The FFT requires that you identify the mapping between the continuous-time world and the discrete-time world. This is done with two parameters, D T and N .

The parameter D T defines the time interval between successive samples of the waveform to be transformed. The samples are evenly spaced . The spacing D T must be small enough to fairly represent the complete signal waveform without loss of significant information. This implies that on a scale of time commensurate with one sample, the signal waveform had better look smooth, not changing very much between samples.

For example, if you are working with a waveform having a 10% to 90% percent rise and fall time of t 10-90 , then D T should be set anywhere from 4 to 100 times smaller than t 10-90 . Gaussian-shaped rising and falling edges require only a 4x oversampling . Parabolic edges (as created by some IBIS simulators) require about 10x. Linear-ramp edges require much higher sampling rates. Plain square edges, no matter how fast you sample them, create special sampling problems that require fantastic sampling speeds.

The requirement for a small D T is roughly equivalent to a requirement that the sampling rate 1/ D T exceed twice the highest frequency contained in the sampled waveform. Precisely defining the highest frequency can be somewhat problematic . The general rule is that if the spectral content of your signal is down to “60 dB below the meat of the signal spectrum at frequency f 1 , and if the spectrum above f 1 plummets at a rate of at least “40 dB from that point onward, then a sampling rate of 2 f 1 should provide a discrete-time conversion accuracy on the order of one part in one thousand.

One simple test of sampling efficacy is to halve the sampling interval and look for any observable change in the computed spectrum. A noticeable change indicates the original sampling rate was inadequate. Factors that dictate very high oversampling rates include outrageous requirements for time-domain accuracy, investigation of high-frequency EMC phenomena occurring at large multiples of the signal bandwidth, and inadequate low-pass filtering in the system you are trying to model. If you see results that change markedly with a small shift in the sampling phase, that's another sign of sample-rate inadequacy.

Another test of sampling sufficiency is made by observing the magnitude of your FFT output. If the magnitude as a function of frequency is not quickly diminishing toward zero at 1/2 the sampling rate, then your output will most certainly suffer noticeable amounts of aliasing (see Section 4.7, "Effect of Inadequate Sampling Rate").

When working with simulated signals that have linear ramps or square edges, I always include in the simulation a model of a scope probe with limited bandwidth. The scope probe model limits the spectral content of the observed signals (just as in real life), so you can safely use a finite sampling rate to process the simulated waveforms.

The parameter N defines the length of the time-domain sample vector. In conjunction with D T it defines the total window of time ( N · D T ) available within which the FFT must do its work. Always provide an N large enough to allow your simulated system to come to a steady-state condition before the end of the FFT time window. This condition comes about because FFT time, unlike time in the real world, wraps back around onto itself in a loop. Any residual tail in your waveform that " falls off the end of time" doesn't really fall off, but wraps back onto the beginning of time, superimposing onto and distorting your view of the initial circuit conditions.

In rare cases the wrapping helps. For example, when analyzing a clock waveform, if you establish the parameters such that N · D T equals a precise multiple of the clock period, then the tail of the last clock cell overlaps with the beginning of the first period in a manner reminiscent of how the circuit would actually work in continuous time. Most often, though, the wrapping is a bothersome artifact of the FFT approximation and must be dealt with by providing sufficient additional time at the end of the FFT time window to allow your system to fully stabilize to within a negligible tolerance. When working with channels subject to the skin-effect the full stabilization time can be quite long.

When you plot discrete-time waveforms, it is handy to precompute a horizontal axis vector t . The horizontal time axis t n associated with each point x n in the sampled time-domain vector depends on the index n and the step interval D T .

**Equation 4.5 **

POINTS TO REMEMBER

- The FFT requires two parameters: a sample interval D T and a sample vector length N .
- The spacing D T must be small enough to fairly represent the complete signal waveform without loss of significant information.
- Always provide an N large enough to allow your simulated system to come to a steady-state condition before the end of the FFT time window.

Fundamentals

- Impedance of Linear, Time-Invariant, Lumped-Element Circuits
- Power Ratios
- Rules of Scaling
- The Concept of Resonance
- Extra for Experts: Maximal Linear System Response to a Digital Input

Transmission Line Parameters

- Transmission Line Parameters
- Telegraphers Equations
- Derivation of Telegraphers Equations
- Ideal Transmission Line
- DC Resistance
- DC Conductance
- Skin Effect
- Skin-Effect Inductance
- Modeling Internal Impedance
- Concentric-Ring Skin-Effect Model
- Proximity Effect
- Surface Roughness
- Dielectric Effects
- Impedance in Series with the Return Path
- Slow-Wave Mode On-Chip

Performance Regions

- Performance Regions
- Signal Propagation Model
- Hierarchy of Regions
- Necessary Mathematics: Input Impedance and Transfer Function
- Lumped-Element Region
- RC Region
- LC Region (Constant-Loss Region)
- Skin-Effect Region
- Dielectric Loss Region
- Waveguide Dispersion Region
- Summary of Breakpoints Between Regions
- Equivalence Principle for Transmission Media
- Scaling Copper Transmission Media
- Scaling Multimode Fiber-Optic Cables
- Linear Equalization: Long Backplane Trace Example
- Adaptive Equalization: Accelerant Networks Transceiver

Frequency-Domain Modeling

- Frequency-Domain Modeling
- Going Nonlinear
- Approximations to the Fourier Transform
- Discrete Time Mapping
- Other Limitations of the FFT
- Normalizing the Output of an FFT Routine
- Useful Fourier Transform-Pairs
- Effect of Inadequate Sampling Rate
- Implementation of Frequency-Domain Simulation
- Embellishments
- Checking the Output of Your FFT Routine

Pcb (printed-circuit board) Traces

- Pcb (printed-circuit board) Traces
- Pcb Signal Propagation
- Limits to Attainable Distance
- Pcb Noise and Interference
- Pcb Connectors
- Modeling Vias
- The Future of On-Chip Interconnections

Differential Signaling

- Differential Signaling
- Single-Ended Circuits
- Two-Wire Circuits
- Differential Signaling
- Differential and Common-Mode Voltages and Currents
- Differential and Common-Mode Velocity
- Common-Mode Balance
- Common-Mode Range
- Differential to Common-Mode Conversion
- Differential Impedance
- Pcb Configurations
- Pcb Applications
- Intercabinet Applications
- LVDS Signaling

Generic Building-Cabling Standards

- Generic Building-Cabling Standards
- Generic Cabling Architecture
- SNR Budgeting
- Glossary of Cabling Terms
- Preferred Cable Combinations
- FAQ: Building-Cabling Practices
- Crossover Wiring
- Plenum-Rated Cables
- Laying Cables in an Uncooled Attic Space
- FAQ: Older Cable Types

100-Ohm Balanced Twisted-Pair Cabling

- 100-Ohm Balanced Twisted-Pair Cabling
- UTP Signal Propagation
- UTP Transmission Example: 10BASE-T
- UTP Noise and Interference
- UTP Connectors
- Issues with Screening
- Category-3 UTP at Elevated Temperature

150-Ohm STP-A Cabling

- 150-Ohm STP-A Cabling
- 150- W STP-A Signal Propagation
- 150- W STP-A Noise and Interference
- 150- W STP-A: Skew
- 150- W STP-A: Radiation and Safety
- 150- W STP-A: Comparison with UTP
- 150- W STP-A Connectors

Coaxial Cabling

- Coaxial Cabling
- Coaxial Signal Propagation
- Coaxial Cable Noise and Interference
- Coaxial Cable Connectors

Fiber-Optic Cabling

- Fiber-Optic Cabling
- Making Glass Fiber
- Finished Core Specifications
- Cabling the Fiber
- Wavelengths of Operation
- Multimode Glass Fiber-Optic Cabling
- Single-Mode Fiber-Optic Cabling

Clock Distribution

- Clock Distribution
- Extra Fries, Please
- Arithmetic of Clock Skew
- Clock Repeaters
- Stripline vs. Microstrip Delay
- Importance of Terminating Clock Lines
- Effect of Clock Receiver Thresholds
- Effect of Split Termination
- Intentional Delay Adjustments
- Driving Multiple Loads with Source Termination
- Daisy-Chain Clock Distribution
- The Jitters
- Power Supply Filtering for Clock Sources, Repeaters, and PLL Circuits
- Intentional Clock Modulation
- Reduced-Voltage Signaling
- Controlling Crosstalk on Clock Lines
- Reducing Emissions

Time-Domain Simulation Tools and Methods

- Ringing in a New Era
- Signal Integrity Simulation Process
- The Underlying Simulation Engine
- IBIS (I/O Buffer Information Specification)
- IBIS: History and Future Direction
- IBIS: Issues with Interpolation
- IBIS: Issues with SSO Noise
- Nature of EMC Work
- Power and Ground Resonance

Points to Remember

Appendix A. Building a Signal Integrity Department

Appendix B. Calculation of Loss Slope

Appendix C. Two-Port Analysis

- Appendix C. Two-Port Analysis
- Simple Cases Involving Transmission Lines
- Fully Configured Transmission Line
- Complicated Configurations

Appendix D. Accuracy of Pi Model

Appendix E. erf( )

Notes

High-Speed Signal Propagation[c] Advanced Black Magic

ISBN: 013084408X

EAN: N/A

EAN: N/A

Year: 2005

Pages: 163

Pages: 163

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