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Kramers-Kronig in Pictures

Posted December 10th, 2009 · 1 Comment · Application Note

Posted by Colin Warwick

The real part of the frequency response comes entirely from the even part of the time response.

The real part of the frequency response comes entirely from the even part of the time response.

Fangyi Rao and I recently wrote an article for EE Times’ RF Design Line about our patented method of creating causal time domain models from bandlimited s-parameters. The title S-parameters without tears is homage to the 1936 comic play French Without Tears by Terence Rattigan.

Our method makes use of a useful but obscure piece of mathematical physics called the Kramers-Kronig relation. The usual proof of this relation involves contour integration in the complex plane of frequency domain. Unless you’re a math genius, you don’t get much insight into what is going on. In the pictorial "proof" linked to below begins in the time domain and gives more insight. It’s based on a treatment in Advanced Signal Integrity for High-Speed Digital Designs by Stephen H. Hall and Howard L. Heck, pp.331-336 ISBN-13 978-0-470-19235-1

Here’s our white paper entitled Understanding the Kramers-Kronig Relation Using A Pictorial Proof (PDF)

Thanks to Eric Bogatin for pointing out the neat treatment in his excellent book review of Hall and Heck’s excellent book.

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