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!

Solving Mystery #1

There are two main mysteries in Benford's law. The first is this: ** Where
does the logarithmic pattern of leading digits come from? Is it some hidden
property of Nature?** We know that

This result can be understood in a simple way, showing how Benford's law resembles a magician's slight of hand. Say you tabulate a list of numbers appearing in a newspaper. You tally the histogram of leading digits and find that they follow the logarithmic pattern. You then wonder how this pattern could be hidden in the numbers. The key to this is realizing that something has been concealed– a big something.

Recall the program in Table 34-1, where lines 400-430 extract the leading digit of each number. This is done by multiplying or dividing each number repeatedly by a factor of ten until it is between 1 and 9.999999. This manipulation of the data is far from trivial or benign. You don't notice this procedure when manually tabulating the numbers because your brain is so efficient. But look at what this manipulation involves. For example, successive numbers might be multiplied by: 0.01, 100, 0.1, 1, 10, 1000, 0.001, and so on.

This changes the numbers in a pattern based on powers of ten, i.e., the
*anti-logarithm*. You then examine the processed data and marvel that it
looks *logarithmic*. Not realizing that your brain has secretly manipulated
the data, you attribute this logarithmic pattern to some hidden feature of
the original numbers. *Voila! The mystery of Benford's law!*