Analysis of the use of CSK for future GNSS signals (797922), страница 2
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The special characteristic of the CSKmodulation with respect to the typical orthogonal M-arysignaling is that each waveform (or symbol representing aset of input bits) is obtained from a different circularcyclic phase shift of a single fundamental PRN sequence.Moreover each circular cyclic phase shift is made by aninteger number of chips [9] and is assumed to be a fullperiod version of the fundamental sequence [11]. Figure 1provides a graphical explanation of the CSK modulation.II.B.
CSK Modulation Mathematical ModelEach single CSK symbol modulates U bits. Thenumber of circularly shifted versions of the fundamentalcode is equal to M, where M = 2U. The CSK fundamentalcode is called cd(t) and has a period length equal to Twhich spans over C chips. C is not necessarily equal to Mand the chip interval is equal to Tc.
From this fundamentalcode cd(t), the modulator generates the M circularlyshifted versions, which are called c0(t) to cM-1(t). Amathematical expression of a generic circularly shiftedversion of the code is shown below:( )[([([ ]])Figure 1: Example of CS K Modulation with M=4 (and U = 2 bits)(1)])(2)Figure 2: CSK modulator blockWhere mx is the integer number representing the codeshift of the xth symbol and mod(x,y) represents themodulus operation of y over x.The received signal at the receiver antenna output, v(t),can be modeled assuming the transmission of a CSKsignal through an AWGN channel as:( )( )()( )(3)Where A is the received signal amplitude and n(t) is theAWG noise with power equal to σ2.
A possible CSKmodulator block scheme is given in Figure 2.II.C. CSK Demodulator Output Mathematical ModelIn order to estimate which CSK symbol is transmitted,a matched filter should be implemented for eachcomponent (symbol) of the signal space basis [3]. For aCSK modulation, since each symbol is a circular shiftversion of the fundamental code, each matched filteroutput is equivalent to the evaluation of the correlationbetween the received signal and the fundamentalspreading sequence at a different shift, the bank ofmatched filters can be replaced by Fourier Transform andInverse Fourier Transform blocks which conduct thecorrelation in the frequency domain [12].
Figure 3 showsthe CSK FT-based demodulator block scheme andequation (4) shows the conducted mathematical operation:(( [ ])( [ ]))Where,is the normalized value of the circularcorrelation function of the fundamental spreadingsequence at point (k-x). Thevalue depends on thenature of the fundamental spreading sequence (Msequence, Gold, Kasami, etc) but it is always fulfilled that.are independent narrow-band Gaussian()noises with power equal to, and,isthe CSK symbol transmission rate.The noisesare assumed to be independent becausethe correlation between two different circular shiftedversions of the fundamental spreading code, is very low,.
In this paper, the cross-correlation value,, isassumed to be 0 for all andvalues with ().II.D. CSK bitsexpressionlikelihoodratiosmathematicalThe general expression of the likelihood ratio of the uthbit of an orthogonal CSK modulation at the ith interval is[13]:(4)The mathematical model of the normalizeddemodulator output at the ith interval (or instant), Yivector, can be modeled as:{Figure 3: CSK FFT-base demodulator block [12]∑((())(6)∑Where, ∑)(()()) represents the addition of all the(5)elements , evaluated by function ( ), which representa combination of bits where the uth bit is equal to b at theith instant ( = b).
( ) is the a-priori probability of :Mapping B: All the bits mapped by an orthogonalM-ary symbol belong to the same codeword. Thiscodeword source mapping provides the worstdemodulation/decoding performance [13] but thefastest access to the codeword.III. ADVANTAGES AND DRAWBACKS OF A CSKMODULATION WITH RESPECT TO A BPSKMODULATIONIn this section, the main advantages and drawbacks of aCSK modulated signal with respect to a BPSK modulatedsignal are presented.III.A.Figure 4: Codeword source mapping A (above) and codewordsource mapping B (below) of a CSK modulation()∏ ()(7)Where,is the value of the zth bit, bz, of the kth CSKthsymbol at the i instant.Equation (6) shows that depending on the a-prioriprobability of the different correlator outputs, thelikelihood ratio of the bits vary.
In fact, depending on thisprobability, more weigh is given to certain correlatoroutputs and thus, if this a-priori probability is reliable, thelikelihood ratio expression should also be more reliable inaverage. This means that a way to improve thedemodulation/decoding performance of a CSK signalconsists in determining reliable symbol a-prioriprobabilities.II.E. Codeword Source MappingThe codeword source mapping of an orthogonal M-arymodulation is defined as the mapping between the bitscarried by an orthogonal M-ary symbol and the bitsbelonging to a codeword.
The codeword source mappingis a very important element of an orthogonal M-arymodulation since it determines the codeword duration andthe signal demodulation performance. In this paper, twotypes of mappings are analyzed since they represent themost extreme cases. Both mappings are represented inFigure 4.Mapping A: Each bit mapped by an orthogonal Mary symbol belongs to a different codeword.Mapping A was shown to provide the bestdemodulation/decoding performance in [13] butrequires more time to access the codeword.AdvantagesThe first and most important advantage is thepossibility of implementing a non-coherent demodulationsince a CSK modulation is a kind of orthogonal M-arymodulation [3][9]. A non-coherent demodulation processconsists in demodulating the received signal withoutestimating the signal carrier phase by means of noncoherently adding the in-phase and quadrature-phasesignal components [3].
Therefore, a non-coherentmodulation may enable CSK signal demodulation inharsh environments (such as mobile channels representingurban or indoor environments) whereas for a BPSK signalthe demodulation would not be possible unless the PLLhas achieved lock. However, the exact gain of thisadvantage must be quantified (through simulations).The second advantage is that the symbol rate, chippingrate and PRN code length of a reference signal must notbe modified when the original coded bit rate is increased:the coded bit rate increase is simply achieved byintroducing a CSK modulation in the reference signal(instead of a BPSK modulation) or by increasing thenumber of bits mapped by a CSK symbol (within a limit).In fact, for a BPSK signal, the only possibility ofincreasing the coded bit rate consists in increasing thesymbol rate (decreasing the symbol period).
Therefore,two scenarios are possible. On one hand, the PRN codelength can remain constant but this implies that the chiprate must be increased. However, if the chip rate isincreased, the total signal bandwidth is increased, whichimplies the generation of interferences on the adjacentbands and the requirement of a wideband receiver withthe consequent increase of the number of operations. Onthe other hand, the PRN code length can be decreased butthis implies a degradation of the PRN code performance:isolation and near/far effect.The third most important advantage is the flexibility ofthe coded bit data rate provided by a CSK modulation: thecoded bit data rate of a CSK modulated signal candynamically change at any moment of the signaltransmission in order to be adapted to the kind ofbroadcasted data and its priorities (slow and thus morerobust for the ephemeris, clock error corrections, etc., andfaster for less essential information such as precisepositioning, etc.).
In fact, the CSK modulation shouldonly change the number of bits mapped by each symbol(or change the number of coherently accumulated PRNcodes as shown in section V.C-1). Moreover, the dynamicchange of the coded bit rate can follow a predeterminedstructure (signal known in advance) or could even bechanged on-the-fly (through some information providedby the signal itself). On the opposite side, the coded bitrate of a BPSK signal is fixed although some flexibilitycould be given by allowing the coherent accumulation ofconsecutive PRN codes.III.B.DrawbacksThe main drawback of a CSK modulated signal is thatthe synchronization process is extremely hard to achieve:due to each different PRN code cyclic shifted versionfound in each received symbol, the receiver cannot knowwhich cyclic shifted local replica must be generated inorder to synchronize the signal.
This means the receiverwould need first to demodulate the CSK signal. But thedemodulation process is not possible without firstacquiring the signal and tracking the code delay.Therefore, from the previous explained reasons, a CSKmodulated signal needs a pilot signal in order to achievethe synchronization required to demodulate (eithercoherently or non-coherently) the signal and to providethe essential pseudo-range measurements.The second drawback is the increase of the receivercomplexity, more specifically the demodulator part: theintroduction of a CSK modulation implies that instead ofonly using one correlator which output is fed to thedecoder/detector block, M correlators are necessary (withthe consequent complexity increase).
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