Advanced global navigation satellite system receiver design (797918), страница 13

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It is shown in [Van Nee and Coenen 1991] that FFT detection can beup to 2000 times faster than the equivalent serial search technique. However,performing FFT detection in the receiver hardware imposes large processing overheadand often requires a dedicated FFT unit in the receiver design. More discussion ofFFT detection for future GNSS signals is given in Chapter 10.Figure 5-8, FFT of GPS C/A code signal83Receiver theoryWhether using a serial search technique or FFT detection the correlation profile of thePSK signal is a single peak as given by Equation 5-9.

A detection threshold must beset within the receiver software above which signal presence is declared. This may bebased on a single search result, ‘single dwell’ or multiple search results, ‘multipledwells’. Taking multiple dwells clearly increases the reliability of detection but alsolengthens the signal search time. Raising the detection threshold reduces the risk offalse alarm but also reduces the probability detecting the signal. The choice ofdetection threshold is strongly dependent on the receiver’s operating environment andapplication.5.2Tracking PSK signalsTo achieve a reasonable estimate of the time delay of the received signal a PSKGNSS receiver must implement a carrier loop to maintain lock on the frequency and /or phase of the incoming carrier wave.

Simultaneously, a code loop must alsomaintain lock on the PRN code sequence present in the received signal. This can onlybe achieved by adjusting the carrier phase estimate, φˆ and the code phase estimate, τˆto track the incoming φ .and τ respectively.For the carrier loop a Frequency Locked Loop (FLL) or Phase Locked Loop (PLL)can be used from which we define an incoherent or coherent system respectively.A FLL delivers more robust tracking in highly dynamical environments and in veryweak signal conditions [Kaplan and Hegarty 2006].

However, in order to enableprecise delta pseudorange, integrated Doppler measurements and carrier phasepositioning a GNSS receiver must track the phase of the incoming carrier. Hence, acoherent Phase Locked Loop (PLL) system is required.The carrier phase error is commonly determined using a Costas decision-directeddiscriminator [Kaplan and Hegarty 2006] as follows.()()eφ = wQI × sgn (wII ) ≈ A × sin φ − φˆ × sqc φ − φˆ × Λ(τˆ − τ )5-1284Receiver theorysqc( ) is a square wave cosine function, shown in Figure 5-9.

Incorporating the signcorrection by hard-limiting the wII correlation in the discriminator allows the PLL tofunction with no knowledge of the navigational data state.sqc( )+1φ − φˆ−4π− 2π2π4πFigure 5-9, Square wave cosine function()It can be seen that eφ tends to zero as sin φ − φˆ tends to zero, which occurs at integern multiples of π radians as φˆ → φ + nπ . This 180º ambiguity is caused signcorrection in the discriminator but can be resolved by inspection of the incomingnavigational data.

The discriminator characteristic is shown in Figure 5-10.10.670.331642024600.33-10.6711Phase error (rads)-40.5-200Phase error (rads)-0.524Code error (chips)-1Figure 5-10, Costas PLL discriminator characteristicThe decision-directed discriminator is dependant on amplitude A, a number of otherPLL discriminators can also be used, each with different dependencies and85Receiver theorycomputational loads. Table 5-1 shows the most commonly used PLL discriminators[Kaplan and Hegarty 2006], their error signals and dependencies.Table 5-1, Costas PLL discriminatorsDiscriminatorDecision-correctedError signaleφ = wQI × sgn (wII )(Dot product2≈Normalised)()≈ A × sin φ − φˆ × sqc φ − φˆ × Λ (τˆ − τ )eφ = wQI × wII[ (()≈ tan φ − φˆeφ = arctan (wQI / wII )(Slope proportionalto A)]A× sin 2 × φ − φˆ × Λ2 (τˆ − τ )2eφ = wQI / wII2 quadrant arctangentDependencySlope proportionalto A≈ φ − φˆ)2Slope independentof amplitudeSlope independentof amplitudeThe PLL is generally updated using a second order loop as follows.f φ ← f φ + eφ5-13φˆ ← φˆ + k1 × f φ + k 2 × eφfφ is the integrated phase error and k1 and k2 are loop gains which can be adjusted bythe designer.An incoherent system holds frequency lock by maintaining a constant or near constantphase difference across the correlation interval.

The rate of change of carrier phasecan be determined by comparing wII and wQI with the correlation results from the nextepoch wII′ and wQI′. The frequency error can then be determined using the crossproduct discriminator as follows.()′ × wII − w′II × wQI = A2 × sin ∆φˆ − ∆φ × Λ2 (τˆ − τ ) × d × d ′eω = wQI5-14The phase difference and estimated phase difference between epochs is86Receiver theory∆φ = φ ′ − φ5-15∆φˆ = φˆ′ − φˆAs eω tends to zero, ∆φˆ − ∆φ tends to zero. This discriminator only gives a truefrequency error when no data transitions have occurred. FLL discriminators onlyperform well when short integration periods are used to restrict the number of datatransitions affecting the error characteristic.

Filtering of the discriminator output isalso required to reduce the impact of the data transitions. It is common to use asecond order loop to update the carrier phase estimate by the frequency error asfollows.eφ ← eφ + eω5-16f φ ← f φ + eφφˆ ← φˆ + k1 × f φ + k 2 × eφeφ is the estimated phase error derived by the integrating frequency error eω , k1 andk2 are loop gains which can be adjusted by the designer.

The cross-productdiscriminator has a dependence on A2, a number of other FLL discriminators can alsobe used, each with different dependencies and computational loads. Table 5-2 showsthe most commonly used FLL discriminators, their error signals and dependencies.Table 5-2, FLL discriminatorsDiscriminatorCross-productError signal′ × wII − w′II × wQIeω = wQI(4 quadrant arctangent(cross-product))≈ A2 × sin ∆φˆ − ∆φ × Λ2 (τˆ − τ )′ × wII − w′II × wQI ,  wQIeω = arctan 2 w′ × w + w′ × w IIQIQI  II≈ ∆φˆ − ∆φ′ × sgn (wII ) − w′II × sgn (wQI ) ,  wQIeω = arctan 2 w′ × sgn (w ) + w′ × sgn (w ) IIQIQI IIˆ≈ ∆φ − ∆φ(4 quadrant arctangent(decisioncorrected)(DependencySlope proportionalto A2Slope independentof amplitude)Slope independentof amplitude)87Receiver theoryCommon practice is for the PRN code sequence to be acquired and tracked using aDelay Locked Loop (DLL).

This can be achieved when implementing eitherincoherent FLL or coherent PLL carrier tracking. However, different correlations anddiscriminators are required for each system. A common discriminator suitable fortracking with both coherent and incoherent systems is the dot-product discriminator,formed as follows.eτ = wIQ × wII + wQI × wQQ = A2 × VΛ (τˆ − τ ) × Λ(τˆ − τ )5-17The ‘ VΛ ’ symbol represents the code tracking error equivalent to subtracting separateearly and late correlations, written as follows.TDC T Λ − Λτˆ − τ + DC V (τˆ − τ ) = Λτˆ − τ −2 5-182 The dot-product discriminator characteristic is shown in Figure 5-11.

It can be seenthat this discriminator has no phase dependence and is therefore suitable for anincoherent system.0.60.40.40.20.21.510.500.5101.5-0.20.2-0.40.41-40.6Code error (chips)0.5-200Phase error (rads)-0.524Code error (chips)-1Figure 5-11, Dot product discriminator characteristicIt is common to use a first order loop to update the code phase estimate.

This is onlymade possible by the use of the Doppler aiding from the carrier tracking loop. Thecarrier loop is generally a second or third order loop, which accurately tracks the88Receiver theorydynamics of the system. Therefore, the system dynamics can be effectively removedby applying the Doppler estimated by the carrier loop to the DLL with sufficientscaling. The first order code loop update equation with carrier Doppler aiding can bewritten as follows.τˆ = τˆ + kφ × f φ + kτ × eτ5-19kω and kφ are constants calculated to provide the necessary open loop correction ofDoppler shift appropriately scaled down to the code rate and kτ is a loop gain whichcan be adjusted by the designer.

The dot-product discriminator is dependant onamplitude A2, a number of other DLL discriminators can also be used, each withdifferent dependencies and computational loads. A number of discriminators used toremove the amplitude dependence require generation of signals individuallymultiplied by early and late replica code sequences as follows.vIE (t )T = vI (t ) × a t − τˆ ± DC vIL (t )2 vQE (t )T = vQ (t ) × a t − τˆ ± DC vQL (t )2 5-20The resulting early and late correlations are written asT≈ A × cos φ − φˆ × Λτˆ m DC − τ wIL2wQET≈ A × sin φ − φˆ × Λτˆ m DC − τ wQL2wIE()()5-21Table 5-3 shows the most commonly used DLL discriminators, their error signals anddependencies.

The choice of discriminator must be made depending on thecomputational capability of the receiver and the signal environments in which it willoperate. Removing the sensitivity of the receiver to signal amplitude provides themost robust solution but maximises the microprocessor loading.89Receiver theoryTable 5-3, DLL discriminatorsDiscriminatorCoherentDot productError signalDependencyeτ = wIQ × wIISlopeproportional(Decision-directed)≈ A2 × cos 2 φ − φˆ × VΛ (τˆ − τ ) × Λ (τˆ − τ )eτ = wIQ × sgn (wII )()≈ A × cos φ − φˆ × Λ (τˆ − τ )VIncoherentDot producteτ = wIQ × wII + wQI × wQQ≈ A2 × VΛ (τˆ − τ ) × Λ(τˆ − τ )Decision-directedeτ = wIQ × sgn (wII ) + wQQ × sgn (wQI )( ( )())≈ A × cos φ − φˆ + sin φ − φˆ × VΛ (τˆ − τ )Power22eτ = wIE2 + wQE− wIL2 − wQL2≈ A2 × VΛ (τˆ − τ )Normalisedenvelopeeτ =≈22wIE2 + wQE− wIL2 + wQL22wIE2 + wQE+ wIL2 − wQLto A2Slopeproportionalto ASlopeproportionalto A2Slopeproportionalto ASlopeproportionalto A2Slopeindependent ofamplitudeΛV (τˆ − τ ) TT Λτˆ − DC − τ  + Λτˆ + DC − τ  22 Although listing discriminators may seem an exhaustive it is essential that receiverdesigners account for incoming signal amplitude variation in the loop designs.

Thesetables provide designers with essential information for providing robust tracking loopdesign. Equivalent tables are given for BOC signals tracking in Chapter 6. AppendixD shows practical examples of how these considerations can be critical to reliablereceiver operation.5.3Searching for BOC signalsA BOC signal is a square wave sub-carrier modulation of the coding sequence andcan be writtenb(t ) = a (t ) × s (t )5-2290Receiver theorywhere s(t) is the sub-carrier, which can have any phasing relative to the codesequence, although sine and cosine are the most common.

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