error performance

7/23/2019 Error Performance

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BENG4 DIGITAL COMMUNICATION

Performance of digital communications sstems !"en distur#ed # Additi$e %"ite Gaussian Noise&A%GN'

Bit error rate

A measure of t"e transmission (ualit of a sstem is t"e num#er of #its recei$ed in error as a

)ro)ortion of t"e total num#er of #its transmitted* T"is is normall measured o$er a s)ecified )eriod

of time+ for e,am)le+ a - "our &routine test'+ or .4 "our test !"ic" ma #e done for ne! installations*

received bitsof number Total

tserrored bi Number of BER

=

/ince digital sstems ma carr information at different #it rates+ t"is )ro$ides a fair measure of

com)arison #et!een sstems*

E,am)le0 If a sstem "as t!o errored #its in 122 , -23 recei$ed+ t"en t"e BE is 4 , -256*

T"is is a measured (uantit* 7o! do !e 8no! !"et"er our measurement is !it"in t"e design

)arameters of t"e sstem under test9

T"e t"eoretical e,)ectation of t"e BE for a gi$en sstem+ 8no!n as t"e )ro#a#ilit of error+ P(e)+ isa function of t"e carrier5to5noise )o!er ratio of t"e sstem+ "o!e$er since t"e num#er of encoding

conditions of t"e carrier &#its:7;' $aries de)ending on t"e t)e of modulation used+ a more s)ecific

ratio is t"e energy per bit-to-noise power density ratio+ denoted2

N

E b*

T"ere are t!o )rimar causes for error )erformance degradation in digital transmission sstems* T"e

first is t"e effect of #and!idt" limitations of t"e transmitter+ transmission c"annel and t"e recei$er!"ic" causes sm#ol <smearing= or inter5sm#ol interference*

Inter-symbol interference (ISI)

T"e transmission c"annel is al!as #and5limited* A #and5limited c"annel s)reads a )ulse !a$eform

)assing t"roug" it &e,am)le s"o!n #elo!'*

Pulse spreading caused by limited bandwidth*

%"en t"e c"annel #and!idt" is muc" greater t"an t"e #and!idt" of t"e )ulse+ t"e s)reading of t"e

)ulse !a$eform !ill #e slig"t* %"en t"e c"annel #and!idt" is close to t"e signal #and!idt"+ t"e

s)reading !ill e,ceed t"e sm#ol duration &T' and cause t"e signal )ulses to o$erla)* T"iso$erla))ing is called inter-symbol interference &I/I'* Li8e an ot"er source of interference+ I/I causes

degradation in t"e sstem )erformance &i*e* "ig"er error rates'* It is a )articularl insidious t)e of

degradation #ecause increasing t"e signal )o!er does not"ing to reduce t"e errors caused # it*

I/I ma #e reduced t"roug" t"e use of )ulse s"a)ing tec"ni(ues* N(uist in$estigated t"e )ro#lem of

s)ecifing a recei$ed )ulse s"a)e so t"at no I/I occurs at t"e detector in t"e recei$er* 7e s"o!ed t"at

t"e t"eoretical minimum sstem #and!idt" needed in order to detect sm#ols:sec+ !it"out I/I+ is:. 7;> in ot"er !ords a transmission sstem !it" #and!idt" 7; can su))ort a transmission rate

of ! sm#ols:second &N(uist #and!idt" constraint' !it"out I/I*

-





Bandwidth efficiency

A fundamental )arameter for digital communication sstems is bandwidth efficiency+ R" + !"ose

units are bits"sec"#$ * As t"e units im)l+ R" re)resents a measure of data t"roug")ut )er 7; of

c"annel #and!idt"+ and t"us measures "o! efficientl an signaling tec"ni(ue &#ase#and or

#and)ass' utili;es t"e #and!idt" resource*

/ince t"e N(uist #and!idt" constraint dictates t"at t"e t"eoretical ma,imum sm#ol rate !it"out I/I

is . sm#ols:sec:7;+ !e mig"t #e led to as8 !"at it sas a#out t"e ma,imum num#er of #its:sec:7;*T"e constraint deals onl !it" sm#ols and t"e a#ilit to detect t"eir $alues !it"out distortion from

ot"er )ulses* To find t"e R" for an modulation sc"eme+ one must 8no! "o! man #its eac"

sm#ol re)resents* Consider t"en a -6 PAM modulation sc"eme> eac" sm#ol is re)resented # oneof -6 )ossi#le am)litudes !"ic" com)rises 4 #its:sm#ol* T"us t"e t"eoretical ma,imum #and!idt"

efficienc for -6 ? ar PAM is @ #its:sec:7;* Note t"is is for #ase#and modulation

Energy per bit to Noise Power Density ratio

T"e second t)e of degradation is as a result of noise introduced # a $ariet of sources* %it" )ro)er )recautions muc" of t"e noise entering t"e recei$er can #e reduced or e$en eliminated+ "o!e$er t"e

one t)e of noise t"at cannot #e eliminated is thermal noise* It is also t"e most easil )redicted t)e

of noise and "ence t"e general assum)tion can #e made t"at t"is is t"e main noise affecting t"esstem*

T"e )rimar s)ectral c"aracteristic of t"ermal noise is t"at its )o!er s)ectral densit is t"e same forall fre(uencies of interest in communication sstems+ i*e* from d*c* to a#out -2-. 7;* T"erefore !e

ma assume t"at a t"ermal noise source emits an e(ual amount of noise )er unit #and!idt" &- 7;'*

%"en t"e noise )o!er "as a uniform s)ectral densit !e refer to it as white noise &in t"e same senseas !"ite lig"t'* /ince t"ermal noise is a )urel random )rocess+ its sam)les are uncorrelated and it is

modeled as a Gaussian )rocess* inall since t"ermal noise is su)erim)osed+ or added+ to t"e signal it

is referred to as %dditive hite &aussian Noise+ or A%GN for s"ort*

ero5mean Gaussian noise is c"aracteri;ed # its variance+ "ence t"is noise model is )articularl

sim)le to use for calculations in a digital communication sstem*

Gi$en0

'TB N = !atts+ !"ere T am#ient tem)erature in el$in+ B /ignal #and!idt" in 7ert;+ and 8

Bolt;mans constant &-*3@ , -25.3

5-

'

T"e noise )o!er densit+ N + is t"e t"ermal noise )o!er normali;ed to a - 7; #and!idt"+ i*e*

'T B

'TB N ==

2 !atts:7;

T"e #it energ+ E #+ is sim)l t"e energ contained in one #it of information*

.





bcarrier b T P E ×= !"ere Pcarrier carrier )o!er &%atts'

T # #it duration &seconds'

bb RT

-

= !"ere # #it rate in #its:sec

T"usb

b R

) E = Foules !"ere C is t"e carrier )o!er

N

*R

N

*T

N

E bbb

==

2

To em)"asi;e> E #:N2 is Fust a $ersion of t"e /ignal5to5Noise ratio normali;ed # #and!idt" and #itrate*

= R

N

*

N

E b

2

E,am)le0 or a P/ sstem !it" t"e )arameters gi$en #elo!+ calculate t"e noise )o!er densit in

dBm+ energ )er #it in dB and E #:N2

C -25-3 %

# 328#)sBand!idt" 6287;

Noise+ N 6*2 , -25-H %

Error performance for binary systems

As mentioned )re$iousl+ !e see8 to estimate t"e actual error )erformance of a digital

communication sstem #ased on t"e signal to noise ratio at t"e recei$er*

Probability of error, P(e)

%e s"all no! consider t"e detection of digital sm#ols in t"e )resence of A%GN* T"e )ro#a#ilit oft"e detector ma8ing an incorrect decision is termed t"e )ro#a#ilit of symbol error (P e )* or #inar

modulation t"is !ill #e t"e same as t"e )ro#a#ilit of bit error (P b )+

(,) oherently detected BP*

3





=

2

.

N

E . P b

b

(!) oherently detected Binary /*

=

2 N

E . P b

b

(0) %* and non coherently detected /*

= $ erfc P e

.

-

.

-

p B N

% $

2

.

=

(1) or ot"er formulae see t"e general "andout

(2) Note t"at t"ere are t!o forms of t"e com)lementar error function ta#les+ &,' and erfc&,'related as follo!s0

( )

=

..

- 3erfc 3.

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error performance

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