




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.

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