application of frequency-domain system identification techniques in ...

2.3. Extension to output-only data 31where X [s] (ω) and X [s],ref (ω) are respectively the (N o × 1) and (N ref × 1) vectorscomputed in (2.54) and (2.55). The time window W (e.g., Hanning window) isused to reduce the negative effect of leakage (S. Lawrence Marple, 1987). Choosinga higher amount of data samples D in each segment, at the expense of thenumber of averages P , will reduce the effect of leakage. Moreover, a higher spectralresolution will be obtained in the frequency-domain. However, the resultingdecrease in the number of averages P leads to a higher stochastic uncertainty onthe estimates. In practice, a trade-off will have to be made between the mentionedcontradicting aspects.The basic idea behind allowing an overlap between the data segments consistsin allowing a better contribution of all samples of the raw time history responsedata to the averaged estimate. If no overlap is considered, the contribution ofsamples near the edges of the segments will be suppressed by the presence of theHanning window.The correlogram approachApart from the periodogram approach, another method can be considered for theestimation of cross power spectra of the response signals. The unbiased discretetimedomain correlation estimate between the signal x [o] (m) (m = 0, . . . , M − 1)of a response o and the signal x ref[i](m) of a reference-response i is given by⎧⎪⎨⎪⎩R [o,i] (k) = 1 M−k−1 ∑M−kx [o] (m + k)x ref[i](m) for 0 ≤ k ≤ M − 1m=0R [o,i] (k) = 1M−|k|−1∑M−|k|x ref[i](m + |k|)x [o] (m) for − (M − 1) ≤ k < 0m=0(2.57)with k the correlation time. The biased correlation estimate uses 1/M rather than1/(M − k). Instead of computing the correlation estimate by means of multiplicationand summation of time samples, a high speed implementation of (2.57) ispossible by applying the FFT to the time signals, cross multiplying the Fouriertransforms and taking the real part of the inverse FFT to the cross products. Theinverse FFT results in a periodic correlation function estimate. The bias errorthat is introduced due to this circular convolution can be avoided by zero-paddingthe original time records (Bendat and Piersol, 1980). The cross power functionestimates can then be obtained by Fourier transforming the correlation functionsobtained from (2.57)S [o,i] (ω) = T sM ∑k=−MW (k)R [o,i] (k)e −iωkTs (2.58)