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  42. <h1>The Scientist and Engineer's Guide to<br />Digital Signal Processing<br /><span class="txtBlue txt26">By Steven W. Smith, Ph.D.</span></h1>
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  86. <h2>Table of contents</h2>
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  88. <ul style="border-top:1px solid #aeaeeb;"><li style="border-top:1px solid #aeaeeb;"><a href="../ch1.htm">1: The Breadth and Depth of DSP</a><ul><li style="border-top:1px solid #aeaeeb;"><a href="../ch1/1.htm">The Roots of DSP</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch1/2.htm">Telecommunications</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch1/3.htm">Audio Processing</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch1/4.htm">Echo Location</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch1/5.htm" style="color:#b4b4e9;">Image Processing</a></li></ul></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch2.htm">2: Statistics, Probability and Noise</a><ul><li style="border-top:1px solid #aeaeeb;"><a href="../ch2/1.htm">Signal and Graph Terminology</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch2/2.htm">Mean and Standard Deviation</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch2/3.htm">Signal vs. Underlying Process</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch2/4.htm">The Histogram, Pmf and Pdf</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch2/5.htm" style="color:#b4b4e9;">The Normal Distribution</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch2/6.htm">Digital Noise Generation</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch2/7.htm">Precision and Accuracy</a></li></ul></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch3.htm">3: ADC and DAC</a><ul><li style="border-top:1px solid #aeaeeb;"><a href="../ch3/1.htm">Quantization</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch3/2.htm">The Sampling Theorem</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch3/3.htm">Digital-to-Analog Conversion</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch3/4.htm">Analog Filters for Data Conversion</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch3/5.htm" style="color:#b4b4e9;">Selecting The Antialias Filter</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch3/6.htm">Multirate Data Conversion</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch3/7.htm">Single Bit Data Conversion</a></li></ul></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch4.htm">4: DSP Software</a><ul><li style="border-top:1px solid #aeaeeb;"><a href="../ch4/1.htm">Computer Numbers</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch4/2.htm">Fixed Point (Integers)</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch4/3.htm">Floating Point (Real Numbers)</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch4/4.htm">Number Precision</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch4/5.htm" style="color:#b4b4e9;">Execution Speed: Program Language</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch4/6.htm">Execution Speed: Hardware</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch4/7.htm">Execution Speed: Programming Tips</a></li></ul></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch5.htm">5: Linear Systems</a><ul><li style="border-top:1px solid #aeaeeb;"><a href="../ch5/1.htm">Signals and Systems</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch5/2.htm">Requirements for Linearity</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch5/3.htm">Static Linearity and Sinusoidal Fidelity</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch5/4.htm">Examples of Linear and Nonlinear Systems</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch5/5.htm" style="color:#b4b4e9;">Special Properties of Linearity</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch5/6.htm">Superposition: the Foundation of DSP</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch5/7.htm">Common Decompositions</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch5/8.htm">Alternatives to Linearity</a></li></ul></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch6.htm">6: Convolution</a><ul><li style="border-top:1px solid #aeaeeb;"><a href="../ch6/1.htm">The Delta Function and Impulse Response</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch6/2.htm">Convolution</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch6/3.htm">The Input Side Algorithm</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch6/4.htm">The Output Side Algorithm</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch6/5.htm" style="color:#b4b4e9;">The Sum of Weighted Inputs</a></li></ul></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch7.htm">7: Properties of Convolution</a><ul><li style="border-top:1px solid #aeaeeb;"><a href="../ch7/1.htm">Common Impulse Responses</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch7/2.htm">Mathematical Properties</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch7/3.htm">Correlation</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch7/4.htm">Speed</a></li></ul></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch8.htm">8: The Discrete Fourier Transform</a><ul><li style="border-top:1px solid #aeaeeb;"><a href="../ch8/1.htm">The Family of Fourier Transform</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch8/2.htm">Notation and Format of the Real DFT</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch8/3.htm">The Frequency Domain's Independent Variable</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch8/4.htm">DFT Basis Functions</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch8/5.htm" style="color:#b4b4e9;">Synthesis, Calculating the Inverse DFT</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch8/6.htm">Analysis, Calculating the DFT</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch8/7.htm">Duality</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch8/8.htm">Polar Notation</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch8/9.htm">Polar Nuisances</a></li></ul></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch9.htm">9: Applications of the DFT</a><ul><li style="border-top:1px solid #aeaeeb;"><a href="../ch9/1.htm">Spectral Analysis of Signals</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch9/2.htm">Frequency Response of Systems</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch9/3.htm">Convolution via the Frequency Domain</a></li></ul></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch10.htm">10: Fourier Transform Properties</a><ul><li style="border-top:1px solid #aeaeeb;"><a href="../ch10/1.htm">Linearity of the Fourier Transform</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch10/2.htm">Characteristics of the Phase</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch10/3.htm">Periodic Nature of the DFT</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch10/4.htm">Compression and Expansion, Multirate methods</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch10/5.htm" style="color:#b4b4e9;">Multiplying Signals (Amplitude Modulation)</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch10/6.htm">The Discrete Time Fourier Transform</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch10/7.htm">Parseval's Relation</a></li></ul></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch11.htm">11: Fourier Transform Pairs</a><ul><li style="border-top:1px solid #aeaeeb;"><a href="../ch11/1.htm">Delta Function Pairs</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch11/2.htm">The Sinc Function</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch11/3.htm">Other Transform Pairs</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch11/4.htm">Gibbs Effect</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch11/5.htm" style="color:#b4b4e9;">Harmonics</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch11/6.htm">Chirp Signals</a></li></ul></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch12.htm">12: The Fast Fourier Transform</a><ul><li style="border-top:1px solid #aeaeeb;"><a href="../ch12/1.htm">Real DFT Using the Complex DFT</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch12/2.htm">How the FFT works</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch12/3.htm">FFT Programs</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch12/4.htm">Speed and Precision Comparisons</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch12/5.htm" style="color:#b4b4e9;">Further Speed Increases</a></li></ul></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch13.htm">13: Continuous Signal Processing</a><ul><li style="border-top:1px solid #aeaeeb;"><a href="../ch13/1.htm">The Delta Function</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch13/2.htm">Convolution</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch13/3.htm">The Fourier Transform</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch13/4.htm">The Fourier Series</a></li></ul></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch14.htm">14: Introduction to Digital Filters</a><ul><li style="border-top:1px solid #aeaeeb;"><a href="../ch14/1.htm">Filter Basics</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch14/2.htm">How Information is Represented in Signals</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch14/3.htm">Time Domain Parameters</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch14/4.htm">Frequency Domain Parameters</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch14/5.htm" style="color:#b4b4e9;">High-Pass, Band-Pass and Band-Reject Filters</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch14/6.htm">Filter Classification</a></li></ul></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch15.htm">15: Moving Average Filters</a><ul><li style="border-top:1px solid #aeaeeb;"><a href="../ch15/1.htm">Implementation by Convolution</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch15/2.htm">Noise Reduction vs. Step Response</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch15/3.htm">Frequency Response</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch15/4.htm">Relatives of the Moving Average Filter</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch15/5.htm" style="color:#b4b4e9;">Recursive Implementation</a></li></ul></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch16.htm">16: Windowed-Sinc Filters</a><ul><li style="border-top:1px solid #aeaeeb;"><a href="../ch16/1.htm">Strategy of the Windowed-Sinc</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch16/2.htm">Designing the Filter</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch16/3.htm">Examples of Windowed-Sinc Filters</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch16/4.htm">Pushing it to the Limit</a></li></ul></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch17.htm">17: Custom Filters</a><ul><li style="border-top:1px solid #aeaeeb;"><a href="../ch17/1.htm">Arbitrary Frequency Response</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch17/2.htm">Deconvolution</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch17/3.htm">Optimal Filters</a></li></ul></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch18.htm">18: FFT Convolution</a><ul><li style="border-top:1px solid #aeaeeb;"><a href="../ch18/1.htm">The Overlap-Add Method</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch18/2.htm">FFT Convolution</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch18/3.htm">Speed Improvements</a></li></ul></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch19.htm">19: Recursive Filters</a><ul><li style="border-top:1px solid #aeaeeb;"><a href="../ch19/1.htm">The Recursive Method</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch19/2.htm">Single Pole Recursive Filters</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch19/3.htm">Narrow-band Filters</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch19/4.htm">Phase Response</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch19/5.htm" style="color:#b4b4e9;">Using Integers</a></li></ul></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch20.htm">20: Chebyshev Filters</a><ul><li style="border-top:1px solid #aeaeeb;"><a href="../ch20/1.htm">The Chebyshev and Butterworth Responses</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch20/2.htm">Designing the Filter</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch20/3.htm">Step Response Overshoot</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch20/4.htm">Stability</a></li></ul></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch21.htm">21: Filter Comparison</a><ul><li style="border-top:1px solid #aeaeeb;"><a href="../ch21/1.htm">Match #1: Analog vs. Digital Filters</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch21/2.htm">Match #2: Windowed-Sinc vs. Chebyshev</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch21/3.htm">Match #3: Moving Average vs. Single Pole</a></li></ul></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch22.htm">22: Audio Processing</a><ul><li style="border-top:1px solid #aeaeeb;"><a href="../ch22/1.htm">Human Hearing</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch22/2.htm">Timbre</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch22/3.htm">Sound Quality vs. Data Rate</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch22/4.htm">High Fidelity Audio</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch22/5.htm" style="color:#b4b4e9;">Companding</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch22/6.htm">Speech Synthesis and Recognition</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch22/7.htm">Nonlinear Audio Processing</a></li></ul></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch23.htm">23: Image Formation & Display</a><ul><li style="border-top:1px solid #aeaeeb;"><a href="../ch23/1.htm">Digital Image Structure</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch23/2.htm">Cameras and Eyes</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch23/3.htm">Television Video Signals</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch23/4.htm">Other Image Acquisition and Display</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch23/5.htm" style="color:#b4b4e9;">Brightness and Contrast Adjustments</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch23/6.htm">Grayscale Transforms</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch23/7.htm">Warping</a></li></ul></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch24.htm">24: Linear Image Processing</a><ul><li style="border-top:1px solid #aeaeeb;"><a href="../ch24/1.htm">Convolution</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch24/2.htm">3x3 Edge Modification</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch24/3.htm">Convolution by Separability</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch24/4.htm">Example of a Large PSF: Illumination Flattening</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch24/5.htm" style="color:#b4b4e9;">Fourier Image Analysis</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch24/6.htm">FFT Convolution</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch24/7.htm">A Closer Look at Image Convolution</a></li></ul></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch25.htm">25: Special Imaging Techniques</a><ul><li style="border-top:1px solid #aeaeeb;"><a href="../ch25/1.htm">Spatial Resolution</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch25/2.htm">Sample Spacing and Sampling Aperture</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch25/3.htm">Signal-to-Noise Ratio</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch25/4.htm">Morphological Image Processing</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch25/5.htm" style="color:#b4b4e9;">Computed Tomography</a></li></ul></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch26.htm">26: Neural Networks (and more!)</a><ul><li style="border-top:1px solid #aeaeeb;"><a href="../ch26/1.htm">Target Detection</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch26/2.htm">Neural Network Architecture</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch26/3.htm">Why Does it Work?</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch26/4.htm">Training the Neural Network</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch26/5.htm" style="color:#b4b4e9;">Evaluating the Results</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch26/6.htm">Recursive Filter Design</a></li></ul></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch27.htm">27: Data Compression</a><ul><li style="border-top:1px solid #aeaeeb;"><a href="../ch27/1.htm">Data Compression Strategies</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch27/2.htm">Run-Length Encoding</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch27/3.htm">Huffman Encoding</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch27/4.htm">Delta Encoding</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch27/5.htm" style="color:#b4b4e9;">LZW Compression</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch27/6.htm">JPEG (Transform Compression)</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch27/7.htm">MPEG</a></li></ul></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch28.htm">28: Digital Signal Processors</a><ul><li style="border-top:1px solid #aeaeeb;"><a href="../ch28/1.htm">How DSPs are Different from Other Microprocessors</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch28/2.htm">Circular Buffering</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch28/3.htm">Architecture of the Digital Signal Processor</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch28/4.htm">Fixed versus Floating Point</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch28/5.htm" style="color:#b4b4e9;">C versus Assembly</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch28/6.htm">How Fast are DSPs?</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch28/7.htm">The Digital Signal Processor Market</a></li></ul></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch29.htm">29: Getting Started with DSPs</a><ul><li style="border-top:1px solid #aeaeeb;"><a href="../ch29/1.htm">The ADSP-2106x family</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch29/2.htm">The SHARC EZ-KIT Lite</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch29/3.htm">Design Example: An FIR Audio Filter</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch29/4.htm">Analog Measurements on a DSP System</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch29/5.htm" style="color:#b4b4e9;">Another Look at Fixed versus Floating Point</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch29/6.htm">Advanced Software Tools</a></li></ul></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch30.htm">30: Complex Numbers</a><ul><li style="border-top:1px solid #aeaeeb;"><a href="../ch30/1.htm">The Complex Number System</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch30/2.htm">Polar Notation</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch30/3.htm">Using Complex Numbers by Substitution</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch30/4.htm">Complex Representation of Sinusoids</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch30/5.htm" style="color:#b4b4e9;">Complex Representation of Systems</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch30/6.htm">Electrical Circuit Analysis</a></li></ul></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch31.htm">31: The Complex Fourier Transform</a><ul><li style="border-top:1px solid #aeaeeb;"><a href="../ch31/1.htm">The Real DFT</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch31/2.htm">Mathematical Equivalence</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch31/3.htm">The Complex DFT</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch31/4.htm">The Family of Fourier Transforms</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch31/5.htm" style="color:#b4b4e9;">Why the Complex Fourier Transform is Used</a></li></ul></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch32.htm">32: The Laplace Transform</a><ul><li style="border-top:1px solid #aeaeeb;"><a href="../ch32/1.htm">The Nature of the s-Domain</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch32/2.htm">Strategy of the Laplace Transform</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch32/3.htm">Analysis of Electric Circuits</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch32/4.htm">The Importance of Poles and Zeros</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch32/5.htm" style="color:#b4b4e9;">Filter Design in the s-Domain</a></li></ul></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch33.htm">33: The z-Transform</a><ul><li style="border-top:1px solid #aeaeeb;"><a href="../ch33/1.htm">The Nature of the z-Domain</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch33/2.htm">Analysis of Recursive Systems</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch33/3.htm">Cascade and Parallel Stages</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch33/4.htm">Spectral Inversion</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch33/5.htm" style="color:#b4b4e9;">Gain Changes</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch33/6.htm">Chebyshev-Butterworth Filter Design</a></li><li style="border-top:1px solid #aeaeeb;"><a href="../ch33/7.htm">The Best and Worst of DSP</a></li></ul></li><li class="open" style="border-top:1px solid #aeaeeb;"><a href="../ch34.htm" style="color:#b4b4e9;">34: Explaining Benford's Law</a><ul><li style="border-top:1px solid #aeaeeb;"><a href="1.htm">Frank Benford's Discovery</a></li><li style="border-top:1px solid #aeaeeb;"><a href="2.htm">Homomorphic Processing</a></li><li style="border-top:1px solid #aeaeeb;"><a href="3.htm">The Ones Scaling Test</a></li><li style="border-top:1px solid #aeaeeb;"><a href="4.htm">Writing Benford's Law as a Convolution</a></li><li style="border-top:1px solid #aeaeeb;"><a href="5.htm" style="color:#b4b4e9;">Solving in the Frequency Domain</a></li><li style="border-top:1px solid #aeaeeb;"><a href="6.htm">Solving Mystery #1</a></li><li style="border-top:1px solid #aeaeeb;"><a href="7.htm">Solving Mystery #2</a></li><li style="border-top:1px solid #aeaeeb;"><a href="8.htm">More on Following Benford's law</a></li><li style="border-top:1px solid #aeaeeb;"><a href="9.htm">Analysis of the Log-Normal Distribution</a></li><li style="border-top:1px solid #aeaeeb;"><a href="10.htm">The Power of Signal Processing</a></li></ul></li>
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  112. <div class="breadcrumbs"><a href="../ch34.htm">Chapter 34 - Explaining Benford's Law</a> / Solving in the Frequency Domain</div><h2>Chapter 34: Explaining Benford's Law</h2><div class="subTitle">Solving in the Frequency Domain</div><p><div style="text-align: justify"><p>Figure 34-5 is what we have been working toward, a systematic way of
  113. understanding the operation of Benford's law. The left three signals, the
  114. <b>logarithmic domain</b>, are <i>pdf(g)</i>, <i>sf(g)</i> and <i>ost(g)</i>. The particular examples
  115. in this figure are the same ones we used previously (i.e., Fig. 34-4).
  116. These three signals are related by convolution (Eq. 34-3), a mathematical
  117. operation that is not especially easy to deal with. To overcome this we
  118. move the problem into the <b>frequency domain</b> by taking the Fourier
  119. transform of each signal. Using standard DSP notation, we will represent
  120. the Fourier transforms of <i>pdf(g)</i>, <i>sf(g)</i>, and <i>ost(g)</i>, as <i>PDF(f)</i>, <i>SF(f)</i>, and
  121. <i>OST(f)</i>, respectively. These are shown on the right side of Fig. 34-5.</p>
  122.  
  123. <div style="text-align: center; margin: 20px;"><img src="../graphics/F_34_5.gif" border="0" alt=""></img></div>
  124.  
  125. <p>By moving the problem into the frequency domain we replace the
  126. difficult operation of convolution with the simple operation of
  127. multiplication. That is, the six signals in Fig. 34-5 are related as follows:</p>
  128.  
  129. <div style="text-align: center; margin: 20px;"><img src="../graphics/E_34_4.gif" border="0" alt=""></img></div>
  130.  
  131. <p>A small detail: The Fourier transform of <i>pdf(g)</i> is <i>PDF(f)</i>, while the
  132. Fourier transform of <i>pdf(-g)</i> is <i>PDF*(f)</i>. The star in <i>PDF*(f)</i> means it is
  133. the complex conjugate of <i>PDF(f)</i>, indicating that all of the phase values
  134. are changed in sign. However, notice that Fig. 34-5 only shows the
  135. magnitudes; we are completely ignoring the phases. The reason for this
  136. is simple&#8211; the phase does not contain information we are interested in for
  137. this particular problem. This makes it unimportant if we use <i>pdf(g)</i> vs.
  138. <i>pdf(-g)</i>, or <i>PDF(f)</i> vs. <i>PDF*(f)</i>.</p>
  139.  
  140. <p>Notice how these signals represent the key components of Benford's law.
  141. First, there is a group of numbers or a probability density function that
  142. can generate a group of numbers. This is represented by <i>pdf(g)</i> and
  143. <i>PDF(f)</i>. Second, we modify each number in this group or distribution by
  144. taking its leading digit. This action is represented by convolving <i>pdf(g)</i>
  145. with <i>sf(g)</i>, or by multiplying <i>PDF(f)</i> by <i>SF(f)</i>. Third, we observe that the
  146. leading digits often have an unusual property. This unusual characteristic
  147. is seen in <i>ost(g)</i> and <i>OST(f)</i>.</p>
  148.  
  149. <p>All six of these signals have specific characteristics that are fixed by the
  150. definition of the problem. For instance, the value at f=0 in the frequency
  151. domain always corresponds to the average value of the signal in the
  152. logarithmic domain. In particular, this means that <i>PDF(0)</i> will always be
  153. equal to one, since the area under <i>pdf(g)</i> is unity. In this example we are
  154. using a Gaussian curve for <i>pdf(g)</i>. One of the interesting properties of the
  155. Gaussian is that its Fourier Transform is also a Gaussian, one-sided in
  156. this case, as shown in Fig. (d). These are related by &sigma;<sub>f</sub> = 1/(2&pi;&sigma;<sub>g</sub>).</p>
  157.  
  158. <p>Since <i>sf(g)</i> is periodic with a period of one, <i>SF(f)</i> consists of a series of
  159. spikes at f = 0, 1, 2, 3, ..., with all other values being zero. This is a
  160. standard transform pair, given by Fig. 13-10 in chapter 13. The zeroth
  161. spike, SF(0), is the average value of <i>sf(g)</i>. This is equal to the fraction of
  162. the time that the signal is in the high state, or log(2) - log(1) = 0.301.
  163. The remaining spikes have amplitudes: SF(1) = 0.516, SF(2) = 0.302,
  164. SF(3) = 0.064, and so on, as calculated from the above reference.</p>
  165.  
  166. <p>Lastly we come to <i>ost(g)</i> and <i>OST(f)</i>. If Benford's law is being followed,
  167. <i>ost(g)</i> will be a flat line with a value of 0.301. This corresponds to
  168. OST(0) = 0.301, with all other values in <i>OST(f)</i> being zero. However, if
  169. Benford's law is not being followed, then <i>ost(g)</i> will be periodic with a
  170. period of one, as show in Fig. (c). Therefore, <i>OST(f)</i> will be a series of
  171. spikes at f = 0, 1, 2, 3, ..., with the space between being zero.</p></div></p>Next Section: <a href="6.htm">Solving Mystery #1</a>
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