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A new identification technique for a class of weakly nonlinear systems whose behavior is adequately characterized in terms of a finite Volterra functional series is presented. Application of the identification technique results in a complete specification of the nonlinear impulse responses which describe a weakly nonlinear system. The identification technique is a "black box" procedure in that only measurements at the system input and output terminals are required. A functional form for the second-order impulse response, , is derived for a nonlinear system with a finite number of power-law devices. The identification of is accomplished by exciting the system with a sum of exponentially decaying signals and appropriately processing the input and output signals using the pencil-of-functions system identification approach. This results in a complete set of linear equations involving all the parameters of . Solution of these equations uniquely determines . An example of the practical application of the technique to a common emitter amplifier is presented.