The frequency-domain model discussed in this chapter is based on the Fourier transform. The Fourier transform is one of many transformations that establish relations between the time domain and the frequency domain (Figure 4.2). With it, you can turn time-domain functions into frequency-domain functions, and vice versa. The most important aspect of the Fourier transform is that it changes time-domain convolution (difficult to compute) into frequency-domain multiplication (easy to compute).

Figure 4.2. The Fourier transform maps time-domain functions to frequency-domain functions, turning time-domain convolution into frequency-domain multiplication.

Given a signal x ( t ) and the impulse response of a filter h ( t ), you can compute the output y ( t ) by first Fourier-transforming both x and h to the frequency domain. Then, for all possible frequencies w , form the product X ( w ) H ( w ) and transform the resulting frequency-domain function Y ( w ) back to the time domain using an inverse Fourier transform.

The Fourier transform is defined as

**Equation 4.1 **

**Equation 4.2 **

Unfortunately, evaluation of the Fourier transformation requires the calculation of continuous-time integrals, which, unless a closed-form solution is available for your particular signal, is generally impossible to perform. Therefore, a shortcut has been developed for approximating Fourier transformations. The shortcut is called the DFT (discrete Fourier transform).

The DFT is a discrete-time approximation to the Fourier transform. It operates on a vector x n with n 0,1..( N “ 1), translating x n into a discrete-frequency domain according to the following finite sum.

**Equation 4.3 **

**Equation 4.4 **

The DFT [4.3] is closely related to the Fourier transform [4.1] with the exception that both time and frequency have been rendered in a discrete form, changing the continuous integrations of [4.1] into discrete summations. With a suitable mapping between the continuous-time domain and the discrete-time domain, the DFT can successfully approximate the Fourier transform.

The fast-Fourier transform (FFT) is nothing more than a very clever implementation of equation [4.3]. It works only for particular values of N . The most common variety is the Cooley-Tukey FFT, which works only for N equal to a power of two. The FFT algorithm arranges the calculation so that the total effort required to accomplish the sum grows not in proportion to N 2 (as would a simple-minded matrix multiplication) but in proportion to N log 2 N . Its efficacy is so great that the DFT practically never appears in its direct form ”the FFT universally has replaced it. Except for some very subtle differences involving coefficient quantization and sensitivity to rounding errors, the mathematical properties of the FFT and DFT are identical ( [32] , [33] , [34] and [35] ).

NOTE: There is some disagreement in the literature about whether the factor 1/ N in equation [4.3] belongs in the forward DFT, in the inverse DFT, or perhaps split equally as in both transformations. If you want the forward and inverse transformations to match so that for any time-domain vector x , you get DFT “1 [ DFT ( x )] = x , then a factor of 1/ N has to go somewhere , but it doesn't really matter where. Here I have shown the factor of 1/ N in the forward transform as implemented in the MathCad function FFT( ). If your tool defines the FFT otherwise , then small adjustments will have to be made in the normalizing constants used with your FFT algorithm (see Section 4.5, "Normalizing the Output of an FFT Routine").

POINTS TO REMEMBER

- The DFT is a discrete-time approximation for the Fourier transform.
- The popular Cooley-Tukey FFT algorithm is a clever, highly efficient implementation of the DFT that works only for N equal to a power of two.

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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