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)]

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.

sinusoids

complex exponentials

• 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 , 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

, 0ramp u u

0 , 0

tt tt 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 theunit 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

0 0g 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 ata 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 integerTn

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

tri0 , 1

t tt

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

sinsinc

tt

t

The unit sinc function is related to the unit rectangle function through the Fourier transform.

The Dirichlet Function

sin

drcl ,sin

Ntt N

N t

The Dirichlet function is the sum of infinitely manyuniformly-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 FunctionsLet 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 t0

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 inversionmeans flipping the curve 1800 with the g axis as the rotation axis of the flip.

0g gt t

t Aa

4. Multiple transformations

0

amplitudescaling, / 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

0

amplitudescaling, / 0

0 0g g g g gt t tA t t a t ttt A t A t t A t A

a a

The order of the changes is important. For example, if weexchange the order of the time-scaling and time-shifting operations, we get:

Amplitude scaling, time scaling and time shifting can be appliedsimultaneously.

g t Agt t0

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 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 FunctionsEven Functions Odd Functions

g t g t

g t g t

Even and Odd Parts of Functions

g gThe of a function is g

2e

t tt

even part

g gThe of a function is g

2o

t tt

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 function

Function 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

An Even Function and an Odd Function

Cont…

Two Odd Functions

Cont…

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

0

g 2 ga a

a

t dt t dt

g 0a

a

t dt

Signal Energy and Power

2

x xE 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 square of 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 caseit is more convenient to deal with average signal power.

/ 2

2

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

1x

TP t dt

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 discrete-

time signal

x x where is the time between sampless sn nT T

Complex Exponentials and Sinusoids

• DT signals can be defined in a manner analogous to their continuous-time counter partg[n] = A cos (2Пn/No+θ)

= A cos (2ПFon+ θ) = A cos (Ωon+ θ)

g[n] = Aeβn

= Azn

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.

sinusoids

complex exponentials

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 Sinusoids4 discrete-time sinusoids

DT SinusoidsAn Aperiodic Sinusoid

A discrete time sinusoids is not necessarily periodic

DT SinusoidsTwo 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 n is discrete time and therefore an integer),

01 02cos 2 cos 2A 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.

The Impulse Function

1 , 0

0 , 0

nn

n

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,

0 0x xn

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 , 0u

0 , 0

nn

n

The Unit Ramp Function

, 0ramp u 1

0 , 0

n

m

n nn m

n

The Rectangle Function

1 ,

rect , 0 , an integer0 ,w

w

N w w

w

n Nn N N

n N

The Periodic Impulse Function

Nm

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 integration is accumulation (or summation).

Even and Odd Functions

g gg

2e

n nn

g g

g2o

n nn

g gn n g gn n

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 and odd Even or Odd Even or odd Odd 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

1

g g 0 2 gN N

n N n

n n

g 0N

n N

n

Signal Energy and Power

The signal energy of a signal x[n] is

2

x xn

E n

Signal Energy and PowerSome signals have infinite signal energy. In that caseIt is usually more convenient to deal with average signal power. 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

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.