The multiplication of two complex numbers is one of the most common functions performed in digital signal processing. It's mandatory in all discrete and fast Fourier transformation algorithms, necessary for graphics transformations, and used in processing digital communications signals. Be it in hardware or software, it's always to our benefit to streamline the processing necessary to perform a complex multiply whenever we can. If the available hardware can perform three additions faster than a single multiplication, there's a way to speed up a complex multiply operation [16].

The multiplication of two complex numbers, a + jb and c + jd, results in the complex product

We can see that Eq. (13-14) requires four multiplications and two additions. (From a computational standpoint we'll assume a subtraction is equivalent to an addition.) Instead of using Eq. (13-14), we can calculate the following intermediate values

Equation 13-15

We then perform the following operations to get the final R and I

Equation 13-16

The reader is invited to plug the k values from Eq. (13-15) into Eq. (13-16) to verify that the expressions in Eq. (13-16) are equivalent to Eq. (13-14). The intermediate values in Eq. (13-15) required three additions and three multiplications, while the results in Eq. (13-16) required two more additions. So we traded one of the multiplications required in Eq. (13-14) for three addition operations needed by Eq. (13-15) and Eq. (13-16). If our hardware uses fewer clock cycles to perform three additions than a single multiplication, we may well gain overall processing speed by using Eq. (13-15) and Eq. (13-16) instead of Eq. (13-14) for complex multiplication.


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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

Understanding Digital Signal Processing
Understanding Digital Signal Processing (2nd Edition)
ISBN: 0131089897
EAN: 2147483647
Year: 2004
Pages: 183 © 2008-2020.
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