Implementation of Frequency-Domain Simulation

The example code in Table 4.2 uses MathCad syntax, although it could easily be rewritten into any mathematical spreadsheet notation. The MathCad symbol := means the variable on the left is assigned the value of the expression on the right.

The example in Table 4.2 simulates a pulse of length N /2, with Gaussian rising and falling edges having 10% to 90% rise/fall time equal to 40 ps (4 D T ), and a delay equal to 4096 ps ((1/10) N D T ). These factors were implemented using definitions PulN , Gaus , and Dly from Table 4.1. Such a specification might well represent the differential output of a driver with 1-V amplitude and 40-ps rise/fall time (see Figure 4.4).

Figure 4.4. The time-domain signal x n shows a pulse of length (1/2) N D T , offset by a delay of 4096 ps = (1/10) N D T . The inset reveals a Gaussian rising edge with a 10% to 90% risetime of 40 ps (four samples).

Assuming the differential driver is connected in a system configuration as shown in Appendix C, "Two-Port Analysis," the system gain G may be computed, sampled on the dense grid of frequencies w k to produce a frequency-domain vector G k , and then multiplied point-by-point times the vector X k . The frequency-domain result, once inverse-transformed to the time domain, represents the response of system G to the stimulus X . The time-domain vector y will show the effects of all resistive losses, dielectric losses, bulk transport delay, and reflections within the transmission environment defined by G .

Equation 4.17

Equation 4.18

 

Table 4.2. Example Code Showing FFT Simulation

Item	Expression	Units
Sampling resolution	D T := 10 “11	sec
Length of sample vector	N := 4096	a power of two
Index to time points	n := 0,1..( N “1)	integer
Horizontal axis for time-domain plots	t n := n D T	sec
Index to frequency points	k := 0,1..( N /2)	integer
Horizontal axis for frequency-domain plots	f k := k/ ( N D T )	Hertz
Frequencies used to sample Fourier transform functions	w k := 2 p f k , k 0,1..( N /2)	rad/sec
Pulse of width ( N/ 2) D T		vector
Delay operator (delays by amount t )		vector
Gaussian LPF with 10% to 90% rise/fall time equal to 4 D T		vector
Example definition of signal in the frequency domain	X k := PulN k · Gaus k · Dly k	vector
Inverse transformation of frequency-domain vector X to produce time-domain vector x (see Figure 4.4)		vector