Note added 2/18/2010: A more comprehensive treatment of this topic is given in a later posting Kramers-Kronig in Pictures
Consider if you will the following MATLAB code which models a lossless delay by its amplitude and phase frequency response, then applies an inverse discrete Fourier transform, then plots one period of the time domain response:
close all clear all npts = 256; delta_t = 1e-9; % s t = 0:delta_t:delta_t*(npts-1); f = linspace(-(npts-1)/(2*npts*delta_t),... 1/(2*delta_t),npts); % Hz amplitude = ones(size(f)); delay = 10.5e-9; %s phase = -2 * pi * f * delay; fresp = amplitude .* exp(j * phase); tresp = ifft(ifftshift(fresp)); plot(t,tresp)
The resulting plot shows that this method cannot be used to create an accurate impulse response:
The pulse is spread out so badly that the skirt of the next period leaks into the end of this one.
The fundamental issue is that to get an impulse response, you have to do an inverse Laplace transform or an inverse Fourier integral, not an inverse discrete Fourier transform. (The output of an inverse Fourier series isn’t an impulse response at all: it’s one period of the repeated pulse train response.) “But,” you say, “I don’t have the frequency response in the complex plane, I only have the steady state response on a grid of points the upper half of the f = j ω line. And they only go up to 10 GHz!” Kramers-Kronig relation to the rescue! This relation says that if you have a real physical system i.e. a causal system, it is possible to construct the impulse response (causal of course) from the band-limited, gridded, steady-state data alone.
If you don’t want to do the math (and I don’t blame you for that), don’t worry we’ve done it for you in ADS (Patent Pending 20080281893. Note: USPTO’s web server seems to serve “fragile” links, so if the link is broken, you can try their search page, and searching for application number 20080281893. Also, the patent office page scans are in TIFF format, so you might need something like the AlternaTIFF browser plugin, gratis with registration.)