1. 1 Acquisition, Tracking and Position Calculations (Module 4) This module will provide the necessary theory behind acquisition, tracking, satellite position calculations and receiver position calculations. The theory behind receiver position calculation is based on the method of least-squares is described.

2. 2 Acquisition and Tracking Acquisition is the process of locking onto a satellite’s C/A code and P code. A receiver acquires all available satellites when it is first powered up, then acquires additional satellites as they become available and continues tracking them until they become unavailable. Tracking is a planned or intended horizontal path of travel with respect to the Earth rather than the air or water. The track is expressed in degrees from 000° clockwise through 360° (true, magnetic, or grid).

3. 3 Acquisition The purpose of acquisition is to determine visible satellites and coarse values of carrier frequency and code phase of the satellite signal. The satellites are differentiated by the 32 different PRN sequences. The code phase is the time alignment of the PRN code in the current block of data. The code phase in useful to be able to generate a local PRN code that is perfectly aligned with the incoming code. The third parameter is the carrier frequency which corresponds to the IF. The IF should be known from the L1 carrier frequency of 1575.42 MHz and from the mixer in the down-converter. Take Doppler effect into consideration.

4. 4 Doppler Frequency Shift This phenomenon was discovered by Christian Johann Doppler (1803-1853), an Austrian scientist. Doppler is the apparent change in wavelength (or frequency) of an EM or acoustic wave when there is relative movement between the transmitter (or frequency source) and the receiver. In everyday life this effect is commonly noticeable when a whistling train or police siren passes you. Audio Doppler is depicted here, however Doppler can also affect frequency of a radar carrier wave, or even light waves causing an apparent shift of color Frequency increases Frequency decreases

5. 5 Doppler Frequency in GPS Maximum 5 kHz Doppler shift due to satellite motion occurs with satellite moving directly towards/away from receiver. Maximum 5 kHz Doppler shift due to receiver motion occurs with a very fast airoplane. Total maximum Doppler shift 10 kHz (excluding accuracy of the local oscillator). Commonly used Doppler frequency bins for acquisition is 500 Hz. This gives a total of 41 different frequencies to be tested

6. 6 This is done by measuring the shift in frequency of a wave caused by an object in motion: Transmitter in motion Reflector in motion Receiver in motion All three For a closing relative velocity –Wave is compressed –Frequency is increased For an opening relative velocity –Wave is stretched –Frequency is decreased.

7. 7 Search for Signal Why is a signal search necessary? –GPS signals are spread-spectrum signals in which the C/A or P- code spread the total signal power over a wide bandwidth. The signals are therefore undetectable unless they are despread with a replica code in the receiver which is aligned with the received code. –A relatively narrow post-despreading bandwidth (100-1000 Hz) is required to raise the signal-to-noise ratio to detectable and/or usable levels. –GPS receiver must conduct a two-dimensional search in order to find each satellite signal, where the dimensions are C/A-code delay and carrier frequency.

8. 8 Search in Frequency The range of frequency uncertainty that must be searched is a function of the receiver reference oscillator, how well the approximate user position is known, and the accuracy of the receiver’s built-in real-time clock. The first step in the search is to use stored almanac data to obtain an estimate of the Doppler shift of the satellite signal. An interval [ f 1, f 2 ] of frequencies to be searched is then established. The center of the interval is located at f c + f d where f c is the L 1 or L 2 frequency and f d is the estimated carrier Doppler shift. The frequency search is conducted in N discrete frequency steps that cover the entire search interval. N = (f 2 – f 1 ) /  f where  f is the spacing between adjacent frequencies (bin width).

9. 9 Serial Search Acquisition This method is one of the first methods used in acquisition in code division multiple access (CDMA) system. The algorithm is based on multiplication of locally generated PRN code sequences and locally generated carrier signals. Total number of combinations to search 41 different carrier frequencies 1023 different C/A code phases Total 41 x 1023 = 41943 combinations

10. 10 Parallel Frequency Space Search Acquisition Very time consuming procedure. The technique utilizes the Fourier transform to perform a transformation from the time domain to frequency domain. Total number of combinations to search: 1023 Each of the combinations is quite demanding because of the use of a Fourier transform.

11. 11 Parallel Code Phase Search Acquisition Parallelizes the code space dimension. Multiply the incoming signal with a locally generated sine and cosine. Perform a Fourier transform of this signal combining I and Q into a complex signal. Perform a Fourier transform and a complex conjugate of the locally generated C/A code. Multiply the signals in the frequency domain. Perform an inverse Fourier transform to obtain the output in the time domain.

12. 12 Sequential Versus Parallel Search Nethods Almost all current GPS receivers are multichannel units in which each channel is assigned a satellite and processing in the channels is carried out simultaneously. Therefore the simultaneous searches can be made for all usable satellites when the receiver is turned on. Because the search in each channel consists of sequencing through all possible frequency and code delay steps, it is called sequential search. Certain applications (mostly military) demand that the satellites be acquired much more rapidly (few seconds). This can be accomplished by using a parallel search technique.

13. 13 Tracking After the acquisition, the frequency and code offset parameters of a satellite signal are roughly known. The main purpose of tracking is to refine these values and keep track and demodulate the navigation data from the specific satellite

14. 14 Demodulation Signal received from the satellite is represented by powers of signals with C/A or P code sequences and navigation data sequence. The two carrier frequencies are L1 and L2. Because of the filter and down-conversion in the front end, the output is described in terms of the intermediate frequency. This signal is sampled by the A/D converter, but because of the narrow bandpass filter around the C/A code, the P code is distorted and filtered out and it can not be demodulated.

15. 15 Code Tracking The aim of a code tracking loop is to keep track of the code phase of a specific in the signal. The output of such a code tracking loop is a perfectly aligned replica of the code. The code tracking loop in the GPS receiver is a delay lock loop (DLL).

16. 16 Code tracking. Three local codes are generated and correlated with the incoming signal.

17. 17 DLL with Six Correlators

18. 18 Carrier Tracking

19. 19 Complete Tracking Block: DDL and PLL loops

20. 20 Tracking Channel on the GPS Receiver

21. 21 Satellite Position Calculations In a simplified approach, each satellite is sending out signals with the following content: I am satellite A, my position is X and this information was sent at time t. In addition to its own position, each satellite sends data about the position of other satellites. These orbit data (ephemeris and almanac data) are stored by the GPS receiver for later calculations. The ephermeris parameters refers to the Keplerian orbit elements, so to be able to use the satellite positions in ground navigation, these elements has to be transformed into the Earth-centered Earth fixed (ECEF) coordinate system. For the determination of its position on Earth, the GPS receiver compares the time when the signal was sent by the satellite with the time the signal was received. From this time difference the distance between receiver and satellite can be calculated.

22. 22 If data from other satellites are taken into account, the present position can be calculated by trilateration (meaning the determination of a distance from three points). This means that at least three satellites are required to determine the position of the GPS receiver on the earth surface. The calculation of a position from 3 satellite signals is called 2D- position fix (two-dimensional position determination). It is only two dimensional because the receiver has to assume that it is located on the earth surface (on a plane two-dimensional surface). By means of four or more satellites, an absolute position in a three dimensional space can be determined. A 3D-position fix also gives the height above the earth surface as a result.

23. 23 Keplerian Orbit Elements

24. 24 Ephemeris and Almanac Data When calculating positions of GPS satellites some more accurate orbit values are needed. These values are included in the so called broadcast ephemerides, which are included in the navigation data transmitted from the GPS satellite.

25. 25 The data signal is divided into 25 frames, each having a length of 1500 bit (meaning an interval of 30 seconds for transmission). The 25 frames are divided into subframes (300 bit, 6 sec.), which are again divided into 10 words each (30 bit, 0.6 sec). The first word of each subframe is the TLM (telemetry word). It contains information about the age of the ephemeris data. The next word is the HOW (hand over word), which contains the number of counted z-epoches. These data contain the time since last “restart” of the GPS time on the previous Sunday 0:00 o’clock. As the P-code is 7 days long, the HOW is used by military receivers to locate their access to the P- code.

26. 26 The rest of the first subframe contains data about status and accuracy of the transmitting satellite as well as clock correction data. The second and third subframes contain ephemeris parameters. Subframes 4 and 5 contain the so-called almanac data which include information about orbit parameters of all satellites, their technical status and actual configuration, identification number and so on. Subframe 4 contains data for the satellites number 25 – 32, ionospheric correction data, special information and UTC time information; subframe 5 contains almanac data for the satellites 1 – 24 as well as time and the number of the GPS week.

27. 27 The first three subframes are identical for all 25 frames. Every 30 seconds the most important data for the position determination are transmitted with these three subframes. From the almanac data the GPS receiver identifies the satellites that are likely to be received from the actual position. The receiver limits its search to these previously defined satellites and hence this accelerates the position determination. When ephemeris and almanac data are stored in the GPS receiver, it depends on their actuality how long the GPS needs for the first position determination. If the receiver has not had any contact to the satellites for long time, the first position determination will take longer.

28. 28 If the contact has only been interrupted for a short time (e.g. when driving through a tunnel), the position determination is restarted instantly and we speak of reacquisition. If position and time are known and the almanac and ephemeris data are up-to-date, we speak of a hot start. This is the case when the receiver is turned on at approximately the same position within 2 – 6 hours after the last position determination. In this case a position fix can be obtained within approximately 15 seconds. If the almanac data are available and the time of the receiver is correct but the ephemeris data are outdated, this is called a warm start. In this case it takes about 45 seconds to actualize the ephemeris data and obtain a position fix. Ephemeris data are outdated when more than 2 – 6 hours have elapsed since the last data reception from the satellites in view. The more new satellites have come into view since the last position determination, the longer the warm start takes.

29. 29 Receiver Position Calculations The most commonly used method for GPS calculations is based on the least squares method. This method is used when there are more equations than unknown variables.

30. 30 Using the ephemerides, the satellite clock offset dt k and the exact position of the satellite (X k, Y k, Z k ) can be calculated.

31. 31 Linearization of the Observation Equation The first step of the linearization of the equation is to find an initial approximated position for the receiver called X i,0, Y i,0, Z i,0. Using these approximated positions leads to

32. 32

33. 33 Using the Least Squares Method

34. 34

35. 35 References Global Positioning Systems by MS Grewal, LR Weill, AP Andrews, Wiley, 2001. Peter Rinder and Nicolaj Bertelsen, Design of a Single GPS Software Receiver, Aalborg University 2004. The Internet.