is This pair and its dual are shown in Figure 1. Since
has a pole at its origin, its Fourier transform integral diverges there as has a transform only in a limiting sense. (It can be evaluated using contour integration). Likewise, the Fourier integral of sgn (x), in computing either the forward or inverse transform, does not exist in the conventional sense because sgn (x) is not absolutely integrable. The transform of sgn (x) can be defined by considering a sequence of transformable functions that approaches sgn (x) in the limit. We do not let these mathematical inconveniences deter us any more than they did in our previous discussion of functions and sinusoids, for the pair Slatex (1/t)$ — sgn has some intriguing properties.
Since is real and odd, its Fourier transform is odd and pure imaginary. But, more interestingly, its magnitude spectrum is obviously constant. (Could there be a delta function lurking nearby?). The interest in this transform pair comes from convolving a function with in the time domain. This convolution integral, called the Hilbert Transform is as follows:
This transform arises in many applications in Digital Communications and Mathematical Physics. The Cauchy principal value, which is a kind of average familiar to those knowledgeable in contour integration, is to be used over singularities of the integrand. This mathematical inconvenience is avoided in the frequency domain where we can easily visualize the effect of the Hilbert transform.
Multiplication by in the frequency domain produces a phase shift at all frequencies. The phase of is advanced constant for all positive frequencies and retarded a constant for all negative frequencies. The magnitude spectrum of is unchanged since the spectrum of is flat. The Hilbert transform operation in the frequency domain is summarized in Fig 2. The Fourier transform of the given function has its phase shifted , in opposite directions for positive and negative frequencies, then, the inverse Fourier transform produces the time domain result.
The exact phase shift [including the dependence] occurs in several different instances in wave propagation: the reflection of electromagnetic waves from metals at a certain angle of incidence involves an exact phase shift independent of frequency; special teleseismic ray paths produce the same type of phase shifts, and the propagation of point sources for all types of waves includes a phase shift for all frequencies in the far field.
More about Hilbert Transforms later…