1_Signals and Systems Classifications

7/29/2019 1_Signals and Systems Classifications

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Signals and Systems

Classifications

By Dr. Samer Awad

Assistant professor of biomedical engineering

The Hashemite University, Zarqa, Jordan

[email protected]

Last update: 24 September 2012
mailto:[email protected]:[email protected]


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Definitions

Signals: Quantitative representation of avariable changing in time &/or space. They

can be functions of 1 or more independent

variables. Examples: force, voltage, stock

market and attendance


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Definitions

Systems: An operator acting upon signals toextract useful information

from them

Example: ECG system

Example: Passive LPF

x(t) y(t)System x(t) y(t)


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Definitions

Systems: Operator acting upon signals toextract useful information

Example: Ultrasound imaging system

Transducer processor display

summation

amplification

filtering


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Definitions

Studying signals & systems involve some sortof mathematical/numerical approach to

model/approximate the behavior of a given

system

Why study signals & systems?

1) Betterinsight better design

2) Compare model with reality3) Save time, money, stress & injuries


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Classification of Signals

Different analysis methods

1) Periodic vs non-periodic (a-periodic)

- A signal is periodic if there is a number T s.t.

x(t) = x(t+T)

- The smallest positive T is the period

- The fundamental freq fo=1/T

- Example: ECG (approximately), x(t)=cos(t)- The sum of periodic signals is periodic iff the

ratios of the periods are ratios of integers


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2) Random vs nonrandom- A signal is random if there is some degree of

uncertainty before the signal actually occurs.

Example: stock market, noise, ECG?

- A signal is non-random if there is no

uncertainty before it occurs

Example: x(t) = e-t

x(t) = cos(t)

- The amplitude for a random variable is

estimated using

n

xx

RMS

2)(


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3) Power vs Energy signalsIt is useful to estimate the size of the signal

- A signal is an energy signal if x(t) satifies:

Example: x(t) = e-t u(t)

- Power is a time average of energy. "Power

signals" have finite and non-zero power

Example: x(t) = sin(t)


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3) Power vs Energy signals

- Note that power is the time average of

energy.

- Finite energy zero power

- Finite power infinite energy

- A power signal has infinite energy and

everlasting non-physical

- A signal can be neither energy nor power.

Example: x(t) = r(t)


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4) Continuous time vs discrete time- x1(t) has values for every t [0,1]

- x2(n) has values at discrete points in time

- t & n are related by sampling intervalt or

sampling fresquency fs = 1/t

- x2(n) = x2(nt )X1(t) continuous

X2(n) discrete


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5) Analog vs digital- x1(t) has amplitude values [a,b]

- x3(n) has amplitude values finite set of

numbers

X1(t) continuous

X3(n) discrete


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X1(t): Continuous time & analogX2(n): Discrete time & analogX3(t): Continuous time & digital

X4(n): Discrete time & digital


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6) Even vs Odd- A signal is even if xe(t) = xe(-t) Ex: cos(t)

- A signal is odd if xo(t) = -xo(-t) Ex: sin(t)

- odd + odd = odd

- even + even = even

- odd + even = neither

- odd x odd = even- even x even = even

- odd x even = odd


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6) Even vs OddTo find out whether the signal x(t) is odd,

even, or neither:

a) Flip x(t) around the y-axis x(t)

x(t)


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even, or neither:


If x(t) = x(t) x(t) is even

x(t)


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even, or neither:



b) If not, flip x(t) around x-axisx(t)

x(t)


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even, or neither:



b) If not, flip x(t) around x-axisx(t)

If x(t) = x(t) x(t) is odd

c) If not, x(t) is neither odd, nor even

x(t)


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6) Even vs OddEven and odd components of a signal:

Any signal can be expressed as the sum of an

even part and odd part

x(t) = xe(t) + xo(t) ----------(1)

x(-t) = xe(-t) + xo(-t)

x(-t) = xe(t) - xo(t) ------(2)

(1) + (2) xe(t) = [ x(t) + x(-t) ]

(1) - (2) xo(t) = [ x(t) - x(-t) ]

Remember:

xe(t) = xe(-t)

xo(t) = -xo(-t)


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7) Odd Half-wave symmetryOnly for periodic signals

Can pertain to odd or even functions

Odd half wave symmetry (OHWS)

We are left with only ODD harmonics.

Even harmonics disappear.


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Even + OHWS

Odd + OHWS

Neither Odd

nor Even

+ OHWS



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Classification of Systems

1) Causal vs. non-causal- Causal: present o/p doesnt depend on future

values of i/p. It only depends on present &/or

past values

- Non-causal: o/p depends on future values of

i/pnon-physical

x(t)

y(t)

x(t)

y(t)

x(t)

y(t)


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1) Causal vs. non-causalAn example of a non-causal system. Will be

explained later when convolution is

presented

x(t) y(t)h(t)

zero


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2) Linear vs. non-linear

The system h(t) is linear if:

a)Additivity rule holds:

b)Homogeneity rule holds:

c)Superposition rule holds:

x2(t) y2(t)h(t)

x1(t) y1(t)h(t)

x1(t)+x2(t) y1(t)+y2(t)h(t)

kx(t) ky(t)h(t)

ax1(t)+bx2(t) ay1(t)+by2(t)h(t)


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2) Linear vs. non-linearShow that y(t) y(t) + 3y(t) = x(t) is non-linear

Create x1 y1 (eq1), x2 y2 (eq2) & show that:

a(eq1) + b(eq2)

x(t)=ax1+bx2 y(t)=ay1 + by2

y1(t) y1(t) + 3y1(t) = x1(t) -------- eq1

y2(t) y2(t) + 3y2(t) = x2(t) -------- eq2

{ ay1 y1 + by2 y2 }+ 3{ ay1 + by2 } = ax1 + b x2{ay1+ ay2} {ay1+ by2} + 3{ ay1 + by2 } = ax1 + b x2

LHS of the last two equations is not equal

system is nonlinear


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2) Linear vs. non-linearVin(t) Vout(t)h(t)

Vs - 2V

Vin

Vout

x1 x2


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3) Time-invariant vs. time varyingc/s doesnt change with time

x(t-) y(t-)h(t)

x(t) y(t)h(t)

x(t)

y(t)

x(t-)

y(t-)


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HW#1: Classification of systems:

- Linear vs. non-linear- Time-invariant vs. time varying


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- Most systems that we will deal with are lineartime invariant (LTI) systems continuous and

discrete

4) Continuous time vs. discrete time

- A system is continuous time if i/p & o/p are

continuous time signals

- A system is discrete time if i/p & o/p arediscrete time signals


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5) Instantaneous vs. dynamic- Instantaneous: depends only on present

values of the i/p memoryless system

Always causal

Example: resistor network

- Dynamic: depends on i/p from a period of

time

Can be causal or non-causalExample: passive LPF

Vin

Vout


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System Representations

1) Differential equations:


)()(

)(

])()([1

)(

)()(

txtydt

tdy

RC

tytxR

ti

dt

tdyCti


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SystemRepresentations

2) Transfer function H(s) Laplace transform:



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3) Frequency response H() - Fourier transform:



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4) Impulse response h(t):



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Singularity Functions

Singularity: a notation used to describediscontinuous functions

1)Impulse function:

- VERY IMPROTANT

- Delta, Dirac delta

- Infinite amplitude with zero width

a)

b) (t-to

)=0 every where tto

5


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1)Impulse function:

c)

d) Sifting/sampling property

Given that f(t) is continuous @ to

)()()( oo tfdttttf


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1)Impulse function:

e)

a/2

a/2


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2) Unit step function: u(t)

3) Unit ramp function: r(t)

1

1


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4) Rect function

rect(t) = u(t+) - u(t-)

-

1

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1_Signals and Systems Classifications