The DFT expression [4.3] doesn't have a clue about the original sampling rate used to form the discrete-time vector; therefore, it can't really compute the correct value of the Fourier transform without auxiliary scale factors. If you want your Fourier transformations to produce results with the correct amplitude, you must apply the scale factors every time you use either the forward or inverse transformation.

The mathematical spreadsheet package I use, called MathCad, provides a forward-transform FFT and its inverse IFFT defined as in [4.3] and [4.4]. When using these transformations in an environment with a sample interval of D T and a discrete-vector length of N , the appropriate scale factors work like this:

**Equation 4.6 **

**Equation 4.7 **

Other tools may define their FFT routines differently, requiring different scaling constants (see Section 4.10, "Checking the Output of Your FFT Routine").

MathCad routines FFT( ) and its inverse IFFT( ) are specialized for working with real-valued data sequences (as opposed to sequences with complex values). A real-valued data sequence x n has the property that the FFT of that sequence, X k , is complex-conjugate-symmetric around the origin, which means in practical terms that if you know the output point X i , then you can trivially compute the output point X N-i . The MathCad functions FFT( ) and IFFT( ) therefore don't bother to generate, store, or use values of X k for k greater than N /2. Since the top half of each frequency-domain vector associated with FFT( ) and IFFT( ) is unnecessary, the length of X is truncated to ( N /2) + 1. The length of the time-domain vector x remains N .

4.5.1 Deriving the DFT Normalization Factors

When used as an approximation to the Fourier transform, the DFT requires a scaling factor. This scaling factor derives from [4.1], substituting for the integral an approximate summation carried out using the available samples taken at points in time t n = n · D T .

**Equation 4.8 **

Presuming that you wish only to compute values of the output on a dense grid of frequencies w k = 2 p k/(N · D T) , where k 0,1..( N “1),

**Equation 4.9 **

The sampled values of a ( t ) and A ( w ) may be represented in vector notation:

**Equation 4.10 **

**Equation 4.11 **

which simplifies the appearance of [4.9]:

**Equation 4.12 **

Going in the other direction, suppose you have an expression for the Fourier transform A ( w ) of a waveform a ( t ). To approximate a ( t ) using the inverse DFT, you first sample the frequency-domain function A ( w ) according to [4.11], apply the inverse DFT to X , and then multiply by the scaling factor 1/(N · D T ).

**Equation 4.13 **

The scaling factors apply to a DFT of any length, although when using the Cooley-Tukey FFT algorithm the length N will always be a power of two.

POINT TO REMEMBER

- Most FFT routines require external scale factors that depend on the sample interval D T and sample vector length N .

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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