Automatic Position Calibration of Multiple Microphones

Vikas Chandrakant Raykar | Ramani DuraiswamiPerceptual Interfaces and Reality Lab. | University of Maryland, CollegePark

Motivation

Multiple microphones are widely used for applications like source localization, tracking and beamforming.

Most applications need to know the precise locations of the microphones.

Small uncertainity in the sensor location could make substantial contribution to the overall localization error.

In ad-hoc deployed arrays it is tedious and often inaccurate to manually measure using a tape or a laser device.

In this paper we describe a method to automatically determine the three dimensional positions of multiple microphones.

Automatically fix a coordinate system

X

Y

Z

If we know the positions of 3 speakers….

Distances are not exact

Need atleast 3 speakers in 2D. Can use more speakers

X

Y

?

Find the intersection inthe least square sense

If positions of speakers unknown…

Consider M Microphones and S speakers.

What can we measure?Distance between each speaker and all microphones.

Or Time Of Flight (TOF)

MxS TOF matrix

Assume TOF corrupted by Gaussian noise.

Can derive the ML estimate.

Calibration signal

Nonlinear Least Squares..More formally can

derive the ML estimateusing a Gaussian

Noise model

Find the coordinates of both the microphones as speakers which minimizes

speed of sound

Maximum Likelihood (ML) Estimate..

we can define a noise modeland derive the ML estimate i.e. maximize the likelihood ratio

Gaussian noise

If noise is Gaussianand independentML is same asLeast squares

observationparameters to beestimated

model

Reference Coordinate SystemReference Coordinate system

X axis

Positive Y axis

OriginSimilarly in 3D

1.Fix origin (0,0,0)

2.Fix X axis

(x1,0,0)

3.Fix Y axis

(x2,y2,0)

4.Fix positive Z axis

x1,x2,y2>0

Which to choose? Later…

Nonlinear least squares..

Levenberg Marquadrat method

Function of a large number of parameters [ 3(M+S)-6 ]

Unless we have a good initial guess may not convergeto the minima.

Approximate initial guess required.

If we have M microphones and S speakers

[ 3M+3S–6 ] parameters to estimate. [ MS ] TOF observations

[ MS ] >= [ 3M+3S – 6 ] If M=S=K then K>=5

Why do we consider M=S ? Later..

Closed form Solution.. Say if we are given all pairwise distances between N points

can we get the coordinates.

1 2 3 41 X X X X2 X X X X3 X X X X4 X X X X

Classical Metric Multi Dimensional Scaling

dot product matrixSymmetric positive definiterank 3

Say given B can you get X ?....Singular Value Decomposition

Same asPrincipal component Analysis

One hitch.. we can measureonly the pairwise distance matrix

How to get dot product from the pairwise distance matrix…Cosine Law

k

ijd

kjd

kid

i

j

• If given pairwise distances between cities we can build a map.

• Instead of pairwise distances we can use pairwise “dissimilarities”.

• When the distances are Euclidean MDS is equivalent to PCA.

• Eg. Face recognition, wine tasting

• Can get the significant cognitive dimensions.

MDS...

Steyvers, M., & Busey, T. (2000). Predicting Similarity Ratings to Faces using Physical Descriptions. In M. Wenger, & J. Townsend (Eds.), Computational, geometric, and process perspectives on facial cognition: Contexts and challenges. Lawrence Erlbaum Associates

Can we use MDS..

1. We do not have the complete pairwise distances

UNKNOWN

UNKNOWN

s1 s2 s3 s4 m1 m2 m3 m4 m5 m6 m7s1 ? ? ? ? X X X X X X X

s2 ? ? ? ? X X X X X X X

s3 ? ? ? ? X X X X X X X

s4 ? ? ? ? X X X X X X X

m1 X X X X ? ? ? ? ? ? ?m2 X X X X ? ? ? ? ? ? ?m3 X X X X ? ? ? ? ? ? ?m4 X X X X ? ? ? ? ? ? ?m5 X X X X ? ? ? ? ? ? ?m6 X X X X ? ? ? ? ? ? ?m7 X X X X ? ? ? ? ? ? ?

Forming microphone speaker pairs…

Now we know the locations of speakers and microphones close to them.

Problem is essentially same as with position of speakers known.

Can get a closed form solution using least squares technique.

Can refine all the values further by a further ML estimation.

The complete algorithm…

ApproxDistance matrix

Between Microphone

Speaker pairs

Approximation

MDSTOF matrix Approx. microphone

and speakerlocations

Nonlinear minimization

Microphone and speakerlocations

Approx. Microphone

locations

Nonlinear minimization

Exact. microphone and speaker

locations

Sample result in 2D…

Algorithm Performance…

•The performance of our algorithm depends on

•Noise variance in the estimated distances.•Number of microphones and speakers.•Microphone and speaker geometry

•One way to study the dependence is to do a lot of monte carlo simulations.

•Else can derive the covariance matrix and bias of the estimator.

•The ML estimate is implicitly defined as the minimum of a certain error function.

•Cannot get an exact analytical expression for the mean and variance.

•Can use implicit function theorem and Taylors series expansion to get approximate expressions for bias and variance.

Where to place loudspeakers..

Monte Carlo Simulations…

Calibration Signal…

• Compute the cross-correlation between the signals received at the two microphones.

• The location of the peak in the cross correlation gives an estimate of the delay.

• Task complicated due to two reasons 1.Background noise. 2.Channel multi-path due to room reverberations.• Use Generalized Cross Correlation(GCC).

• W(w) is the weighting function. • PHAT(Phase Transform) Weighting

Time Delay Estimation…

Experimental Setup…

Results

Related Previous work…

J. M. Sachar, H. F. Silverman, and W. R. Patterson III. Position calibration of large-aperture microphone arrays. ICASSP 2002

Y. Rockah and P. M. Schultheiss. Array shape calibration using sources in unknown locations Part II: Near-field sources and estimator implementation. IEEE Trans. Acoust.,Speech, Signal Processing, ASSP-35(6):724-735, June 1987.

R. Moses, D. Krishnamurthy, and R. Patterson. A self-localization method for wireless sensor networks. Eurasip Journal on Applied Signal Processing Special Issue on SensorNetworks, 2003(4):348-358, March 2003.

J. Weiss and B. Friedlander. Array shape calibration using sources in unknown locations a maximum likelihood approach. IEEE Trans. Acoust., Speech, Signal Processing , 37(12):1958-1966, December 1989.

Our Contributions…

• Locations of the speakers need not be known.

• Only constraint is that there showld be a microphone close to a loud speaker.

• In a practical setup attach a microphone to a louspeaker.

•Derived the theoretical variance of the estimator.

•Where to place the loudspeakers?

Acknowledgements…

•Dr. Dmitry Zotkin for building the microphone array.

•Dr. Elena Grassi and Zhiyun Li for the data capture boards,

Thank You ! | Questions ?