The math of DSP, part 1: Series, integration, and frequency

Part 2 explains complex numbers.

The heart of DSP is, naturally enough, numbers. More specifically, DSP deals with how numbers are processed. Most texts on DSP either assume that the reader already has a background in numerical theory, or they add an appendix or two to review complex numbers. This is unfortunate, since the key algorithms in DSP are virtually incomprehensible without a strong foundation in the basic numerical concepts.

Since the numerical foundation is so critical, we begin our discussion of the mathematics of DSP with some basic information. This material may be review for many readers. However, we suggest that you at least scan the material presented in this section, as the discussions that follow will be much clearer.

First, let's begin by defining some terms used in this chapter. As you probably remember from beginning calculus, a function is a rule that assigns to each element in a set one and only one element in another set. The rule can be specified by a mathematical formula or by tables of associated numbers. A complex number is a number of the form a + bj , having a "real" part a and an "imaginary" part bj , with j representing the square root of − 1 (although the word "imaginary" doesn't mean that part of the number isn't useful in the real world). A causal signal is a signal that has a value of zero for all negative numbered samples. We'll encounter many other important terms in this chapter, but we'll define those as we use them.

FUNCTIONS
In general, applied mathematics is a study of functions. Primarily, we are interested in how the function behaves directly. That is, for any given input, we want to know what the output is. Often, however, we are interested in other properties of a given function. For example, we may want to know how rapidly the function is changing, what the maximum or minimum values are, or how much area the function bounds.

Additionally, it is often handy to have a couple of different ways to express a function. For some applications, one expression may make our work simpler than another.

Polynomials are the workhorse of applied mathematics. The simplest form of the polynomial is the simple linear equation:

where m and b are constants. For any straight line drawn on an x-y graph, an equation in the form of Equation 4-1 can be found. The constant m defines the slope, and b defines the y -intercept point. Not all functions are straight lines, of course. If the graph of the function has some curvature, then a higher-order function is required. In general, for any function, a polynomial can be found of the form:

which closely approximates the given function, where a, b, and c are constants called the coefficients of f (x).

If polynomials are so powerful and easy to use, why do we turn to transcendental functions such as the sine, cosine, natural logarithm, and so on? There are a number of reasons why transcendental functions are useful to us.

One reason is that the transcendental forms are simply more compact. It is much easier to write:

than it is to write the polynomial approximation:

Another reason is that it is often much easier to explore and manipulate relationships between functions if they are expressed in their transcendental form.

For example, one look at Equation 4-3 tells us that f(x) will have the distinctive shape of a sine wave. If we look at Equation 4-4, it's much harder to discern the nature of the function we are working with. It is worth noting that, for many practical applications, we do in fact use the polynomial form of the function and its transcendental form interchangeably. For example, in a spreadsheet or high-level programming language, a function call of the form:

results in y being computed by a polynomial form of the sine function.

Often, polynomial expressions called series expansions are used for computing numerical approximations. One of the most common of all series is the Taylor series. The general form of the Taylor series is:

Again, by selecting the values of a_n, it is possible to represent many functions by the Taylor series. In this book we are not particularly interested in determining the values of the coefficients for functions in general, as this topic is well covered in many books on basic calculus. The idea of series expansion is presented here because it plays a key role in an upcoming discussion: the z-transform.

A series may converge to a specific value, or it may diverge. An example of a convergent series is:

As n grows larger, the term 1/2ⁿ, grows smaller. No matter how many terms are evaluated, the value of the series simply moves closer to a final value of 2.

A divergent series is easy to come up with:

As n approaches infinity, the value of f (n) grows without bound. Thus, this series diverges.