2286 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 55, NO. 6, DECEMBER 2006 Leakage Reduction in Frequency-Response Function Measurements Johan Schoukens, Yves Rolain, and Rik Pintelon Abstract—This paper analyzes how to reduce leakage errors in frequency-response function (FRF) measurements. First, the na- ture of leakage errors is revealed; next, windowing methods are analyzed, and a new default window is proposed. Finally, a supe- rior Taylor-series-based method is proposed. Index Terms—Frequency-response function (FRF) methods, leakage, windows. I. INTRODUCTION F REQUENCY-RESPONSE FUNCTION (FRF) measure- ments of transfer functions are a basic tool in many engineering fields. For random excitations, these measurements are disturbed by leakage and disturbing noise errors. For these reasons, we strongly advise applying periodic excitation signals whenever it is possible [8]. However, in many applications, for psychological or technological reasons, the users prefer to apply random noise excitations. It is well known that for random noise excitations, the FRF measurements are disturbed by leakage (windowing) errors that are induced by the finite length of the measurement window. This paper uses new insights in the nature of these errors to propose improved FRF-measurement techniques. The classical approach to reduce the leakage errors is based on the use of windows. In the literature, a large number of windows is defined and their properties are intensively studied, keeping essentially spectral analysis applications in mind [2], [6]. In this paper, these properties are analyzed again keeping FRF measurements in mind which leads to new insights, and eventually to the definition of a new window. This allows a reduction of the “leakage errors” on the FRF measurements, while the noise sensitivity is not increased. In the next step, an alternative Taylor-based method is proposed. In its simplest version it reduces to the Hanning window, but with more advanced settings a superior method is found. II. HIDDEN NATURE OF LEAKAGE ERRORS Let us consider a stable, causal, discrete- or continuous-time, single-input–single-output linear, time-invariant system with im- pulse response that is excited with a random input (1) Manuscript received June 15, 2005; revised May 9, 2006. This work was sup- ported by theFund for Scientific Research (FWO-Vlaanderen), the Flemish gov- ernment (GOA-IMMI), and the Belgian government (Interuniversity Poles of Attraction, IUAP V/22). The authors are with the Electrical Measurement Department (ELEC), Vrije Universiteit Brussel, Brussels B1050, Belgium (e-mail: Johan.Schoukens@ vub.ac.be). Digital Object Identifier 10.1109/TIM.2006.887034 with the convolution, the exact input and output signal, and disturbing noise. samples of the input and output are measured at . For notational sim- plicity, we note these measurements as with (2) Those results of measurements are used for estimation of the FRF at frequency . The dis- crete Fourier transform (DFT) of the input–output signal [3] is (3) The following remarkably simple relation holds between the DFT spectra for [8], [7] (4) with and smooth rational functions of the frequency. can be interpreted as a generalized “transient” term. Some of these ideas were already reported in [5]. Due to the DFT defini- tion (3), , and are of order , and the transient is of order [8]. In the absence of disturbing noise , the FRF estimate is given by (5) It is the last term in (5) that causes the leakage in the FRF mea- surements. The ratio has a random behavior and looks like noise in FRF measurements because is random. However, this hides the highly structured nature that is described by the smooth function . Windowing methods exploit this smoothness to reduce the leakage errors. It is common practice to average over multiple measurements [1] (6) where is the spectrum of the signal in the th realiza- tion of the experiment. This estimate converges for to the solution corresponding to if the output noise is not correlated with the input (7) Due to the leakage effects, this limit is still biased as it will be shown later. 0018-9456/$20.00 © 2006 IEEE