DYNAMIC BEAM FORMING USING CHIRP SIGNALS

Transcription

1 BeBeC-018-D04 DYNAMIC BEAM FORMING USING CHIRP SIGNALS Stuart Bradley 1, Lily Panton 1 and Matew Legg 1 Pysics Department, University of Auckland 38 Princes Street, 1010, Auckland, New Zealand Scool of Engineering & Advanced Tecnology, Massey University, Auckland, New Zealand ABSTRACT Beam forming generally gives good lateral spatial resolution control but poorer axial resolution. For active array systems, in wic ecoes are received from a transmitted pulse, swept frequency cirp pulses give very good axial spatial resolution. Tis is because te pase of te eco from a particular range needs to very accurately matc te pase of te reference transmitted signal wit wic it is compared in te matced filter. Te result is an axial resolution inversely proportional to te swept frequency bandwidt, and independent of te pulse duration. We describe ow tis tigt pase requirement also gives tigt lateral resolution for a cirped pulse, because decorrelation in te matced filter occurs rapidly off-axis. Tis gives scope for dynamic beam forming for active arrays based on te pulse design, or on wat part of a swept frequency pulse is included in te matced filter. 1 INTRODUCTION Muc of te origins of beamforming arise from early radar development. Te development by Bell of cirp signal processing [1] was ugely influential since te early interest was in target identification and location. Tis pulse-compression metodology as wide application in remote sensing systems, including in ultrasonic imaging. Te most common cirp is a linear sweep in frequency during te transmitted pulse, s(t), of te form s 0 t < 0 B t sin = sin π f 0 + t 0 t τ (1) τ 0 τ < t ( t) = ( ϕ) were t is time, ϕ is te pase, f0 is te frequency at t = 0, B is te bandwidt, and τ is te pulse duration (Fig. 1a). Te instantaneous frequency is 1

2 7 t Berlin Beamforming Conference 018 Bradley, Panton and Legg 1 dϕ = = f π dt f 0 Resolution transverse to te propagation axis depends on antenna beamforming. Systems are mostly monostatic, wit a common array used for bot transmit and receive, since tis is by far te simplest geometry and makes ardware design easier. Beamforming is terefore similar for transmit and receive. But for array beamforming te frequency content of te signal is generally considered to be constant, rater tan canging in a non-stationary way as wit te linear FM cirps described above. Te dynamic nature of tis beamforming gives rise to a number of interesting questions considered in tis paper. Te eco signals from different distances off- t + B τ () If an eco r(t) = s(t - t) aving te same form is received from a static target time t after pulse transmission, te normalized envelope of te square of te cross correlation of s(t) and r(t) is approximately ( πb[ t t] ) B[ t t] Tis as a peak at t = t, falling to zero at t = t ± 1/B. Te Rayleig criterion is te generally accepted criterion for te minimum resolvable detail in an imaging process, defined as wen te first minimum of te image of one source point coincides wit te maximum of anoter. Here tis gives a time resolution of 1/B (Fig. 1b). Tis metod allows te pulse duration τ to be long, ence providing more power, witout affecting te along-axis spatial resolution. sin χ( t ) = π (3) Fig. 1. An example of a linear cirp signal wit f 0τ =Bτ = 0. Te transmitted signal (a), and (b) te square of te correlation of transmitted and received signals (black), te envelope (red), a delayed eco (blue), and te combination of non-delayed and delayed ecoes (green).

3 7 t Berlin Beamforming Conference 018 Bradley, Panton and Legg axis will ave different frequencies wen tey reac te receiver. How do tese new pase dependencies affect transverse spatial (or angular) resolution? BEAM SHAPING BY CHIRP SIGNALS Beam forming usually involves design of a beam from a transmitting array or design of te angular sensitivity of a receiving array. In tose cases were bot transmitting and receiving occur, it is common to consider te resulting sensitivity as te being due to te product of te transmitting array beam and te receiving array beam. However, tis is not adequate wen te signals are not tonal, since te timing from transmitted signal to target and ten to te micropone is crucial. Te general geometry for planar co-located transmitting and receiving arrays, and a co-planar target, te simplest configuration, is sown in Fig. z target rt rt φt φt θ speaker rs x φs rm micropone y φs rm φm φm Fig.. Te general geometry from transmission from a small element on te z = 0 plane, reflection from a small element on te z = plane, and reception at a small element on te z = 0 plane..1 Beam pattern from a tonal disc In general a sparse array, typically a spiral array, will be used. However, to avoid limiting to a particular array at tis point, we investigate te source being a disc of diameter D a distance from te target, and te receiver also being a sensitive disc of te same diameter. Initially we integrate te effect of small elements on te source disc and compare wit te analytic far-field 3

4 7 t Berlin Beamforming Conference 018 Bradley, Panton and Legg Airy diffraction pattern, for a tonal source, in Fig. 3. Te agreement is close, particularly around te central lobe.. Beam pattern from a cirp disc source Te beam pattern on te target plane wen te source produces a linear FM cirp pulse is timedependent, as sown in Fig. 4. Fig. 3. A ceck on te integration metod (blue) against te analytic Airy diffraction pattern (orange) using a scaled distance k = 400 and were k is te wavenumber. Figure 4 is a plot of intensity on te target plane as a function of normalised time, ct/, and beam zenit angle θ, were c is te sound speed. Te first arrival of sound at te center of te target (θ = 0) is at ct/ = 1. Sound continues to arrive at te center until te last sound from te peripery of te source disc. For off-center positions on te target, te first sound arrives later. In general, for first sound arrival, and ct D = 1+ tan θ (4a) 4

5 7 t Berlin Beamforming Conference 018 Bradley, Panton and Legg c [ t τ] D = 1+ tan θ + (4b) for te last sound arrival. In tis example, f0τ = 0, Bτ = 15, cτ/ = 0.85, and D/ = 0.. Te circular arc limits are clearly seen in Fig. 4. Tere is also muc fine detail caused by interference between te differing frequency components and pases from sound arriving at a point on te target from different points on te source. Te brigtness of te limiting circles arises because initially tere is limited sound arriving from more extended parts of te source, and so little interference, and similarly te last sound arriving does not ave muc interference from oter parts of te source apart from te edges. If a particular time is considered, say ct/ = 1., we see an Airy-like beam pattern versus θ wit deep nulls, but sarply cut off at θ = tan 1 D ct + 1 1/. (5) Fig. 4. Plot of intensity (db) as a function of normalised time ct/ and beam zenit angle θ. 5

6 7 t Berlin Beamforming Conference 018 Bradley, Panton and Legg.3 Reception of cirp pulse signals Given te complexity of Fig. 4, ow do we define beam sape for tis pulsed transmitterreceiver system? Wile it is straigt-forward to accumulate te reflected signals at micropone locations on te sensor plane, tis gives an overall intensity rater tan te zenit-angle dependence, or transmit-receive beam pattern. But accumulating signals from tin rings on te target plane does not give te beam pattern eiter because it does not take into account all te time-dependent interference from positions widely spread on te target. Fig. 4 also does not provide information on te spectra, wic vary wit time and angle. We ave terefore decided to accumulate signals as a time-dependent fluctuation at eac micropone location, and ten do te cross-correlation wit te transmitted signal s(t) as described in Section 1. Tis is, in any case, te way te eco signals are processed in a system like tis. Te zenit angle θ can ten be related to te time in te cross-correlation output χ(t) via θ = cos -1 (/[ct]). As an example, we do tis for a central micropone, making te assumptions tat te reflections are diffuse (or omnidirectional) and tat reflections do not cange te pase (i.e. te pase canges are only due to te time delays in propagation from source to target and from target to micropone). Figure 5 sows tis result, togeter wit te Airy patterns for f0τ = 0 and Bτ = 15. Fig. 5. Plot of intensity as a function of beam zenit angle θ for transmission and reception of a cirp pulse (blue), and te corresponding Airy patterns for te base frequency (orange) and te maximum frequency (yellow). 6

7 7 t Berlin Beamforming Conference 018 Bradley, Panton and Legg Te cirp-determined beam pattern as a different sape from te Airy pattern, being more constant over te bulk of te main lobe. Tis is generally a good feature. However te -3 db widt is only comparable to tat of te base frequency diffraction pattern. 3 CONCLUSIONS Te use of linear FM cirp pulses makes te implementation of pased array systems muc more complex. Here we ave only considered, at tis time, disc-like transmitters and a single central micropone. Also, since tis paper is an introduction to te problem being addressed, we ave not explored te tuning effect of canging te cirp bandwidt B. Furter work will consider ow to optimise te layout of finite numbers of speakers and micropones, and also ow to dynamically tune te beam widt by altering te cirp properties. B bandwidt c speed of sound D diameter of source f frequency f0 cirp frequency at t = 0 distance from array to scattering plane k wavenumber r received signal rm, φm, rm, φm micropone element position and size rs, φs, rs, φs speaker element position and size rt, φt, rt, φt target element position and size s transmitted signal t time t small time difference x, y, z Cartesian coordinates χ square of cross correlation ϕ pase θ azimut angle τ pulse duration ACKNOWLEDGEMENT Tis work is part of te Precision Acoustic Sensors project funded by New Zealand s MBIE. REFERENCES [1] J. R. Klauder, A. C. Price, S. Darlington and W. J. Alberseim. Te Teory and Design of Cirp Radars. Te Bell System Tecnical Journal, XXXIX, ,