Digital Signal Processing

By Steven W. Smith, Ph.D.

- 1: The Breadth and Depth of DSP
- 2: Statistics, Probability and Noise
- 3: ADC and DAC
- 4: DSP Software
- 5: Linear Systems
- 6: Convolution
- 7: Properties of Convolution
- 8: The Discrete Fourier Transform
- 9: Applications of the DFT
- 10: Fourier Transform Properties
- 11: Fourier Transform Pairs
- 12: The Fast Fourier Transform
- 13: Continuous Signal Processing
- 14: Introduction to Digital Filters
- 15: Moving Average Filters
- 16: Windowed-Sinc Filters
- 17: Custom Filters
- 18: FFT Convolution
- 19: Recursive Filters
- 20: Chebyshev Filters
- 21: Filter Comparison
- 22: Audio Processing
- 23: Image Formation & Display
- 24: Linear Image Processing
- 25: Special Imaging Techniques
- 26: Neural Networks (and more!)
- 27: Data Compression
- 28: Digital Signal Processors
- 29: Getting Started with DSPs
- 30: Complex Numbers
- 31: The Complex Fourier Transform
- 32: The Laplace Transform
- 33: The z-Transform
- 34: Explaining Benford's Law

Your laser printer will thank you!

The Power of Signal Processing

Benford's law has never been viewed as a major mathematical problem, only a minor mystery. Nevertheless, many bright and creative people have spent time trying to understand it. The primary goal of this chapter has been to demonstrate the power of DSP in nontraditional applications. In the case of Benford's law this power is clear; signal processing has succeeded where other mathematical techniques have failed.

Nowhere is this more apparent than a review article published in 1976 by
mathematician Ralph Raimi. He examined the many approaches in
explicit mathematical detail, and his paper has become a landmark in the
history of this problem. Buried in the detailed math, Raimi makes the
brief comment: *"...many writers ... have said vaguely that Benford's law
holds better when the distribution ... covers several orders of magnitude."*
As we now know, this is the root of the phenomenon. In one of the most
colorful events of this history, a small error in logic prompted Raimi to
argue that this could not be correct. [Specifically, scaling a distribution
does not change how many orders of magnitude it covers.] While this
slight misdirection probably made no difference, it shows just how little
success had been achieved by traditional mathematics. An understanding
of the basic operation of Benford’s law was nowhere on the horizon.

Lastly, this discussion would be incomplete without mentioning the practical applications of Benford's law Next time you file your income tax return or other financial report, consider what happens to the distribution of leading digits if you fabricate some of the numbers. I'm not going to help you cheat, so I won't give the details away. Simply put, the numbers you make up will probably not follow Benford's law, making your fraudulent report distinguishable from an accurate one. I'll let you imagine who might be interested in this.