Understanding Kramers-Kronig Relation with A Pictorial Proof

In principle, application of the inverse Fourier integral to the complete frequency response of any physical network should always yield a causal time-domain impulse response. Such a response would be useful as a model in the transient mode of a SPICE-like simulator. In practice however, the frequency response information we have in hand is often incomplete (e.g., it’s bandlimited and on a discrete-frequency point grid) and can contain measurement errors. A naïve application of the inverse Fourier integral to such a frequency response almost always yields an incorrect, non-causal time-domain model. The Kramers-Kronig relation1 is very useful in this situation because it allows us to correct the frequency response and build a causal time-domain model. For example, the convolution simulator in Keysight Technologies’ Advanced Design System (ADS) Transient Convolution Element uses this relationship in a patented implementation that builds a passive and delay causal model from bandlimited frequency-domain data like S-parameters.

The first step in understanding the validity of this approach is to examine the math behind the Kramers-Kronig relation. The usual proof involves contour integration in the complex plane of the frequency domain, but it doesn’t afford you much insight into what is going on. The pictorial proof offered here aids understanding. It illustrates a treatment in a textbook by Hall & Heck.2

In essence, the Kramers-Kronig relationship comes about because of several facts: 

• Even functions (cosine-like) in the time domain yield the real parts of the frequency-domain response.

• Odd functions (sine-like) in the time domain yield imaginary parts of the frequency-domain response

• All functions can be decomposed into the sum of an odd and even function. In general, these terms are independent, but unlike the general case, the odd and even terms of a decomposed causal function have a simple, specific dependency on each other. Knowledge of one determines the other.

• This dependency carries through to the real and imaginary parts of the frequency response because of facts 1 and 2.