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 1V amplitude and 40ps rise/fall time (see Figure 4.4).
Figure 4.4. The timedomain 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, "TwoPort Analysis," the system gain G may be computed, sampled on the dense grid of frequencies w k to produce a frequencydomain vector G k , and then multiplied pointbypoint times the vector X k . The frequencydomain result, once inversetransformed to the time domain, represents the response of system G to the stimulus X . The timedomain 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 timedomain plots 
t n := n D T 
sec 
Index to frequency points 
k := 0,1..( N /2) 
integer 
Horizontal axis for frequencydomain 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 frequencydomain vector X to produce timedomain vector x (see Figure 4.4) 
vector 
Fundamentals
Transmission Line Parameters
Performance Regions
FrequencyDomain Modeling
Pcb (printedcircuit board) Traces
Differential Signaling
Generic BuildingCabling Standards
100Ohm Balanced TwistedPair Cabling
150Ohm STPA Cabling
Coaxial Cabling
FiberOptic Cabling
Clock Distribution
TimeDomain Simulation Tools and Methods
Points to Remember
Appendix A. Building a Signal Integrity Department
Appendix B. Calculation of Loss Slope
Appendix C. TwoPort Analysis
Appendix D. Accuracy of Pi Model
Appendix E. erf( )
Notes