1. Correlation (Packet detection)

2. Cross Correlation
A simple, yet powerful technique for detecting known patterns
A key physical layer technique for detecting packets
Detecting cell tower ID based on known sequences
Detecting gene sequences
Other broad applications in - pattern recognition, single particle analysis, electron tomography, averaging, cryptanalysis, and neurophysiology
preamble
payload
channel is idle

3. Cross Correlation Consider a sequence of complex numbers –
𝑍= 𝑠 1 𝑒 𝑖 𝜃 1 , 𝑠 2 𝑒 𝑖 𝜃 2 , 𝑠 3 𝑒 𝑖 𝜃 3 , 𝑠 4 𝑒 𝑖 𝜃 4
Consider its complex conjugate
𝑍 ∗ = 𝑠 1 𝑒 −𝑖 𝜃 1 , 𝑠 2 𝑒 −𝑖 𝜃 2 , 𝑠 3 𝑒 −𝑖 𝜃 3 , 𝑠 4 𝑒 −𝑖 𝜃 4
𝑍 ∗ 𝑍 𝑇 =
= 𝑠 1 + 𝑠 2 + 𝑠 3 + 𝑠 4
𝑠 1 𝑒 𝑖 𝜃 1
𝑠 2 𝑒 𝑖 𝜃 2 𝑠 3 𝑒 𝑖 𝜃 3
𝑠 4 𝑒 𝑖 𝜃 4
𝑠 1 𝑒 −𝑖 𝜃 1 𝑠 2 𝑒 −𝑖 𝜃 2 𝑠 3 𝑒 −𝑖 𝜃 3 𝑠 4 𝑒 −𝑖 𝜃 4

4. Cross Correlation 𝑍 ∗ 𝑍 𝑇 = = 𝑠 1 + 𝑠 2 + 𝑠 3 + 𝑠 4 𝑍 ∗ 𝑁 𝑇 =
𝑠 1 𝑒 −𝑖 𝜃 1 𝑠 2 𝑒 −𝑖 𝜃 2 𝑠 3 𝑒 −𝑖 𝜃 3 𝑠 4 𝑒 −𝑖 𝜃 4
𝑠 1 𝑒 𝑖 𝜃 1
𝑠 2 𝑒 𝑖 𝜃 2 𝑠 3 𝑒 𝑖 𝜃 3
𝑠 4 𝑒 𝑖 𝜃 4
Phase match, the sum is high
𝑍 ∗ 𝑍 𝑇 = = 𝑠 1 + 𝑠 2 + 𝑠 3 + 𝑠 4
𝑍 ∗ 𝑁 𝑇 =
= 𝑠 1 𝑛 1 𝑒 −𝑖( 𝜃 1 + 𝜙 1 ) + 𝑠 2 𝑛 2 𝑒 −𝑖( 𝜃 2 + 𝜙 2 ) + 𝑠 3 𝑛 3 𝑒 −𝑖( 𝜃 3 + 𝜙 3 ) + 𝑠 4 𝑛 4 𝑒 −𝑖( 𝜃 4 + 𝜙 4 )
𝑠 1 𝑒 −𝑖 𝜃 1 𝑠 2 𝑒 −𝑖 𝜃 2 𝑠 3 𝑒 −𝑖 𝜃 3 𝑠 4 𝑒 −𝑖 𝜃 4
𝑛 1 𝑒 𝑖 𝜙 1
𝑛 2 𝑒 𝑖 𝜙 2 𝑛 3 𝑒 𝑖 𝜙 3
𝑛 4 𝑒 𝑖 𝜙 4
Phase mis-match, the sum is low magnitude

5. Cross Correlation
The phase match results in strong detection of the known pattern even in presence of noise.
For example
The sequence Z is a subsequence of another large sequence
Noise has been added to it
Cross correlation helps detects this submerged sequence
𝑎 1 𝑒 𝑖 𝜓 1 , 𝑎 2 𝑒 𝑖 𝜓 2 , 𝑎 3 𝑒 𝑖 𝜓 3 , 𝑎 4 𝑒 𝑖 𝜓 4 , , 𝑠 1 𝑒 𝑖 𝜃 1 , 𝑠 2 𝑒 𝑖 𝜃 2 , 𝑠 3 𝑒 𝑖 𝜃 3 , 𝑠 4 𝑒 𝑖 𝜃 4 , 𝑎 5 𝑒 𝑖 𝜃 1 , 𝑎 6 𝑒 𝑖 𝜃 2 , 𝑎 7 𝑒 𝑖 𝜃 3 , 𝑎 8 𝑒 𝑖 𝜃 4
𝑎 1 𝑒 𝑖 𝜓 1 , 𝑎 2 𝑒 𝑖 𝜓 2 , 𝑎 3 𝑒 𝑖 𝜓 3 , 𝑎 4 𝑒 𝑖 𝜓 4 , 𝑠 1 𝑒 𝑖 𝜃 1 + 𝑛 1 , 𝑠 2 𝑒 𝑖 𝜃 2 + 𝑛 2 , 𝑠 3 𝑒 𝑖 𝜃 3 + 𝑛 3 , 𝑠 4 𝑒 𝑖 𝜃 4 + 𝑛 4 , 𝑎 5 𝑒 𝑖 𝜃 1 , 𝑎 6 𝑒 𝑖 𝜃 2 , 𝑎 7 𝑒 𝑖 𝜃 3 , 𝑎 8 𝑒 𝑖 𝜃 4

6. Detecting patterns
Cross-correlation can be used to detect the submerged sequence
Specifically, it answers following questions
Does the known sequence (Z) exist in the input sequence
Where in the input sequence does Z appear?
𝑎 1 𝑒 𝑖 𝜓 1 , 𝑎 2 𝑒 𝑖 𝜓 2 , 𝑎 3 𝑒 𝑖 𝜓 3 , 𝑎 4 𝑒 𝑖 𝜓 4 , 𝑠 1 𝑒 𝑖 𝜃 1 + 𝑛 1 , 𝑠 2 𝑒 𝑖 𝜃 2 + 𝑛 2 , 𝑠 3 𝑒 𝑖 𝜃 3 + 𝑛 3 , 𝑠 4 𝑒 𝑖 𝜃 4 + 𝑛 4 , 𝑎 5 𝑒 𝑖 𝜃 1 , 𝑎 6 𝑒 𝑖 𝜃 2 , 𝑎 7 𝑒 𝑖 𝜃 3 , 𝑎 8 𝑒 𝑖 𝜃 4

7. Methodology
𝑎 1 𝑒 𝑖 𝜓 1 , 𝑎 2 𝑒 𝑖 𝜓 2 , 𝑎 3 𝑒 𝑖 𝜓 3 , 𝑎 4 𝑒 𝑖 𝜓 4 , 𝑠 1 𝑒 𝑖 𝜃 1 + 𝑛 1 , 𝑠 2 𝑒 𝑖 𝜃 2 + 𝑛 2 , 𝑠 3 𝑒 𝑖 𝜃 3 + 𝑛 3 , 𝑠 4 𝑒 𝑖 𝜃 4 + 𝑛 4 , 𝑎 5 𝑒 𝑖 𝜃 1 , 𝑎 6 𝑒 𝑖 𝜃 2 , 𝑎 7 𝑒 𝑖 𝜃 3 , 𝑎 8 𝑒 𝑖 𝜃 4
Input sequence
𝑍= 𝑠 1 𝑒 −𝑖 𝜃 1 𝑠 2 𝑒 −𝑖 𝜃 2 𝑠 3 𝑒 −𝑖 𝜃 3 𝑠 4 𝑒 −𝑖 𝜃 4
Template sequence

8. Methodology
𝑎 1 𝑒 𝑖 𝜓 1 , 𝑎 2 𝑒 𝑖 𝜓 2 , 𝑎 3 𝑒 𝑖 𝜓 3 , 𝑎 4 𝑒 𝑖 𝜓 4 , 𝑠 1 𝑒 𝑖 𝜃 1 + 𝑛 1 , 𝑠 2 𝑒 𝑖 𝜃 2 + 𝑛 2 , 𝑠 3 𝑒 𝑖 𝜃 3 + 𝑛 3 , 𝑠 4 𝑒 𝑖 𝜃 4 + 𝑛 4 , 𝑎 5 𝑒 𝑖 𝜃 1 , 𝑎 6 𝑒 𝑖 𝜃 2 , 𝑎 7 𝑒 𝑖 𝜃 3 , 𝑎 8 𝑒 𝑖 𝜃 4
𝑠 1 𝑒 −𝑖 𝜃 1 𝑠 2 𝑒 −𝑖 𝜃 2 𝑠 3 𝑒 −𝑖 𝜃 3 𝑠 4 𝑒 −𝑖 𝜃 4

9. Methodology
𝑎 1 𝑒 𝑖 𝜓 1 , 𝑎 2 𝑒 𝑖 𝜓 2 , 𝑎 3 𝑒 𝑖 𝜓 3 , 𝑎 4 𝑒 𝑖 𝜓 4 , 𝑠 1 𝑒 𝑖 𝜃 1 + 𝑛 1 , 𝑠 2 𝑒 𝑖 𝜃 2 + 𝑛 2 , 𝑠 3 𝑒 𝑖 𝜃 3 + 𝑛 3 , 𝑠 4 𝑒 𝑖 𝜃 4 + 𝑛 4 , 𝑎 5 𝑒 𝑖 𝜃 1 , 𝑎 6 𝑒 𝑖 𝜃 2 , 𝑎 7 𝑒 𝑖 𝜃 3 , 𝑎 8 𝑒 𝑖 𝜃 4
𝑠 1 𝑒 −𝑖 𝜃 1 𝑠 2 𝑒 −𝑖 𝜃 2 𝑠 3 𝑒 −𝑖 𝜃 3 𝑠 4 𝑒 −𝑖 𝜃 4

10. Methodology
𝑎 1 𝑒 𝑖 𝜓 1 , 𝑎 2 𝑒 𝑖 𝜓 2 , 𝑎 3 𝑒 𝑖 𝜓 3 , 𝑎 4 𝑒 𝑖 𝜓 4 , 𝑠 1 𝑒 𝑖 𝜃 1 + 𝑛 1 , 𝑠 2 𝑒 𝑖 𝜃 2 + 𝑛 2 , 𝑠 3 𝑒 𝑖 𝜃 3 + 𝑛 3 , 𝑠 4 𝑒 𝑖 𝜃 4 + 𝑛 4 , 𝑎 5 𝑒 𝑖 𝜃 1 , 𝑎 6 𝑒 𝑖 𝜃 2 , 𝑎 7 𝑒 𝑖 𝜃 3 , 𝑎 8 𝑒 𝑖 𝜃 4
𝑠 1 𝑒 −𝑖 𝜃 1 𝑠 2 𝑒 −𝑖 𝜃 2 𝑠 3 𝑒 −𝑖 𝜃 𝑠 4 𝑒 −𝑖 𝜃 4

11. Methodology
𝑎 1 𝑒 𝑖 𝜓 1 , 𝑎 2 𝑒 𝑖 𝜓 2 , 𝑎 3 𝑒 𝑖 𝜓 3 , 𝑎 4 𝑒 𝑖 𝜓 4 , 𝑠 1 𝑒 𝑖 𝜃 1 + 𝑛 1 , 𝑠 2 𝑒 𝑖 𝜃 2 + 𝑛 2 , 𝑠 3 𝑒 𝑖 𝜃 3 + 𝑛 3 , 𝑠 4 𝑒 𝑖 𝜃 4 + 𝑛 4 , 𝑎 5 𝑒 𝑖 𝜃 1 , 𝑎 6 𝑒 𝑖 𝜃 2 , 𝑎 7 𝑒 𝑖 𝜃 3 , 𝑎 8 𝑒 𝑖 𝜃 4
𝑠 1 𝑒 −𝑖 𝜃 1 𝑠 2 𝑒 −𝑖 𝜃 𝑠 3 𝑒 −𝑖 𝜃 𝑠 4 𝑒 −𝑖 𝜃 4

12. Methodology
𝑎 1 𝑒 𝑖 𝜓 1 , 𝑎 2 𝑒 𝑖 𝜓 2 , 𝑎 3 𝑒 𝑖 𝜓 3 , 𝑎 4 𝑒 𝑖 𝜓 4 , 𝑠 1 𝑒 𝑖 𝜃 1 + 𝑛 1 , 𝑠 2 𝑒 𝑖 𝜃 2 + 𝑛 2 , 𝑠 3 𝑒 𝑖 𝜃 3 + 𝑛 3 , 𝑠 4 𝑒 𝑖 𝜃 4 + 𝑛 4 , 𝑎 5 𝑒 𝑖 𝜃 1 , 𝑎 6 𝑒 𝑖 𝜃 2 , 𝑎 7 𝑒 𝑖 𝜃 3 , 𝑎 8 𝑒 𝑖 𝜃 4
𝑠 1 𝑒 −𝑖 𝜃 𝑠 2 𝑒 −𝑖 𝜃 𝑠 3 𝑒 −𝑖 𝜃 𝑠 4 𝑒 −𝑖 𝜃 4
board

13. Methodology
𝑎 1 𝑒 𝑖 𝜓 1 , 𝑎 2 𝑒 𝑖 𝜓 2 , 𝑎 3 𝑒 𝑖 𝜓 3 , 𝑎 4 𝑒 𝑖 𝜓 4 , 𝑠 1 𝑒 𝑖 𝜃 1 + 𝑛 1 , 𝑠 2 𝑒 𝑖 𝜃 2 + 𝑛 2 , 𝑠 3 𝑒 𝑖 𝜃 3 + 𝑛 3 , 𝑠 4 𝑒 𝑖 𝜃 4 + 𝑛 4 , 𝑎 5 𝑒 𝑖 𝜃 1 , 𝑎 6 𝑒 𝑖 𝜃 2 , 𝑎 7 𝑒 𝑖 𝜃 3 , 𝑎 8 𝑒 𝑖 𝜃 4
𝑠 1 𝑒 −𝑖 𝜃 𝑠 2 𝑒 −𝑖 𝜃 𝑠 3 𝑒 −𝑖 𝜃 𝑠 4 𝑒 −𝑖 𝜃 4

14. Methodology
𝑎 1 𝑒 𝑖 𝜓 1 , 𝑎 2 𝑒 𝑖 𝜓 2 , 𝑎 3 𝑒 𝑖 𝜓 3 , 𝑎 4 𝑒 𝑖 𝜓 4 , 𝑠 1 𝑒 𝑖 𝜃 1 + 𝑛 1 , 𝑠 2 𝑒 𝑖 𝜃 2 + 𝑛 2 , 𝑠 3 𝑒 𝑖 𝜃 3 + 𝑛 3 , 𝑠 4 𝑒 𝑖 𝜃 4 + 𝑛 4 , 𝑎 5 𝑒 𝑖 𝜃 1 , 𝑎 6 𝑒 𝑖 𝜃 2 , 𝑎 7 𝑒 𝑖 𝜃 3 , 𝑎 8 𝑒 𝑖 𝜃 4
𝑠 1 𝑒 −𝑖 𝜃 1 .𝑠 2 𝑒 −𝑖 𝜃 𝑠 3 𝑒 −𝑖 𝜃 𝑠 4 𝑒 −𝑖 𝜃 4

15. A matlab example
Pattern
Noise

16. Pattern embedded in noise – visually not detectable

17. Correlation can detect the hidden pattern

18. Correlation output when the pattern is absent

19. Dealing with multipath
Received signal is a convolution of transmitted signal and the wireless channel
Consider a preamble transmission 𝑋 𝑝
𝑌 𝑝 = 𝑋 𝑝 ∗𝐻
Correlation between 𝑋 𝑝 and 𝑌 𝑝 would be small because 𝑌 𝑝 is polluted by wireless multipath (𝐻)
Auto-correlation is used to deal with multipath effects

20. Autocorrelation 𝑋 𝑝 𝑋 𝑝 Transmit repeated copies of the preamble 𝑋 𝑝
𝑌 𝑝 = 𝑋 𝑝 ∗𝐻
𝑌 𝑝
Distorted but repeated copies the preamble are received

21. Autocorrelation
𝑌 𝑝
𝑌 𝑝
Cross correlate the two copies to detect the preamble
Strong cross correlation indicates the presence of a preamble
Such repeated sequence does not appear elsewhere in the packet, so no false positives