Keysight Technologies Understanding the Kramers-Kronig Relation Using A Pictorial Proof

Transcription

1 Keysight Technlgies Understanding the Kramers-Krnig Relatin Using A Pictrial Prf By Clin Warwick, Signal Integrity Prduct Manager, Keysight EEsf EDA White Paper

2 Intrductin In principle, applicatin f the inverse Furier integral t the cmplete frequency respnse f any physical netwrk shuld always yield a causal time-dmain impulse respnse. Such a respnse wuld be useful as a mdel in the transient mde f a SPICE-like simulatr. In practice hwever, the frequency respnse infrmatin we have in hand is ften incmplete (e.g., it s bandlimited and n a discrete-frequency pint grid) and can cntain measurement errrs. A naïve applicatin f the inverse Furier integral t such a frequency respnse almst always yields an incrrect, nn-causal time-dmain mdel. The Kramers-Krnig relatin 1 is very useful in this situatin because it allws us t crrect the frequency respnse and build a causal time-dmain mdel. Fr example, the cnvlutin simulatr in Keysight Technlgies Advanced Design System (ADS) Transient Cnvlutin Element uses this relatinship in a patented implementatin that builds a passive and delay causal mdel frm bandlimited frequency-dmain data like S-parameters. The first step in understanding the validity f this apprach is t examine the math behind the Kramers-Krnig relatin. The usual prf invlves cntur integratin in the cmplex plane f the frequency dmain, but it desn t affrd yu much insight int what is ging n. The pictrial prf ffered here aids understanding. It illustrates a treatment in a textbk by Hall & Heck. 2 In essence, the Kramers-Krnig relatinship cmes abut because f several facts: Even functins (csine-like) in the time dmain yield the real parts f the frequency-dmain respnse. Odd functins (sine-like) in the time dmain yield imaginary parts f the frequency-dmain respnse. All functins can be decmpsed int the sum f an dd and even functin. In general, these terms are independent, but unlike the general case, the dd and even terms f a decmpsed causal functin have a simple, specific dependency n each ther. Knwledge f ne determines the ther. This dependency carries thrugh t the real and imaginary parts f the frequency respnse because f facts 1 and Jhn S. Tll, Causality and the Dispersin Relatin: Lgical Fundatins, Physical Review, vl. 104, pp (1956). R. de L. Krnig, On the thery f the dispersin f X-rays, J. Opt. Sc. Am., vl. 12, pp (1926). H.A. Kramers, La diffusin de la lumiere par les atmes, Atti Cng. Intern. Fisica, (Transactins f Vlta Centenary Cngress) Cm, vl. 2, p (1927). 2. Advanced Signal Integrity fr High-Speed Digital Designs by Stephen H. Hall and Hward L. Heck, pp ISBN

3 T get a better understanding f these facts, let s take a clser lk at each ne step by step. The Furier integral-frequency respnse f an arbitrary functin h( is defined as: jwt = e h( dt = cs( ω h( jsin( h( dt H ( ω) ω Think abut the subset f functins that are real-valued and causal (ne example is shwn in Figure 1): h( = 0 fr t < 0, h( is real fr t >=0 Figure 1. Example f a causal impulse respnse, namely a damped 30-MHz sine wave. Let s see hw this class f functins cnstrains H (ω). Fr reasns that yu ll see belw, we re ging t build ur causal impulse respnse ut f nn-causal even and dd terms. Befre we d that thugh, let s assemble a mini tlbx f prperties f even and dd functins. The first tl is the relatinship between an dd impulse respnse (an example is shwn in Figure 2) and its Furier integral, where the dd impulse respnse is defined by: h ( h ( 3

4 Figure 2. Example f a nn-causal dd impulse respnse, an increasing then damped 30 MHz sine wave. Tl 1: The Furier integral f a (nn-causal) dd impulse respnse is pure imaginary. T see why this is s, remember that csines are even and sines are dd. Fr an dd functin, the dd-even prducts cs(ωh ( integrate ut t zer because the left and right halves have equal magnitudes but ppsite signs and therefre, always cancel each ther. Cnsequently, the nly finite terms are the j sin(ωh ( dd-dd terms which are pure imaginary. The secnd tl is the relatinship between an even impulse respnse and its Furier integral, where the impulse respnse is defined by: h ( h ( e e Tl 2: The Furier integral f a (nn-causal) even impulse respnse is purely real. The even-dd prducts j sin (ω)h e ( must integrate t zer fr the same reasn as previusly stated. Only the even-even prducts cs (ωh e ( are finite and thse are purely real. Nw, let s decmpse a causal functin int even and dd parts, and then apply these tls t each part. Yu can cnstruct any causal r nn-causal h( ut f a sum f sme linear cmbinatin f even and dd cmpnents, using: h( h( h( h( h( 2 2 In the general case, this cnstructin isn t particularly useful r interesting. Hwever, an interesting thing happens when yu cnstruct a causal functin ut f even and dd cmpnents. 4

5 Cnsider ur dd functin h (, then multiply it by the signum functin illustrated in Figure 3 and defined as: signum( 1if t 0 and signum( 1if 0 t The signum functin gives the left hand half f h ( an up-dwn flip and this yields a new even functin h e = signum(h (, an example f which is shwn in Figure 4. Figure 3: The signum functin is simply the sign (but nt the sine) f its argument. Figure 4: Example f a nn-causal even impulse respnse, created by multiplying the functins in Figures 2 and 3. 5

6 Nw, think abut what happens when we add the dd functin and the even functin we derived frm it (Figure 5). In this case: h( h ( h ( signum( h ( h ( e Figure 5. Example f a causal impulse respnse, created by adding the functins in Figures 2 and 4. Impulse respnses cnstructed in this way are necessarily causal since the left hand half f even exactly cancels the left hand half f dd. It might seem strange t g t all this truble t cnstruct an impulse respnse, but the beauty is that we can nw see what the Furier integral lks like. Befre diving int the specifics, ntice frm Tls 1 and 2 that the even h e = signum ( h ( will yield a real respnse, while the dd h ( will yield a pure imaginary respnse (Figure 6). Bth depend n the same functin h (. Figure 6. The imaginary part f the frequency respnse cmes entirely frm the dd part f the time respnse (Figure 2, in this case). Nte that fr aesthetic reasns, the curve was flipped up-dwn. The imaginary part is actually negative. Physically, the peak crrespnds t severe damping at the resnant frequency (30 MHz). 6

7 We can see immediately that the causality cnstraint means that the real and imaginary parts are related and cntain the same infrmatin. But what is the exact relatinship? Multiplicatin in the time dmain is equivalent t cnvlutin in the frequency dmain, s: H ( ω) SIGNUM ( ω) H ( ω) H ( ω) Here, upper case functins dente the Furier integral f the crrespnding lwer case functin and dentes cnvlutin. The recipe fr cnvlutin in wrds and pictures is: flip the kernel left-right using a dummy variable, slide it ver the ther term, multiply, integrate ver the dummy variable, rinse, and repeat. Breaking that dwn, take the cnvlutin kernel (called the Hilbert kernel) in a dummy variable, flip it left-right SIGNUM ( ω ), slide it ver by ω t give SIGNUM (ω ω ), multiply by H (ω) and integrate: SIGNUM ( ω) H ( ω) SIGNUM ( ω ω ) H ( ω ) dω What des the Hilbert kernel lk like? Well, signum( is dd s we knw that SIGNUM (ω) must cme frm the sine waves and be pure imaginary (Tl 1). Figure 7 shws ne pint f the Furier integrals we must d. Imagine Furier integratin as being a multiply and add peratin n the red and green curves. We get the respnse at, in this case, 30 MHz. Nte that nly the tw shaded half perids immediately t the left and right f the sign change at the rigin give nncanceling prducts. Every ther pair f half perids cancel each ther ut because, away frm the rigin, signum( is either cnstant +1 r cnstant 1. Figure 7. Pictrial representatin f ne frequency pint f the Furier integral f the signum( functin. 7

8 The area under the nn-canceling, shaded area is prprtinal t wavelength and s inversely prprtinal t perid ω. Actually, pictures aside, it s nt t hard t d the Furier integral because we can split it int ne half frm t 0 and ne half frm 0 t. Yu can quickly cnvince yurself that the Hilbert kernel is: SIGNUM ( ω) 2 jω Figure 8 illustrates what it wuld lk like. Figure 8. The imaginary part f the Hilbert kernel. The real part is zer. The bttm line is that the Hilbert transfrm in the frequency dmain is equivalent t multiplicatin by signum( in the time dmain. We can therefre, rewrite the real part f ur frequency respnse as the Hilbert transfrm f the imaginary part: H 2 j( ω ω ) e ( ω) SIGNUM ( ω) H ( ω) H ( ω ) dω Nte that here H (ω ) is pure imaginary and j times pure imaginary is purely real, as expected. Next, imagine cnvlutin as running the Hilbert kernel (Figure 8) ver the imaginary part (Figure 6) f the frequency respnse. What yu end up with is shwn in Figure 9. Thus, the real part f frequency-dmain respnse f a causal impulse respnse can be calculated knwing nly the imaginary part. Yu can apply a similar prf starting with an even functin and shw that the imaginary part can be calculated knwing nly the real part. 8

9 Figure 9. The real part f the example frequency respnse cmes entirely frm the even part f the time respnse (Figure 2, in this case). Physically, the switchback crrespnds t a damped resnatr that can respnd t a stimulus belw its resnant frequency (30 MHz), but that turns ff abve it, because it can n lnger respnd quickly enugh. In summary, yu can decmpse any causal impulse respnse as an dd functin plus signum times the same functin. This secnd term is even and the left hand part f it exactly cancels the left hand part f the first dd term, thereby ensuring causality. The even and dd decmpsitin f the casual impulse respnse yield the real and imaginary parts f the frequency respnse, respectively. Using the fact that multiplicatin by the signum functin in the time dmain is equivalent t the Hilbert transfrm in the frequency dmain, we can calculate the real part slely frm knwledge f the imaginary part r visa versa. The Kramers-Krnig relatins give a cnditin that is bth necessary and sufficient, s even befre applying an inverse Furier integral, yu can determine whether a given frequency respnse will yield a causal r a nn-causal impulse respnse. If the real and imaginary parts are Hilbert transfrms f each ther, the impulse respnse is causal, and nt therwise. This fact is very useful because it allws us t test whether r nt a frequency respnse is causal withut ever having t leave the frequency dmain. Keysight s ADS Transient Cnvlutin Element cntains a cnvlutin simulatr that uses the Kramers-Krnig relatinship as part f its algrithm t build passive and delay causal mdels frm frequency-dmain data like S-parameters. Fr mre infrmatin, refer t the article S-parameters Withut Tears in RF DesignLine at: and the ADS Signal Integrity brchure: 9

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