On Generalized Signal Waveforms for Satellite Navigation (797942), страница 51
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We can recognize that the integration in the numerator is takenas circular (auto) correlation over an interval of Tc given the repetitive nature of ci (t ) .Moreover we can see that this integration time tends to infinity. Since in reality the integrationintervals are longer than the code period but not infinite, we propose an alternative version ofthe autocorrelation function.
For the even case, this can be expressed as follows:MTc∫ c (t ) c (t + τ )dtiγMeven(τ ) =i0TcM ∫ ci (t ) dt2valid for τ ≤ MTc(6.15)0where for simplicity a total coherent integration equal to a multiple M of the code repetitioninterval was assumed. Additionally, since now the code sequence is only defined in theobservation interval MTc, no circular correlation is needed any more.230Spectral Separation Coefficients with data and non ideal codesFollowing the thoughts of previous chapter, let us assume now an ideal random code of lengthTc (or in a relaxed form, the average of the codes of the set). The linear autocorrelationsequence in this case is not periodic and adopts the following form:⎧M − n⎪Mfor τ = nTc and n integer with n ≤ M(6.16)γ even (τ ) = ⎨ M⎪⎩ 0for all other τThis is shown graphically in the next figure for the specific case that the code repeats 20 timeswithin one data chip as is the case with the GPS C/A Code.Figure 6.19.
GPS C/A Code Autocorrelation FunctionSince the power spectral density is the Fourier Transform of the autocorrelation, we have:MT∞M ⎧M − n ⎫− jωτ− jωτMMΓeven (ω ) = ∫ γ even (τ ) edτ = ∫ ∑ ⎨dτ(6.17)⎬ δ (τ − nTc ) eM ⎭−∞− MT n = − M ⎩This expression can be further expanded into three parts since the integral above can beevaluated at values of τ = nTc , including n = 0 , as follows:⎧0− jωτdτ⎪ ∫ δ (τ ) e⎪0 −⎪ 0 − −1⎧M − n ⎫⎪M− jωτdτΓ even (ω ) = ⎨+ ∫ ∑ ⎨⎬ δ (τ − nTc ) eM⎭⎪ − MT n = − M ⎩⎪ MT M⎧M − n ⎫⎪+δ (τ − nTc ) e − jωτ dτ∑∫⎪ + n =1 ⎨⎩ M ⎬⎭⎩ 0+MT M −1(6.18)M −1⎧M − n⎫⎧M − n⎫− jωτjωτ−+=+δ(τnT)eedτ12⎨⎬⎨⎬ cos(nωTc )∑c∫+ ∑M ⎭⎭n =1 ⎩ M0 n =1 ⎩(6.19)where we have assumed positive values of n with n ≤ M . It is important to note that when theMΓeven(ω ) = 1 +{}code rate and the data rate are equal, M =1 and the code spectrum is flat.MIf we take a closer look at equation (6.19), we can clearly recognize that the function Γeven(ω )is not square integrable and would thus be difficult to use alone.
As we know, a real orcomplex function is square-integrable if the integral of the square of its absolute value over231Spectral Separation Coefficients with data and non ideal codesthe interval is finite and thus belongs to the Hilbert space L2. This is a necessary condition toMapply the Fourier theory with correctness. Accordingly Γeven(ω ) can not, on its own, representa PSD since the integral over frequency is not finite.
However, as we will see next, used incombination with other functions offering appropriate frequency behaviour, we will beallowed to use it.The origin of this effect lies in the definition of the ACF for periodic functions. There areways to circumvent this problem using, for example, the ACF of m-sequences.
Indeed, thesecan be taken as an approach to purely random sequences since the ACF could be arranged tohave an integrated value of zero, that is 1 in phase and -1/n out of phase. This could also bearranged for the periodic ACF by ensuring that the average of the ACF over all delays isexactly 0 and there is thus no DC component. The result would be then a function that wouldbe square integrable but the conclusions would remain unaffected.M(ω ) with M = 20 , is shown in the next figure. It must beThe power spectral density for Γevennoted that it is not normalized and thus the maxima have a value of 10log10(20).Figure 6.20.
Power Spectral Density (not normalized) of an ideal code sequence thatrepeats 20 times within one data bitIt is interesting to note that the obtained power spectral density is the result of convoluting aDirac comb with separation 1 kHz, with a sinc of f d = 50 Hz. Indeed, equation (6.19) can berewritten as shown next:2⎡ ⎛ πf ⎞ ⎤⎢ sin ⎜⎜ ⎟⎟ ⎥M −1∞⎛⎞2πMnnf−⎧⎫M⎢ ⎝ f d ⎠ ⎥ ⊗ δ ( f − l Δf ) (6.20)⎜⎟( f ) = 1 + 2∑ ⎨cosΓevenf=⎬∑dc⎜ f⎟⎢ (πf ) ⎥⎭n =1 ⎩ Ml = −∞c⎝⎠⎥⎢⎥⎦⎢⎣being Δf the separation of the deltas of the Dirac comb or 1 kHz in our particular example.Next, the Dirac comb is shown in detail.232Spectral Separation Coefficients with data and non ideal codesFigure 6.21.
Power spectral Density for ideal code sequence with M=20As we can see, the frequency components repeat at intervals of 1/Tc, being Tc = 1 ms sincethis is the period of the C/A code. As it is trivial to show, this Dirac frequency comb is theFourier transformation of a similar comb in the time domain according to the followingexpression:l ⎞⎧ ∞⎫ 1 ∞ ⎛(6.21)F ⎨ ∑ δ (τ − nTc )⎬ =δ ⎜⎜ f − ⎟⎟∑Tc ⎠⎩l =−∞⎭ Tc l =−∞ ⎝More important even, the magnitude of each line component is the same, unlike in actual C/Asequences where the magnitude of the lines can be larger or smaller than the average valuedue to specific non-zero values of the autocorrelation function for τ ≠ 0 .A very important conclusion can also be obtained from Figure 6.21. In fact, The Dirac combwith separation Tc, being Tc the duration of the C/A code and thus 1 millisecond, results fromthe convolution of the linear autocorrelation function of Figure 6.19 with a train of Diracdeltas separated by 20 milliseconds.
This value comes from the fact that M=20 is the numberof times that the C/A code repeats within one data bit. This is shown in the next figure:Figure 6.22. Convolution between the linear even correlation and the even train of Diracpulses to form the even periodic correlation233Spectral Separation Coefficients with data and non ideal codesSince this function is the key to form the PSD of the signal modulated with data and a shortcode in the even correlation case, it is important to remember it when we work with its oddcounterpart.
Moreover, since the train of Dirac deltas in the case of the odd correlation will beslightly different, we denote this train in the even case as even train of Dirac deltas.In addition, although for both even and odd correlations we have to work with periodiccorrelations, we have shown in previous lines that the linear correlation in conjunction withthe particular train of Dirac deltas contains all the necessary information to derive thecorrelation that is seen by the receiver at the output. Indeed, the previous Figure 6.22 reflectsnothing else than the following mathematical identity:∞∞M ⎧M − n ⎫δ(τ−nT)⊗δ(τ−MnT)=(6.22)∑⎨ M ⎬∑∑ δ (τ − nTc )ccn=− M ⎩n = −∞n = −∞⎭being thus the result of the convolution a constant discrete function with pulses separated 1millisecond. As in previous chapters, the operator ⊗ refers to the convolution.6.2.1.3Odd Autocorrelation Function of Quasi Ideal CodesAs we have seen in Figure 6.18, the even correlation occurs when the data bits do not changeof sign during the integration, while the odd corresponds to the case that the bits flip duringthe coherent integration.
This case would happen indeed, when the data bits are not perfectlyaligned at receiver level.In the previous pages we have shown that the even correlation can be expressed by means ofthe linear correlation of the code sequence that results from replicating the primary codetwenty times one after each other with the same data sign.
This is so because the interferingsignal is assumed not to flip any of its data bits during the correlation integration as thecorrelation is even.Now, for the case of the odd correlation, the situation is similar but additional care has to bepaid since the odd correlation implies by definition that the data bit flips within the integrationtime. Given the fact that the primary C/A code repeats twenty times within one data bit, theposition of the data flip can adopt in principle any of the twenty possible positions.Accordingly, the linear correlation can also adopt twenty different shapes.
As a conclusion,since the distribution of the data flip over the 20 possible locations p is uniformly distributed,the odd linear correlation will be the average of the 20 linear correlations that are possibledepending on where the data flip actually occurs.The twenty possible cases are summarized in the following table:234Spectral Separation Coefficients with data and non ideal codesTable 6.7.
Relative position of the data flip of the interfering signal with respect to thedesired replicap1234567891011121314151617181920s (τ = 1)11111111111111111111s (τ = 2)1111111111111111111-1s (τ = 3)111111111111111111-1-1s (τ = 4)11111111111111111-1-1-1s (τ = 5)1111111111111111-1-1-1-1s (τ = 6)111111111111111-1-1-1-1-1s (τ = 7)11111111111111-1-1-1-1-1-1s (τ = 8)1111111111111-1-1-1-1-1-1-1s (τ = 9)111111111111-1-1-1-1-1-1-1-1s (τ = 10)11111111111-1-1-1-1-1-1-1-1-1s (τ = 11)1111111111-1-1-1-1-1-1-1-1-1-1s (τ = 12)111111111-1-1-1-1-1-1-1-1-1-1-1s (τ = 13)11111111-1-1-1-1-1-1-1-1-1-1-1-1s (τ = 14)1111111-1-1-1-1-1-1-1-1-1-1-1-1-1s (τ = 15)111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1s (τ = 16)11111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1s (τ = 17)1111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1s (τ = 18)111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1s (τ = 19)11-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1s (τ = 20)1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1Delay τwhere s denotes the data sequence of the interfering signal.
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