# An Improved Circumferential Fourier Fit (CFF) Method for Blade Tip Timing Measurements
## 1 Introduction  
Rotating blade vibration measurements are of great significance for any turbomachinery research and development program [1; 2; 3; 4]. As a non-contact measurement technique, the blade tip timing (BTT) technique has been widely used in blade vibration measurements, such as in aero-engines, turbines, and compressors [5; 6; 7; 8; 9]. This technique uses the time of arrival (ToA) of the blade tip passing the casing-mounted probes to give the blade vibration. Compared with traditional strain gauges [10], BTT has the advantages of low equipment cost, easy installation, and monitoring vibration of each blade that passes through the probe [11]. However, BTT signals are typically under-sampled. It brings difficulty for blade vibration analysis. In BTT analysis, the character of the blade response is usually grouped into two distinct classes, namely, asynchronous vibration and synchronous vibration. Asynchronous vibration mainly occurs in the abnormal vibrations, such as rotating stall, surge, flutter, and bearing vibration [12; 13; 14]. In these cases, the blade vibration frequency is a non-integer multiple of the rotating frequency and the phase of the response can be arbitrary, which is shown in Figure 0(a). Synchronous vibration is excited by the multiples of the rotating frequency [15; 16]. At constant speed, the probe can only monitor the blade response at a fixed point since the phase of the response remains fixed relative to a stationary datum at a given speed. The under-sampling of synchronous vibration is shown in Figure 0(b). Therefore,the degree of under-sampling is much higher for synchronous vibration, which leads to more difficultly in identifying the blade vibration parameters from raw data. Up to now, the identification methods for BTT signals are divided into two categories; one is spectrum analysis and the other is curve fitting. The spectrum analysis method generally maps BTT signals to frequency domain space to obtain information about blade vibration. This type of method usually requires that the sampling frequency is kept constant, that is, the signal is sampled on a constant rotating speed. Spectrum analysis methods mainly include a traveling wave analysis (TW) [17], the \"5 + 2\" method [18], the minimum variance spectrum estimation (MVSE) [19], non-uniform discrete Fourier transform (NUDFT) [20], cross-spectrum estimation (CSE) [21], sparse reconstruction (SR) method [22], etc. The spectrum method mainly is used to analyze asynchronous vibration. However, the results of spectrum analysis are always corrupted with aliases and replicas of the true frequency components. Consequently, the real spectrum recovery needs to further make use of the knowledge of the blade's dynamic properties, which is usually obtained based on the finite element method (FEM) [23]. The curve fitting method usually uses sine functions to fit the blade vibration displacements, which are obtained by speed sweeping through the resonance region. The curve fitting method mainly used to analyze synchronous vibration. These methods mainly include the single-parameter method [24], the two-parameter plot method [25], autoregressive (AR) method [26; 27], the circumferential Fourier fit (CFF) method [28], the method without once per revolution [29; 30], and so on. The accuracy of these curve fitting methods is often related to the amount of fitted data and the inter-blade coupling of the mistuned blade. The CFF method is highly recommended for synchronous vibration analysis by Hood Technology Corporation, which is a commercial vendor of BTT systems [28]. This method can identify the amplitude and phase of vibration on the conditions that the engine order (EO) is known. [PERSON] proposed a traversed EO method based on CFF, which is no longer limited to known EO [31]. In this paper, we refer to the Ouyang CFF as the traditional CFF (TCFF).