Nonlinear dynamics for signal identification T. L. Carroll Naval Research Lab

Transcription

1 Nonlinear dynamics for signal identification T. L. Carroll Naval Research Lab

2 Multiple radars: how many transmitters are there?

3 Specific Emitter Identification Older transmitters Modern transmitters Transients on leading and trailing edges: use linear signal processing to get signature No transients: linear signal processing can t find signature

4 Good test problem for NLD approach: Radar reflection goes as 1/R 4, so S/N large at target R Radar amplifier runs at high power- very nonlinear All amplifiers based on nonlinear devices consider problem as driven nonlinear system

5 Specific Emitter ID: Data Driven Amplifier is driven nonlinear system Embedding of original signal sk+2!) p 1 p 2 p k Digitize output signal Create vectors: p 1 & s1) # = 1 )! s + '!, s1 + 2' )!" p 2 = & s2) # 2 )! s + '! s2 + 2' )!", sk) sk+!) Has same geometrical properties as original attractor

6 Specific Emitter ID: Data Driven If 2 dynamical systems identical, same point in phase space should have same derivatives Identical systems Non-identical systems Quantify this difference: average over signal Average = <">

7 Simple numerical example ) = sin "t )! t dx A dt dy A dt ' =# A & g &!) * x A dx =# & A A dt * y & A ) = x + + A x 3 g x Amplifier model: ' ) ) ' ) x A i+!), x B i+!) x A, x B ) = sin "t )! t dx B dt dy B dt g x ' =# B & g &!) * x B dx =# & B B dt * y & B ) = x + + B x 3 ' ) ) ' ) Set different parameters for A, B Embed x A, x B in 2-d phase space Derivative difference # = [x A i+1)-x A i)] - [x B i+1)-x B i)] Take average for many points

8 Simulation results: Different time constant, linear or nonlinear <#> A = 1 A = B = 0 linear) A = B = 0.1 nonlinear) <#> & Same time constant, different nonlinearity A = B = 1 A = B

9 Specific Emitter ID: Data Driven Experiment 3 nominally identical op amps driven with same signal A B C in out A out B out C Save a signal from each amp as a reference Take an unknown signal - compare to each reference Which reference gives smallest phase space difference?

10 Experiment Embed reference Embed unknown Pick index point u j on unknown Nearest reference point is v k reference unknown u j u j+1 v k v k+1 Derivatives u j+1 -u j, v k+1 -v k

11 Real Data: complications v i+1 reference unknown v i u j u j+1 v k v k+1 Frequency modulation: reference may have 2 very different derivatives search for pairs of points v i, v i+1 )

12 Real Data: complications v i+2 v i+1 v i u j+1 u j reference unknown u j-1 Unknown signal sampled at different phase than reference Draw line between v s z j is closest approach of line to u j z j is estimate for reference point v i difference in derivatives proportional to distance between u and z: normalize by this distance

13 Phase space statistic: k is vector component!= d ) k=1 d ) k=1 *,#,,,, +, # Reference derivative zk j+1 "z j k & " u j+1 ' & ' # 2 zk j+1 "uk j+1 Unknown derivative { { k # "u j k & ' + zk j "uk j Average " over signal: <"> & ' 2 - / / / / /./ distance

14 Summary of algorithm 1. Record reference signals from several amplifiers 2. Record unknown signal 3. Embed unknown and reference signals 4. For each reference: A. Pick index pair on unknown: search for nearest reference pair. B. Interpolate reference point to correct for phase error C. Compute unknown and reference derivatives: take difference D. Normalize difference by distance between unknown and reference pairs E. Average over trajectory.

15 Experiment: Drive 3 op amps with 25 khz sine, amplitude 1V harmonics 0-20 P db) f Khz) Divide signal into many segments Measure <"> for each Compute histogram

16 p<!>) Experiment Histogram of phase space difference Compare unknown to: Reference A Reference B Reference C <!> unknown came from amplifier A)

17 Experiment: frequency modulated FM) signals Randomly shift signal between 22.5 khz and 27.5 khz Compare unknown to: Reference A Reference B Reference C Drive amplitude: p<!>) V reference amplitude = 1V) <!> p<!>) V nonlinear effects larger) reference amplitude = 2V) unknown came from A) <!>

18 What about signals with different amplitudes? Need information from all parts of phase space Step up amplitude of driving signal: sine wave input signal output signal

19 At each point in phase space, find output derivative as function of input derivative d y! x!! " # &!! =a dt 1 x +a!! x d x!!! +a " # & 2 " # & dt 3 x d x! + " # & 1 dt 2 sine wave driving) Store coefficients as function of x locations This is the phase space model

20 Using phase space model Record unknown signal output signal u Assume unknown input signal was sine Fit sine to u: fitted sine is assumed input signal v unknown u fitted sine v Amplitude of real driving signal not known

21 Rescale assumed drive signal v s ='v Use phase space model: estimate unknown du/dt based on v s d u!! E # " dt! vs &!! =a vs +a!! vs d v! 1 # " & 2 # & s!! +a vs d v! " dt 3 # & s + " dt 1 2 v s du E /dt Find value of ' to minimize 'du/dt - du E/ dt = "

22 Unknown amplitudes: results- sinusoidal driving Use phase space model from A, B, or C histogram P<">) Phase space model: A B C Drive amp = 1V <"> P<">) Same model Drive amp = 2V Unknown was from A <">

23 Amplifier ID based on phase space methods is possible larger nonlinearity- easier to ID Phase space modeling: need better models- should reflect physics of problem need adjustable parameters for different amplifiers of same class.