Derivation of the impulse response of ultrasonic transducers by experimental system identification

An important stage in realistic simulations of ul- trasound imaging systems is the employment of a model for the electroacoustic behaviour of the transducer. State of the art models based on equivalent circuits, finite element modelling or measured impulse responses suffer from miscellaneous practical deficiencies. A method for an experimental estimation of the impulse response based on linear parametric system identification and carefully chosen input signals is presented. Appropriate measurement conditions are derived from numerical simulations of diffraction effects. The approach is shown to provide a way to accurately determine the impulse response of a transducer without any advance information about the physical properties of the piezoelectric crystal. I. INTRODUCTION A piezoelectric ultrasound transducer in pulse echo imaging is used to transform an electrical signal produced by some driving circuit into an acoustical wave and vice versa to transform a pressure wave which impinges onto the transducer into an electrical signal. Simulation of ultrasound pulse echo systems thus require a realistic description of the relation be- tween electric and acoustic signals of the simulated transducer. The electromechanical behaviour can either be included into the simulation by the application of some kind of analytic model for the coupling between electrical and acoustical quantities or it can be treated as a black box within the system model whose behaviour is derived from measurements prior to the simulation. Within the analytic approach most frequently simple linear models have been employed which assume either proportionality between the acoustic and electric quantities (1) or relate them by an impulse response which has the form of a modulated sine function (2). While this might be sufficient for a coarse analysis of ultrasound systems it lacks the capability to simulate real systems thus making comparisons between simulations and experiments impossible. Equivalent circuits or finite element simulations have suc- cessfully been used to acquire more realistic models for the electromechanical behaviour of piezoelectric transducers (3), (4). These approaches require knowledge of the geometry and material properties of the piezoelectric crystals, their backing and matching layers. Moreover, in many cases these models are either not very accurate or they involve high numerical effort. Linear models based on measured impulse responses avoid problems associated with analytical approaches and are precise enough to be successfully included into pulse echo simulations (5)-(8). The impulse response of a piezoelectric ultrasound trans- ducer is most commonly been measured from reflections of a flat metal plane due to an excitation of the piezoelectric element with an electric pulse signal. It is well known that for non focussing transducers even a single point scatterer gives rise to a complex echo signal with overlapping contributions from plane and edge wave components due to diffraction from the transducer. It is therefore problematic to consider the echo signal to be the electromechanical impulse response. Removing the diffraction effects has successfully be done by numerical deconvolution (5). Since the diffraction depends on the geometry of the transducer and its oscillation modes, deconvolution is only applicable to situations in which detailed knowledge of the geometrical and mechanical properties of the transducer is available. It is advantageous to find a measure- ment set-up in which the diffraction proves to be negligible. By employing the work of Rhyne (9) some authors pointed out that the diffraction effects between a circular non focussing transducer with pistonlike behaviour and an infinite perfectly reflecting plane can be neglected if the plane is placed in the nearfield of the transducer (5), (7), (8). In this case the assumption that the measured echo signal due to an impulse excitation resembles the electromechanical impulse response is thus true. There is some evidence that this holds also for circular transducers having a Gaussian velocity distribution (10). We will show by numerical computations that the result is also valid for rectangular transducers and non uniform movement of the transducer surface. This gives strong support that the measurement principal is applicable even if the exact properties of the transducer are unknown. Direct measurement of the impulse response belongs to the class of non parametric system identification methods (11). It is afflicted with practical difficulties due to the inability to create pulses of sufficiently short duration, transient behaviour of the transducer, and a poor signal to noise ratio of the measured impulse response. We therefore suggest the use of parametric system identification methods based on ARX- system models and linear frequency modulated excitation signals. The usage of specific broadband excitation signals to identify mathematical models is quite common in control engineering but have to our knowledge not yet been applied to ultrasound transducers. The approach potentially overcomes the mentioned drawbacks associated with needle shaped sig- nals.