Asymptotic limits of Negative Group Delay (NGD) in linear causal media satisfying Kramers-Kronig relations are investigated. Even though there is no limit on the NGD-bandwidth product of a linear medium, it is shown that the out-of-band to center frequency amplitude ratio, or out-of-band gain, increases with the NGD-bandwidth product, and is proportional to the amplitude of undesired transients when waveforms with defined "turn on/off" times propagate in the media. The optimal causal dispersion characteristic exhibiting NGD is obtained through Kramers-Kronig relations, which maximizes the NGD-bandwidth product as a function of the out-of-band gain. It is shown that the NGD-bandwidth product has an upper asymptotic limit proportional to the square root of the logarithm of the maximum out-of-band gain. The derived NGD-bandwidth upper asymptotic limit of the optimally engineered causal dispersion characteristic is validated with two examples of physical media, a Lorentzian dielectric medium, and an artificially fabricated loaded transmission line medium.
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