Combining secondary code correlations for fast GNSS signal ... · 1. Introduction The receiver...

Jérôme Leclère

René Jr Landry

Combining secondary code correlations for

fast GNSS signal acquisition

Content

1. Introduction

2. How to extend the coherent integration time ?

3. Solution : Combining secondary code correlations

4. Application to the GPS L5 signal

5. Conclusion

24/04/2018 IEEE/ION PLANS 2018, Monterey, CA, USA 2

Content

1. Introduction

A GNSS signal contains three essential elements :

Data (ephemeris, corrections, …)

Pseudorandom code (distance measurement, spectrum sharing)

Carrier

Modern signals include new features :

Pilot channel (allows long coherent integration time)

Subcarrier (improves multipath mitigation)

Secondary pseudorandom code


1.2 GNSS signals

1. Introduction

Model of a GNSS signal transmitted :

Model of a GNSS signal received :


1.2 GNSS signals

𝑠t 𝑡 = 𝑎t,d 𝑐d 𝑡 𝑑 𝑡 cos 2𝜋𝑓L𝑡 + 𝜑t,d

+ 𝑎t,p 𝑐p 𝑡 sin 2𝜋𝑓L𝑡 + 𝜑t,p

𝑠r 𝑡 = 1𝐿 𝑠t 𝛼 𝑡 − 𝜏 + 𝜂 𝑡

= 𝑎r,d 𝑐d 𝛼 𝑡 − 𝜏 𝑑 𝛼 𝑡 − 𝜏 cos 2𝜋 𝑓L + 𝑓D 𝑡 + 𝜑r,d

+ 𝑎r,p 𝑐p 𝛼 𝑡 − 𝜏 sin 2𝜋 𝑓L + 𝑓D 𝑡 + 𝜑r,p losses

Doppler effect traveling delay

noise

1. Introduction

The receiver wants to :

decode the data

synchronize with the code

For this, the receiver needs to :

estimate the Doppler frequency 𝑓D to remove the carrier

estimate the code delay 𝜏 to synchronize with the code and

remove it

This is the goal of the acquisition.


1.2 Goal of the acquisition

1. Introduction

Acquisition is basically a correlation + Doppler search.

Actual implementations use high parallelization or fast Fourier

transforms (FFTs).


1.3 How do we do acquisition ?

repeated for different and

Σ incoming

signal|· |

local code replica of delay

local carrier replica of

frequency 𝜏 𝑓 D

𝜏 𝑓 D

cross ambiguity function (CAF)

1. Introduction

If 𝑓 D = 𝑓D and 𝜏 = 𝜏 (both replicas match the input) ⇒ high CAF

If 𝑓 D ≠ 𝑓D or 𝜏 ≠ 𝜏 (at least 1 replica does not match) ⇒ low CAF


1.3 How do we do acquisition ?

Example:

L1 C/A signal

𝑓D = 2000 Hz

𝜏 = 200 chips

1 ms integration

1. Introduction

The other challenge of the acquisition is the noise.

To reduce the noise as much as possible the integration time

should be as long as possible.


1.4 Why the integration time is important ?

1 ms 10 ms 100 ms

1. Introduction

But there are limitations : data, dynamics, XO, freq. resolution…

One way to overcome these drawbacks is to continue integrating

after taking the magnitude :



repeated for different and

Σ |· |

local code replica of delay

local carrier replica of

frequency 𝜏 𝑓 D

𝜏 𝑓 D

cross ambiguity function (CAF)

Σ incoming

signal

coherent integration

non-coherent integration

1. Introduction

However, in terms of detection, coherent integration is more

efficient than non-coherent integration.



1 ms 100×1 ms 10×10 ms

1. Introduction

Modern GNSS signals include a secondary code.

Advantages :

Easier synchronization with the data

Better cross-correlation between satellites/constellations

Better interferences mitigation

Drawbacks :

Complicates the primary code correlation for FFT-based

implementations (sign transition possible anytime)

Complicates the extension of the coherent integration time


1.5 What do modern signals change to acquisition ?

1. Introduction




5. Conclusion


Content


Mainly two possibilities to extend the coherent integration time

in presence of a secondary code :

Use short coherent integration time, 𝑇C < 𝑇SC

Use long coherent integration time, 𝑇C ≥ 𝑇SC

𝑇C : coherent integration time (s)

𝑇SC : period of the secondary code (s)


2.1 Possibilities

Note added after conference : In fact there is a third method,

namely the computation of a partial correlation using less

than one secondary code period as input. This method was

discussed in [12]. For more details, and for a comparison of

the method proposed by the authors with this third method,

see J. Leclère, R. Jr Landry, “Galileo E5 signal acquisition using

intermediate coherent integration time”, INC, Nov 2018, and

Journal of Navigation, 2019.

https://doi.org/10.1017/S0373463318001054


If have 𝑇C < 𝑇SC, all the possible sign sequences should be tested.

Drawback : number of possibilities grows exponentially

limited to 4 or 5 primary code periods.


2.2 Short coherent integration time

Parallel implementation with coherent integration only : (𝑁C = 3)

Primary code

correlation

xi ri r0 + r1 + r2

Σ NP

+ + +

Σ NP

+ + –

Σ NP

+ – –

Σ NP

+ – +

r0 + r1 – r2

r0 – r1 + r2

r0 – r1 – r2

r0

r1

r2


If non-coherent integration is used too, more possibilities

Drawback : number of possibilities grows double exponentially



Parallel implementation with coherent and non-coherent integrations : (𝑁C = 2, 𝑁NC = 2)

Primary code

correlation

xi ri

r0 + r1,

r2 + r3

r0 – r1,

r2 – r3

|r0 + r1|

+ |r2 + r3|

|r0 + r1|

+ |r2 – r3|

|r0 – r1|

+ |r2 + r3|

|r0 – r1|

+ |r2 – r3|

Σ NP

+ +

Σ NP

+ –

| · |

| · |

Σ NP

+ +

Σ NP

+ +

Σ NP

+ +

Σ NP

+ +

r0

r1

r3

r2

Number of accumulators explodes with non-coherent integration

Non-coherent integration not usable

Stuck to low sensitivity




Number of coherent accumulators

= 𝟐𝑵𝐂−𝟏

𝑵𝐍𝐂

𝑵𝐂 1 2 3 4 5

1 1 1 1 1 1

2 2 2 2 2 2

3 4 4 4 4 4

4 8 8 8 8 8

5 16 16 16 16 16

Number of non-coherent accumulators

= 𝟐𝑵𝐂−𝟏 𝑵𝐍𝐂

𝑵𝐍𝐂 𝑵𝐂

1 2 3 4 5

1 0 1 1 1 1

2 0 4 8 16 32

3 0 16 64 256 1024

4 0 64 512 4096 32 768

5 0 256 4096 65 536 1 048 576


To have 𝑇C ≥ 𝑇SC , correlation with the secondary code is needed


2.3 Long coherent integration time

Primary code

correlation

xi ri

si

y0

Σ NP

NS – 1

i = 0Σ

NP

NNC

k = 0

| · |z0

si – 1

y1

Σ NP

NS – 1

i = 0Σ

NP

NNC

k = 0

| · |z1

si – (N – 1)

yN –1

Σ NP

NS – 1

i = 0Σ

NP

NNC

k = 0

| · |

S

SzN –1S

···

···

···

···

Primary code

correlation

xi ri

si – k

yk

Σ NP

NS – 1

i = 0Σ

NP

NNC

i = 0

| · |zk

Parallel implementation :

Serial implementation :

Drawback :

Huge computa-

tional burden

Huge

resources or long

acquisition time.

𝑁S : secondary code length



2.4 Solution : intermediate coherent integration time

Method

Use of

non-coherent

integration ?

Sensitivity Complexity

long 𝑇C ✓ high high

short 𝑇C ✗ low low to moderate

proposed solution

with intermediate 𝑇C ✓ moderate to high low to moderate

1. Introduction




5. Conclusion


Content


Combining secondary code correlations reduce the complexity :

There are less delays to search

Shorter processing time for the serial implementation

Less accumulators for the parallel implementation

Inputs may be combined

Less primary code correlations to compute

Shorter local secondary code

The local modified secondary code may contain zeros

Less operations for the secondary code correlation


3.1 Why combining secondary code correlations ?


Not every combination gives the same complexity reduction !

Let’s see a simple example for a 4-chip code.

Traditional correlation :


3.2 How combining secondary code correlations ?

𝒚0

𝒚1

𝒚2

𝒚3

=

𝑠0 𝑠1 𝑠2 𝑠3

𝑠3 𝑠0 𝑠1 𝑠2

𝑠2 𝑠3 𝑠0 𝑠1

𝑠1 𝑠2 𝑠3 𝑠0

𝒓0

𝒓1

𝒓2

𝒓3

Primary code

correlation

xi ri

si

y0

Σ NP

NS – 1

i = 0Σ

NP

NNC

k = 0

| · |z0

si – 1

y1

Σ NP

NS – 1

i = 0Σ

NP

NNC

k = 0

| · |z1

Σ NP

NS – 1

i = 0Σ

NP

NNC

k = 0

| · |

si – 2

y2

Σ NP

NS – 1

i = 0Σ

NP

NNC

k = 0

| · |z2

si – 3

y3 z3

i = 0, 1, 2, 3

computed

4 times


Combination of one result with the next one :



𝒚0 + 𝒚1

𝒚2 + 𝒚3=

𝑠0 + 𝑠3 𝑠1 + 𝑠0 𝑠2 + 𝑠1 𝑠3 + 𝑠2

𝑠2 + 𝑠1 𝑠3 + 𝑠2 𝑠0 + 𝑠3 𝑠1 + 𝑠0

𝒓0

𝒓1

𝒓2

𝒓3

Primary code

correlation

ri

si + si –1

Σ NP

NS – 1

i = 0Σ

NP

NNC

k = 0

| · |

si –1 + si – 2

Σ NP

NS – 1

i = 0Σ

NP

NNC

k = 0

| · |

xi y0 + y1

y2 + y3

i = 0, 1, 2, 3

z0 + z1

z2 + z3still

computed

4 times


Combination of one result with the next but one :



𝒚0 + 𝒚2

𝒚1 + 𝒚3=

𝑠0 + 𝑠2 𝑠1 + 𝑠3 𝑠2 + 𝑠0 𝑠3 + 𝑠1

𝑠3 + 𝑠1 𝑠0 + 𝑠2 𝑠1 + 𝑠3 𝑠2 + 𝑠0

𝒓0

𝒓1

𝒓2

𝒓3

=𝑠0 + 𝑠2 𝑠1 + 𝑠3

𝑠3 + 𝑠1 𝑠0 + 𝑠2

𝒓0 + 𝒓2

𝒓1 + 𝒓3

xi +2

xi

ri + ri+2

si + si –2

Σ NP

NS – 1

i = 0Σ

NP

NNC

k = 0

| · |

si –1 + si – 3

Σ NP

NS – 1

i = 0Σ

NP

NNC

k = 0

| · |

y0 + y2

y1 + y3

Primary code

correlation

i = 0, 1

z0 + z2

z1 + z3computed

2 times !


Not every combination gives the same performance !

Let’s see a simple example for the 6-chip code [1 1 1 –1 1 –1].

Autocorrelation : 𝑦0 𝑦1 𝑦2 𝑦3 𝑦4 𝑦5 = [6 –2 2 –2 2 –2]

Combination 1 : 𝑦0 + 𝑦2 + 𝑦4 𝑦1 + 𝑦3 + 𝑦5 = [10 –6]

Combination 2 : 𝑦0 + 𝑦1 + 𝑦2 𝑦3 + 𝑦4 + 𝑦5 = [6 –2] or [2 2]

The maximum depends on the delay of the incoming code !




Rules :

1. Combine 𝑁RC secondary code correlation results separated by a delay 𝑁S

𝑁RC

2. Sign of the combinations : only +, or alternatively + and –

Then :

Maximum autocorrelation identical regardless of the delay of the incoming

secondary code

Number of secondary code delays to test is divided by 𝑁RC

𝑁RC inputs can be combined

Complexity is approximately divided by 𝑁RC2




This does not come for free :

Result modified loss in the signal-to-noise ratio (SNR)

Using 𝑇SC of data, instead of a SNR gain of 𝑁S, the gain is

Easy to express through an equivalent coherent integration time.


3.3 Impact on the SNR

𝐺 = 𝑠𝑖−𝑘1

𝑠𝑖−𝑚S

𝑁S−1𝑖=0 ± 𝑠𝑖−𝑘2

𝑠𝑖−𝑚S

𝑁S−1𝑖=0 ± ⋯ ± 𝑠𝑖−𝑘𝑁RC

𝑠𝑖−𝑚S

𝑁S−1𝑖=0

2

𝑠𝑖−𝑘1± 𝑠𝑖−𝑘2

± ⋯ ± 𝑠𝑖−𝑘𝑁RC

2𝑁S−1𝑖=0

1. Introduction




5. Conclusion


Content

4. Application to the L5 signal

Autocorrelation of the L5 pilot secondary code (20 chips)


4.1 Secondary code autocorrelation

Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Auto-correlation 20 0 0 0 0 0 –4 0 4 0 –4 0 4 0 –4 0 0 0 0 0


Combination 𝑵𝐑𝐂 Resulting code and correlation Numerator

of 𝑮

Denomin

ator of 𝑮

SNR

gain 𝑮

1 𝒚10𝑘

1

𝑘=0

2 2 0 2 0 2 0 0 0 –2 0

162 = 256 32 8 16 0 4 0 –4 0 –4 0 4 0

2 −1 𝑘𝒚10𝑘

1

𝑘=0

2 0 2 0 2 0 –2 2 2 0 –2

242 = 576 48 12 ±24 0 ∓4 0 ±4 0 ∓4 0 ±4 0

3 𝒚5𝑘

3

𝑘=0

4 2 0 2 –2 2

162 = 256 64 4 16 –4 4 4 –4

4 −1 𝑘𝒚5𝑘

3

𝑘=0

4 2 0 2 2 2

162 = 256 64 4 ±16 ±4 ±4 ∓4 ∓4

5 𝒚4𝑘

4

𝑘=0

5 1 –3 3 3

282 = 784 140 5.6 28 0 -12 0


4.2 Possible combinations


Combination 𝑵𝐑𝐂 Resulting code and correlation Numerator

of 𝑮

Denomina

tor of 𝑮

SNR

gain 𝑮

6 𝒚2𝑘

9

𝑘=0

10 4 0

162 = 256 160 1.6 16 0

7 −1 𝑘𝒚2𝑘

9

𝑘=0

10 –2 –6

402 = 1600 400 4 ±40 0

8 𝒚𝑘

19

𝑘=0

20 4

162 = 256 320 0.8 16

9 −1 𝑘𝒚𝑘

19

𝑘=0

20 4

162 = 256 320 0.8 ±16


4.2 Possible combinations



4.3 Proposed method vs short coherent integration time

Number of 𝑵𝐑𝐂 𝑻𝐂 𝑵𝐒

𝑵𝐑𝐂 𝑵𝐒,𝐍𝐙

′ Hardware

Software Conclusion Parallel Serial

memory

- 4 - - 8 1 5

Prop. sol.

better

10 4 2 2 2 1 4

- 5 - - 16 1 6

5 5.6 4 4 4 1 6

primary

code

correlation

- 4 - - 4 11 4

Prop. sol.

better

10 4 2 2 2 4 2

- 5 - - 5 20 5

5 5.6 4 4 4 16 4

additions

- 4 - - - - 10

Prop. sol. less

good

10 4 2 2 - - 24

- 5 - - - - 19

5 5.6 4 4 - - 36

Moreover, the proposed solution allows non-coherent integration !



4.4 Proposed method vs long coherent integration time

Number of 𝑵𝐑𝐂 𝑻𝐂 𝑵𝐒

𝑵𝐑𝐂 𝑵𝐒,𝐍𝐙

′ Hardware

Software Conclusion Parallel Serial

memory

- 20 - - 20 1 21 Prop. sol.

better 2 12 10 6 10 1 12

2 24 10 6 10 1 12

primary code

correlation

- 20 - - 20 400 20 Prop. sol.

better or

similar

2 12 10 6 10 60 10

2 24 10 6 20 120 20

additions

- 20 - - - - 380 Prop. sol.

better 2 12 10 6 - - 80

2 24 10 6 - - 160

significant complexity reduction for smaller 𝑇C

or slight complexity reduction for slightly longer 𝑇C (it’s a win-win)



4.4 Proposed method vs long coherent integration time

Primary code correlations

+ secondary code correlation Secondary code correlation only

1. Introduction




5. Conclusion


Content

5. Conclusion

The secondary code complicates the acquisition : Short coherent integrations limited to low sensitivities Long coherent integrations have high computational load

A solution has been proposed fitting between current solutions : Combine secondary code correlation results Intermediate coherent integration times + non-coherent integration usable Allow moderate to high sensitivity Flexible due to the several possible combinations

Drawbacks : Use more data than the integration time Require several simultaneous accesses to input data Still an ambiguity in the secondary code delay


5.1 Conclusions

Laboratory of Space Technologies,

Embedded Systems, Navigation and Avionic

Thank you for your attention !

[email protected]

http://lassena.etsmtl.ca

École de Technologie Supérieure

Combining secondary code correlations for fast GNSS signal ... · 1. Introduction The receiver...

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