Priiples of Commuiatios Leture 1: Noise i Modulatio Systems Chih-Wei Liu 劉志尉 Natioal Chiao ug Uiversity wliu@twis.ee.tu.edu.tw
Outlies Sigal-to-Noise Ratio Noise ad Phase Errors i Coheret Systems Noise i Agle Modulatio hreshold Pheomeo Noise i Pulse Code Modulatio Commu.-Le1 wliu@twis.ee.tu.edu.tw
Sigal-to-Noise Ratio SNR is oe of the most importat parameters i ommuiatios. (he other is badwidth). N / SNR Sigal power Noise power Chael Model: Additive, White, Gaussia Noise (AWGN). Idepedet of sigal. Compare differet reeivers (modulatios): Same SNR at A, ompare SNR at B. = sigal (otaiig message) at a ertai poit + A B Reeiver + (t) oise message Commu.-Le1 wliu@twis.ee.tu.edu.tw 3
Noise i Basebad System he power spetral desity is. (Some books use N ) Physial meaig: N is the average oise power per uit N badwidth (sigle-sided psd) at the frot ed of the reeiver. Commu.-Le1 wliu@twis.ee.tu.edu.tw 4
Basebad model: a basis for ompariso (behmark) Sigal power = P watts. (trasmitted power, modulated sigal) B 1 P Filter iput oise power = Ndf = NB, ad ( SNR) i =. B NB W 1 P Filter output oise power= Ndf = NW, ( SNR) o =. W NW SNR ehaemet ( SNR) ( SNR) o i = B. (he filter redues part of oise.) W his is a basi operatio i may omm systems We filter out the out-of-bad oise. his filterig does ot hage sigal (power). Commu.-Le1 wliu@twis.ee.tu.edu.tw 5
A. SB-SC System Assume oheret detetio. os(ω t +θ) x () t = Am()os( t ω t+ θ ) + () t r e () t = Am()os( t ω t+ θ) + ()os( t ω t+ θ) ()si( t ω t+ θ) Badpass filterig Badpass oise s Commu.-Le1 wliu@twis.ee.tu.edu.tw 6
1 1 Noise power: () t = () t + s() t = NW. A rasmitted sigal power: m ( t ). Am Pre-detetio SNR: ( SNR) =. (For SB-SC) 4WN Passbad e () t = e () t os( ω t+ θ) 3 = Am(){1 t + os(( ω t+ θ))} + ( t){1 + os( ( ω t+ θ))} ( t)si(( ω t+ θ)). hrough LPF y ( t) = Am( t) + ( t). s Commu.-Le1 wliu@twis.ee.tu.edu.tw 7
= Sigal power: Am. (Sigal power "doubled") Noise power: ( t) NW. (Noise power remais the same.) Am Post-detetio SNR: ( SNR) =. (For SB-SC) NW Δ ( SNR) etetio gai = = ( = 3 db). ( SNR) We assume oheret detetio, i.e., the LO sigal has the same freq ad phase as the arrier. oes this detetio (demodulatio) provide a 3dB gai? Not quite. Compare the equivalet basebad system. Commu.-Le1 wliu@twis.ee.tu.edu.tw 8
What is the equivalet basebad system? P γ = trasmitted power = NW Am = = ( SNR) NW 1 A m Why? he basebad oise is N W (BW=W). he passbad oise is N W (BW=W). he detetio gai aels the oise BW irease. Commu.-Le1 wliu@twis.ee.tu.edu.tw 9
B. SSB System Assume oheret detetio he reeived sigal ( + for the upper; for the lower sidebad): x () t = A[ m()os( t ω t+ θ) ± mˆ ()si( t ω + θ)] + () t r where mˆ : the Hilbert trasform of m( t). -w w f os(ω t +θ) Commu.-Le1 wliu@twis.ee.tu.edu.tw 1
Key: SSB BW = W ot W. Hee, the reeived oise power is redued. Noise ompoet: (eompose ito I ad Q ompoets) t () = ()os( t ω t+ θ) ()si( t ω t+ θ) Let N s BP oise hus, e () t = [ Am() t + ()]os( t ω t+ θ) ± [ Amˆ () t m ()]si( t ω t+ θ) e 3 s = = = N W os( ω t + θ) s oheret LPF y ( t) = Am( t) + ( t) oise term. N / f -w do't are f f Commu.-Le1 wliu@twis.ee.tu.edu.tw 11
Assume that mt ( ) is idepedet of = = = Post-detetio sigal power: S A( m ). Post-detetio oise power: N ( ) NW. Am Post-detetio SNR: SNR = NW What is the pre-detetio SNR? he trasmitted sigal power (t).. (For SSB-SC) S = { A[ m( t)os( ω t+ θ) mˆ ( t)si( ω t+ θ)]} = A {[ m( t)os( ω t+ θ)] + [ mˆ ( t)si( ω t+ θ)] mtmt ( ) ˆ ( )os( ωt+ θ)si( ωt+ θ) ( mt ( ) ad mt ˆ ( ) are orthogoal.) ( os( ω t) ad si( ω t) are orthogoal.) Commu.-Le1 wliu@twis.ee.tu.edu.tw 1
S = A { m ( t) os ( ω t+ θ) + mˆ ( t) si ( ω t+ θ) 1 1 1 1 + os(ωt+ θ) os( ωt+ θ) 1 1 { ( ) ˆ ( ) = A m t + m t } (Notie that m ( t) = mˆ ( t).) = A ( m ( t)) = S! he detetio gai is ( SNR) ( SNR) S N ( W ) N = = 1. hat is, SNR = SNR. S N ( = N W) N / -f -f +w f -w No gai! No loss ( Q N = N )! (I SB, N = N ) f f Commu.-Le1 wliu@twis.ee.tu.edu.tw 13
C. AM System -- Coheret Assume oheret detetio Reeived sigal x ( t) = A [1 + am ( t)]os( ω t + θ). a: modulatio idex x r (t) = x (t) + (t) m : ormalized message arrier BPF M (f) e (x) X os(ω t + θ) e 3 (x) LPF -W W y () t = Aam () t + () t + A. ( A is removed.) Commu.-Le1 wliu@twis.ee.tu.edu.tw 14
A is removed. Q1) assume mt ( ) =, we a thus remove d term; ) I reality, we aot reover d term of mt ( ) ayway. hus we simply remove it. Post-detetio sigal power: S = ( Aam ( t)) = Aa m Post-detetio oise power: N = = NW. Aa m Post-detetio SNR: ( SNR) =. (For AM) NW. N ( W ) = N W Commu.-Le1 wliu@twis.ee.tu.edu.tw 15
Now, let's alulate the pre-detetio SNR. he x sigal power: S = { A[1 + am ( t)]os( ω t+ θ)} = A os ( ω t+ θ) + A a m ( t) os ( ω t+ θ). ( m( t) = m, os ( ωt+ θ) = + os( ωt+ θ).) 1 1 P = S = A + Aa m. 1 1 A + Aa m Badpass oise power N = NW. ( SNR) =. NW ( SNR) A a m a m he detetio gai = = = < 1. 1 ( SNR) ( A ) 1 + A am a m + am Reall, (power) effiiey E ff =. 1+ am ( SNR) ( SNR) (.5).1 : If.1 ad.5. ff.44. Ex m = a = E = = 1 + (.5).1 etetio gai =.488! It is quite low! Commu.-Le1 wliu@twis.ee.tu.edu.tw 16 = E. ff
. AM System: Evelope etetio Evelope detetor ioheret he reeived sigal e () t = x () t + () t x ():modulated t sigal ( t) : arrow-bad oise = A[1 + am ( t)]os( ω t+ θ) + ( t)os( ω t+ θ) ( t)si( ω t+ θ). s e () t = r()os[ t ω t+ θ + φ()]. t Noise: ( ) = ( s) = NW. rt = A + am t + t + t 1 s() t φ() t = ta. A[1 + am( t)] + ( t) () { [1 ()] ()} s(). Commu.-Le1 wliu@twis.ee.tu.edu.tw 17
y ( t) = r( t) amplitude (evelope) ' remove d (iludig the arrier) y () t = rt () rt () rt ():average Case 1: Whe ( SNR) is large (i.e., small oise) A[1 + am ( t)] + ( t) >> ( t) s rt () A[1+ am( t)] + ( t) most of the time y () t Aam () t + () t zero mea same as oheret detetio Commu.-Le1 wliu@twis.ee.tu.edu.tw 18
Case : Whe ( SNR) is small he badpass oise: r ( t)os( ω t + φ ( t)). he eevelope detetor iput ( θ is ot importat): e () t = A [1 + am ()]os( t ω t + θ) + r ()os( t ω t + φ () t + θ) = rt ( )os( ω t+ ψ + φ ( t)). Assume that A [1 + am ( t)] << r ( t) for most of time. rt ( ) = [ r( t) + A[1 + am( t)]os φ ( t)] + [ A(1 + am( t)]si φ ( t)] r () t + A [1 + am ()]os t φ () t remove "d" y () t r () t + A [1 + am ()]os t φ () t rt () Radom atteuatio; the message is lost! Commu.-Le1 wliu@twis.ee.tu.edu.tw 19
AM Noise isussios (t) = s (t) r (t) f R (r ) where (t) ad s (t) are all idepedet zero mea Gaussia (t) r r r ( t) = Now, r( t) ~ r ( t) + s ( t) + ( t) = Rayleigh - distr. A [1 + am ( t)]osφ ( t) message is lost! osφ ( t) : radom What we try to show here is that whe oise > (message) sigal, oise beomes the domiate output ompoet. I the output sigal, o message sigal is proportioal to the message sigal. (f. Coheret detetio: ) Commu.-Le1 wliu@twis.ee.tu.edu.tw
hreshold Effet his is the threshold effet. (Reall: he aalysis i iterferee.) ( he evelope detetor is oliear, the message is lost whe SNR< threshold ) Remark: It s diffiult to alulate the exat (SNR). ef. of threshold: A value of the arrier-to-oise ratio (or SNR) below whih the oise performae of a detetor deteriorates muh more rapidly tha proportioately to the arrier-to-oise ratio (or SNR). (Hayki&Moher, Comm Systems, p.15, 1) Commu.-Le1 wliu@twis.ee.tu.edu.tw 1
E. AM System: Square-Law etetio A example of simple oliear detetor that we a alulate ad thus the threshold regio a be more preisely determied. e () t = x + () t ():badpass t oise ' ' 3 = A [1 + am ( t)]os( ω t) + ( t)os( ω t) ( t)si( ω t). e () t = e () t s Commu.-Le1 wliu@twis.ee.tu.edu.tw
e () t = e () t = { A[1 + am ()] t + ()}os( t ω t) 3 A am t t t ω t ω t t ω t { [1 + ( )] + ( )} s( )os( ) si( ) + s( )si ( ) 1 1 = { A[1 + am( t)] + ( t)} [ + os( ωt)] LPF 1 1 { A[1 + am( t)] + ( t)} s()si( t ωt) + s()[ t os( ωt)]. (si( ω t), os( ω t): high freq. ompoets) y () t { A[1 + am ()] t + ()} t + () t ' s = r ( t) ( = evelope ) = A + Aam( t) + Aam( t) + A( t) + Aam( t ) ( t) + () t + (). t s y t y t m t = t = t = ' remove d () (), (), () s() Commu.-Le1 wliu@twis.ee.tu.edu.tw 3
y t = Aam t + Aa m t m t + A t + () () [ () ()] () message ( ) ( ) ( ) ( ) s( ) s( ). 4 Post-detetio sigal power: 4 ( ). Post-detetio oise power: N = A 4 A am t t + t t + t t S = A a m a [( m ( t) m ) ] + 4 A [1 + am ( t)] + 4 4 4 ( ) ( ) + ( s) ( s). 4 4 ( x x ) = ( x ) x x + ( x ) = ( x ) ( x ) ( Assume the ross terms are either zero or a be egleted) = 4 Note : ( ) 3 ( ), if is Gaussia. his a be show usig the momet geerator. N = A a [( m m ) ] + 4 A (1 + a m ) σ + σ + σ. 4 4 4 4 4 S 4Aa m ( SNR) = =. N Aa ( m m ) + 4 A (1 + a m ) σ + 4σ 4 4 4 Commu.-Le1 wliu@twis.ee.tu.edu.tw 4
Example: message = siusoidal Let m ( t) = os( ω t). ot a radom sigal 1 1 1 ( m( t) m) = [os ( ωmt) ] = [ os( ωmt)] =. 8 Assume this part i N a be egleted. ( Will be disussed.) 4 he, N = 4 A(1 + a ) + 4. m 1 σ ' Note: If we examie y( t) ad y( t) σ y ( t) = A + A aos( ω t) + A a [os ( ω t)] ' m m + ω + + remove d + A[1 aos( mt)] () t () t s() t 1 1 ( os ωmt = + os ωmt) 1 y( t) = Aaos( ωmt) + Aa os( ωmt) message A + a ω t t + t t + t t + [1 os( m )] ( ) ( ) ( ) s( ) s( ) he term ( )os( ω t) is alled harmoi distortio.--ot radom oise. 1 1 Its power: = A a ( = a S ) 8 16 m 4 4 Commu.-Le1 wliu@twis.ee.tu.edu.tw 5
Example (oti.) 1 = + + = Aa 4 4 Now, N 4 A(1 a ) σ 4 σ. Ad S Aa. 4 Aa σ 4 + a σ + σ σ ( SNR) = =. A ( ) 4 ( + a ) + ( ) A hreshold effet illustratio: 1 1 1 he total x power is P = { A[1 + am( t)] os( wt)} = A(1 + a m) = A (1 + a ). P a NW P ( SNR) ( ). (Basebad systems: ( ) = SNR.) = + a 1 + ( N W / P ) NW Case 1: If 1, ( SNR) ( ) E ( Coheret) NW + a NW NW Case : If 1, ( SNR) ( ) ( ). ( ( ) NW + a NW P a P P P a P P NW ( 1 ) Q NW NW + P P Commu.-Le1 wliu@twis.ee.tu.edu.tw 6 )
Example (oti.) For a 1 st approximatio, the performae of liear evelope detetor ~ square-law detetor For high SNR ad a=1, the performae of evelope detetor is better by ~1.8 db. Commu.-Le1 wliu@twis.ee.tu.edu.tw 7
isussios ( m m ) a 4 Geeral form: (assume is small) ( SNR) ( SNR) 4 4 + a m σ + σ 1 P = A(1 + a m) ( = { A[1 + am( t)] os( ωt)} ) am E = 1 + 4Aa m = 4 A (1 ) 4 am 4 4Aa m 1 (1 ) A E A E + am 4 1 4 1 + a m σ + σ 1 4 σ + σ σ + σ (1 + am) 4 P = = = 4 A (1 ) 4 1 A E = P = N W N W am P P a m NW + (1 + ) NW 1+ Commu.-Le1 wliu@twis.ee.tu.edu.twp 8 P
isussios (oti.) Speial ases: If P >> N W, ( SNR) a (1 + a m m ) P N W If P << N W, ( SNR) ( ) (1 + a m ) N W a m P Example: siusoidal message (previous example) = m 1 Commu.-Le1 wliu@twis.ee.tu.edu.tw 9
Summary Commu.-Le1 wliu@twis.ee.tu.edu.tw 3