Printed Circuit Design & Fab - August 2008 - (Page 20)

signal integrity applications of mathcad, Part 3 Mathcad allows Fourier transformation, convolution, SI modeling and simulation of PCB topologies. As an example, Figure 9 presents a complex time domain vector gn with real part [Re (gn)] and imaginary [Im (gn)]. There are M data points at sample spacing P (with reciprocal fp). n represents an index/integer and ω0 is angular frequency. tn and frqn are horizontal axes for time and frequency plots respectively. Mathcad CFFT is executed to generate the vector Gn in frequency space. Furthermore, ICFFT can be operated on Gn to reproduce gn. This implies the Fourier transform pair: G := CFFT(g) and g := ICFFT(G). Fourier transform possesses remarkable applications in high-speed digital design. Digital signals can be approxi mated by square waves,18 having wide bandwidths or many frequency components. Fourier expansion can evaluate the spectral content (or harmonics) of such periodic time domain waveforms. TDR measurements are related19 to corresponding VNA measurements via Fourier transform. Another notable feature of Fourier transform is that convolution in time domain, often difficult to compute, is converted into a simple multiplication20 in frequency domain. A discrete-time approximation to the Fourier transform called DFT forms a foundation for fundamental techniques and applications of digital signal processing (DSP). Johnson, et al4 have simulated pulse propagation by applying Mathcad’s FFT/IFFT feature. Norte 3 has also illustrated Mathcad’s simulation of single-ended and differential high-speed topologies. Consider the single-ended topology in Figure 10, which includes a driver U1, a transmission line T1 and a receiver U2. T1 has bandwidth (Bw) and characteristic impedance (Zo). U2 can be modeled by parasitic resistance (Rr) in series with capacitance (Cr). U1 outputs a series of pulses having bit period (Tb) and launches a voltage waveform (Vp) at input of T1. These signaling schemes can be simulated using various tools, including SPICE or SI tools from companies such as Mentor Graphics, Cadence, Ansoft and others. It can be Forward Fourier transForm and its inverse furnish a means for data conversion from time to frequency domain and vice versa. Mathcad makes available two Fourier transform pairs: FFT/ IFFT and CFFT/ICFFT. The former pair is applicable if the data values in the time domain are real and data vector DR. ABE has 2^m elements (m being an integer, (ABBAS) RIAzI m > 0). For all other cases, the CFFT/ ICFFT pair is required. This is because, for real data, the second half of the transform is just the conjugate of the first. The FFT/IFFT takes advantage of this fact-saving memory and time (by discarding the second half of the result vector). The CFFT/ICFFT pair does not assume symmetry in the transform and should be used for complex valued data. The CFFT/ICFFT can be also employed for real numbers, which are just a subset of complex domain. !Figure 9. Complex data gn and its Fourier transform Gn. ! Figure 10. a single-ended topology. 20 PRINTED CIRCUIT DESIGN & FAB AUGUST 2008

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Printed Circuit Design & Fab - July 2008 Contents Our Line Market Watch Around the World Happenings ROI Tip Jar Software Performance Interconnect Strategies Final Finish Forum Product Development Challenges in a Global Market Innovative Modeling Supports Co-Design of the Power Supply Chain, Part 2 Low-Loss Fluoropolymer Copper Clad Laminate Qualifying PCBs Outsourced in Asia Copper Plating and Microvia Fill for Advanced PCBs Off the Shelf Marketplace Ad Index BGA Bulletin

Printed Circuit Design & Fab - August 2008

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