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7/29/2019 Ssp Pt Signal Transmission Through Linear Systems
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Presented by APN Rao, Dept ECE, GRIET, Hyderabad. Sep 2011 1
Signal Transmission ThroughLinear Systems
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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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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
Definitions of System Classification
Presented by APN Rao, Dept ECE, GRIET, Hyderabad. Sep 2011
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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Time Response of LTI Systems
Presented by APN Rao, Dept ECE, GRIET, Hyderabad. Sep 2011
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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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)
Presented by APN Rao, Dept ECE, GRIET, Hyderabad. Sep 2011
t
f(t)
f()
Time-responseof an LTIsystem equalsthe input
convolvedwithimpulseresponse ofthe system
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Interpretation of System Characteristicsfrom Impulse Response
Presented by APN Rao, Dept ECE, GRIET, Hyderabad. Sep 2011
Causality: ( ) 0 0
Time-invariance: ( ) ( )
Memorylessness: ( ) ( )
Stability: ( )
h t t
t h t
h t k t
h t dt
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Step Response
Presented by APN Rao, Dept ECE, GRIET, Hyderabad. Sep 2011
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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Frequency Response of LTI Systems
Presented by APN Rao, Dept ECE, GRIET, Hyderabad. Sep 2011
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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Frequency Domain Response of LTI Systems
Presented by APN Rao, Dept ECE, GRIET, Hyderabad. Sep 2011
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
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Bandwidth - Rise time Relation
Presented by APN Rao, Dept ECE, GRIET, Hyderabad. Sep 2011
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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Conditions for Distortionless Transmission
Presented by APN Rao, Dept ECE, GRIET, Hyderabad. Sep 2011
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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Ideal Filter Characteristics
Presented by APN Rao, Dept ECE, GRIET, Hyderabad. Sep 2011
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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Physical Realizability of Ideal Filters
Presented by APN Rao, Dept ECE, GRIET, Hyderabad. Sep 2011
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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Realizability in Frequency Domain: Paley - Wiener Criterion
Presented by APN Rao, Dept ECE, GRIET, Hyderabad. Sep 2011
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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( )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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