Sampling Rate Synchronisation in Acoustic Sensor Networks with a Pre-Trained Clock Skew Error Model

Transcription

1 in Acoustic Sensor Networks with a Pre-Trained Clock Skew Error Model Joerg Schmalenstroeer, Reinhold Haeb-Umbach Department of Communications Engineering - University of Paderborn Computer Science, Electrical Engineering and Mathematics Communications Engineering Prof. Dr.-Ing. Reinhold Häb-Umbach

2 Table of Contents 1 Introduction 2 Clock Frequency Offset Estimation 3 Error Model & Kalman Filter 4 Experimental Results Hardware Platform Wired connection Wireless connection 5 Conclusions R. Haeb-Umbach 1 / 11

3 Introduction Acoustic sensor network Consists of sensor nodes connected by wire or wirelessly Nodes record (multi-)channel audio signals and process them Applications: ad-hoc teleconferencing, monitoring, surveillance Problem statement Sampling clock oscillators at nodes differ typically by ±50 ppm Time base of nodes diverges rapidly Precludes application of certain algorithms (e.g. TDOA estimation across sensor nodes) Our approach Estimate clock frequency and phase offset via time stamp exchange protocol Improve estimate by Kalman Filter with dedicated error model Adjust sampling frequency through direct digital synthesis (DDS) on hardware platform R. Haeb-Umbach 2 / 11

4 Clock Frequency Offset Estimation t A,k+1 R,k+1 t t A,k t R,k ξ B A,k ξ ξ R,k+1 ξ A,k+1 R,k A t R,k t A,k t R,k+1 t A,k+1 t t Time stamp exchange Times at nodes A and B t R,k = (t R,k +ξ R,k ) ω +ϕ t A,k = (t A,k ξ A,k ) ω +ϕ ω: clock freq. offset (ω = 1: perfect sync.) Clock Frequency Offset Estimation [Chaudhari 2012] ( 1+ ξ R,k+1+ξ A,k )ω and t R,k+1 t A,k t + t + = t R,k+1 t A,k t R,k+1 t A,k = t t = ( 1 ξ R,k+ξ A,k+1 t R,k t A,k+1 ω = t + t ( t + t = 1+ (ξ ) R,k+1 ξ R,k )+(ξ A,k ξ A,k+1 ) ω (t R,k+1 t R,k )+(t A,k+1 t A,k ) ) ω Influence of transmission times vanishes with increasing temporal distance between k-th and (k +1)-st time stamp exchange R. Haeb-Umbach 3 / 11

5 Transmission Error Model Probability Histogram Gaussian Mixture Model Frequency offset estimation error [ppm] Node A MPU DDS MPU Experiment: Two nodes connected to a single crystal oscillator Transceiver Node B Transceiver Measured observation error distribution p(ˆω ω = 0) (6 hours of data) Two types of errors Largecscale: Packet losses, protocol dependent wait states, medium access control (caused by network) Small-scale: Estimation error of time stamp exchange protocol, MPU hardware interrupts, I/O latencies Gaussian mixture model (GMM) to approximate measured histogram p(v o) = K p(k)p(v o k) = K γ k N(v o;µ k,o,σk,o 2 ) k=1 k=1 GMM trained offline Time stamps R. Haeb-Umbach 4 / 11

6 Kalman Filter (1/2) Kalman filter Simple kinematic model to model oscillator frequency drifts (x = [ω, ω] T ) [ ] [ ] [ ] 1 T x(n +1) = x(n) v s(n) }{{}}{{}}{{} F G v s Measurement equation: z(n) := ω(n) = [ 1 0 ] x(n)+v o(n) }{{} H T Minimum mean square error (MMSE) estimate of the system state ( x(n n) = x(n n 1)+K(n) z(n) H T x(n n ) 1) E[v o(n)] Predicted MMSE estimate of frequency offset ω (KF) (n) = H T x(n n 1) R. Haeb-Umbach 5 / 11

7 Kalman filter (2/2) Assumption Kalman filter prediction ω (KF) (n) is close to true value ω ω (KF) (n) ω δ, Probability δ Histogram Gaussian Mixture Model δ = min µk,o k,l µ l,o Frequency offset estimation error [ppm] Removal of large-scale observation errors Contributions of large scale effects to the observation error can be identified ˆk = argmin ω (KF) (n) µ k,o = argmax p(v o k) k k Large-scale observation error can be removed: ( ) x(n n) = x(n n 1)+K(n) (z(n) µˆk,o ) HT x(n n 1)] R. Haeb-Umbach 6 / 11

8 Hardware platform Mics Node A ADC DDS Crystal Oscillator Network DSP MPU Transceiver Time stamps Network DSP MPU Transceiver Node B ADC DDS Crystal Oscillator Mics Hardware platform Network-connected multi-channel acoustic sensor nodes (own development) ADC with an oversampling factor of 512 to generate a 16kHz sampling rate Direct Digital Synthesis (DDS): Generates arbitrary frequencies with sub-hertz resolution: Hz ˆ= ppm@16 khz Time stamps: MPU counts oscillations Time stamp exchange via wireless link (IEEE MAC & physical layer) Stacked on top: BeagleBoard XM (DSP & Ethernet connection) R. Haeb-Umbach 7 / 11

9 Experiment on Wired Connection Difference in oscillations η(t) Time t [h] Setup: Wired connection, DDS of slave node adjusted USART connection between two sensor nodes for time stamp exchange Absence of large-scale errors: GMM turns into single Gaussian distribution t Difference in oscillations: η(t) = (f M (τ) f (A) S (τ))dτ f M (t): Frequency of master node 0 f (A) S (t): Adjusted frequency of slave node Maximum difference was kept below 250 oscillations Maximum sampling error below a half sample (oversampling factor of 512!) R. Haeb-Umbach 8 / 11

10 Experiment (1/2) on Wireless Connection Frequency offset [ppm] Kalman ˆω (KF) Ground Truth ω Time t [h] Setup: wireless connection, DDS not adjusted Wireless connection (ZigBee) Comparison between Ground truth ω(n) (measured by extra hardware device) Kalman filter estimate ω (KF) (n) Mean square error was measured to be ppm R. Haeb-Umbach 9 / 11

11 Experiment (2/2) on Wireless Connection Difference in oscillations η Time t [min] Setup: wireless connection, DDS adjusted Wireless connection (ZigBee) Difference between data streams remains below 180/512 = 0.35 samples Network load caused by exchange of 64-Bit time stamps each 10 s: 4 64Bit/10s = 25.6Bit/s R. Haeb-Umbach 10 / 11

12 Conclusions and Outlook Conclusions Clock frequency synchronisation of distributed sensor nodes by time stamp exchange protocol Improved clock frequency offset estimates by post filter Kalman filter which exploits the characteristics of the estimation error Implementation on microprocessor units Communication via an IEEE wireless network Low network load by time stamp exchange Long term experiments: Difference between two data streams is kept below a maximum of half a sample Outlook Improved feedback control Online error model estimation R. Haeb-Umbach 11 / 11

13 Thank you for your attention! Questions? Prof. Dr.-Ing. R. Haeb-Umbach University of Paderborn Department of Communications Engineering nt.uni-paderborn.de Computer Science, Electrical Engineering and Mathematics Communications Engineering Prof. Dr.-Ing. Reinhold Häb-Umbach