Embedded Systems September 2000 Vol13_10

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ROBERT SCOTT Doing Hartley Smartly Think the fast Fourier transform is fast enough? Think again. Meet the fast Hartley transform. ransforming a series of samples taken in the time domain into a frequency domain representation is useful in a wide variety of signal processing applications. Thi includes direct analysis of the frequency spectrum as well as operations like convolu- tions, which are u ed in digi tal filtering. The usual tool for per- forming this transformation is the Fourier transform, or to be more specific, the fast Fourier transform (FFT) . But for most applications, there is an even faster method called the Hartley transform. This article reviews some published data on the fast Hartley transform (FHT) and then develops an optimized implementation of this algorithm. A typical FFT application starts with a series of samples taken in the time domain. For example, suppose you wan t to determine the vibration char- acteristics of an automobile engine. You could attach an accelerometer to the engine block and sample the output of the accelerometer at regular intervals. Suppose you take 4,096 samples over a one-second period. These 4,096 samples would then be transformed, using an FFT, into an array of 4,096 elements where each element specifies the vibrational energy at a particular frequency. The frequencies represented in the FFT array run from OHz to 4,095Hz in steps of 1Hz. So if you needed to determine the total energy in the frequency band from 200Hz to 300Hz, you could add up the amplitudes of FFT entries 200 through 300. Although this example is described using the FFT, the FHT works just as well. When I first saw an article about tl1e Hartley transform, l must admit tl1at I didn 't give it serious consideration. The formulae looked similar to the ones for the Fourier transform, except that the imaginary, i, was mis - ing. I thought there must be some catch. But then, years later, I ran across

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