Formation of a discrete-time representation from a data set

The description o f a high-frequency interconnect network in terms o f s- parameters is very useful since 5-parameters depend only on the networks’ electrical characteristics and are not influenced by voltages at terminations. Secondly, as previously stated, their accurate measurement at very high frequencies is possible. Thirdly, since any s- parameter is the ratio o f reflected/incident power, the magnitude o f

a ¿-parameter is always less than 1, i.e. scattering parameters remain bounded and

stable. On the other hand, admittance (y) or impedance (z) parameters can become singular at the resonant frequencies o f the network in question. Therefore, the s- parameters are chosen as a preferred description o f an arbitrary complex interconnect network at high-frequencies.

5.3.1. Enforcement of causality conditions

The values o f ¿-parameters are frequency-dependant values due to skin effect, proxim ity effect and edge effects. Hence, from this point forward, the ¿-parameter data set will be assumed to be in the form o f a set o f frequency-domain values where H(co) denotes the value at the frequency co.

For the case o f data provided by measurement, it is necessary to ensure that errors due to noise or systematic errors do not lead to a non-causal impulse response. Non-causality indicates non-physical behaviour and is inappropriate for interconnect

If the network is connected to a load impedance ZL equal to reference impedance Z0

(5.17) a,

CHAPTER 5 Modelling of interconnects from a tabulated data set

models. Consequently, for a measured frequency response, it is necessary to

precondition the data. To this end, Perry and Brazil [PB98] proposed the Hilbert

Transform relationship:

( W = — (5.18)

This relationship relates the phase response o f a positive real filter to its magnitude response. By enforcing this relationship, causality o f the impulse response is ensured. However, because the frequency response is only known over a narrow range o f frequencies (between co, and coh), a reduction in the limits o f integration is required.

= — j ^ d l ; (5.19)

t t J r n — r

71 ■’ 0 0 - £

CD I ~

The integral m ay be interpreted as a convolution:

tp(co) = a(co) * —- (5.20)

n co

Equation (5.20) m ay be implemented num erically in an efficient manner using the Fast Fourier Transform as described in [PB97]:

</>(co) = IF F T { F F T ( o f c o ))(- js ig n ( v))} . (5.21)

</>((o) is the phase o f the tabulated data set. \H{co)\ is the magnitude response o f the measured frequency domain data and a(cd) = ln|//(&>)|. v is the new transform-domain variable and { - jsign(v)} is the analytical Fourier Transform o f the - 1 / nco term.

As stated above, the Hilbert Transform applies to positive real systems. However, scattering parameters are bounded between -1 and +1 and reflection scattering parameters are rarely positive real numbers. Hence, the relationship in (5.18)

may not be directly applied. To overcome this, the remedy presented in [PB98] is

employed whereby an offset o f one is applied to the scattering parameters. The

resultant offset parameters are thus positive real functions. The phase o f the s- parameters is then determined from (5.21) and the offset is removed. In this manner, it is possible to bound the parameters to ensure that a causal impulse response is obtained and that passivity is maintained or enforced.

5.3.2. Determination of the impulse response

Having ensured that the initial set o f frequency-dom ain data describes a physically realisable (causal) system, the next stage involves determining the

CHAPTER 5 Modelling of interconnects from a tabulated data set

corresponding impulse response. To this end, the following discrete-time Fourier Transform pair [B95] is proposed for use:

where com is maximum frequency o f interest and T = 7t/com.

Two points are worth noting in relation to (5.22) and (5.23). Firstly, note the

change in scaling factors is introduced to enable h(nT) to limit to the continuous impulse response as com tends to infinity. Secondly, an exponent sign-change is introduced. This sign-change is necessary to maintain causality o f the time-domain samples (the opposite sign in the exponent would lead to anti-causal behaviour in the time domain, i.e. samples in the time domain would be zero-valued for positive time).

Let the m easured response consist o f (N +\) equally-spaced samples o f //(co) in the frequency range [0, com]. The first sample corresponds to co0 = 0 and the last sample corresponds to coN = com. In order to ensure a real-valued time-domain response, the condition o f Hermitean symmetry is assumed, i.e.

The integral in (5.25) on the interval [0, com] may be written as a sum o f integrals on intervals [a>k-i, a>k\ as:

(5.22)

oo

H ((o ) = T Y , h ( n T y Jntar (5.23)

change in the scaling factors when compared to the traditional Fourier Series. The

Thus the formula in (5.22) may be written as:

(5.24)

(5.25)

F (co) = H Y<d) e JnaT + H ( co)e jm>T. (5.26)

(5.27)

CHAPTER 5 Modelling of interconnects from a tabulated data set

To numerically calculate the integrals in (5.27), the trapezoidal rule o f integration given as: % A r J f ( x ) d x = y P T * , ) + f ( x , ) ] ~ 0( ( A x f ) (5.28) *1 m ay be applied yielding 1 N 1 N A m h ( n T ) ^ Y . f + 2,1 2 n ‘-' 2 (5.29) = ^ E [ f K . J + F('roH y)]. 471 k=i

Thus inserting (5.26) into equation (5.29) gives:

h ( n T ) = ^ ¿ [ » Y co,_, )e -Jm“- T + Jej™-'r + H ’( a t ) e ‘- r + H ( a k )e J- T]

4k “ l _k=\ j

(5.30) This enables the calculation o f 2N samples o f the impulse response h(nT). The developed formula in (5.30) relates a continuous periodic function o f frequency to a discrete real-valued function in the time domain up to some specified boundary frequency com. This frequency is the highest frequency at which the ^-parameters were measured/simulated. It is very important to choose the frequency com such that the spectral energy beyond com is relatively small. If this is not the case, the errors will arise in the simulated transient response.

5.3.3. Formation of the ^-domain representation

The determination o f an FIR filter corresponding to the impulse response is a trivial task as it is well-known that the coefficients o f an FIR filter correspond to its impulse response, i.e.

2 N - \

H ( z ) = Y J h ( k T) z ~ k (5.31) 4 = 0

Hence, an FIR filter representation for each element o f the descriptor matrix may be directly determined.