CIICT 2009 Proceedings

j 2 πβND = e(2)The frequency dependent CSPE function can be written asCSPE = Fwws0F*ws1 * n2D Fw( D )a 2= ( ) +2Re2 −n+ D F ( )wDj2bn * −n{ e DF ( D ) ⊗ F ( D )}The windowed transform requires multiplication of the timedomain data by the analysis widow, and thus the resultingtransform is the convolution of the transform of the windowfunction, w f , with the transform of a complex sinusoid.Since the transform of a complex sinusoid is a pair of deltafunctions in the positive and negative frequency positions,the result of the convolution is merely a frequencytranslatedcopy of w f centred at + and -. Consequently,with a standard windowing function, the ||F w (D n )|| term isonly considerable when k and it decays rapidly when kis far from . Therefore, the analysis window must bechosen carefully so that it decays rapidly to minimize anyspectral leakage into adjacent bins. If this is so it will renderthe interference terms, i.e. the second and third terms, to benegligible in Eq.(3). Thus, the CSPE for the positivefrequencies gives:w2w(3)2an2−1CSPEw ≈ Fw( D ) D(4)4From Eq. (4). we find the CSPE frequency estimate− N∠(CSPEw)f =2π2 an2− N∠ Fw( D ) D= 42πCSPE w(5)2a− N∠(= 4F ( Dw2πn)−12π2 − j βNe2π) − N ( − β )= N = β2πThe frequency dependent function as illustrated in Equation(4) produces a graph with a staircase-like appearance wherethe flat parts of the graph indicate the exact frequencies ofthe components. The width of the flat parts is dependent onthe main-lobe width of window function used to select thesignal before FFT processing. An example of the output ofthe CSPE algorithm is shown in Figure 2. Consider thesignal S 1 which contains components with frequency values(in Hz) of 17, 293.5, 313.9, 204.6, 153.7, 378 and 423. Thesampling frequency is 1024 HZ. A frame of 1024 samples inlength is windowed using a Blackman window and ispadded using 1024 zeros. The frequency dependent CSPEfunction is computed as per Equation (5). As shown inFigure 2, each component can be calculated and these areidentified with an arrow in the graph. The largest erroramong all the estimates of the components frequencies isapproximately 0.15 Hz.Frequency value50045040035030025020015010050Frequency estimation by improved CSPE00 100 200 300 400 500 600 700 800 900 1000bin indexFig. 2 Frequency estimation of S 1 by CSPENotice too in Figure 2 that at the flat sections in the graph ofthe CSPE result, the width of flat sections where the arrowspoint are related to the width of the window’s main-lobe inthe frequency domain.In addition, with CSPE, we can get the amplitude and phaseof the kth frequency component using the followingequations, where W(-fcspe(k)) is the Fourier Transform ofwindow function which has been shifted to fcspe(k) infrequency domain.AmpPhasekk2 * Fw s0=W ( ω − fcspe(k)) 2 * Fw s0= ∠W( ω − fcspe(k))2.2 EXPERIMENTAL EVALUATION OF CSPEExperiments were designed to evaluate the performance ofthe CSPE algorithm in correctly identifying frequencycomponents within a multiple-component signal. In each setof experiments, a total of 500 signals with SamplingFrequency 44100 Hz and containing components across the(6)(7)36