Faster GPS via the Sparse Fourier Transform Haitham Hassanieh Fadel - - PowerPoint PPT Presentation

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Faster GPS via the Sparse Fourier Transform Haitham Hassanieh Fadel - - PowerPoint PPT Presentation

Apr 08, 2024 134 likes •696 views

Faster GPS via the Sparse Fourier Transform Haitham Hassanieh Fadel Adib Dina Katabi Piotr Indyk GPS Is Widely Used Faster GPS benefits many applications Faster GPS benefits many applications How Do We Improve GPS? Need to Improve GPS

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SLIDE 1

Faster GPS via the Sparse Fourier Transform

Haitham Hassanieh Fadel Adib Dina Katabi Piotr Indyk

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SLIDE 2

Faster GPS benefits many applications Faster GPS benefits many applications

GPS Is Widely Used

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SLIDE 3

How Do We Improve GPS? Need to Improve GPS Synchronization

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SLIDE 4

100s of millions of multiplications

[Team, Kaplan]

100s of millions of multiplications

[Team, Kaplan]

GPS Synchronization

Synchronization is locking onto a satellite’s signal

  • Consumes 30%‐75% of GPS receiver’s power

[ORG447X datasheet, Venus 6 datasheet] GPS signals are very weak, less than ‐20dB SNR

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SLIDE 5

Goal

Faster Synchronization Algorithm

Reduce number of operations

Reduction in power consumption and delay

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SLIDE 6

Rest of this Talk

  • GPS Primer
  • Our GPS Synchronization Algorithm
  • Empirical Results
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SLIDE 7

How Does GPS Work?

Compute the distance to the GPS satellites

d1 d3 d2

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SLIDE 8

How Does GPS Work?

Compute the distance to the GPS satellites

d1 d3 d2

distance = propagation delay speed of light distance = propagation delay speed of light

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SLIDE 9

How to Compute the Propagation Delay?

Satellite Transmits CDMA code

CDMA code

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SLIDE 10

How to Compute the Propagation Delay?

CDMA code delay

Code arrives shifted by propagation delay

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SLIDE 11

How to Compute the Propagation Delay?

CDMA code delay

Receiver knows the code and when the satellite starts transmitting

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SLIDE 12

How to Compute the Propagation Delay?

Correlation

delay

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SLIDE 13

How to Compute the Propagation Delay?

Correlation

delay

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SLIDE 14

How to Compute the Propagation Delay?

Correlation

delay

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SLIDE 15

How to Compute the Propagation Delay?

Correlation Spike

delay

Spike determines the delay

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SLIDE 16

GPS Synchronization is a convolution with CDMA code

Convolution in Time Multiplication in Frequency : Number of samples in the code

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SLIDE 17

GPS Synchronization is a convolution with CDMA code

Convolution in Time Multiplication in Frequency : Number of samples in the code

State of the art GPS synchronization algorithm:

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SLIDE 18

Rest of this Talk

  • GPS Primer
  • Our GPS Synchronization Algorithm
  • Empirical Results
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SLIDE 19

QuickSync

  • Fastest GPS synchronization algorithm to date
  • Analytical complexity:

– for any SNR – for moderately low SNR

  • Empirical Results:

–Evaluated on real GPS signals –Improves performance by 2.2x

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SLIDE 20

How can we make GPS synchronization faster than FFT‐Based synchronization?

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SLIDE 21

FFT‐Based GPS Synchronization

Received Signal

FFT

Signal in Freq. FFT of Code

IFFT

Output

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SLIDE 22

FFT‐Based GPS Synchronization

Received Signal

FFT

Signal in Freq. FFT of Code

IFFT

FFT Stage IFFT Stage

Output

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SLIDE 23

FFT‐Based GPS Synchronization

Received Signal

FFT

Signal in Freq. FFT of Code

IFFT

FFT Stage IFFT Stage

Output

Each stage takes  need to reduce complexity of both stages

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SLIDE 24

FFT‐Based GPS Synchronization

Received Signal

FFT

Signal in Freq. FFT of Code

IFFT

FFT Stage IFFT Stage

Output

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SLIDE 25

FFT‐Based GPS Synchronization

Received Signal

FFT

Signal in Freq. FFT of Code

IFFT

FFT Stage IFFT Stage

Output Output

Sparse

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SLIDE 26

FFT‐Based GPS Synchronization

Received Signal

FFT

Signal in Freq. FFT of Code

IFFT

FFT Stage IFFT Stage

Output Output

Sparse

Sparse IFFT

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SLIDE 27

QuickSync

A Sparse IFFT algorithm customized for GPS

  • Exactly One Spike  Simpler algorithm
  • Extends to the FFT‐stage which is different

(will discuss later)

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SLIDE 28

1‐ Bucketize

∑

2‐ Estimate

QuickSync’s Sparse IFFT

Divide output into a few buckets Estimate the largest coefficient in the largest bucket

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SLIDE 29

1‐ Bucketize

∑

2‐ Estimate

QuickSync’s Sparse IFFT

Divide output into a few buckets Estimate the largest coefficient in the largest bucket

So how can we bucketize and estimate efficiently?

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SLIDE 30

How to Bucketize Efficiently?

IFFT IFFT

  • utput samples

input samples

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SLIDE 31

How to Bucketize Efficiently?

IFFT IFFT IFFT IFFT

  • utput samples

input samples

Subsamples Buckets

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SLIDE 32

How to Bucketize Efficiently?

IFFT IFFT IFFT IFFT

  • utput samples

input samples

Subsamples Buckets

Efficient since: small IFFT of size equal to the number buckets Efficient since: small IFFT of size equal to the number buckets

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SLIDE 33
  • Keep largest bucket; ignore all the rest
  • Out of the samples in the large bucket,

which one is the spike?

How to Estimate Efficiently?

Largest bucket

Buckets

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SLIDE 34
  • Keep largest bucket; ignore all the rest
  • Out of the samples in the large bucket,

which one is the spike?

How to Estimate Efficiently?

Largest bucket

The spike is the sample that has the maximum correlation Buckets

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SLIDE 35
  • Keep largest bucket; ignore all the rest
  • Out of the samples in the large bucket,

which one is the spike?

How to Estimate Efficiently?

Largest bucket

The spike is the sample that has the maximum correlation Buckets Efficient since: compute correlation only for few samples in the largest bucket

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SLIDE 36

Estimation:

QuickSync’s Sparse IFFT

  • is number of samples
  • samples per bucket 

buckets

Bucketization:

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SLIDE 37

QuickSync Synchronization

Input

FFT

Signal in Freq. FFT of Code

IFFT

FFT Stage

Sparse IFFT

Output

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SLIDE 38

QuickSync Synchronization

Input

FFT

Signal in Freq. FFT of Code

IFFT

FFT Stage

Sparse IFFT

Output

Output is not sparse Cannot Use Sparse FFT

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SLIDE 39

QuickSync Synchronization

Input

FFT

Signal in Freq. FFT of Code

IFFT

FFT Stage

Sparse IFFT

Output

Input to next stage

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SLIDE 40

QuickSync Synchronization

Input

FFT

Signal in Freq. FFT of Code

IFFT

FFT Stage

Sparse IFFT

Output

Subsampled FFT

IFFT samples its input Need only few samples of FFT output

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SLIDE 41

QuickSync Synchronization

Input

FFT

Signal in Freq. FFT of Code

IFFT Sparse IFFT

Output

Subsampled FFT

FFT and IFFT are dual of each other

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SLIDE 42

QuickSync Synchronization

Input

FFT

Signal in Freq. FFT of Code

IFFT Sparse IFFT

Output

Subsampled FFT

Subsampling IFFT Bucketization

FFT

Bucketization

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SLIDE 43

QuickSync Synchronization

Input

FFT

Signal in Freq. FFT of Code

IFFT Sparse IFFT

Output

Subsampled FFT

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SLIDE 44

QuickSync Synchronization

Input

Subsampled FFT

Subsampled FFT of Code

Sparse IFFT

Correct delay

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SLIDE 45

Formal Analysis

Theorem: (informally restated) For any SNR QuickSync achieves the same accuracy as FFT‐Based synchronization and has a complexity of where is the number of samples in the code For moderately low SNR (i.e. noise is bounded by

  • ), QuickSync has

complexity

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SLIDE 46

Rest of this Talk

  • GPS Primer
  • Our GPS Synchronization Algorithm
  • Empirical Results
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SLIDE 47
  • Traces are collected both US and Europe
  • Different locations: urban – suburban
  • Different weather conditions: cloudy – clear

Setup

SciGe GN3S Sampler USRP Software radios

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SLIDE 48
  • QuickSync Synchronization
  • FFT‐Based Synchronization

Compared Schemes

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SLIDE 49
  • Hardware implementations
  • Software implementations

Metrics

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SLIDE 50

2.1x

Multiplication Gain

3x 1.3x

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SLIDE 51

2.1x

QuickSync provides an average gain of 2.1x Multiplication Gain

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SLIDE 52

QuickSync provides an average gain of 2.2 × FLOPS Gain

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SLIDE 53

Does the Gain Depend on the GPS SNR? QuickSync improves over FFT‐Based for the whole range of GPS SNRs

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SLIDE 54
  • Past work on GPS [NC91, SA08, RZL11]

– QuickSync presents the fastest algorithm to date

  • Sparse FFT Algorithms [GMS05, HKIP12a, HKIP12b]

– QuickSync’s bucketization leverages duality  reduces the complexity of both stages in GPS

Related Work

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SLIDE 55

Conclusion

  • Fastest GPS synchronization algorithm

– for any SNR – for moderately low SNR

  • Empirical results show an average 2x gain
  • QuickSync applies to general synchronization

tasks beyond GPS