The Scientist and Engineer's Guide to
Digital Signal Processing
By Steven W. Smith, Ph.D.

DSP without Complex Numbers?

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.