Gibbs phenomenon in Hilbert transform

I have just read that the Gibbs phenomenon can cause artifacts in the use of Hilbert transforms for audio processing applications, but the reference contained no more specific information. Can anyone expand on this remark? When does the Gibbs phenomenon cause problems for the Hilbert transform, and how does one solve this problem? Specific references to books or articles, and specific examples of signals with difficulties with Gibbs phenomenon for the Hilbert transform, would be appreciated. Thanks.

Bruce

The Gibbs phenomenon occurs wherever a discontinuous function is approximated by a truncated (Fourier) series. For the design of systems, the Gibbs phenomenon occurs when approximating ideal brick wall filter responses (like low-pass, band-pass, etc.), but also for all-pass systems such as the Hilbert transformer because of its discontinuity at $$\omega=0$$:

$$H(e^{j\omega})=-j\textrm{ sign}(\omega),\quad |\omega|<\pi\tag{1}$$

The impulse response of the ideal discrete-time Hilbert transformer is

$$h(n)=\left\{\begin{array}{lc}\frac{2\sin^2(\pi n/2)}{\pi n},&n\neq 0\\ 0,& n=0\end{array}\right.\tag{2}$$

A naive design approach would be to simply truncate the impulse response given by (2) to obtain an FIR approximation of a Hilbert transformer. The magnitude of the resulting frequency response for an FIR filter with 23 taps is shown by the blue curve in the following figure: The point '1' on the x-axis is the Nyquist frequency. One can clearly see the Gibbs phenomenon with large ripples near the discontinuity. Note that with an even length filter the approximation would be better around the Nyquist frequency (because such a filter doesn't have a zero at Nyquist), but the Gibbs phenomenon around 0 would remain. The green curve in the figure is the magnitude of the frequency response when the truncated impulse response is windowed by a Hamming window. Clearly, the effect of the Gibbs phenomenon is greatly reduced, but this comes at the expense of a wider transition band.

Using more sophisticated methods for designing FIR Hilbert transformers, such as the Parks McClellan algorithm, weighted least squares, or constrained least squares, the Gibbs phenomenon can be avoided. Of course, there will always remain an approximation error, but it can be made sufficiently small by choosing an appropriate filter length. Note that this is not the case when simply truncating the ideal impulse response. In this case, the maximum ripple size near the discontinuity remains the same, regardless of the filter length. Also note that with FIR filters the phase approximation error can be made exactly zero by choosing an anti-symmetric impulse response. The only error is the error in the magnitude of the response, as shown in above figure. This is different from IIR approximations, which can achieve an exact magnitude of 1, but there will always be a phase approximation error. You can find some basics about the design of Hilbert transformers in the textbook Discrete-time Signal Processing by Oppenheim and Schafer.