CN102680985A - Phase discrimination method for suppressing multipath signals in global navigation satellite system - Google Patents

Phase discrimination method for suppressing multipath signals in global navigation satellite system Download PDF

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CN102680985A
CN102680985A CN2012101673457A CN201210167345A CN102680985A CN 102680985 A CN102680985 A CN 102680985A CN 2012101673457 A CN2012101673457 A CN 2012101673457A CN 201210167345 A CN201210167345 A CN 201210167345A CN 102680985 A CN102680985 A CN 102680985A
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CN102680985B (en
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杨东凯
刘德荣
李伟强
张波
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SHANDONG GUANG'AN CHELIAN TECHNOLOGY CO.,LTD.
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Beihang University
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Abstract

The invention discloses a phase discrimination method for suppressing multipath signals in the global navigation satellite system and belongs to the technical field of global navigation satellite systems. The method includes: receiving GNSS (global navigation satellite system) signals containing the multipath signals and performing analog-digital conversion after down-conversion is performed to generate low-intermediate frequency signals; and performing carrier stripping for the GNSS signals subjected to the analog digital conversion to obtain signals of a path I and a path Q and performing code stripping for the signals of the path I and the path Q. In a code tracking loop during GNSS signal processing, after cross-correlation power of the multipath signals is obtained, the multipath signals can be suppressed effectively by adopting the phase discrimination method based on precise TK (teager-kaiser) sample intervals, and precision improvement of a GNSS receiver can be achieved.

Description

The phase detecting method that in GLONASS, suppresses multipath signal
Technical field
The present invention relates to the GLONASS technical field, particularly a kind of phase detecting method that in GLONASS, suppresses multipath signal.
Background technology
In the application of GLONASS (GNSS, Global Navigation Satellite System), the raising of receiver performance is the prerequisite of promotion and application industry.Be the receptivity that improves navigation signal, need filtering via different reflectings surface or multipath signal through repeatedly reflecting.Choking coils, the narrow technology such as relevant of adopting suppress multipath signal more in fields such as mapping; And, adopt the special left-handed antenna that receives reflected signal that the reflected signal that receives is handled, and then remote sensing is carried out in ocean, land etc. in fields such as remote sensing.
Multipath is one of main source of error among the GNSS, reduces and postpones estimation technique although developed multiple multipath, and short to postpone the application of multipath in high precision, indoor and intensive urban environment still serious.In these zones, the quantity in medium path is not only many, and has by the compact arranged caused dynamic of natural forms fixing and mobile reflection, for example people, plant, building structure and the vehicle around the receiving antenna.Therefore, need a kind of high-precision disposal route to come effectively to suppress the GNSS multipath signal, and then reach the effect that improves GNSS receiver precision.
Summary of the invention
In view of this, the object of the present invention is to provide a kind of phase detecting method that in GLONASS, suppresses multipath signal, be used for effectively suppressing the GNSS multipath signal, reach the effect that improves GNSS receiver precision.
The invention provides a kind of phase detecting method that in GLONASS, suppresses multipath signal, comprising:
Reception comprises the global navigation satellite system GNSS signal of multipath signal, carries out carrying out analog to digital conversion after down coversion generates the Low Medium Frequency signal; To pass through GNSS signal after the analog to digital conversion and carry out carrier wave and peel off, obtain the signal on I road and Q road, and said I road and Q road signal carried out sign indicating number peel off.
The present invention proposes a kind of phase detecting method that in the GNSS signal, suppresses multipath signal; In the code tracking loop of GNSS signal processing; After obtaining the simple crosscorrelation power of multipath signal; Employing can suppress multipath signal effectively based on the phase detecting method in accurate TK SI, can reach the purpose that improves GNSS receiver precision.
Description of drawings
Fig. 1 is the structural principle synoptic diagram of GNSS signal receiver in the embodiment of the invention;
The phase detecting method process flow diagram that in GNSS, suppresses multipath signal that Fig. 2 provides for the embodiment of the invention;
Fig. 3 is for carrying out the circuit theory synoptic diagram that sign indicating number is peeled off to I road and Q road signal in the embodiment of the invention;
Fig. 4 is the code tracking error envelope diagram that uses the TK-EML5 phase detector in the embodiment of the invention;
Fig. 5 is the code tracking error envelope diagram that uses the EML phase detector in the embodiment of the invention.
Embodiment
For making the object of the invention, technical scheme and advantage clearer, the present invention is made further detailed description below in conjunction with accompanying drawing.
The embodiment of the invention has proposed a kind of phase detecting method that in the GNSS signal, suppresses multipath signal; In the code tracking loop of GNSS signal processing; After obtaining the simple crosscorrelation power of multipath signal; Employing can suppress multipath signal effectively based on the phase detecting method in accurate TK SI, can reach the purpose that improves GNSS receiver precision.
As shown in Figure 1, the GNSS signal receiver mainly is made up of antenna, radio-frequency front-end, high-speed AD converter, FPGA correlator, DSP digital signal processor and storer.The method of reseptance process flow diagram of the high-precision GNSS signal that Fig. 2 provides for present embodiment comprises:
Step 201, antenna receive the GNSS signal.The signal that the GNSS receiver antenna receives comprises direct signal and multipath signal, and direct signal is directly to pass to the signal of receiver from satellite; Multipath signal is that antenna is except receiving from the GNSS satellites transmits after the electromagnetic wave signal of rectilinear propagation; Also possibly receive one or more by this electromagnetic wave through around signal behind the clutter reflections, and each reflected signal maybe be through reflection back arrival antenna once or repeatedly.
Step 202, the GNSS signal that comprises multipath signal get into radio-frequency front-end and carry out down coversion, generate the Low Medium Frequency signal, and carry out analog to digital conversion.
GNSS signal after step 203, the process analog to digital conversion carries out carrier wave to be peeled off, and obtains the signal on I road and Q road.Because the GNSS signal is in generative process; At first be that navigation information and C/A sign indicating number are modulated (multiplying each other); The information that generates is modulated with carrier wave again; So will carry out demodulation (peeling off of carrier wave is exactly the demodulation to carrier wave in fact) to carrier wave after receiving, to will obtain the signal on I, Q road in the demodulating process of carrier wave because of phase relation.
Step 204, I road and Q road signal are carried out sign indicating number peel off.Referring to Fig. 3, sign indicating number is peeled off specifically and may further comprise the steps:
Step 2041, duplicate one through the C/A code generator and have the C/A coded signal of same phase with I road and Q road signal; And be 5 road C/A sign indicating numbers of 1/4 chip through the distance that postpones to obtain between the adjacent two-way C/A, be respectively: Very Early road, Early road, Prompt road, Late road, Very Late road.
Step 2042,5 road C/A sign indicating numbers and said I road and Q road signal are carried out related calculation.Specifically comprise:
Because the signal that receives of GNSS receiver comprises direct signal and multipath signal, ignore that (s (t) is the GNSS signal, s under the situation of noise d(t), s m(t) represent direct signal and multipath signal in the GNSS signal respectively):
s(t)=s d(t)+s m(t)
=Ag[(1+ξ)t-γ]exp{t[(ω dc)t+φ 1]}+αAg[{(1+ξ)t-γ-Δτ m]exp{t[(ω dc)t+φ 1+β]}
(1)
Wherein, ω cBe carrier frequency,
Figure BDA00001685520100031
ω wherein dBe Doppler shift, φ 1Be carrier phase.A is the amplitude of GNSS direct signal, and g (t) is ± 1, the value of expression C/A sign indicating number, and α is the ratio of multipath signal amplitude and direct signal amplitude, Δ τ mTime delay for the relative direct signal of multipath signal.β=(ω d+ ω c) Δ τ m+ Δ θ mBe the multipath signal phase compensation that causes because of reflection, Δ θ mBe the phase deviation that causes because of reflection, t is the time.J is the imaginary number scalar, and γ is the time delay of original signal, is to be caused to the distance the receiver by the GNSS satellite.
After through the radio-frequency front-end down coversion, following formula becomes:
s db(t)=Ag[(1+ξ)t-γ]exp(jψ)+αAg[(1+ξ)t-γ-Δτ m]exp[j(ψ+β)]
(2)
Wherein,
Figure BDA00001685520100041
is the phase differential between direct signal phase place and the multipath signal phase place.
Figure BDA00001685520100042
is the phase place of multipath signal.
After result's process analog to digital conversion of formula (2) and carrier wave are peeled off, will generate I road and Q road signal, the equally spaced 5 road C/A sign indicating numbers synthetic with this locality (are respectively Very Early roads; The Early road; The Prompt road, Late road, Very Late road) carrying out related calculation obtains following formula:
I VE = A 2 R ( τ + Δ ) cos ( ψ ) + αA 2 R ( τ + Δ - Δ τ m ) cos ( ψ + β )
I E = A 2 R ( τ + Δ 2 ) cos ( ψ ) + αA 2 R ( τ + Δ 2 - Δτ m ) cos ( ψ + β )
I P = A 2 R ( τ ) cos ( ψ ) + αA 2 R ( τ - Δτ m ) cos ( ψ + β )
I L = A 2 R ( τ - Δ 2 ) cos ( ψ ) + αA 2 R ( τ - Δ 2 - Δτ m ) cos ( ψ + β )
I VL = A 2 R ( τ - Δ ) cos ( ψ ) + αA 2 R ( τ - Δ - Δ τ m ) cos ( ψ + β )
Q VE = A 2 R ( τ + Δ ) sin ( ψ ) + αA 2 R ( τ + Δ - Δ τ m ) sin ( ψ + β )
Q E = A 2 R ( τ + Δ 2 ) sin ( ψ ) + αA 2 R ( τ + Δ 2 - Δτ m ) sin ( ψ + β )
Q P = A 2 R ( τ ) sin ( ψ ) + αA 2 R ( τ - Δ τ m ) sin ( ψ + β )
Q L = A 2 R ( τ - Δ 2 ) sin ( ψ ) + αA 2 R ( τ - Δ 2 - Δτ m ) sin ( ψ + β )
Q VL = A 2 R ( τ - Δ ) sin ( ψ ) + αA 2 R ( τ - Δ - Δ τ m ) sin ( ψ + β ) - - - ( 3 )
Wherein, A is the amplitude of direct signal, and R is the cross correlation function of two-way C/A sign indicating number, and τ is the code delay difference between direct signal and the multipath signal, and Δ is the distance between Early road and the Late road.
The value of step 2043, related operation is through getting into the code phase Discr. behind integration-remover, whether the amplitude that the code phase Discr. detects the value of related operation reaches maximum, and therefrom estimates the phase differential that duplicates between C/A sign indicating number and the reception C/A sign indicating number.
Present embodiment adopts TK-EML5 (Teager-Kaiser Early-Minus-late 5; Leading and the hysteresis road of Teager-Kaiser; 5 expressions all need 5 correlators for I road and Q road) the phase demodulation mode, TK-ELM5 phase demodulation mode and general ELP (Early-Late Power Discriminatory subtracts after-power type phase detector in advance) phase demodulation mode are compared; Can suppress multipath signal effectively, and then reach the effect that improves GNSS receiver precision.
Want to obtain accurately final navigation message, just should fall C/A sign indicating number perfect restitution, according to the character of C/A sign indicating number; Want to fall the demodulation of C/A sign indicating number fully; Just require local synthetic C/A sign indicating number in full accord with the C/A phase place that receives, and can not be in full accord in the actual conditions, the unanimity of can only trying one's best.The output valve of TK-EML5 be exactly the C/A sign indicating number that duplicates with respect to the phase difference value that receives the C/A sign indicating number, specifically comprise:
Carry out following computing for discrete signal x (n) arbitrarily:
Ψ d [ x ( n ) ] = x ( n ) x * ( n ) - 1 2 [ x ( n - 1 ) x * ( n + 1 ) + x ( n + 1 ) x * ( n - 1 ) ] - - - ( 4 )
Wherein, x *(n) conjugate function of expression x (n), x (n-1) and x (n+1) are respectively x (n) the move to right function of a unit, the unit that moves to left, in like manner x *(n+1) and x *(n-1) be respectively the conjugate function of x (n+1) and x (n-1).
Code phase Discr. based on TK-EML5 is output as:
D TKE(τ)=Ψ EL (5)
Wherein, Ψ EAnd Ψ LBe respectively that (concrete here is a setting with the cross correlation function that receives C/A sign indicating number in the signal respectively for the C/A sign indicating number on local Early road of synthesizing and the C/A sign indicating number on Late road; The character decision of C/A sign indicating number; Two-way C/A sign indicating number multiplies each other; What obtain is exactly a related function, a triangular waveform) through the result behind the code phase Discr..Adopt 5 correlators respectively for I road signal and Q road signal in the present embodiment, adopted accurate sampling clock T here t=Δ/2, Δ are the distance between Early road and the Late road, are the half-chip length of C/A.
Local synthetic Early road C/A sign indicating number and the cross correlation function
Figure BDA00001685520100061
that receives signal and local synthetic Late road and the cross correlation function
Figure BDA00001685520100062
(result that two-way C/A sign indicating number is relevant can use R (τ) to represent) that receives signal computing through (4) formula in the code phase Discr. obtain following result:
ψ E ( τ ) ∝ R ( τ + Δ 2 ) R * ( τ + Δ 2 ) - 1 2 [ R ( τ ) R * ( τ + Δ ) + R * ( τ ) R ( τ + Δ ) ] - - - ( 6 )
ψ L ( τ ) ∝ R ( τ - Δ 2 ) R * ( τ - Δ 2 ) - 1 2 [ R ( τ ) R * ( τ - Δ ) + R * ( τ ) R ( τ - Δ ) ] - - - ( 7 )
In the present embodiment, having adopted 5 correlators respectively for I road signal and Q road signal, is respectively Very Early road, the Early road, and the Prompt road, the Late road, Very Late road, the distance between the adjacent two-way is Δ/2, just 1/4 C/A chip lengths; Distance between Early road and the Late road is a Δ, just 1/2 C/A chip lengths; Distance between Very Early road and the Very Late road is 2 Δs, just 1 C/A chip lengths.So Ψ EAnd Ψ LCan be expressed as following formula:
ψ E ( τ ) = ( I E + j Q E ) ( I E - j Q E ) - 1 2 [ ( I P - j Q P ) ( I VE + j Q VE ) + ] ( I P + j Q P ) ( I VE - j Q VE ) - - - ( 8 )
So ψ E ( τ ) = I E 2 + Q E 2 - ( I VE * I P + Q VE * Q P ) - - - ( 9 )
ψ L ( τ ) = ( I L + j Q L ) ( I L - j Q L ) - 1 2 [ ( I P - j Q P ) ( I VL + j Q VL ) + ] ( I P + j Q P ) ( I VL - j Q VL ) - - - ( 10 )
So ψ L ( τ ) = I L 2 + Q L 2 - ( I VL * I P + Q VL * Q P ) - - - ( 11 )
So the code phase Discr. is output as:
D TKE ( τ ) = ( I E 2 + Q E 2 ) - ( I L 2 + Q L 2 ) - ( I VE * I P + Q VE * Q P ) + ( I VL * I P + Q VL * Q P ) - - - ( 12 )
The value of formula (12) is the phase difference value between time channel C/A sign indicating number and the input signal C/A sign indicating number videlicet.
Behind step 2044, the code phase difference process sign indicating number ring wave filter, through assisting of carrier loop, as the control input of C/A yardage controlled oscillator; Last C/A code generator is in time regulated frequency of duplicating the C/A sign indicating number and the phase place of being exported under the driving of yardage controlled oscillator, make to duplicate that the C/A sign indicating number is consistent constantly in C/A sign indicating number and the input signal.
More than through duplicate one with receive signal in pseudo-code (the being defaulted as the C/A sign indicating number here usually) pseudo-code that phase place is consistent; Let then acknowledge(ment) signal with duplicate pseudo-code and multiply each other relevant; Peeling off the pseudo-code that receives in the signal, and therefrom obtain GNSS and locate this important measured value of necessary pseudorange.
In GNSS receiver code track loop; The TK-EML5 phase detecting method is based on a kind of phase detecting method of TK algorithm; It is compared with general ELP phase detecting method, can under the situation of not knowing multipath signal information, estimate the multidiameter distance of multipath signal with respect to direct signal; And then in the phase demodulation process, effectively suppressed multipath signal, and then improved the precision of GNSS receiver.
Fig. 4 and Fig. 5 are respectively the TK-EML5 phase detector that emulation comes out and the multipath error envelope diagram of ELP phase detector.The multipath error envelope diagram is to show the figure that is caused code tracking limit of error value by multipath.Horizontal ordinate among the figure is a multidiameter, and ordinate is a tracking error.Solid line among the figure is the code phase measuring error of the spacing between leading road (Early) and hysteresis road (Late) when being 1/4 chip; Dotted line is the code phase measuring error of the spacing between leading road (Early) and hysteresis road (Late) when being 1/2 chip, and dotted line is the code phase measuring error of the spacing between leading road (Early) and hysteresis road (Late) when being 3/4 chip.Code phase measuring error under the effect of homophase reflection wave is that the figure intermediate value is a positive broken line; Code phase measuring error under the effect of anti-phase reflection wave is that value is a negative broken line; And when the transmitted wave carrier phase was worth for other, the code phase measuring error was then at this up and down between two broken lines.Can find out that by Fig. 4 and Fig. 5 for same multipath situation, if the correlator spacing that the sign indicating number ring is adopted is narrow more, then code phase multipath measuring error is just more little.Can know by Fig. 4 and Fig. 5 contrast; Under the square one; Because TK-EML5 phase detector code tracking error envelope is littler than ELP phase detector code tracking error envelope; Therefore aspect the inhibition of multipath signal, multi-path signal inhibition aspect especially to delaying time short, the TK-EML5 phase detector has better effect than ELP phase detector.
In a word, the above is merely preferred embodiment of the present invention, is not to be used to limit protection scope of the present invention.

Claims (6)

1. a phase detecting method that in GLONASS, suppresses multipath signal is characterized in that, comprising:
Reception comprises the global navigation satellite system GNSS signal of multipath signal, carries out carrying out analog to digital conversion after down coversion generates the Low Medium Frequency signal; To pass through GNSS signal after the analog to digital conversion and carry out carrier wave and peel off, obtain the signal on I road and Q road, and said I road and Q road signal carried out sign indicating number peel off.
2. the phase detecting method that in GLONASS, suppresses multipath signal according to claim 1 is characterized in that, saidly I road and Q road signal are carried out a yard step of peeling off specifically comprises:
Duplicate one through the C/A code generator and have the C/A coded signal of same phase with said I road and Q road signal; And be 5 road C/A sign indicating numbers of 1/4 chip through the distance that postpones to obtain between the adjacent two-way C/A, be respectively: Very Early road, Early road, Prompt road, Late road, Very Late road;
Said 5 road C/A sign indicating numbers and said I road and Q road signal are carried out related calculation;
The value of said related operation gets into the amplitude that the code phase Discr. detects this value after through integration-remover and whether reaches maximum, and therefrom estimates the phase differential between the said C/A of duplicating sign indicating number and the reception C/A sign indicating number;
With of the input of said phase differential as C/A yardage controlled oscillator, regulate said frequency and phase place of duplicating the C/A sign indicating number, the said C/A of the duplicating sign indicating number and the moment that receives the C/A sign indicating number are consistent.
3. the phase detecting method that in GLONASS, suppresses multipath signal according to claim 2 is characterized in that, is T to the sampling clock of said I road and the employing of Q road signal t=Δ/2, Δ are the distance between Early road and the Late road, and Δ equals the half-chip length of C/A sign indicating number.
4. according to claim 2 or the 3 described phase detecting methods that in GLONASS, suppress multipath signal, it is characterized in that the step of said related operation specifically comprises:
The GNSS signal that receives comprises direct signal and multipath signal:
s(t)=s d(t)+s m(t)
=Ag[(1+ξ)t-γ]exp{t[(ω dc)t+φ 1]}+αAg[(1+ξ)t-γ-Δτ m]exp{t[(ω dc)t+φ 1+β]}
Wherein, s (t) is the GNSS signal, s d(t), s m(t) represent direct signal and multipath signal in the GNSS signal respectively, ω cBe carrier frequency, ω dBe Doppler shift,
Figure FDA00001685520000011
A be the GNSS direct signal amplitude, g (t) is ± 1, the value of expression C/A sign indicating number, φ 1Be carrier phase, α is the ratio of multipath signal amplitude and direct signal amplitude, Δ τ mBe the time delay of the relative direct signal of multipath signal, β=(ω d+ ω a) Δ τ m| Δ θ mBe the multipath signal phase compensation that causes because of reflection, Δ θ mBe the phase deviation that causes because of reflection, t is the time, and j is the imaginary number scalar, and γ is the time delay of original signal;
After the radio-frequency front-end down coversion, obtain:
s db(t)=Ag[(1+ξ)t-γ]exp(jψ)+αAg[(1+ξ)t-γ-Δτ m]exp[j(ψ+β)]
Wherein,
Figure FDA00001685520000021
is the phase differential between direct signal phase place and the multipath signal phase place, and
Figure FDA00001685520000022
is the phase place of multipath signal;
After process analog to digital conversion and carrier wave are peeled off, generate I road and Q road signal, carrying out related calculation with synthetic equally spaced 5 road C/A sign indicating numbers in this locality obtains:
I VE = A 2 R ( τ + Δ ) cos ( ψ ) + αA 2 R ( τ + Δ - Δ τ m ) cos ( ψ + β )
I E = A 2 R ( τ + Δ 2 ) cos ( ψ ) + αA 2 R ( τ + Δ 2 - Δτ m ) cos ( ψ + β )
I P = A 2 R ( τ ) cos ( ψ ) + αA 2 R ( τ - Δτ m ) cos ( ψ + β )
I L = A 2 R ( τ - Δ 2 ) cos ( ψ ) + αA 2 R ( τ - Δ 2 - Δτ m ) cos ( ψ + β )
I VL = A 2 R ( τ - Δ ) cos ( ψ ) + αA 2 R ( τ - Δ - Δ τ m ) cos ( ψ + β )
Q VE = A 2 R ( τ + Δ ) sin ( ψ ) + αA 2 R ( τ + Δ - Δ τ m ) sin ( ψ + β )
Q E = A 2 R ( τ + Δ 2 ) sin ( ψ ) + αA 2 R ( τ + Δ 2 - Δτ m ) sin ( ψ + β )
Q P = A 2 R ( τ ) sin ( ψ ) + αA 2 R ( τ - Δ τ m ) sin ( ψ + β )
Q L = A 2 R ( τ - Δ 2 ) sin ( ψ ) + αA 2 R ( τ - Δ 2 - Δτ m ) sin ( ψ + β )
Q VL = A 2 R ( τ - Δ ) sin ( ψ ) + αA 2 R ( τ - Δ - Δ τ m ) sin ( ψ + β )
Wherein, R is the cross correlation function of two-way C/A sign indicating number, and τ is the code delay difference between direct signal and the multipath signal, and Δ is the distance between Early road and the Late road.
5. the phase detecting method that in GLONASS, suppresses multipath signal according to claim 4 is characterized in that said code phase Discr. adopts the phase detecting method of TK-EML5.
6. the phase detecting method that in GLONASS, suppresses multipath signal according to claim 5 is characterized in that, the said step that estimates the phase difference between the said C/A of duplicating sign indicating number and the reception C/A sign indicating number specifically comprises:
Carry out following computing for discrete signal x (n) arbitrarily:
Ψ d [ x ( n ) ] = x ( n ) x * ( n ) - 1 2 [ x ( n - 1 ) x * ( n + 1 ) + x ( n + 1 ) x * ( n - 1 ) ]
Wherein, x *(n) conjugate function of expression x (n), x (n-1) and x (n+1) are respectively the move to right function of a unit, the unit that moves to left of x (n); x *(n+1) and x *(n-1) be respectively the conjugate function of x (n+1) and x (n-1);
Code phase Discr. based on TK-EML5 is output as:
D TKE(τ)=Ψ EL
Wherein, Ψ EAnd Ψ LBe respectively local synthetic Early road C/A sign indicating number and Late road the C/A sign indicating number respectively with the reception signal in the cross correlation function of C/A sign indicating number;
Local synthetic Early road C/A sign indicating number after the process computing, obtains in the code phase Discr. with the cross correlation function
Figure FDA00001685520000033
that receives signal with the cross correlation function
Figure FDA00001685520000032
and the local synthetic Late road of reception signal:
ψ E ( τ ) ∝ R ( τ + Δ 2 ) R * ( τ + Δ 2 ) - 1 2 [ R ( τ ) R * ( τ + Δ ) + R * ( τ ) R ( τ + Δ ) ]
ψ L ( τ ) ∝ R ( τ - Δ 2 ) R * ( τ - Δ 2 ) - 1 2 [ R ( τ ) R * ( τ - Δ ) + R * ( τ ) R ( τ - Δ ) ]
Ψ then EAnd Ψ LFor:
ψ E ( τ ) = ( I E + j Q E ) ( I E - j Q E ) - 1 2 [ ( I P - j Q P ) ( I VE + j Q VE ) + ] ( I P + j Q P ) ( I VE - j Q VE )
So ψ E ( τ ) = I E 2 + Q E 2 - ( I VE * I P + Q VE * Q P )
ψ L ( τ ) = ( I L + j Q L ) ( I L - j Q L ) - 1 2 [ ( I P - j Q P ) ( I VL + j Q VL ) + ] ( I P + j Q P ) ( I VL - j Q VL )
So ψ L ( τ ) = I L 2 + Q L 2 - ( I VL * I P + Q VL * Q P )
The phase difference value of code phase Discr. output is:
D TKE ( τ ) = ( I E 2 + Q E 2 ) - ( I L 2 + Q L 2 ) - ( I VE * I P + Q VE * Q P ) + ( I VL * I P + Q VL * Q P ) .
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