Aliasing and Leakage

The aliasing and leakage effects come from the manipulation of finite number of data points measured at a finite sampling rate. First, the aliasing effect is demonstrated. Consider a sinusoidal signal with frequency = 1.0 Hz x t) = sin (2nt), which is shown by the solid line 

Bayesian Methods for Structural Dynamics and Civil Engineering 

It will be shown later in this example that utilizing the whole range of frequencies of displacement measurement in Equation (3.61) (i.e., /C = 1, 2. N qy — 1 ) leads to bias and unreasonably small variances, especially for the damping ratio and spectral intensity. In order to obtain reasonable results, an appropriate frequency range is necessary and the recommendation in Section 3.3.3 can be utilized. 

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Chapters 3 and 4 introduce two recently developed Bayesian methods for updating the mathematical models of dynamical systems. Chapter 3 presents the Bayesian spectral density approach. The spectral density estimator is defined to take into account of the aliasing and leakage effect. The statistical properties of the spectral density estimator are examined and... 

The Bayesian spectral density approach approximates the spectral density matrix estimators as Wishart distributed random matrices. This is the consequence of the special structure of the covariance matrix of the real and imaginary parts of the discrete Fourier transforms in Equation (3.53) [295]. Another approximation is made on the independency of the spectral density matrix estimators at different frequencies. These two approximations were verified to be accurate at the frequencies around the peaks of the spectmm. The spectral density estimators in the frequency range with small spectral values will become dependent since aliasing and leakage effects have a greater impact on their values. Therefore, the likelihood function is constructed to include the spectral density estimators in a limited bandwidth only. In particular, the loss of information due to the exclusion of some of the frequencies affects the estimation of the prediction-error variance but not the parameters that govern the time-frequency structure of the response, e.g., the modal frequencies or stiffness of a structure. 

Full details of the origin and effects of leakage and aliasing may be found in several texts.35-37... 

In summary, the sampled waveform should contain a whole number of periods to avoid leakage and the sampling should be with at least the Nyquist frequency or faster to avoid aliasing. In EIS practice, a waveform containing a predetermined number of frequencies and whole number of periods of waveforms is used and sampling is synchronized (Chap. 3.7). 

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