A Repertoire of DSP transforms and esp. Hilbert transforms and their significance— part 3

hilbertxform1 hilbertxform2The Fourier transform of (1/t) is essentially a step function. We shall see that the convolution with (1/t) leads to an important integral transform. Specifically, the Fourier transform of

( -1/(\pi t)) is i .sgn (\omega) This pair and its dual are shown in Figure 1. Since (1/t)

has a pole at its origin, its Fourier transform integral diverges there as (1/t) 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 \delta functions and sinusoids, for  the pair Slatex (1/t)$ — sgn has some intriguing properties.

Since (1/t) 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 (-1/\pi t) in the time domain. This convolution integral, called the Hilbert Transform is as follows:

\mathcal{H}[f(t)]=-(1/\pi)\int_{-\infty}^{\infty}\frac {f(t^{'})dt^{'}}{t-t^{'}} Equation I

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 i.sgn (\omega) in the frequency domain produces a 90 \deg phase shift at all frequencies. The phase of F(\omega) is advanced constant 90 \deg for all positive frequencies and retarded a constant 90\deg for all negative frequencies. The magnitude spectrum of F(\omega) is unchanged since the spectrum of i.sgn(\omega) 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 90 \deg, in opposite directions for positive and negative frequencies, then, the inverse Fourier transform produces the time domain result.

The exact 90 \deg phase shift [including the sgn (\omega) 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 90 \deg 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 90\deg phase shift for all frequencies in the far field.

More about Hilbert Transforms later…

Nalin Pithwa

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