Advanced global navigation satellite system receiver design (797918), страница 36

Просмтор этого файла доступен только зарегистрированным пользователям. Но у нас супер быстрая регистрация: достаточно только электронной почты!

Текст из файла (страница 36)

The worse case of many trials identifies a threshold of 8 (see Figure E-3) whichcorresponds to the analysis and threshold choice in [Fine and Wilson 1999].E-1Choosing a bump jumping thresholdx := −0.75y := 10z := −0.750(x := rnorm N , x , σ00vl := 0σ :=0110(y := rnorm N , y , σ)ve := 0N := 100000)(z := rnorm N , z , σ0)k := 1 .. N − 1f( ve , x, y , z) :=ve ← ve + 1 if ( x > y ) ∧ ( x > z )ve ← ve − 1 if ( y > x ) ∧ ( y > z )ve ← 0 if ve < 0veg ( vl , x, y , z) :=vl ← vl + 1 if ( z > y ) ∧ ( z > x )vl ← vl − 1 if ( y > x ) ∧ ( y > z )vl ← 0 if vl < 0vl(ve := f vek,xk− 1 k− 1,y(vl := g vlk,z),x,yk− 1 k− 1k− 1 k− 1,zk− 1 k− 1)10Count valuesµE := mean( ve )ve kvl kµE = 0.6785µL := mean( vl)8µL = 0.48502004006008001000kNoise samplesFigure E-2, Simulation of BJ threshold with uncorrelated noise samplesE-2Choosing a bump jumping threshold10Count values8ve kvl k58−10200400600k180010001000Noise samplesFigure E-3, Worse case VE and VL count values for BOC(2×fC, fC) with uncorrelated signal tonoise of 10 dB per correlationHowever, as identified by Dr Hodgart, the noise samples are in fact stronglycorrelated which actually improves the performance of the BJ algorithm.

ForBOC(2×fC, fC) the correlation coefficient between noise separated by TS and thereforebetween VE and P and between VL and P is ρ = −0.75 . The correlation coefficientof noise between the VE and VL samples is ρ = −0.5 . For simulation an easy way tocreate the three correlated noise samples, nx, ny,and nz, from three uncorrelated noisesamples u,v and w is by linear combination asn x = +u − v − wE–1n y = +u + v + wn z = −u + v − wwhereu2=σ28v2=σ28w23σ 2=4E–2The Mathcad simulation program for uncorrelated noise samples is shown in FigureE-4. The resulting worse case noise induced count values for correlated noise areshown in Figure 5-26, which finds a required threshold of 5.E-3Choosing a bump jumping thresholdx := −0.75y := 10z := −0.750u := rnorm N , 0 ,vl := 00σ :=01108v := rnorm N , 0 ,σve := 0N := 100008σw := rnorm N , 0 , σ⋅34k := 1 ..

N − 1x := u − v − w1y := u + v + w1x ⋅ y  = −0.768∑ ( k k)⋅2σ ⋅ ( N − 1)kx := x − 0.75∑ (yk⋅zk) = −0.756⋅k∑ (xk⋅ zk) = 0.511⋅2σ ⋅ ( N − 1)ky := y + 1f( ve , x, y , z) :=2σ ⋅ ( N − 1)1z := −u + v − wz := z − 0.75ve ← ve + 1 if ( x > y ) ∧ ( x > z )ve ← ve − 1 if ( y > x ) ∧ ( y > z )ve ← 0 if ve < 0veg ( vl , x, y , z) :=vl ← vl + 1 if ( z > y ) ∧ ( z > x )vl ← vl − 1 if ( y > x ) ∧ ( y > z )vl ← 0 if vl < 0vl(ve := f vek,xk− 1 k− 1,y(vl := g vlk,z),x,yk−1 k− 1k− 1 k− 1),zk− 1 k− 16µE := mean( ve )ve k 4µE = 0.19vl k52µL := mean( vl)µL = 0.217002004006008001000kFigure E-4, Simulation of BJ threshold with correlated noise samplesE-4Choosing a bump jumping threshold6Count values5ve k 4vl k5−1201200400600k80010001000Noise samplesFigure E-5, Worse case VE and VL count values for BOC(2×fC, fC) with correlated noise samples,signal to noise of 10 dB per correlationE-5FDigital Noise SynthesisThis appendix was kindly supplied by Dr Hodgart and provides the formulasnecessary to synthesise additive noise in GNSS signal generators.F.1IntroductionThe aim is to create a simulation on a transmitter of electrical noise to be added to anencoded GNSS signal and then together to be converted into a representation of ananalogue signal at a given intermediate frequency f0 .

The proposed system in thefollowing analogue representation isa(t)b(t)c(t)CODEGENERATORv(t)NOISEGENERATORFigure F-1, Noise synthesisPrecision to the concept is provided by an arithmetic/algebraic representation. Wetake the GPS transmission known as ‘L1’ as an example, which generates a ‘codechip’ every 1/1.023 µsec. Notation here for each code chip is a[k], countingcontinuously in k. Allowed values of a[ ] acknowledge its bipolarity i.e. twopossible integer values −A or +A.The chip sequence is periodic in a count 1023 .

So a[k] = a[n ×1023 +k] where n isany integer. This periodicity is not relevant in the following analysis.F-1Digital Noise SynthesisWe use the example of the SSTL Galileo signal generator, which creates an i.f offrequency f0 = 60 ×1.023 = 61.38 MHz .

Therefore, in the absence of synthetic addednoise, a multi-rate sequence must be created as inb[k ] = a[k / 120]F–1c[k ] = b[k ] × (− 1)kwhere the [ ] notation always means ‘integer’ or ‘floor’ to what may be a realnumber. Soa[119/120] = a[0]buta[121/120] = a[1]So herec[0] = a[0]F–2c[1] = − a[0]up toc[119] = − a[0]thenc[120] = a[1]F–3and so onEach code chip in this proposal is transmitted therefore in a sequence of alternating120 samples - so creating the ‘digital’ equivalent of an IF at 60 times the code rate.We will define a sampling rate fS = 2f0 = 122.76 MHz.F.2Adding noiseThe proposal is to add an integer noise sequence.

The particular suggestion here is tocreate this noise and add it to the signal before multiplication to u[k]. The sequence ofsamples is u[0], u[1] ... u[k] where each sample is uncorrelated to any other sample.F-2Digital Noise SynthesisThe probability distribution is quasi-Gaussian with a known standard deviation σ(which need not be integer).The aim is to create a pseudo noise sequence which will reproduce a preciselycalculated C/N0 in a bench test transmission to a receiver.The proposal here is to create a variable rate noise sequence to be added to the codebut before multiplication.

So write for exampleb[k ] = a[k / 120] + u[k / 2]F–4c[k ] = b[k ]× (− 1)kwhich equation says to add noise on every sample but only update it every othersample - i.e. at the rate of 61.38 MHz.At the other extreme we could writeb[k ] = a[k / 120] + u[k / 120]F–5c[k ] = b[k ] × (− 1)kwhich equation says to add noise on every sample but only update it on a new code biti.e. at a rate of 1.023 MHz.Define a sampling rate to code rate ratio KSC = fS/fC and in general writeb[k ] = (a[k / K SC ] + v[k / K N ])F–6c[k ] = b[k ]× (− 1)kwhere KN is a decimation factor (and must be divisible into the sampling to code rateratio (which here is KSC =120).Define also a noise update ratefN =fSKNF–7and sincef S = K SC f CF–8F-3Digital Noise SynthesisKff N =  SC CKNF–9Examples(61.38 = 120F.32)×1.023(6.138 = 12020)× 1.023(1.023 = 120120)× 1.023Achieved C/N0On every independent sample then the signal to noise isΓ2 =A2σF–102where σ is the rms value of the synthesised noise .

For an 8-bit representation theoptimum value will be around σ = 40 units allowing deviations of near ± 3σ . For anupdate rate fN then in an observation interval TL there will be an effective number ofaverages = fNTL and the achieved signal to noise will beγ2 =A2σ2× f N TLF–11From standard theory it can be shown that in an equivalent analogue world with awhite noise of density N0 the achieved signal to noise in a coherent receiverγ2 =2 A 2T LC= 2×× TLN0N0F–12Therefore the effective carrier to noise density ratioCN0=2 A2 f NσF–132or in terms of the sampling frequency fS and decimation factor KNCA2 f S2=× 2N0 KσNF–14F-4Digital Noise SynthesisF.4Effective v.

actual carrier to noise density ratioThis formula (F–13) computes a value which is double that worked out by FourierAnalysis in a first draft of this note. Allowing for up conversion and a comparison ofthe carrier power against the noise density at the carrier frequency it is found thatCN0=A2 f NσF–152The question then is which formula is correct? The answer is that both are! EquationF–15 is the actual C/N0 after up-conversion while Equation F–13 is the effective C/N0after up-conversion. The explanation of these different meanings is that the noisewhich is synthesised by this method is not a true synthesis of normal thermal noise because upper and lower sidebands are inherently correlated.F.5Example implementationSet a minimum possible value to A = 1 and a likely maximum to σ = 40 (allowingGaussian noise with an excursion of ± 3σ in range ± 128).

Характеристики

Список файлов учебной работы