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 = e–j2pm/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 = e–j2pm/N.

Equation B-8

confirming Eq. (B-2).

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Chapter One. Discrete Sequences and Systems

Chapter Two. Periodic Sampling

Chapter Three. The Discrete Fourier Transform

Chapter Four. The Fast Fourier Transform

Chapter Five. Finite Impulse Response Filters

Chapter Six. Infinite Impulse Response Filters

Chapter Seven. Specialized Lowpass FIR Filters

Chapter Eight. Quadrature Signals

Chapter Nine. The Discrete Hilbert Transform

Chapter Ten. Sample Rate Conversion

Chapter Eleven. Signal Averaging

Chapter Twelve. Digital Data Formats and Their Effects

Chapter Thirteen. Digital Signal Processing Tricks

Appendix A. The Arithmetic of Complex Numbers

Appendix B. Closed Form of a Geometric Series

Appendix C. Time Reversal and the DFT

Appendix D. Mean, Variance, and Standard Deviation

Appendix E. Decibels (dB and dBm)

Appendix F. Digital Filter Terminology

Appendix G. Frequency Sampling Filter Derivations

Appendix H. Frequency Sampling Filter Design Tables

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Understanding Digital Signal Processing
Understanding Digital Signal Processing (2nd Edition)
ISBN: 0131089897
EAN: 2147483647
Year: 2004
Pages: 183
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