Fourier transform leakage

In principle, Fourier transformation allows you to be much more specific, and to pick out or reject, e.g., a single frequency or a specific set of frequencies. For example, one can tune in to a particular frequency (the digital equivalent of a sharply tuned filter, or of a lock-in amplifier, see section 8.4), or selectively remove noise at, say, 60,120, and 180 Hz, while leaving signals at other frequencies unaffected. However, this requires that the selected frequency or frequencies precisely coincide with those used in the Fourier transformation, in order to avoid the so-called leakage to be described in section 7.4. 

In Fig. 7.4-5 the cosine wave fits exactly eight times, and this shows in its transform, which exhibits a single point at f = 8 X (1/128) = 0.0625. On the other hand, the cosine wave in Fig. 7.4-6 does not quite form a repeating sequence, and its frequency, 77/(0.4 X 128) —0.06136, likewise does not fit any of the frequencies used in the transform. Consequently the Fourier transform cannot represent this cosine as a single frequency (because it does not have the proper frequency to do so) but instead finds a combination of sine and cosine waves at adjacent frequencies to describe it. This is what is called leakage the signal at an in-between (but unavailable) frequency as it were leaks into the adjacent (available) analysis frequencies. 

The result is that the Fourier transform of a monochromatic source is not an infinitely narrow line, but has the shape of the (sinx)/jc function. As shown in Fig. 5.3a right, this function is centred about v = 0 and intersects the v axis at V = n/2l, where n = 1, 2, 3,..., so that the first intersection occurs at a wave-number 1/2/. Obviously, the main maximum at v = 0 has a series of negative and positive side lobes or feet with diminishing amplitudes. These side lobes cause a leakage of the spectral intensity, i.e. the intensity is not strictly... 

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. 

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