**Note added February 11, 2010:** We polished up the draft below and made it into a white paper called *Understanding the Kramers-Kronig Relation Using A Pictorial Proof*

The usual proof of the Kramers-Kronig
relation involves contour integration in the complex plane of the frequency
domain. Unless you’re a math genius, you don’t get much insight into what is
going on. The pictorial “proof” here begins in the time domain and gives more
insight. It illustrates 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 .

In outline, the 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 1) and 2) above.

Let’s look at these step by step.

The Fourier integral frequency response of an arbitrary
function *h(**t)* is defined as:

Let’s think about the subset of functions that are real-valued and causal:

*h(**t) = 0* for *t < 0*, *h(t)* is real
for *t >=0*

*Figure 1: Example of a
causal impulse response, namely a damped 30 MHz sine wave.*

Let’s see how this class of functions constrains_{}.

For reasons that you’ll see below, we’re going to build our causal impulse response out of non-causal even and odd terms. So before we do that build up, let’s assemble a mini- toolbox of relationships to do with even and odd functions.

The first tool is the relationship between an odd impulse response, defined by:

_{}

…and its Fourier integral.

*Figure 2: Example of a
non-causal odd impulse response, an increasing then damped 30 MHz sine wave*

**Tool 1:** The Fourier
integral of a (non-causal) odd impulse response is pure imaginary.

To see why this is so remember cosines are even, sines are odd. For an odd function, the odd-even products_{}integrate out to zero because the left and right halves have
equal magnitudes but opposite signs and so they always cancel each other. So the
only finite terms are the _{}odd-odd terms which are pure imaginary.

The second is the relationship between an even impulse response, defined by:

_{}

…and its Fourier integral.

**Tool 2:** The
Fourier integral of a (non-causal) even impulse response is purely real.

The even-odd products_{}must integrate to zero for the same reason as before. Only
the even-even products_{}are finite and those are purely real.

Let’s decompose a causal function into even and odd parts, then apply these tools to each part.

You can construct any causal or non-causal *h(**t)* out of a sum of some linear combination
of even and odd components:

_{}

…but it’s not particularly useful or interesting.

However an interesting thing does happens
when you construct a *causal* function
out of even and odd components.

Let’s start with our odd function *h _{o}(*

_{}if _{}and_{}if _{}

*Figure 3: The signum function is simply the sign (but not the sine!) of
its argument.*

The signum function gives the left
hand half of *h _{o}(*

*Figure 4: Example of a
non-causal even impulse response, created by multiplying the functions in
Figures 2 and 3.*

Now think about what happens when we add the odd function and the even function we derived from it.

_{}

*Figure 5: Example of a
non-causal impulse response, created by adding the functions in Figures 2 and
4.*

Impulse responses constructed in this way are necessarily causal because the left hand half of even exactly cancels the left hand half of odd. It might seem strange to go to all this trouble to construct an impulse response, but the beauty is we can now see what the Fourier integral looks like.

Before we dive into the specifics, notice that (from Tools 1
and 2 above) the even* _{}*will yield a real response and the odd

*Figure 6: The
imaginary part of the frequency response comes entirely from the odd part of
the time response (Figure 2 in this case). Note: For aesthetic reason I flipped
the curve up-down. The imaginary part is actually negative. Physically, the
peak corresponds to severe damping at the resonant frequency (30 MHz).*

We can immediately see that the causality constraint means that the real and imaginary parts are related and contain the same information. But what is the exact relationship?

Multiplication in the time domain is equivalent to convolution in the frequency domain, so:

_{}where upper case functions denotes
the Fourier integral of the corresponding lower case function, and _{}denotes convolution. The recipe for convolution in words and pictures
is “flip the kernel left-right using a dummy variable, slide it over the other
term, multiply, integrate over the dummy variable, rinse, repeat”

Breaking that down, take the convolution kernel (called the
Hilbert kernel) in a dummy variable, flip it left-right_{}, slide it over by _{}to give _{}, multiply by _{} and integrate:

_{}

But what does the Hilbert kernel look like? Well *signum**(**t)* is odd so we know _{}must come from the sine waves and be pure imaginary (Tool 1).

Figure 7 shows one point of the Fourier integrals we must
do. Imagine Fourier integration as being a multiply and add operation on the
red and green curves. We get the response at, in this case, 30 MHz. Note that
only the two shaded half periods immediately to the left and right of sign
change at the origin give a non-canceling products. Every other pair of half
periods cancel each other out because, away from the origin, *signum**(**t)* is either constant +1 or constant -1.

*Figure 7: Pictorial
representation of one frequency point of the Fourier integral of the *signum(t)* function*

The area under the non-canceling, shaded area is
proportional to wavelength and so inversely proportional to period_{}. Actually, pictures aside, it’s not too hard to do the
Fourier integral because we can spilt it into one half from _{}to 0 and one half from 0 to _{}. You can quickly convince yourself that the Hilbert kernel
is_{}.

Here’s what it looks like:

*Figure 8: The
imaginary part of the Hilbert kernel. The real part is zero.*

The bottom line is that the Hilbert transform in the
frequency domain is equivalent to multiplication by *signum**(**t)* in the time domain.

We can re-write the real part of our frequency response as the Hilbert transform of the imaginary part:

_{}

Note that_{}is pure imaginary, and *j* times pure imaginary is purely real as expected.

Imagine convolution as running the Hilbert kernel (Figure 8) over the imaginary part (Figure 6). Here’s what you get:

*Figure 9: The real
part of the frequency response comes entirely from the even part of the time
response (Figure 2 in this case). Physically, the switchback corresponds to a damped resonator that can respond to a stimulus below its
resonant frequency (30 MHz), but that “turns off” above it, because it can no
longer respond quickly enough.*

Thus the real part of frequency domain response of a causal impulse response can be calculated knowing only the imaginary part.

You can apply a similar proof starting with an even function and show that the imaginary part can be calculated knowing only the real part.

In summary, you can decompose any causal impulse response as an odd function plus signum times the same function. This second term is even, and the left hand part of it exactly cancels the left hand part of the first, odd term, thus ensuring causality. The even and odd decomposition of the casual impulse response yield the real and imaginary parts of the frequency response, respectively. Using the fact that multiplication by the signum function in the time domain is equivalent to the Hilbert transform in the frequency domain we can calculate the real part solely from knowledge of the imaginary part or vise versa.

The Kramers-Kronig relations give a condition that is both necessary and sufficient, so even before applying an inverse Fourier integral, you can determine whether a given frequency response will yield a causal or a non-causal impulse response. If the real and imaginary parts are Hilbert transforms of each other, the impulse response is causal, and not otherwise.

This fact is very useful because we can test whether or not a frequency response is causal or not without leaving the frequency domain.

*Click the link to go
back to the signal integrity
blog.*