ssp pt signal transmission through linear systems

7/29/2019 Ssp Pt Signal Transmission Through Linear Systems

1/16

Presented by APN Rao, Dept ECE, GRIET, Hyderabad. Sep 2011 1

Signal Transmission ThroughLinear Systems

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2/16

What is a System?

2Presented by APN Rao, Dept ECE, GRIET, Hyderabad. Sep 2011

A collection of components interconnected in such a way as toperform some specific function

Has a physical or abstractboundary, separating it from the

external world. Inputs crossboundary inward, and outputscross boundary outwards.

System operates on an input,and produces an output or

response. Input-output relationsdepend on the systemcharacteristics.

Systeminputoutput

Our interest is to mathematically describe the processing of signalwaveforms by communication systems.

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3/16

3

Classification of Systems

Presented by APN Rao, Dept ECE, GRIET, Hyderabad. Sep 2011

Linear/ non-linear Time-invariant/time-variant

Lumped/distributed

Analog/ discrete/digital

Causal/ non-causal

Stable/ unstable

Static/ dynamic

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4/16

4

Definitions of System Classification


Linearity: A linear combination of inputs produces the same linearcombination of the inputs.

Time-invariance: A system is time-invariant (or stationary) if the response isindependent of the time of application of input.

Causality: A system is causal if output does not precede the input. That is,output at an instant is not due to future inputs. All physical systems are causalby nature and can not produce noncausal responses.

Memory: A system has memory if it can store energy and has responsedependent both on present and past inputs. A memoryless systems responseat any instant is only due to input at that instant. System with memory is said

to be dynamic, and without memory static.

Stability: A system is stable if any bounded input causes only a boundedoutput; that is, no bounded input causes a response which is unbounded. Thisis said to be in bounded-input bounded-output (BIBO) sense. In an unstablesystem, the response may become independent of input.

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5/16

5

Time Response of LTI Systems


LTI System

Input f(t)

(Excitation,Driving function,Forcing function)

Output

(Response)

r(t)

Total response = Natural Response + Forced Response

Forced Response: is that part of the response which is only due to the input. It is

absent when input is removed. It is responsible for the steady state portion ofresponse.

Natural Response: is that part of the output which is only due to the systemcharacteristics. It is responsible for the transientportion of the response.

Impulse Response h(t): System response for unit impulse input. h(t) representssystems natural response for t>0 and specifies the manner in which the systemreturns to initial state after a momentary disturbance. h(t) completely characterizesa LTI system in time-domain.

LTI System(t) h(t)

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6/16

Let f(t) be input to an LTI system with impulse response ( ).

( ) ( ) for

For very small, f(t) can be approximated by a

train of impulses such that ( ) ( ) ( )f t

h t

f t f t

f t

0

Input Output

( ) ( )

( ) ( )

( ) ( ) ( ) ( )

lim ( ) (

t h t

t h t

f t f h t

f

0) lim ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

t f h t

f t d f h t d

f t t f t f t h t

6

Time Response of LTI Systems to arbitrary input f(t)


t

f(t)

f()

Time-responseof an LTIsystem equalsthe input

convolvedwithimpulseresponse ofthe system

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7/167

Interpretation of System Characteristicsfrom Impulse Response


Causality: ( ) 0 0

Time-invariance: ( ) ( )

Memorylessness: ( ) ( )

Stability: ( )

h t t

t h t

h t k t

h t dt

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8/168

Step Response


t0 0 t

h(t)=e-at u(t), a>0

s(t)=(1-e-at )u(t)/a, a>0

1 1/a

( ) output for a unit step input, is a practically

obtainable system characteristic,

( ) ( ) ( ) ( ) h(t) = '( )

Speed of respon

unlike impulse response.

; and

se:

t

Step response

s t u t h t h d s t

Rise Ti

t

m

s

e

may be defined as the time

taken by system step-response from zero initial state to reach

steady state value, or ( ), if the (initial) maximum rate of rise

is maintained. Faster systems have smalle

rT

s

r rise times.

Example

Tr

Max rate of rise = s(t)max

= s(0) = 1; therefore rise time Tr= 1/a

Steady state output

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9/169

Frequency Response of LTI Systems


LTI System

h(t) j tf t e

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

( ) ( ) completely characterizes a LTI system in frequency domain,

and is called , or

j t j t j t j j t

Transfer

r t e h t h e d e h e d e

Function Filter Charact

H

eristic Fr

H

e

h t

F

of the system.

( ) is complex-valued; its magnitude is the frequency dependent and

its angle is the frequency dependent given to the input signal.

retains

itsj t

quency Response

gain

phase shift

H

e

form when passing through LTI systems; is hence an

. For the same reason, sinusoidal signals also are eigen functions; their

waveshape is retained except for a change in amplitude and pha

eigen

function

se shift. This

enables experimental measurement of frequency response.

( ) ( )j tr t e H

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10/1610

Frequency Domain Response of LTI Systems


1The synthesis equation ( ) ( ) ( ) can be viewed as a continuous

2

sum of weighted complex exponentials. The corresponding output of an LTI system then

1is given by ( ) ( ) [ ( )

2

j tf t F e d f t

r t F H

1] [ ( ) ( )] .

2

Thus, ; and the same can also be deduced from con( ) ( ) ( ) volution property of

Fourier Transform.

j t j t

R

e d F e

F

H

H

d

System function as a Filter in frequency domain

Gain variation with frequency shows how system acts as a filter in frequency

domain. If maximum gain is at zero frequency, system is a lowpass filter(LPF) ; and if

at a non-zero frequency it is a bandpass filter(BPF).

Half-power (or 3 dB) bandwidth for low pass filter is |0-c|where

For a bandpass filter with maximum gain at c, it is |1 2| where 1


11/1611

Bandwidth - Rise time Relation


System with smaller rise time has faster transient response, narrower impulse

response and consequently wider bandwidth. Thus system bandwidth isinversely related to its rise time; alternatively stated as Bandwidth x Rise time= Constantfor a given system

Signal Distortion

Distortionis the change in signal waveshape while passing through a system.Uniform gain and pure time delay undergone by a signal do not amount todistortion.

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12/1612

Conditions for Distortionless Transmission


0

0

For a system,

( ) ( - ) ( )j t

distortio

h t k

nl

t t H k e

ess

F

0 t0

k

t

h(t)

0

k

|H()|

- t0H()=

In time domain,h(t) should be an impulse, may

be weighted and time shifted

In frequency domain,gain should be independent of

frequency (if not, signal undergoesamplitude distortion)

phase shift should vary linearly withfrequency (if not, signal undergoesphase distortion)

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13/1613

Ideal Filter Characteristics


An ideal filter provides distortionless transmission over some

finite bandwidth.

0

k

|H()|

- t0

H()

0

k

|H()|

0

- t0

H()

Ideal LPF characteristic Ideal BPF characteristic

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14/1614

Physical Realizability of Ideal Filters


0

k

|H()|

0

- t0

H()

Ideal LPF characteristic

t00

W- W

kW/

t

h(t)F

h(t)=(kW/) Sa[W(t-t0)]

Impulse Response of Ideal LPF

Physical devices must be causal, and therefore filters with noncausal impulseresponses are not realizable. Observe that ideal lowpass filter has a noncausal

impulse response, and hence is not realizable. Similarly, ideal bandpass filter is alsonot realizable. Practical designs can be approximated by ignoring noncausal portionsof the actual impulse response.

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15/1615

Realizability in Frequency Domain: Paley - Wiener Criterion


Paley-Wiener Theorem: A necessary and sufficient condition for a filtergain function |H()| to be realizable is that

2

ln ( )

1

Hd

A realizable gain characteristic can not have too great a total attenuation.

eg. is not realizable.

A realizable filter characteristic may have zero gain for discrete set offrequencies, but cannot have zero gain over a band of frequencies. eg. An

ideal filter characteristic is not realizable.

Non-realizability of filter characteristics which do not meet Paley-Wienercriterion is not just a practical difficulty, but is a theoretical impossibility.

2

( )H e

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16/16

( )fP 2

( ) ( )fH P

2( )

( )2f

FS

2 2

2 2( ) ( ) ( )( ) ( ) ( ) ( )

2 2r f

H F FS H H S

16

Effect of System on Energy Spectral Density

Presented by APN Rao Dept ECE GRIET Hyderabad Sep 2011

LTI SystemH()

Energy spectral density of an input energy signal is scaled by the squared gainof the system.

Effect of System on Power Spectral Density

LTI SystemH()

Power spectral density of an input power signal is scaled by the squared gainof the system.

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ssp pt signal transmission through linear systems

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