Signal Transmission Through Linear Systems

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

Signal Transmission Through

Linear Systems

What is a System?

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

A collection of components interconnected in such a way as to

perform some specific function

Has a physical or abstract

boundary, separating it from the

external world. Inputs cross

boundary inward, and outputs

cross boundary outwards.

System operates on an input,

and produces an output or

response. Input-output relations

depend on the system

characteristics.

Systeminput output

Our interest is to mathematically describe the processing of signal

waveforms by communication systems.

3

Classification of Systems

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

Linearity

Linear/ non-linear

Time-variance

Time-invariant/

time-variant

Finite propagation time

Lumped/

distributed

Continuity of processing

Analog/ discrete/

digital

Causality

Causal/ non-causal

Stability

Stable/ unstable

Memory

Static/ dynamic

4

Definitions of System Classification


Linearity: A linear combination of several inputs produces the same linear

combination of the corresponding outputs.

Time-invariance: A system is time-invariant (or stationary) if the response is

independent 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 causal

by nature and can not produce noncausal responses.

Memory: A system has memory if it can store energy and has response

dependent both on present and past inputs. A memoryless system’s response

at 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 bounded

output; that is, no bounded input causes a response which is unbounded. This

is said to be in bounded-input bounded-output (BIBO) sense. In an unstable

system, the response may become independent of input.

5

Time Response of LTI Systems


LTI SystemInput 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 of

response.

Natural Response: is that part of the output which is only due to the system

characteristics. It is responsible for the transient portion of the response.

Impulse Response h(t): System response for unit impulse input. h(t) represents the

natural response for t>0 and specifies the manner in which the system returns to

initial state after a momentary disturbance. h(t) completely characterizes a LTI

system in time-domain.

LTI Systemδ(t) h(t)

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(τ)

τΔτ

Time-response

of an LTI

system equals

the input

convolved with impulse

response of

the system

7

Interpretation of System Characteristics

from Impulse Response


Causality: ( ) 0 0

Time-invariance: ( ) ( )

Memorylessness: ( ) ( )

Stability: ( )

h t t

t h t

h t k t

h t dt

8

Step Response


t0 0 t

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

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

11/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 ( ), if maximum rate of rise is maintained.

Faster systems have smaller rise t e

im s.

rt

s

Example

tr

Max rate of rise = s’(t)max = s’(0) = 1; therefore rise time tr = 1/a

Steady state output

9

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

its j 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

10

Frequency Domain Response of LTI Systems


1The synthesis equation ( ) ( ) ( ) can b e viewed as a continuous

2

sum of weighted complex exponentials. Th e 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 frequencydomain. If maximum gain is at zero frequency, system is a lowpass filter (LPF) ; and ifat a non-zero frequency it is a bandpass filter (BPF).

Half-power (or 3 dB) bandwidth for low pass filter is |0-ωc|whereFor a bandpass filter with maximum gain at ωc, it is |ω1 – ω2| where ω1 <ω< ω2 and

( ) (0) / 2cH H

1 2( ) ( ) ( ) / 2cH H H

11

Bandwidth-Rise time Relation


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

impulse response and consequently wider bandwidth. Thus system bandwidth is

inversely related to its rise time; alternatively stated as Bandwidth x Rise time

= Constant

Signal Distortion

Distortion is the change in signal wave shape while passing through a system.

Uniform gain and pure time delay undergone by a signal do not amount to

distortion.

12

Conditions for Distortionless Transmission


0

0

For a distortionless system,

( ) ( - ) ( ) F

j th t k t t H k e

0 t0

k

t

h(t)

0

k

ω

|H(ω)|

- ωt0H(ω)=

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, amplitude distortion

is said to be caused)

phase shift should vary linearly withfrequency (if not, phase distortion is

said to be caused)

13

Ideal Filter Characteristics


An ideal filter provides distortionless transmission over some

finite bandwidth.

0

k

ω

|H(ω)|

- ωt0

H(ω)

0

k

ω

|H(ω)|

0 ω

- ωt0

H(ω)

ω

Ideal LPF characteristic Ideal BPF characteristic

14

Realizability of Ideal Filters


0

k

ω

|H(ω)|

0 ω

- ωt0

H(ω)

Ideal LPF characteristic

t00

W- W

kW/π

t

h(t)F

h(t)=(kW/π) Sa[W(t-t0)]

Impulse Response of Ideal LPF

Physical devices are causal by nature, and therefore filters with noncausal impulse

responses are not realizable. Observe that ideal lowpass filter has a noncausal

impulse response, and hence is not realizable. Similarly, ideal bandpass filter is also

not realizable. Practical designs can be approximated by ignoring noncausal portions

of the actual impulse response.

15

Realizability in Frequency Domain: Paley - Wiener Criterion


Paley-Wiener Theorem: A necessary and sufficient condition for a filter

gain function |H(ω)| 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-Wiener

criterion is not just a practical difficulty, but is a theoretical impossibility.

2

( )H e

( )fP 2

( ) ( )fH P

2( )

( )2

f

FS

2 2

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

2 2r f

H F FS H H S

16

Effect of System on Energy Spectral Density


LTI System

H(ω)

Energy spectral density of an input energy signal is scaled by the squared gain

of the system.

Effect of System on Power Spectral Density

LTI System

H(ω)

Power spectral density of an input power signal is scaled by the squared gain

of the system.

Documents

Signal Transmission Through Linear Systems