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At the beginning of Section A.3, we said that the choice of using the rectangular versus the polar form of representing complex numbers depends on the type of arithmetic operations we intend to perform. It's interesting to note that the rectangular form has a practical advantage over the polar form when we consider how numbers are represented in a computer. For example, let's say we must represent our complex numbers using a four-bit sign-magnitude binary number format. This means that we can have integral numbers ranging from –7 to +7, and our range of complex numbers covers a square on the complex plane as shown in Figure A-4(a) when we use the rectangular form. On the other hand, if we used 4-bit numbers to represent the magnitude of a complex number in polar form, those numbers must reside on or within a circle whose radius is 7 as shown in Figure A-4(b). Notice how the four shaded corners in Figure A-4(b) represent locations of valid complex values using the rectangular form, but are out of bounds if we use the polar form. Put another way, a complex number calculation, yielding an acceptable result in rectangular form, could result in an overflow error if we used polar notation in our computer. We could accommodate the complex value 7 + j7 in rectangular form but not its polar equivalent, because the magnitude of that polar number is greater than 7.

Figure A-4. Complex integral numbers represented as points on the complex plane using a four-bit sign-magnitude data format: (a) using rectangular notation; (b) using polar notation.

Although we avoid any further discussion here of the practical implications of performing complex arithmetic using standard digital data formats, it is an intricate and interesting subject. To explore this topic further, the inquisitive reader is encouraged to start with the references.

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[1] Plauger, P. J. "Complex Math Functions," Embedded Systems Programming, August 1994.

[2] Kahan, W. "Branch Cuts for Complex Elementary Functions, or Much Ado About Nothing's Sign Bit," Proceedings of the Joint IMA/SIAM Conference on the State of the Art in Numerical Analysis, Clarendon Press, 1987.

[3] Plauger, P. J. "Complex Made Simple," Embedded Systems Programming, July 1994.

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Appendix B. Closed Form of a Geometric Series

In the literature of digital signal processing, we often encounter geometric series expressions like


Equation B-2

Unfortunately, many authors make a statement like "and we know that," and drop Eqs. (B-1) or (B-2) on the unsuspecting reader who's expected to accept these expressions on faith. Assuming that you don't have a Ph.D. in mathematics, you may wonder exactly what arithmetic sleight of hand allows us to arrive at Eqs. (B-1) or (B-2). To answer this question, let's consider a general expression for a geometric series such as

Equation B-3

where n, N, and p are integers and a and r are any constants. Multiplying Eq. (B-3) by r, gives us

Equation B-4

Subtracting Eq. (B-4) from Eq. (B-3) gives the expression


Equation B-5

So here's what we're after. The closed form of the series is

Equation B-6

(By closed form, we mean taking an infinite series and converting it to a simpler mathematical form without the summation.) When a = 1, Eq. (B-6) validates Eq. (B-1). We can quickly verify Eq. (B-6) with an example. Letting N = 5, p = 0, a = 2, and r = 3, for example, we can create the following list:


arn = 2 · 3n


2 · 30 = 2


2 · 31 = 6


2 · 32 = 18


2 · 33 = 54


2 · 34 = 162


The sum of this column is

Plugging our example N, p, a, and r values into Eq. (B-6),

Equation B-7

which equals the sum of the rightmost column in the list above.

As a final step, the terms of our earlier Eq. (B-2) are in the form of Eq. (B-6) as p = 0, a = 1, and r = ej2pm/N.[] So plugging those terms from Eq. (B-2) into Eq. (B-6) gives us

[] From the math identity axy = (ax)y, we can say e–j2pnm/N = (e–j2pm/N)n, so r = ej2pm/N.

Equation B-8

confirming Eq. (B-2).