In general a causal real valued signal in time has negative frequencies, when a Fourier transform is applied. To overcome this, a complex complement can be calculated to compensate the negative frequency spectrum. The result is called analytic signal or analytic function, which provides a one sided spectrum.
analyticFunction(x)real valued data vector
Complex valued analytic function
An analytic function \(xa\) is composed of the real valued signal representation \(y\) and its Hilber transform \(H(y)\) as the complex complement $$xa(t) = x(t)+i H(x(t))$$. In consequence, the analytic function has a one sided spectrum, which is more natural. Calculating the discrete Fourier transform of such a signal will give a complex vector, which is only non zero until the half of the length. Every component higher than the half of the sampling frequency is zero. Still, the analytic signal and its spectrum are a unique representation of the original signal \(x(t)\). The new properties enables us to do certain filtering and calculations more efficient in the spectral space compared to the standard FFT approach. Some examples are:
because the spectrum is one sided, the user must
only modifiy values in the lower half of the vector. This strongly
reduces mistakes in indexing.
See filter.fft
Since the Hilbert transform is a perfect phase shifter
by pi/2, the envelope of a band limited signal can be calculated.
See envelope
Deriving and integrating on band limited discrete data becomes
possible, without taking the symmetry of the discrete Fourier transform into
account. The secound example of the spec.fft function calculates
the derivative as well, but plays with a centered spectrum and its corresponding
"true" negative frequencies
A slightly different approach on the analytic signal can be found in R. Hoffmann "Signalanalyse und -erkennung" (Chap. 6.1.2). Here the signal \(x(t)\) is split into the even and odd part. According to Marko (1985) and Fritzsche (1995) this two parts can be composed to the analytic signal, which lead to the definition with the Hilbert transform above.
R. Hoffmann, Signalanalyse und -erkennung: eine Einfuehrung fuer Informationstechniker, Berlin; Heidelberg: Springer, 1998.
H. Marko, Systemtheorie: Methoden und Anwendungen fuer ein- und mehrdimensionale Systeme. 3. Aufl., Berlin: Springer, 1995.
G. Fritzsche, Signale und Funktionaltransformationen - Informationselektronik. Berlin: VEB Verlag Technik, 1985