ece 4710: lecture #5 1 linear systems linear system input signal x(t) output signal y(t) h(t) h( f...

ECE 4710: Lecture #5 1

Linear Systems

Linear System

Input Signal

x(t)

Output Signal

y(t)h(t) H( f )

)(

)(

)(

f

R

fX

x

x

P

)(

)(

)(

f

R

fY

y

y

P

Voltage Spectrum (via

FT)

AutoCorrelation Function

Power Spectral Density

Useful Signal Characterizations


Linear Systems

Linear Time Invariant (LTI) Conditions: Linear Superposition holds Time Invariant Shape of system response, H( f ), is

same no matter when input is applied to system» Does not apply for most mobile (wireless)

communication channels Impulse Response = h(t)

h(t) = 0 for t < 0 causal y(t) = h(t) when x(t) = (t) delta impulse function at input

can measure system response


System Output

Output is result of convolution integral between input and impulse response function

Convolution integral is difficult to evaluate Preferred approach is to find system transfer

function H( f ) Convolution in time is multiplication in frequency

)()()()()( thtxdthxty

)()()()()()( fHfXfYthtxty

)(

)()( that so

fX

fYfH


System Transfer Function

H( f ) is the FT of h(t) and is called the frequency response or the transfer function of the system

In general H( f ) is a complex function with magnitude and phase response:

Magnitude response is even function in frequency Positive and negative frequencies have same amplitude

Phase response is an odd function in frequency

)()( fHth

)(exp|)(|)( fj HfHfH


System Transfer Function

H( f ) can be measured by using sinusoidal test input signal and sweeping the frequency over the desired range Spectrum analyzer uses this approach

How is power content of input signal affected by the system?

??)( iswhat )(and)(Given ffHf yx PP

)(|)(|)( 2 ffHf xy PP


Power Transfer

Power Transfer Function

Example: RC Low Pass Filter (LPF) Find Gh( f )

2|)(|)(

)()( fH

f

ffG

x

yh

P

P

+

x(t)

+

y(t)

RC

i(t)


RC LPF

KVL around loop: Capacitor current related to voltage drop:

Table 2-1, pg. 52 :

Take FT of both sides:

Solving for transfer function:

)()()( tytiRtx

)()(

)()()(

)()( tydt

tdyRCtx

dt

tdyC

dt

tdvCtiti C

)()2()(

fWfjdt

twd nn

n

])2(1[)(

)()()2()(

RCfjfY

fYfYfjRCfX

fRCjfX

fYfH

)2(1

1

)(

)()(


RC LPF

Table 2-2, pg. 64:

So

where = RC is the time constant of the LPF

Tfj

T

t

te Tt

2100

0/

00

01

)()2(1

1)(

/

t

tethfRCj

fHt

1

RC

t

e-1

RC

= RC

h(t) = Impulse Response Function of ILPF


RC LPF

Power Transfer Function:

Define fo = cutoff frequency = 1 / 2RC so

At f = fo Gh( fo ) = 0.5 Power @ fo attenuated by half half power or 3 dB BW

)()2(1

1

)2(1

1

)2(1

1 |)(|

so )( )( 52 pageon 1-2 Table from

and )()(|)(|)(

22

*

*2

fGRCffRCjfRCj

fH

fHfH

fHfHfHfG

h

h

2)/(1

1)(

oh ff

fG


Distortionless Transmission

Distortionless channel is very desirable in a communication system Output is simply delayed replica of input : y(t) = A x (t - Td)

where A : channel loss (A < 1)

Td : time delay in channel

In frequency domain a distortion free response is

Thus, a distortion free channel has

dTfjefXAfY 2)()(

)(2

)(

)()( fjTfj eAeA

fX

fYfH d

Note that there is no frequency

dependence for amplitude but there

is for phase


Distortionless Transmission

LTI system will have no distortion if

1) Amplitude response is flat 2) Phase response is linear function of

frequency Distortion classified as either 1) amplitude or 2)

phase distortion Looking at phase distortion from time delay

standpoint: Time delay of channel/system must be independent of

frequency, otherwise phase distortion will occur

AfH |)(|

dTff 2)(

)( alueconstant v fTT dd


RC Filter Distortion

Filter Transfer Function: Amplitude response:

Phase Response:

Time Delay:

Not a constant value

fRCjfH

)2(1

1)(

)/(1

1|)(| so

)/(1

1|)(|)(

222

ooh

fffH

fffHfG

)/(tan

)(Re

)(Imtan)( 11

offfH

fHf

)/(tan2

1)(

2

1)( 1

od fff

ff

fT


For f < 0.5 fo the amplitude distortion is < 0.5 dB (~12%)

RC Amplitude Distortion


For f < 0.5 fo the phase distortion is < 2.1 (~8%)

RC Phase Distortion


High frequency signal components have less delay

For fo = 1 kHz delay is ~0.2 msec

RC Time Delay


Distortion

Most communication systems have both amplitude and phase distortion of H( f )

Distortion can be acceptable depending on Distortion type amplitude, phase, or both Distortion magnitude e.g. how bad? Type of information signal audio, video, or data


Audio Distortion

Human ear is sensitive to amplitude distortion in f Audio spectrum is 300 Hz to 15 kHz

15 phase distortion causes time delay of ~ 3 sec Duration of spoken syllable is 10-100 msec 3 sec delay is imperceptible (< 0.05 % of 10 msec syllable)

3 dB amplitude error is very noticeable to ear High-fidelity audio amplifiers

Focus on maintaining flat spectral response Phase distortion is not concern


Video Distortion

For analog video transmission the phase distortion is the primary concern

Amplitude variations will cause variations in image intensity

Phase variations will cause time delays which will cause objects in the image to blur at edges

Human eye is more sensitive to phase variations Analog video filters require excellent phase

linearity


Data Distortion

For digital data both amplitude and phase distortion can have serious affects

Rectangular data pulse train:

Pulse smearing into adjacent symbol time slots Inter-Symbol Interference (ISI) Increase probability that bit errors will occur (BER )

Special filters designed for digital data to minimize impact of ISI

0 1 0 1 0

Ts

BandlimitedCommunication

System or Channel

0 1 0 1 0

ece 4710: lecture #5 1 linear systems linear system input signal x(t) output signal y(t) h(t) h( f...

Documents