International Global Navigation Satellite Systems Society IGNSS Symposium 2006 Holiday Inn Surfers Paradise, Australia 17 – 21 July 2006

Phase Adaptive Integration for GNSS Signals Zhengdi Qin Nokia Corporation, Tampere Finland Tel: +358 503729041 Email: [email protected]

Harri Valio Nokia Corporation, Tampere Finland Tel: +358 503137515 Email: [email protected]

Presenter Name: Qin Zhengdi ABSTRACT In the GNSS signal acquisition operation, long time coherent integration is a problem if the coherence of the baseband signal is not guaranteed due to the frequency error or the oscillator drift. A phase adaptive method is introduced in which the phase of the signal is adaptively corrected before the integration. The phase estimation and the signal grouping are based on the signal shape distribution. Computer simulations show that the performance of the method is between that of the coherent integration and the noncoherent integration and it converges fast when the SNR goes up. For nominal level signal acquisition, the performance of the method is comparable to that of the coherent integration while the data structure is as simple as that for the non-coherent integration. The method is also robust for long integration times, especially when there is a frequency error or any irregular oscillator drifts. KEYWORDS: Phase-adaptive, CDMA, Integration, GNSS acquisition.

1. INTRODUCTION In the GNSS CDMA signal acquisition after the de-spreading process, the signal level is still often too weak for the acquisition. In many cases, further integration is required for a higher

sensitivity. It is preferable to use a coherent integration, but the coherent integration time cannot be long. The problem for the long-time coherent integration is the non-coherence of the signal itself. If the oscillator frequency is drifting, the phase angle of the signal is also changing. For instance we cannot coherently integrate a signal for over 100 milliseconds if there is a 10 hertz frequency drifting from the oscillator. If we know the drifting frequency, it is easy to compensate. If the frequency drifting is linear (stable), we can use more frequency bins to 'test' it. Unfortunately, the frequency drift is usually not predictable and even worse; mostly it is not linear and stable during the time of concern. To deal with such kind of problem, we can only do partial coherent integration during which the coherency of the signal is guaranteed. After that, several time periods of the results are combined together to improve the acquisition sensitivity. The most popular way is to use a non-coherent integration. In the non-coherent integration only the amplitude of the signal is used. The price paid for it is the so-called 'square-root loss' in the process due to the loss of the phase information. The situation is getting worse for the weak signal acquisition when the signal-to-noise ratio is below the nominal level. In this paper, a phase-adaptive method is presented for the signal integration. The method adapts the shape as well as the phase of the signal to be acquired. It is not like the pure coherent integration that simply integrates the signal without taking care of the phase change, especially the irregular phase change. When dealing with the signals from different time intervals, it first makes an estimation of the phase change and then the phase error is corrected before the integration. The method can be used to combat frequency drift, and of course, the Doppler frequency as well.

2. SHAPE AND PHASE ADAPTIVE PROCESS FOR CDMA SIGNALS 2.1 Signal Shape Modelling and Grouping

First, the shape of the signal to be integrated is modelled. Let’s take a typical CDMA signal as an example. After the de-spreading process (e.g., MF), the output signal looks like a triangle. If the sampling rate is two-samples-per-chip, the effective coverage of the signal is 3 samples in the delay profile (see figure 1).

xn-1

xn

xn+1

Fig. 1, output signal shape after MF.

The acquisition task is trying to find the signal peak (or peaks) in the delay profile in the noise environment. Let’s suppose that the searching range in the delay profile is N samples, X = (x1, x2, x3, …, xn, …, xN)

(1)

We make 3 sequences out from it. Take the original delay profile as Y1 and then shift it by one sample as Y2 and by two samples as Y3: ⎧Y 1 = ( x1 , x 2 , x3 , L , x N ) ⎪ Y = ⎨Y 2 = ( x 2 , x3 , x 4 , L , x N +1 ) ⎪Y 3 = ( x , x , x , L , x ) 3 4 5 N +2 ⎩

(2)

In each sequence, we take every consecutive 3 samples as a group to represent the signal: k =1 4 8 ⎧ ⎛ 647 1 ⎜ ⎪ Yk = x1 , x 2 , x3 , ⎜ ⎪ ⎝ ⎪ k =1 8 ⎪ 2 ⎛⎜ 6474 Y = ⎨Yk = x 2 , x3 , x 4 , ⎜ ⎪ ⎝ k =1 ⎪ 647 4 8 ⎛ ⎪Y 3 = ⎜ x , x , x , ⎪ k ⎜ 3 4 5 ⎝ ⎩

k =2 k =3 647 4 8 647 4 8 ⎞ x 4 , x 5 , x 6 , x 7 , x 8 , x 9 , L⎟ ⎟ ⎠ k =2 k =3 6474 8 64748 ⎞ x5 , x6 , x7 , x8 , x9 , x10 ,L⎟ ⎟ ⎠ k =2 k =3 647 4 8 647 48 ⎞ x6 , x7 , x8 , x9 , x10 , x11 , L⎟ ⎟ ⎠

(3)

In the phase adaptive process, the phase estimation is based on that the signal peak is supposed to be in the middle of each group. Since the signal shape covers 3 samples, only 3 sequences are needed in order have a full coverage of the signal. If the signal shape covers more chips, more sequences are needed with a shift by 1 sample for each sequence. The number of the groups in each sequence depends on the length of the delay profile or the acquisition searching range. 2.2 Phase Estimation

In each of the 3-sample groups, we suppose that the signal is in the middle of the group (xn). In order to correct the phase of the signal before the integration with other copies of the delay profile, blind phase estimation is introduced for each 3-sample group. The estimation is based on the principle of maximum ratio combination for all the samples in the group. In this example, the phase estimation for the sequence s (s = 1, 2,3), group k at time T is defined as:

ψ Ts , k = angle[xn + ξ (xn−1 + xn+1 )]

(4)

where ξ is the combination factor that depends on the signal shape and the SNR. Another possible phase estimation is to make the estimation for each sample and then add them together:

ψ Ts , k =

angle( x n ) + ζ * [angle( xn−1 ) + angle( xn +1 )] (1 + 2ζ )

(5)

where ζ is another combination factor that depends on the signal shape and the SNR.

If the phase change of the signal is not too fast during the integration, the phase estimation process can be extended to cover more time periods to get even a better phase estimation. If the signal shape is not a triangle or the samples-per-chip is not 2, the phase estimator in (4) and (5) should be modified for the actual signal shape and the data type. 2.3 Signal Rotation

The purpose of the signal rotation is to rotate the signal phase to be zero so that the adaptive integration can be done over different time intervals. The rotation of the signal is performed for all the samples in each group.

{

x m' = x m • exp − ψ Ts , k

}

{m = n − 1, n, n + 1}

(6)

After the signal rotation, only the real part of the signal is of concern while the imaginary part can be thrown away. The data in each sequence can be kept in a real array and ready for the adaptive integration. ⎧Y 1 ' = real (x1' , x 2' , x3' , L , x N' ) ⎪ Y' = ⎨Y 2 ' = real (x 2' , x3' , x 4' L , x N' +1 ) ⎪Y 3 ' = real(x ' , x ' , x ' L , x ' ) 3 4 5 N +2 ⎩

(7)

2.4 Adaptive Integration

The phase of the signal is corrected at each time T and then summed up together for a better SNR.

⎛ 1 ⎞ ⎛ z11 ' , z 12 ' , z 31 ' ,L , z 1N ' ⎞ ⎜ ∑ YT ' ⎟ ⎜ ⎟ ⎜ T ⎟ Z' = ⎜ z12 ' , z 22 ' , z 32 ' ,L, z N2 ' ⎟ = ⎜ ∑ YT2 ' ⎟ ⎜ z 3 ' , z 3 ' , z 3 ' ,L, z 3 ' ⎟ ⎜ T 3 ⎟ N ⎠ ⎝ 1 2 3 ⎜ ∑ YT ' ⎟ ⎠ ⎝T

(8)

Since the integration is adaptive, the integration time T is not limited. This of course means that long time integration times are possible. Also, as the phase error is corrected during the process, no frequency bins are needed no matter how long the integration time T is. The frequency estimation can be done by the phase factor in equation (4) or (5). This decreases the memory storage needed for the acquisition process. In order to combine the three sequences for the final acquisition, the samples of the sequences should be shifted back to the original position. That is, shift sequence Z2 by 1 sample back and Z3 by 2 samples back. The final delay profile for the acquisition is then the combined sequence z: 2

z j = ∑ z ss+ j ' (j = 1,2,3,...,N) s =0

(9)

3. COMPUTER SIMULATIONS

As we know, if there is no phase error and no frequency drift, the best way is to use coherent integration for the signal acquisition. If the signal coherency cannot be kept long enough, the non-coherent integration can be followed anyway. It means that to improve the acquisition sensitivity, coherent integration is the ceiling and the non-coherent integration is the floor between which we could try. Simulations have been performed by using the new method and the results are used to compare with those by using the coherent and the non-coherent integration. The acquisition sensitivity is chosen as the criteria for the performance. The GPS L1 channel C/A signal is used for the simulations. The sample rate of the baseband signal is two-samples-per-chip. The full code for the delay coverage is 2046 samples as the code length of C/A code is 1023. The baseband signal first goes through the matched filter. The output is a 1-millisecond de-spreaded signal with a bandwidth of 1 kilohertz. If it is necessary, the signal can be further integrated coherently before the adaptive process. In our simulations, 1 millisecond signal after the matched filter is used directly. The total integration time is 100 milliseconds for three different methods. Figure 2 shows one of the simulation results. From the figure, we see that if there is no frequency error (Df = 0), the best is the coherent integration (blue line) and the worst is the non-coherent integration (green line), while the phase adaptive integration (red line) is between the roof and the floor performances. This means that as we expected, the phase adaptive integration is better than the non-coherent integration, but not as good as the coherent integration. 100

Acquisition Probability (%)

90

80

70

Coherent Integration Df = 0 Hz Non-coherent Integration Phase Adaptive Integration Coherent Integration Df = 6 Hz

60

50

Initial cohe-time = 1 ms Total Integration time = 100 ms Searching range = 1 ms / 2046 samples

40

30

-138 dBm

-136 dBm

-134 dBm

-132 dBm

-130 dBm

-128 dBm

-126 dBm

Signal level (GPS C/A)

Figure 2, simulation results by using 3 different integration methods.

Next a small frequency drift is introduced (in this example, Df = 6 Hz), the performance of the coherent integration deteriorated very much (dotted blue line) while other two methods stay at the same level as if there is no frequency drift. This is understandable since 6 hertz is too much for the coherent integration over 100 milliseconds but it is negligible for other two methods where the coherent length required is only 1 millisecond. In this case, the phase adaptive method is much better than the coherent integration. As we can see from the results, if the signal level is higher than –130 dBm, the performance of the phase adaptive integration is almost as good as that of the coherent integration. We call it as a comparable level. If the initial coherent time is increased, this level can be further reduced. For instance if we perform a 10-milliseconds coherent integration after the despreading before the adaptive process, the comparable level is then reduced to -140 dBm. Furthermore, the simulations show that the phase adaptive process is convergent for the nominal signal acquisition. It means that longer integration times are possible for higher acquisition sensitivity without having to take care of the frequency error or the oscillator drift. This is also true for the navigation data message, as the phase change of BPSK for the navigation data bits is already been taken into account in the phase adaptive process. The bit synchronization as well as the bit estimation can also be done by the phase factor in equation (4) and (5).

4. CONCLUSIONS

A phase adaptive method is introduced for the signal integration of the GNSS acquisition operation. The method is based on the shape and the phase change of the signal after an initial coherent integration. The phase error is adaptively corrected before further integration. The new method is robust against frequency drift and it is good for long integration times. The method can be best described as a complement for the integration process for GNSS signal acquisition. In the acquisition process within a given bandwidth, the phase adaptive method does not need memory storage for frequency bins, as the frequency error is corrected during the process. The hardware requirements are as simple as for the non-coherent integration. For nominal signals, its performance is close to that of the coherent integration. It would be beneficial if the method were combined with other methods. The most computation power needed is the phase angle estimation and the signal rotation. In a hardware realization, the task can be fast done by, for example, the Cordic algorithm. REFERENCES Qin Zhengdi (2000) Optimum gain in the process of the GPS signal acquisition in Proceeding of ION GPS 2000, Salt Lake City, USA: 891-894. Bradford W. Parkinson, James J. Spilker Jr. (1996) Global positioning system: theory and applications volume I., ISBN 1-56347-107-8 Don H. Johnson, Dan E. Dudgeon (1993) Array signal processing: concept and techniques: concepts and techniques, ISBN 0-13-048513-6 Simon Haykin (1991) Advances in spectrum analysis and array processing volume II, Prentice-Hall, ISBN 0-13-008574-X