We continue our exploration of the Hilbert transforms…
For a more elementary example of the Hilbert transform type of phase shift, we turn to the sine and cosine functions that are frequently said to be 90 degree phase shifts of each other.However, the actual phase shift required to transform between sines and cosines is not this simple 90 degree phase shift; rather, it is again, the somewhat curious 90 degree phase shift of the Hilbert transform where positive and negative frequencies are treated oppositely. Fig 6.12a shows that the phase shift applied to the sine spectrum produces the cosine spectrum. Likewise Fig 6.12b shows that the Hilbert transform phase shift applied to the cosine spectrum produces the negative of the sine spectrum. Thus, sines and cosines are Hilbert transform pairs:
This manipulation of the spectra of line spectra of sines and cosines can be done in yet another way, which again relates to the Hilbert transform. In Fig 6.13, we show that the cosine spectrum plus i times the sine spectrum is equal to the complex exponential”s spectrum. That is, we observe that
Additional significance of this familiar result can now be seen. While both the sine and cosine have two-sided spectra, the complex exponential only contains one positive frequency. We see now why the complex exponential form is so suitable in describing rotors in ac circuit theory; it represents a rotor moving counterclockwise with only one frequency. On the other hand, a sine or cosine represents a superposition of two rotors moving in opposite directions, making phases impossible to track.
Using the fact that the sine is the negative Hilbert transform of the cosine, we can write the complex exponential as
This generation of a complex spectrum from the real function can be similarly to applied any function:
and then is called the analytic signal of . Sometimes,
is called the quadrature function or, equivalently, the allied function of . This analytic signal, like the exponential, is one sided in the frequency domain — it only contains positive frequencies. This is easily seen, for the Fourier transform of Equation 6.29 is
. Hence, we get
for Equation 6.30
If we turn this idea around by generating analytic signals in the frequency domain, they will be one-sided, that is, causal, in the time domain. Thus, Hilbert transforms are intimately related to causality. We explore that relationship later in these blogs.
Our last application of the Hilbert transform is the envelope function given by
This non-linear function of has several interesting properties. First, the envelope is tangent to at points where and meet. Since the envelope is clearly greater than everywhere, the envelope circumscribes giving the envelope its name. An example of this property is shown in Fig 6.14
As a consequence of this tangency property, the envelope can arise as a so-called singular solution to certain nonlinear, first order differential equations. A family of solutions is generated by using integration constants. But the envelope although not a member of the family, still satisfies the differential equation because it has the same derivative at each point that members of the family have.
The envelope becomes particularly interesting for signal processing, when we identify the family of functions that form the envelope in terms of the frequency domain. We can show that a constant frequency phase rotation of the spectrum of
, a generalized Hilbert transform, leaves the envelope of unchanged. Thus, the envelope outlines all the possible functions that are obtainable from a given
by rotating its phase spectrum through an arbitrary frequency independent angle. This characteristic of the envelope is useful in describing the effects that filters of unknown phase spectra may have had on data.
Examples come from instrumentation transducers and seismic prospecting. It may be difficult or impossible to measure the transfer characteristic of a transducer because the measurement of the input may require yet another transducer. In the seismic case, the source signal is generally impossible to measure. Frequently, however, the relevant magnitude spectrum can be determined, leaving only the phase spectrum in question. Then, the effects of the transducer’s (or, seismic source’s) magnitude spectrum can be removed by division in the frequency domain. A display of the resulting envelope then provides a result that is independent of the unknown phase shifts. The assumption is made that the unknown phase shift is constant across the bandwidth or the signal, a reasonable assumption for our example, where it is known that
is at least slowly varying over a narrow bandwidth.
In these blogs of four parts, we gave a brief introduction to continuous Fourier transform theory with examples devoted to signal processing. More often than not, digital signals are derived from continuous ones, hence, a sound understanding of continuous theory is necessary. Fortunately, the knowledge of basic symmetries, properties and five transform pairs of our repertoire will suffice for pursuing our study of Digital Signal Processing.