power electronics ppt

Chapter 1 Systems and Signals

Continuous-Time and Discrete-Time Signals

Classification of Signals

Transformations of the Independent Variables

Exponential and Sinusoidal Signals

Unit Impulse and Unit Step Functions

Continuous-Time and Discrete-Time Systems

Basic System Properties

Summary

1. Signals

Signals

Any physical quantity that varies with time, space or any other

independent variable.

Signals are represented mathematically as functions of one

or more independent variables.

In this course, time is usually the only independent variable.

Continuous-time signals are defined for every value of time.

Discrete -time signals are defined at discrete values of time.

2. Classification of Signals

Periodic Signals


Even and Odd Signals

Even signal: x(-t)=x(t) or x[-n]=x[n]

Odd signal: x(-t)=-x(t) or x[-n]=-x[n]

Any signal can be broken into a sum of an even signal

and an odd signal

x[n] = xe[n] + xo[n]

])[][(2

1][ nxnxnxe

])[][(2

1][ nxnxnxo


Complex-valued Signals

Conjugate symmetric signal: x*(-t)=x(t) or x*[-n]=x[n]

Conjugate antisymmetric signal :x*(-t)=-x(t) or x*[-n]=-x[n]

Decomposition

Conjugate symmetric-antisymmetric decomposition: Any

signal may be expressed as the sum of a conjugate-

symmetric component and a conjugate antisymmetric

component as

x[n] = xe[n] + xo[n]

x n x n x ne [ ] ( [ ] *[ ]) 1

2

x n x n x no [ ] ( [ ] *[ ]) 1

2


Instantaneous Power across a resistor R

Energy

Average power


The total energy is defined as

Time Averaged, Power is defined as

dttxdttxE

T

TT)()(lim 2

2/

2/

2

2/

2/

2 )(1

limT

TTdttx

TP


Signal Energy and Power

Energy signal: A signal for which 0

3. Basic Operation on Signals Operations Performed on Dependent Variables

Amplitude Scaling

Addition

Multiplication

Differentiation

Integration

3. Basic Operation on Signals--Transformations of the Independent Variables

Time Shift

3. Basic Operation on Signals


Time Reversal

3. Basic Operation on SignalsTransformations of the Independent Variables

Time Scaling

3. Basic Operation on SignalsTransformations of the Independent Variables

4. Exponential and Sinusoidal Signals

Real Exponential Signals


DT real exponential signals: x[n] = Brn, where B and r are real

numbers

Decaying in amplitude: 00; alternating in sign: r


Definition

Xa(t) = A cos( t+ ), - *


CT Complex Exponential Signals

Eulers Identity


Rectangular Form vs Polar Form


Discrete-Time Form


Discrete-Time Sinusoidal Signals

x(n) = A cos( n+ ), n =1, 2, ...

A is the amplitude of the sinusoid

is the frequency in radians per sample

is the phase in radians

f= /2p is the frequency in cycles per sample or hertz

X(n) = A cos( n+ )


A discrete-time sinsoidal is periodic only if its frequency f is a rational number

x(n+N) = x(n), N=m/f, where m, N are integers.

Discrete-time sinusoidals where frequencies are separated by an integer multiple of 2p are identical

x1(n) = A cos( 0 n)

x2(n) = A cos( (0 2p) n) The highest rate of oscillation in a discrete-time

sinusoidal is attained when =p or (=-p), or equivalently f=1/2.

X(n) = A cos(( 0+p)n) = -A cos((0+p)n

Discrete-Time Sinusoidal SignalsX(n) = A cos( n+ )

5. Unit Impulse and Unit Step Functions

Unit Impulse Signals


Dirac Delta function, (t), often referred to as the unit impulse or delta function, is the function that defines the

idea of an unit impulse.

This function is one that is infinitely narrow, infinitely tall,

yet integrates to unity,


Unit Step Signals


Perhaps the simplest way to visualize this is

as a rectangular pulse from a/2 to a+/2 with a height of 1/.

As we take the limit of this, lim 0, we see that

the width tends to zero and the height tends

to infinity as the total area remains constant

at one.

The impulse function is often written as (t) .

)(lim)(0

txt

)(tx

2/ 2/


Properties


Properties

Shifting property

Time-Scaling

)(1

)(lim)(0

ta

atxat

6. Continuous-Time and Discrete-Time

Systems

7. Basic System Properties

Systems

Mathematically a transformation or an operator that maps

an input signal into an output signal

Can be either hardware or software.

Such operations are usually referred as signal processing.

E.x.

Discrete-Time System H

n n

y n x k x k x n y n x nk

n

k

n

( ) ( ) ( ) ( ) ( ) ( )

1 1


Types of Systems

CT systems: input and output are CT signals

DT systems: input and output are DT signals

Mixed systems: CT-in and DT-out (e.g., A/D converter), DT-in

and CT-out (e.g., D/A converter)


Time-Invariant versus Time-Variant Systems

A system H is time-invariant or shift invariant if and only if

x(n) ---> y(n)

implies that

x(n-k) --> y(n-k)

for every input signal x(n) x(n) and every time shift k.

Causal versus Noncausal Systems

The output at any time depends on values of the input at only the present and past time.

y(n) = F[x(n), x(n-1), x(n-2), ...].

where F[.] is some arbitrary function.


Linear versus Nonlinear Systems

A system H is linear if and only if

H[a1x1(n)+ a2 x2 (n)] = a1H[x1 (n)] + a2H[x2 (n)]

for any arbitrary input sequences x1(n) and

x2(n), and any arbitrary constants a1 and a2.

Multiplicative or Scaling Property

H[ax(n)] = a H[x(n)]

Additivity Property

H[x1(n) + x2 (n)] = H[x1 (n)] + H[x2 (n)]

Linear systems

y(n)

u1(n)

+

u2(n)

a

b

Linear systems

u1(n)

u2(n)

Linear systems

+y(n)

a

b


Stable versus Unstable Systems

An arbitrary relaxed system is said to be bounded-input-

bounded-output (BIBO) stable if and only if every bounded

input produces a bounded output.

Ex.

y(t)=tx(t)

y(t)=ex(t)


Memory versus Memoryless Systems

A system is referred to as memoryless system if the output for

each value of the independent variable depends only on the

input at the same time.

Examples

Memory

y[n] = x[n-1] (Delay system)

y[n] = y[n-1]+x[n] (Accumulator)

Memoryless

y(t)=Rx(t)


Inverse (or Invertible) System

If distinct inputs lead to distinct outputs

8. Matlab

Periodic Signals,1.38

Exponential Signals, 1.39

Sinusoidal Signals, 1.40

Exponentially Damped Sinusoidal Signals, 1.41

Step, Impulse and Ramp Functions

User Defined Function

9. Remarks

Continuous-Time and Discrete-Time Signals

Classification of Signals


Exponential and Sinusoidal Signals

Unit Impulse and Unit Step Functions

Continuous-Time and Discrete-Time Systems

Basic System Properties

Summary

Homeworks

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