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!

** It sounds like heresy!**
Signal processing is regarded as one of the most mathematical areas of electrical engineering. If you open a standard textbook, you will find page after page of equations, mathematical notation, and unfamiliar symbols. This is the language of those that specialize in DSP. It is very abstract and theoretical, but also has tremendous power. This holds great appeal to those striving to expand human knowledge, rather than simply trying to solve a particular problem at hand.

Unfortunately, difficult mathematics is a barrier to learning DSP. This is especially true for scientists and engineers wanting DSP as a * tool*, rather than the focus of their * careers*. Digital signal processing is incredibly powerful- but if you can't understand it- you can't use it!

** A better way to learn DSP**
The __ Scientist and Engineer's Guide to DSP__ overcomes this problem in two ways. First, the algorithms and techniques are * explained*, rather than just proven to be true through a mathematical derivation. The mathematics and programs are included, but they are not used as the primary means of conveying the information. Nothing beats a few well written paragraphs supported by good illustrations!

Second, complex numbers are treated as an * advanced* topic, something to be learned after the fundamental principles are understood. Chapters 1-27 explain all the basic techniques using only algebra, and in rare cases, a small amount of elementary calculus. Chapters 28-31 show how complex math * extends* the power of DSP; presenting techniques that cannot be implemented with real numbers alone.

** Here is an example**
The most important waveforms in signal processing are the sine and cosine waves. Using ordinary algebra, a sine wave is expressed by:

where *f* is the frequency of the sine wave (in cycles/second), and *t* is time (in seconds). This should be very familiar from you classes in math, science, and electronics. Now let's look at how this same sine wave is expressed using complex numbers:

where *e* is the base of the natural logarithm (2.7183) and *j* is the square-root of -1 (an imaginary number). Even though it looks like a jumble of symbols, this expression is mathematically identical to the more familiar sine wave expression. Writing sine and cosine waves in this way is the basis for using complex math in DSP. This has several advantages, but is so complicated that most scientists and engineers can't spare the time to learn or use it.

Traditional DSP textbooks are filled with complex math, often starting from the first chapter. Concepts are "explained" by mathematically proving they are true. __ The Scientist and Engineer's guide to DSP__ is different. Mathematical derivations are replace by * clear explanations*. Complex techniques are presented, but in their proper context: a way of making the basic DSP methods even more powerful.

**It's not such a strange idea**
To understand the future of DSP education, think about another technology: *electronics*. If this is your main field, you probably took dozens of classes on the subject; everything from the operation of transistors to the internal design of integrated circuits. This is a well organized and detailed presentation of the material, designed to make you a leader in the field.

However, if electronics is not your specialty, your education will have been very different. You probably took one or two classes in * applied* electronics. You learned ohm's law, how to use op amps, the design of simple filters, and other practical techniques. You know nothing about electron-hole physics in semiconductors, and you don't care! You use electronics as a *tool* to further your research or design activities. For every expert in electronics, there are 100 scientists and engineers that have a basic familiarly with the practical applications. This is the future of DSP.