On Generalized Signal Waveforms for Satellite Navigation (797942), страница 52
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As we can see, τ =1 denotes thatthe flip transition takes place in the very last primary code and consequently the sequence sconsists of only 1s.As we can recognize from the previous table, we can make use of symmetry properties tosimplify the number of linear autocorrelations that we have to consider. In fact, there are only9+2 different linear autocorrelations since it is trivial to prove thatACFlinear {s(τ = 21 − i )} = ACFlinear {s(τ = i + 1)}(6.23)with i ∈ (1,2,...,9) . Moreover, ACFlinear {s (τ = 1)} and ACFlinear {s (τ = 11)} repeat only one time.Taking this into account, it can be shown that the average autocorrelation function for the oddcase adopts the following form:235Spectral Separation Coefficients with data and non ideal codesFigure 6.23.
Average odd linear autocorrelationOnce we have the average linear autocorrelation for the odd case in all its possible cases, wecan follow the same logic as for the even case. To do so, we correlate next the average linearodd autocorrelation with the odd train of Dirac pulses. As we remember, the even train ofDirac pulses was characterized by having all pulses of the same normalized amplitude +1since there was no change in data bit. The odd case, however, is actually characterized by thefact that the data bit alternates from one bit to the other in order to always have oddcorrelation.
Accordingly, the odd train of Dirac pulses adopts the following form:∞∑ (− 1) δ (τ − MnT )n = −∞nc(6.24)such that pulses of amplitude +1 and -1 alternate in a separation of M=20 milliseconds. As aresult, the convolution between the odd train of Dirac pulses and the average odd correlationadopts the following form:Figure 6.24. Convolution between the linear odd correlation and the odd train of Diracpulses to form the odd periodic correlation236Spectral Separation Coefficients with data and non ideal codesAs a result, the convoluted function adopts the following form:Figure 6.25.
Normalized periodic odd correlation with M=20If we calculate now the Fourier transform of this signal, we can observe that while thespectrum of the even correlation allocates all its power around frequencies multiple of 1 kHz,most of the power is concentrated in the case of the odd correlation at ± 50 Hz aroundfrequencies multiple of 1 kHz.
We can also recognize that the Odd Code Spectrum also haspower at other frequencies ± 50 k with k = ±[2,...10] but of considerably lower amount.Indeed, the spectral lines at ± 50 Hz amount slightly more than 81% of the total power, beingthe rest distributed among the other spectral lines. Accordingly, we can state that while theEven Code spectrum places all its power at multiples of 1 kHz, the Odd concentrates most ofit at ± 50 Hz around frequencies multiple of 1 kHz.Figure 6.26. Normalized Even and Odd Code Power spectral Density for M=20The spectral coefficients of the previous figure can be easily obtained if we recall that the oddautocorrelation function for GPS C/A is periodic with period T0 = 40Tc, being Tc the period ofthe primary code, that is 1 millisecond for the C/A code.
Indeed, the odd autocorrelation canbe expressed by means of a Fourier series as shown next:1 ∞(6.25)ACFodd (τ ) =a k e jkω0τ∑T0 k = −∞237Spectral Separation Coefficients with data and non ideal codeswhere ω0 = 2π T0 = π 20Tc since T0 = 40Tc . Accordingly, the complex coefficients of theFourier series are shown to adopt the following form:ak = ∫20Tcτ = −20TcACFodd (τ )e − jkω0τSince the odd autocorrelation function is shown to be expressed as follows:M −1 M − 2 nACFodd (τ ) = ∑δ (τ − nTc )Mn=− Mwith M = 20, the spectral coefficients are then further shown to simplify to:9π ⎤4l⎡a k = 2 + ∑ cos ⎢k (10 − l ) ⎥20 ⎦⎣l =1 10(6.26)(6.27)(6.28)for odd values of k, where k adopts values between -19 and 19, with ak = a-k.
In general:M−12ak = 2 + ∑l =18lπ ⎤⎡kcos ⎢ (M − 2l ) ⎥MM⎦⎣2(6.29)It is also trivial to show that all the even coefficients are of value 0. Normalizing now thecoefficients to the total power of the signal,99π ⎤π ⎤4l4l⎡⎡2 + ∑ cos ⎢k (10 − l ) ⎥ 2 + ∑ cos ⎢k (10 − l ) ⎥20 ⎦20 ⎦⎣⎣l =1 10l =1 10(6.30)ak ==1940∑ akk = −19k oddwe obtain the normalized spectrum.
For a generic value M, this expression is shown to adoptthe following form:M−12M−12π ⎤π ⎤8l8l⎡k⎡k2 + ∑ cos ⎢ (M − 2l ) ⎥ 2 + ∑ cos ⎢ (M − 2l ) ⎥20 ⎦20 ⎦⎣2⎣2l =1 Ml =1 M=ak =M2M∑ ak(6.31)k =− Mk oddConvoluting now the previous odd code spectrum with the ideal data sinc of 50 Hz, we obtainthe following shape for the odd code spectrum with data:Figure 6.27. Odd Power Spectral Density [dB] of an ideal code sequence that repeats 20times in one data bit238Spectral Separation Coefficients with data and non ideal codes6.2.1.4Combined Spreading Waveform PSDNow that we have obtained the expressions for the chip waveform PSD and for the codespectrum, the PSD of a BPSK(fc) with ideal PRN codes will be:Γ( f ) = Γ M ( f ) ΓS ( f )(6.32)where we have seen that:ΓS ( f ) = G BPSK ( f c )⎛ πfsin 2 ⎜⎜⎝ fc= fc(πf )2⎞⎟⎟⎠(6.33)and⎡ ⎛ πf⎢ sin ⎜⎜fΓM ( f ) = fd ⎢ ⎝ d⎢ (πf )⎢⎣⎢6.2.22⎞⎤⎟⎟ ⎥∞⎠ ⎥ ⊗ δ (ω − 2 π l Δf )∑⎥l = −∞⎥⎦⎥(6.34)Spectral Separation coefficients for short PRNcodesAs we have seen at the beginning of this chapter, the approximation of using the inner productto calculate the spectral separation coefficient is only valid if the code modulating the signal issufficiently long to smooth the spectrum.
Otherwise, the correct definition to use should bethe one derived in (5.8). We recall it to help the understanding in the next pages:Ψid = TI ∫∞−∞∫∞−∞∞Gi ( f1 )Gd ( f 2 − f1 )df1 H ID ( f 2 ) df 2 = TI ∫ Gi ( f 2 ) ⊗ Gd ( f 2 ) H ID ( f 2 ) df 222−∞(6.35)where the filter function is given by:H ID ( f 2 )FT {w(t )}1 ⎡ sin (π f TI ) ⎤== 2⎢⎥TITI ⎣πf⎦222(6.36)Moreover we have assumed a perfect windowing of the incoming signal by means of therectangular function w(t), defined as follows:⎧1 0 ≤ t ≤ TIw(t ) = ⎨⎩0 otherwise(6.37)239Spectral Separation Coefficients with data and non ideal codesGraphically, the filter function adopts the following form:Figure 6.28. Power Spectral Density Convolution of the integration filter functionEqually, we show in the next figure the result of convoluting two BOC(1,1) signals withSVN 1 and SVN 2.Figure 6.29. Power Spectral Density Convolution of two BOC(1,1) signals modulatedrespectively by SVN 1 and SVN 2In addition, if we plot the convolution density function together with the filter function of 1second of coherent integration, we obtain a sinc function with a frequency rate of 1 Hz asshown in the next figure.
As we can recognize, the bandwidth of this last signal comparedwith that of the convolution function is significantly narrower.240Spectral Separation Coefficients with data and non ideal codesFigure 6.30. Power Spectral Density of the Convolution of two BOC(1,1) signals withSVN 1 and SVN 2 and a receiver filter function of 1 ms of coherent integrationFurthermore, if we integrate the product of the convolution function shown above and theintegration filter of 250 Hz (4 ms of integration), we obtain an SSC value of -64.9505 dB-Hzwhich is very close to the one that we would obtain applying the SSC simplified model of(6.38).
Indeed, when we integrate for relatively long periods of time, the filter function getsnarrower tending to a Dirac delta. Accordingly, (6.35) converges to the value we would obtainif we would take the value of the convolution at zero offset. In such a case, the SSC generaldefinition from above simplifies to the well known expression that we used in chapter 5:∞Ψid ≈ TI ∫ Gi ( f )Gd ( f ) H ID ( f ) df−∞∞2f I =0= ∫ Gi ( f )Gd ( f ) df−∞(6.38)Equally, for the C/A code we show in the next figure the convolution density functiontogether with the filter transfer function for an integration of 1 second. As we can recognize,the sinc function is already very narrow and (6.38) is a good approximation to the real SSCcomputed using (6.35).
In fact, the real SSC takes a value of -61.5915 dB-Hz while theapproximation of (6.38) yields -61.8597 dB-Hz. It is important to note that no normalizationwas made. The result of convoluting two C/A Code spectra with different codes is shown inthe following figure together with the filter transfer function:Figure 6.31. Power Spectral Density Convolution of two BPSK(1) signals modulatedrespectively by SVN 1 and SVN 2 and receiver filter function241Spectral Separation Coefficients with data and non ideal codesThe C/A Code is a particular case due to its low data rate. If we now account for the effect ofthe data, similar figures can be obtained:Figure 6.32.
Power Spectral Density Convolution of two BPSK(1) signals modulatedrespectively by SVN 1 and SVN 2 with ideal random data on topIf we use now a filter of 1 kHz (integration of 1 ms instead of 1 s), the resulting spectralseparation coefficient of (6.35) seems to deliver results close to those of the SSCapproximation of (6.38).
We conclude that even for very short integrations, the approximationseems to remain valid. Moreover, the transfer filter has to be multiplied by the factor TIbecause otherwise the filter would amplify or attenuate.Now that we have already studied the effect of data, non-ideal codes and coherent integrationon the power spectral densities, it is time to assess the effect of the Doppler shift between thedesired signal and the interfering signal on the Spectral Separation Coefficients.
To begin weshow in the next figure the self SSC of the C/A Code (SVN 1 and SVN 2) as a function ofDoppler when a data stream of 50 sps is considered. As we can see, the worst cases occur atshifts multiple of 1 kHz.Figure 6.33. C/A Code Self SSC between SVN 1 and SVN 2 as a function of Doppler.
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