## 1 Introduction  
Turbomachinery blades can reach critical vibration amplitude during operating conditions. Undesirable vibrations occur when the blade passes through a resonance. Resonances in the blades can be excited by the fluctuations downstream and upstream of the flow path inducing alternating aerodynamic loadings, which are the major sources of blade excitations. In resonance conditions, blades vibrations can be excessively increased with the risk of incurring a High Cycle Fatigue failure. Measuring the vibration parameters of the rotating blades is therefore of great interest to engineers at two different levels:  
* in the laboratory during the engine test phase, to obtain the natural frequencies and the damping of the blade and blisk (integrally bladed rotor [1]) in order to have experimental parameters to update the numerical models;
* in operation for the control of vibrations and for monitoring the health of the engine. Vibration measurements of rotating blades are traditionally performed by means of strain gauge technology [2]. This method has been developed over time and successfully employed for an extensive range of engineering applications. Currently, it is considered as a well-established technology with the most accurate and reliable results. However the limits of the use of strain gauges are that:they only measure the deformation of the blades on which they are glued, they must therefore be glued on all the blades to control them all;
2. they are connected to telemetry/slip ring; this might require important changes to rotating and stationary parts and implies integration of instrumentation hardware, in order to get accurate vibration response with the best Signal-to-Noise Ratio (SNR), these modifications are costly and time consuming;
3. the adaptations required by the strain gauge installation could lead to constraints for some other turbomachine parameters (for instance, the aerodynamic ones). To minimize the adverse effects of the strain gauges and in response to the high demands for having a non-intrusive blade vibration measurement system, non-contact measurement technology has been developed. Blade Tip-Timing (BTT) is a non-contact measurement method that has attracted considerable attention in previous decades among turbomachinery specialists. This method is still under development in order to enhance its capabilities to become the reference technique for measurement of vibrations in rotating blade assemblies. The main manufacturers of turbo engines have invested in recent years in improving the vibration measurement system based on BTT technology [3]. The main advantage of the BTT method is the simultaneous monitoring of all blades as well as non-intrusive aspects, easy set-up and data transmission. Its major shortcoming is the under-sampling of the signal related to the rotational speed of the rotating disk under study. The research works on BTT in the last years are about the main areas of the sensor development [4; 5], signal processing, vibration parameters identification and the application of this method for condition monitoring [6; 7; 8]. The present paper is focused on the post-processing of the BTT data to estimate the main vibration parameters even in the case of very close or overlapped resonance peaks. One of the earliest works in this area is the well-known single parameter method introduced by [PERSON] and [PERSON] [9] in 1970. This method was proposed based on a single degree of freedom (SDOF or 1 DOF) vibrating system, the maximum vibration amplitude is obtained by deriving the peak to peak values of the deflection signal. Next, [PERSON] and [PERSON] [10] proposed an amendment for the single parameter method by considering the effects of blade vibration on the computation of the arrival time. Then, in 2000, [PERSON] [11] presented a new analysis technique called two-parameter plot method for identifying synchronous resonance vibration by using two tip-timing sensors.