## Spectral analysis parametric and non-parametric digital by Francis Castanié

By Francis Castanié

This ebook offers with those parametric equipment, first discussing these according to time sequence versions, Capon’s strategy and its variations, after which estimators in response to the notions of sub-spaces. in spite of the fact that, the e-book additionally bargains with the conventional “analog” equipment, now known as non-parametric tools, that are nonetheless the main regular in functional spectral research.

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4 summarizes the properties of these PSD. The main role of the autocorrelation function and its Fourier transform, the PSD, is largely justified by the regularity of their behavior in the invariant linear transformations. 4. 3. Higher order representations We are moving away (timidly) from this 2nd order representation for the last few years, because, as has just been explained, it is consistent with a representation of power characteristics of the signal. In these representations, the concepts of phase are lost, and the non-Gaussian nature of the signals is not clearly demonstrated.

We will thus approximate m ( f ) where wo,T(t) is the zero the Fourier transform x(f) of the signal x(t) by xw 0,T signal outside the interval [0, T]. 6. Fourier transform of a sine wave before and after truncation Many truncation windows exist in the literature, all symmetrical in relation to the t = T axis. 3 their Fourier 2 transform. 2. Continuous time truncation windows The first characteristic of a truncation window is the width of the main lobe. This width can be measured by the -3 dB bandwidth, that is B-3dB (T), that is to say the frequency interval for which the Fourier transform module wˆ 0, N −1 ( f ) is greater than its maximum value wˆ 0 divided by 2 (or, similarly, the interval for which the spectral energy density is greater than half its maximum value).

4(d). 4(e). Thus, if f max > 21T , we notice a distortion of the c spectrum close to the Nyquist frequency 1 2Tc . This phenomenon is called spectrum aliasing. 4. 5(a)). The sampled signal is thus a sine wave of 5 Hz frequency. 5(b)). To sample correctly, at least two sampling points per period of the continuous time sine wave are necessary. 5(c), which has a peak at 5 Hz. The spectral aliasing phenomenon also explains why the wheel of a vehicle being filmed seems to rotate slowly in the other direction or even seems stationary when the images are sampled at 24 Hz (25 Hz on television).