# Band-pass and equivalent low-pass signal modeling

Band-pass and equivalent low-pass signal modeling

Contents• Background, motivation and goals• Complex baseband representation of signals• Complex representation of noise• Channel models• Examples of modulated signals• Signal-to-noise considerationsIf timeallows© NOKIA FILENAME.PPT / 13.12.1997 / NN page: 2

Background, motivation and goals(cont’d)• Direct application of sampling theorem• ⇒ required sampling frequency f s> 2 ⋅ f m(??)0f cf mfncs1.drw© NOKIA FILENAME.PPT / 13.12.1997 / NN page: 4

Background, motivation and goals(cont’d)0f bfncs1.drw• For complex bb-representation only f s> 2 . f bis needed© NOKIA FILENAME.PPT / 13.12.1997 / NN page: 5

Useful complex relationships• Define complex number:• Thenz = x +jy• andRe=1 * 1 *{ z } ⋅ Re{ z } = ( z + z ) ⋅ ( z + z )121*( Re{ z z } + Re{ z z })12221121222(1)Re=1 * 1*{ z } ⋅Im{ z } = ( z + z ) ⋅ ( z − z )121* 1*( Im{ z z } − Im{ z z }) = ( Im{ z z } + Im{ z z })1222112122 j212212(2)© NOKIA FILENAME.PPT / 13.12.1997 / NN page: 6

Useful complex relationships (cont’d(cont’d)• andand alsoIm=1 * 1*{ z } ⋅ Im{ z } = ( z − z ) ⋅ ( z − z )121*( − Re{ z z } + Re{ z z })1222 j11212 j22(3)Re{ z z } = x x − y y = Re{ z } Re{ z } − Im{ z } { z }1 2 1 2 1 21 21Im2(4)(5)Im{ z z } = x y + y x = Re{ z } Im{ z } + Im{ z } { z }1 2 1 2 1 21 21Re2© NOKIA FILENAME.PPT / 13.12.1997 / NN page: 7

Complex representation of bandpasssignalAssume bandpass signalthis is equivalent tovt () = At ()cos( ω t+φ())tjωct+jφ()t jωct= { } = { }~ j ct j ct[ zte ()ω ~ zt ()* − ωe ]vt () Re Ate () Re ~ z()te1= +2where the complex envelope is definedc(6)(7)~ () ()()zt Ate j φ=t(8)© NOKIA FILENAME.PPT / 13.12.1997 / NN page: 8

Alternative real representationBP-signal of (7) may be written alsovt () = xt ()cos( ω t) −yt ()sin( ω t)cc(9)where complex envelope has been given usinginphase (I) and quadrature (Q)components(10)~ () () ()zt xt jyt = +© NOKIA FILENAME.PPT / 13.12.1997 / NN page: 9

Spectral considerations• If v(t) is bandpass => A(t) and φ(t) are slowly varying lowpass type• If ω cis the center frequency of the bandpass signal and ω c> ω b=>complex envelope is unique-f cV(f)01/2f cf c+ f bfncs2.drw© NOKIA FILENAME.PPT / 13.12.1997 / NN page: 10

Spectral considerations (cont’d(cont’d)Fourier transform of (7) gives1~ ~V( f)Z ( f f ) Z ( f f )*= − + − −[ ]ccso the bandpass spectrum can be given using onlyinformation of~ ( ) Z f2(11)!© NOKIA FILENAME.PPT / 13.12.1997 / NN page: 11

Spectrum of the complex low-passsignal∼Z(jω)10f bfNotice the spectrum height!© NOKIA FILENAME.PPT / 13.12.1997 / NN page: 12

Recovery of quadrature component xFourier transforming we get− jωct 11j2ωct{ } = 2 { } + 2 { }1 j2ωct ∗ − j2ωct4 [ ]vt ( )Re e Re ~ z() t Re ~ z()te1= xt () + ~ zte () + ~ zt () e2(12)121~ ~X( f)+ 4 Z ( f − 2f ) + Z ( − f − 2f)∗[ ]cc(13)The transform components are bandlimited Õlowpassfiltering the result we get 1/2 X(f) or[ {− ω}]c2xt () = vt ()Ree j t LP(14)© NOKIA FILENAME.PPT / 13.12.1997 / NN page: 13

Recovery of quadrature component yFourier transforming we get− jωct 11j2ωct{ } = 2 { } − 2 { }1 j2ωct ∗ − j2ωct4 j [ ]vt ( )Im e Im ~ z() t Im ~ zte ()1= yt () − ~ z () te −~ z () t e212~ ~Y( f)+ Z ( f − f ) − Z ( − f − f )∗[ ]14 j 2c2c(15)(16)and hence− ω c{ }[ 2]yt () = vt ()Ime j t LP(17)© NOKIA FILENAME.PPT / 13.12.1997 / NN page: 14

Spectra of low-pass componentsStarting from (14) Fourier-transforming, using (11) we get~ ~X( f)=1Z ( f ) + Z ( − f )∗2[ ](18)and similarly~ ~Y( f)=1Z ( f ) − Z ( − f )∗2 j[ ](19)The following figure demonstrates above.(Note: x(t) and y(t) real => X(f) and Y(f)conjugate symmetric i.e. X(-f)=X(f)* )© NOKIA FILENAME.PPT / 13.12.1997 / NN page: 15

Derivation of X(f) from BP spectrum V(f)1/2∼Z(-f)*∼Z(f)1-fcV(f)0 f cf0fX(f)⇒10fncs3.drw© NOKIA FILENAME.PPT / 13.12.1997 / NN page: 16

Linear filtering of BP signalsAssume BP-filter impulse responseh( t)= Re~{ h t ej ct }ω2 ( )(20)Let input be v(t) − then the output w(t) is∫wt () = ht ( −τ)( vτ)dτ{j t jht e z e }1~ccRe ( )ω ( −τ)~ ( )ω τ= ∫ 2 2 − τ τ dτ+{~ j t jht e z e }1ccRe ( )ω ( −τ)− ~∫( )∗ − ω τ2 2 τ τ dτ~c= Re ( − ) ~ ( ){ ht z d e }jωtτ τ τ∫(21)© NOKIA FILENAME.PPT / 13.12.1997 / NN page: 17

(cont’d)and hence the complex envelope of the output w(t) is~ ~ wt () = ( ) ~∫ ht−τ z( τ ) d τ(22)Note the the use of of number 2 in in definingthe the complex envelope of of filter impulse response[However, in simulations this is usually not critical;it is only constant scaling for signals and noise]© NOKIA FILENAME.PPT / 13.12.1997 / NN page: 18

Bandpass random signals and noiseAssume wide sense stationary (WSS) BP random signalor for simplicityj{ ω ct}nt () = Re nte ~ () = n()cos( t ω t) −n()sin( t ω t)c c s cnt () = xt ()cos( ω t) −yt ()sin( ω t)cc(23)(24)Autocorrelation functionRn( τ) = n( t) n( t+τ)∗ jωτ c 1jωτ c + j2ωct{ nt nt τ e } 2 { ntnt τ e }1= Re ~ () ~ ( + ) + Re ~ () ~ ( + )2(25)Due to WSS => second term must be ZERO© NOKIA FILENAME.PPT / 13.12.1997 / NN page: 19

Bandpass random signals and noise (2)DefineThenWSS condition givesR~( ) n~ ()∗τ = t n~ ( t+τ )n= xtxt () ( + τ) + ytyt () ( + τ) + jxtyt [ () ( + τ) − ytxt () ( + τ)= R ( τ) + R ( τ) + j[ R ( τ) − R ( τ)]x y xy yx{jωτ}c~ τR ( τ) Re R ( ) en= 1 2ntnt ~ () ~ ( + τ ) = R( τ ) − R( τ ) + jR [ ( τ ) + R ( τ )] = 0x y xy yxn(26)(27)(28)which gives general properties for the inphase and quadrature correlations© NOKIA FILENAME.PPT / 13.12.1997 / NN page: 20

Bandpass random signals and noise (3)and henceRRxxy( τ ) = R ( τ )y( τ) =−R( τ)R ~ ( τ ) = 2R( τ ) + 2 jR ( τ )nxyxxy(29)(30)Note that the autocorrelation of the complex envelope may be complex (but thespectrum is real!)Fourier transforming (27) we get1S ( f) = [ S~ ( f − f ) + S~( −f − f )]n 4 n c n c(31)As before we find low-pass spectrum(U(f) is the "unilateral step function")© NOKIA FILENAME.PPT / 13.12.1997 / NN page: 21

Bandpass random signals and noise (4)S~( f) = 4S ( f + f ) U( f + f )n n c c(32)S ( f)n-f cNo/20 f cfS ( f)∼n2No0ncs4.drwf© NOKIA FILENAME.PPT / 13.12.1997 / NN page: 22

Spectra of quadrature componentsx(t) and y(t)From (30) we getand F-transform gives∗R ( τ) = [ R~ ( τ) + R~( τ) ]x n n∗R ( τ) = [ R~ ( τ) −R~( τ) ]xy j n n(33)(34)1414)]SSxxy( f ) =( f ) =14[ S14 jn~[ S( f ) + Sn~n~( f ) − S( − f )]n~( − f[see following picture]© NOKIA FILENAME.PPT / 13.12.1997 / NN page: 23

Spectra and cross-spectra of low-passcomponents x and yS ∼ (-f)n2NoS (f)∼nS (f)∼n2NoNo0f-S ∼(-f)nfS (f)x2No3No/4jS (f)xy2NoNo/40fncs5.drwf© NOKIA FILENAME.PPT / 13.12.1997 / NN page: 24

Conclusions about low-pass spectraNote that if BP-spectrum is symmetric (ref f c) thenSR1S ( f) = S~( f) = 2S ( f + f ) U( f)Rxyxyx 2 n n cx( f)≡ 0( τ ) ≡ 01( τ) = R~( τ)2n(35)So• low-pass components are uncorrelated at any t• low-pass spectrum is twice the (two-sided) BP spectrum• Hey, keep eye on those damned "engineer’s 2s"© NOKIA FILENAME.PPT / 13.12.1997 / NN page: 25

Symmetric spectraS (f)n-f cNo/20 f cfS (f)n ∼2NoS (f)x0Nof0ncs6.drwf© NOKIA FILENAME.PPT / 13.12.1997 / NN page: 26

Power of the BP-processPower considerationsFrom (27) we havePn= R ( 0)=n2σR~( 0) = 2R( 0)= 2σ2nn(36)(37)From (29), (30) and (37) we conclude (real variables!)Rxy( 0)= 012R ( 0) = R~( 0)= σ = P = Px 2 n x y(38)© NOKIA FILENAME.PPT / 13.12.1997 / NN page: 27

Remarks• Power of each component is the same as total BP power (general result)• In general case inphase and quadrature components are uncorrelated at theSAME time instant (but not necessarily if t ≠ 0)• For Gaussian random process samples taken at the SAME instant are thenindependent• Only for symmetric BP Gaussian process are also x(t) and y(t) independentGaussian processes (at any time)So, if if you are are dealing with unsymmetric casesyou might consider including correlation into your model© NOKIA FILENAME.PPT / 13.12.1997 / NN page: 28

Complex white Gaussian processAssume that• BP spectrum is constant No/2 (two-sided), Gaussian• ideal rectangular filtering, bandwidth W• system filtering much narrower than Wthen the complex noise envelope is characterized (may be approximated as)S~( f) = 2N , f ≤W/ 2nRn~o( τ) = 2Nδ( τ)o(39)© NOKIA FILENAME.PPT / 13.12.1997 / NN page: 29

Complex white Gaussian process (2)and the joint density function for x(t) and y(t) isp ( xyx , y )= 1exp ⎧− x + y22πσ⎨ 2⎩ 2σ2 2⎫⎬⎭(40)where σ 2 is the filtered noise power (e.g. N oW). This can also be written in theform:p n ~n~ )( 22⎪⎧ n~exp⎨−⎪⎩2σ=12πσ2⎪⎫⎬⎪⎭(41)EXTENSION RESULT:General N-dimensional complex Gaussian© NOKIA FILENAME.PPT / 13.12.1997 / NN page: 30

N-dimensional complex GaussiandensityAssume zero mean and stationarityE[ ~ x]= 0TE[ ~~ ∗xx ] ≡TE[ ~~ xx ] = 02Λ~x(42)where bold x means (column) vector, T stands for transpose and Λ forcovariance matrix.© NOKIA FILENAME.PPT / 13.12.1997 / NN page: 31

N-dimensional complex Gaussian density(2)Probability density function for these joint complex Gaussian variables iswherep~x( ~ 1x)= exp − x xN( 2π) det Λ~x1 ~ T∗−1~{ 2 Λ ~x }(43)[ x1 x2 xN−1xN]~ Tx = ~ , ~ ,... ~ ~Λ~x=⎡⎢⎢⎢⎢⎣⎢σ1xx ~~1 ∗ 22 2 1 2::2 12 1 2~~ ∗xx ...σ⎤⎥⎥⎥⎥2σN ⎦⎥(44)© NOKIA FILENAME.PPT / 13.12.1997 / NN page: 32

Channel modelingWe use complex low-pass representation• if the signal BP bandwidth is W only samples at rate W are needed• system filtering may be represented using complex samples 1/W apart• also channel may be represented with complex samples1/W apartIn practice somewhat denser sampling is preferableNote also that time varying channel expands bandwidth (Doppler spread) anddenser tapping is needed© NOKIA FILENAME.PPT / 13.12.1997 / NN page: 33

Sampled channel~h 01/W 1/W 1/W 1/W~h 1~h 2....+ncs7.drw• Generally coefficients h iare complex numbers (possibly time varying)• Often channel is assumed "freezed" i.e. fixed for a number of transmittedsymbols• Often coefficients complex Gaussian© NOKIA FILENAME.PPT / 13.12.1997 / NN page: 34

ExamplesFixed radio link channel• normally only two taps needed• freezing can be assumed• main component may have nonzero mean i.e.~ ~h0 = h0m= h0me~h1= 0~ ~ jϕ1h = h e1 1jϕo(45)Note: magnitude of h 1follows Rayleigh law in fading if it is zero mean complexGaussian© NOKIA FILENAME.PPT / 13.12.1997 / NN page: 35

Mobile radio channel• normally a few taps needed (e.g. 5 or so)• channel varies relatively fast compared to symbol lengths => Doppler spreads• time variations are important => time varying coefficients often used in models• in many cases all taps may be zero mean complex processes• but also it may be possible that one or more taps have nonzero mean(specular reflections)© NOKIA FILENAME.PPT / 13.12.1997 / NN page: 36

Modulated signals with complexenvelopesMany digital modulations can be represented in a general form with complexenvelope~ z ( t )~ z ( t)=nT )∑ anu(t − nT ) + j∑bnw(t −where u(t) and w(t) are the basic pulse forms.Example:QPSKa ngets values +1+j and b nis zerou(t) is, e.g., half Nyqvist pulsenn(46)© NOKIA FILENAME.PPT / 13.12.1997 / NN page: 37

Signal-to-noise ratioLet us consider QPSK signal with u(t) rectangular pulse [0,T]⎧jωt⎫crt () = vt () + nt () = Re ⎨∑anEut ( −nTe ) ⎬ Re nte ~ ()⎩ n⎭ +Assume ideal T-integrator in the receiverThe output of the integrator at t=(k+1)T (compl.env)Tjωct{ }~ 1gkT ( ) ( ) ~ [( ) ]T a Eu T d 1= ∫T n k T dk− τ τ + ∫ + 1 −τ τT0 0= a E + n~k1T(47)(48)© NOKIA FILENAME.PPT / 13.12.1997 / NN page: 38

Signal-to-noise ratio (2)Signal power is nowand noise powera E = 2Ek~2n1 n ~ [( k ) T ] n ~ [( k ) T ]∗= + 1 − τ + 1 −τ d τ d τ1T1= 2Ndτ= 2NTT∫0∫∫o1oNote that variance of each noise component (I and Q) is now N o.21 2 1 2(49)(50)© NOKIA FILENAME.PPT / 13.12.1997 / NN page: 39

Probality of symbol error (appr) is nowProbability of errorP Q E 2Ebe= ( ) = Q( )N Noo(51)where E bis the bit energy and E is the real energy of one pulse.Q√ EIncs8.drw© NOKIA FILENAME.PPT / 13.12.1997 / NN page: 40

Finally• complex envelopes reduce simulation complexity• complex envelopes make conceptually simple analysis of I-Q signals• some care is needed when calculating signal-to-noise ratios (and may besignal-to-interference also) ["engineer’s 2’s"]• WARNING: if system contains nonlinearities, one should watch what signalrepresentations are valid (bandwidth expansion!)• also fast time varying channels may require special treatment© NOKIA FILENAME.PPT / 13.12.1997 / NN page: 41

References1. Benedetto, Biglieri, Castellani: Digital Transmission Theory.Prentice Hall International Inc. New Jersey 1987, 639 p.2. vanTrees H.,L.: Detection, Estimation and Modulation Theory,Part III. John Wiley & Sons, Inc., New York 1971, 626p.© NOKIA FILENAME.PPT / 13.12.1997 / NN page: 42

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