Advanced global navigation satellite system receiver design (797918), страница 34

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However, care wasalways taken to insure the sampling frequency is not a multiple of the code rate,which is clearly detrimental to receiver operation.Time in the simulation in engineering units of quarter chips. Notice the familiarnotation of the integration results, wII , wIQ , wQI (same as chapter 5). The carrier PLLis second order and the code DLL is first order.B-2Mathcad simulations of PSK and BOCPSK SIMULATIONN := 4T := 15∆ :=δ ≡ 0.0716114k := 201TS :=6N⋅ 1.023⋅ 10prn :=r ←1code I( t ) :=3for i ∈ 0 .. 14τ←tx← r ⊕ r0j− 1←r)τ ← τ − T⋅ N1sub Q( t) := 1 − 2 mod floor t +while τ < 0for j ∈ 1 ..

3rsub I( t ) := 1 − 2( mod( floor( t ) , 2)while τ ≥ T⋅ Nτ ← τ + T⋅ Nj1 , 22τ Nr ←xj ← floora ←xprn3ija ← 2⋅ a − 1code Q( t) := code I t +∆⋅ N 2 ∆⋅ N  − code I t −2 CN0 := 6 + k⋅ 0.1φC := 0CN0CN0 := 10σN :=(1σN = 0.5352⋅ δ⋅ TS⋅ CN0)Λ tT , tRC, φR :=wII ← 0wIQ ← 0wQI ← 0while t RC ≤ 15⋅ N() ( ( )) + rnorm(1, 0, σN)u Q ← sin ( φC − φR) ⋅ ( code I( tT) ) + rnorm( 1 , 0 , σN)u I ← cos φC − φR ⋅ code I tT( )( )v I ← code I t RCv Q ← code Q tRCwII ← wII + u I⋅ v IwIQ ← wIQ + u I⋅ v QwQI ← wQI + u Q⋅ v ItT ← tT + δt RC ← t RC + δδwII ← wII⋅N⋅ TδwIQ ← wIQ⋅N⋅ TδwQI ← wQI⋅N⋅ Tt T ← tT − N⋅ Tt RC ← t RC − N⋅ T( wIIwIQ wQI tT tRC φR)TB-3Mathcad simulations of PSK and BOCk0C ≡ 0.114k0φ ≡ 0.2()Φ SwIQ, SwQI , tT , tRC, φR :=k1φ ≡ 0.005()X ← Λ tT , tRC, φR( 0)d ← sign XwIQ ← X ⋅ d1wQI ← X ⋅ d2SwIQ ← SwIQ + wIQSwQI ← SwQI + wQItT ← X3tRC ← X + k0C⋅ wIQ4φR ← X + k0φ⋅ wQI + k1φ⋅ SwQI5( SwIQk := 1 ..

1500SwQI t T t RC φRSwIQ := 00)TtT := 2tRC := 000SwQI := 00φR 0 := 1 SwIQk  SwQIk  tTk  := Φ SwIQk− 1 , SwQIk− 1 , tTk− 1 , tRCk− 1 , φRk−1 tRCk  φR  k ()21.5tT k−tRC k1φR k− φC0.501002003004005000.5kB-4CAnalogue and digital DLL formulasThis appendix was kindly supplied by Dr Hodgart and provides the formulas used tosimulate additive noise to GNSS signals.C.1IntroductionNecessary formulas for the triple loop PLL + SLL + DLL are presented derived byinference from real time pulse detectionSuggested nomenclaturePLL phase lock loop for carrier (may be replaced by FLL )SLL subcarrier lock loopDLL delay lock loop – for tracking the codeThe loop creates two time estimates - one from the DLL effectively ignoring the subcarrier modulation,We simplify first by just looking at the low pass waveforms –carrier phase analysis isignored.Instead of a code sequence a(t) we have a BOC sequenceb(t − τ) = a(t − τ ) × s (t − τ )C-1where s(t) is a squared up sine wave ( or an actual sinewave in LOC)Because of necessity to describe noise at various points we have had to changesymbols to describe input.Our actual signal input is the in-phase channel only herey I (t ) = Ab(t − τ) + v I (t )C-2which the SLL and the DLL then test with independent delay estimates( )z I τˆ * , τˆ =1TGTG∫0(( s t − τˆ *y I (t ) × ~* s t − τˆ) ×  a(t − τˆ )' dt)  a~(t − τˆ )C-3whereC-1Analogue and digital DLL formulas T   T a~ (t ) = a t + D  − a t − D 2  2 C-4The correlationTG1TGC-5∫ a(t − τ)a(t − τˆ )dt = Λ(τ − τˆ )0where Λ has usual width +/- TC and unit height Also formthe correlation1TGTGC-6~Λ∫ a(t − τ)a (t − τˆ )dt = V (τ − τˆ )0and gives a discriminator characteristicWith conventional gate width TD = TC then discriminator slope = 2/TCLooking at actual outputsz IIQTG1~∫ a(t − τ)a (t − τˆ )dt + TG0A=TGTGΛ~∫ u(t )a (t − τˆ )dt = A V (τ − τˆ ) + wIIQC-70where ΛV has slope 2/TC after setting gate width TD = TC1z I τˆ ,τˆ =TG()*1+TG(( s t − τˆ*()()ττatst−×−×∫0~ s t − τˆ*TG(( s t − τˆ*u(t)×∫0~ s t − τˆ*TG) ×  a(t − τˆ )' dt)  a~(t − τˆ )C-8) ×  a(t − τˆ) ' dt)  a~(t − τˆ )Knowing that the loops are able to proceed independently we can in theory set^^τ* = τ and consider variation of above as only between τ and τz III(τˆ *)A= τ, τˆ =TGTG1∫ a(t − τ) × a(t − τˆ )dt + TG0TG∫ u (t ) × a(t − τˆ )dtC-90or()z III τˆ * = τ, τˆ = AΛ (t ) + wIIIz IIQ(τˆ *)A= τ, τˆ =TGTG1~∫ a(t − τ) × a (t − τˆ )dt + TG0C-10TG~∫ u(t ) × a (t − τˆ )dtC-110C-2Analogue and digital DLL formulasOr()z IIQ τˆ * = τ, τˆ = A ΛV (t ) + wIIQC-12which allows the evaluation of error exactly as if this was PSK transmission.τyI(t) =A (t) + v(t)TCzIII(t) = A Λ(t) + wIII(t)zIIQ(t) = AΛV(t) + wIIQ(t)τ + TDThe system representation may be shown asAzIIQ(t)=A ΛV(t)+wIIQ(t)zIII(t)= A Λ(t)+wIII(t)(t)++filterh(t)∝ (t)+−delay ΤDηwhitenoisepeakdetect&gateC-3Analogue and digital DLL formulasC.2Linear system equivalentA (t)TCA ΛV(t)AA1/TC∆h(t)TCz(t) =whitenoiseA ΛV(t) + wIIQ(t)output = signal + noiseu(t)T D = TCWe specifically identify a rectangular pulse of amplitude A and width TC in presenceof white noise of one-sided density η.

We also set TD = TCThis filter representation of the processing system allows an immediate evaluation oftiming error. A noise sample w = wIIQ converts to a timing error ε by the elementaryrelation of reflection though the slope dz/dt on the zero crossing in z.ε=wdzdtC-13and the mean square timing jitter2Σ = ε2w2=C-14(dz dt )2The basic system theory which is needed is simply to know that the mean square noiseout of the filterη× ∫ ∆h(t )2 dt2η= × ∫ (h(t ) − h(t − TC ))2 dt2w2 = 1η= × 2 × 2 TCC-152 TCThenw2 =ηTCC-16and since dz/dt = 2A/TC we getC-4Analogue and digital DLL formulasΣ2 =TC η4AC-172The custom exists in the literature of normalising the timing jitter to a chip length.(This practice can lead into difficulties when it comes to comparing PSK with BOC ).However to ensure consistency with this approach let us now define and now expressa timing jitter normalised to the chip length2σ =Σ2=TC211× 24TC A / ηC-18A reduced delay time TD < TC may be able to achieve a reduced timing jitter .

Then2σ =Σ2=TC21∆× 24TC A / ηC-19where by usual convention the normalised delay∆=TDTCC-20which may be less than unity (but not greater).C.3Signal averagingLet there be signal averaging over K pulses . Then the timing error reduces toσ2 =1∆× 24 KTC A / ηC-21Now the product KTC = the total transmission time – the total period over which thenotional pulse transmitter is equivalent to an averaging or processing time = Tσ2 =C.41∆× 24T A / ηC-22Digital loop analysisA practical system employs sampling at an interval we shall call here δ.In simulation and reality let there be samples yI[l] at sample intervals δ .

Codecorrelation time TGC-5Analogue and digital DLL formulasNumber of samples per codenG =TδC-23Sample number nG is not an integerAdd noise samples v[l] = vI[l] = with an rms value σ 2v =η2δWe synthesise thereforey I [l ] = Ab(t l − τ) + v I [l ]C-24We need also to identify a code integration time TG ( G for GOLD) this being span ofminimum time in which to run a correlation .We shall ignore carrier recovery – which we can.Ignoring presence and interaction of SLL is allowed in the limiting case with smalldeviationsz III =1nGnG∑ y I [l ]a(tl − τˆ )C-25l =1which approximates the correlationz III =1TGTG∫ y I (t )a(t − τˆ )dtC-260As shown abovez III = AΛ (τ − τˆ ) + vC-27nGC-28z IIQ =1nG∑ xl a~(tl − τˆ )l =1which approximates the correlationz IIG1=TGTG~∫ x(t )a (t − τˆ )dtC-290C-6Analogue and digital DLL formulasC.5Loop operationSimplify noise notation wk = wIIQ[k]On completion of one correlation (in time TG) the loop estimate of time delay isupdated()τˆ k +1 = τˆ k + A ΛV (τ k − τˆ k ) + wk × k 0CC-30or in linear region 2Aτˆ k +1 = τˆ k + ×(τ k − τˆ k ) + wk k 0C TCC-31Let actual time delay tk = 0 then we are looking at an errorε k +1 = (1 − ρ C )ε k + k 0C wkC-32after definingρC =2A× k 0CTCC-33this is a classic 1st order IIR filterThe mean square absolute timing error2Σ 2DLL= ε2ρCT =×  C  × w22 − ρC  2 A C-34or after normalising and identifying2σ 2DLL = ε 2 =ρCη 1 × ×2 − ρ C  2 A  TGC-35comparison of the various formulas then identifies an overall averaging time2 − ρCTG =ρC2−2 Ak 0C2 Ak 0CTCTC TTG =  C − 1TG Ak 0CC-36Substituting in (C-35) after inverting and with ∆ = 1 then1σ 2DLL TA2= 4×× TG ×  C − 1η Ak 0CC-37C-7Analogue and digital DLL formulasThen standard theory finds for the input noiseσ 2v =η2δC-38Therefore given in loop simulation A, σv, δ , TC, TG, k0C then1σ 2DLLC.6= 2×A2σ v2×TG  TC× − 1δ  Ak 0CC-39Conversion from RF systemA× (t − τ)cos(φ) + uI(t)√(2C)× (t)cos(ω0t)cos(ω0t + φ)sin(ω0t + φ)whitenoiseN0A× (t − τ)sin(φ) + uI(t)In down conversion from an r.f channel there are by convention in phase andquadrature channels .

We may generalise toxI(t) = A (t − τ)cos(φ) + uI(t)C-40xQ(t) = A (t − τ)sin(φ) + uQ(t)C.7Informal comparison with coherent DLL systemsIn a coherent system we can assume φ = 02Our result is directly convertible into RF equivalent of input carrier power C ∝ Aand noise density N0 ∝ ησ 2DLL =1∆×4T C / N 0C-41We can match the above to well-established analyses.C-8Analogue and digital DLL formulasIt is in the nature of CDMA using the Gold code to be able to achieve the same effectas sending one pulse by a continuous code. So in L1 the effect of the transmission of1023 chips organised in a code over 1 msec is to achieve K = 1023 in just T = 1msec.

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