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3. Dynamic Range: Distortion and Noise > The Fast Fourier Transform (FFT) - Pg. 166

166 Valve Amplifiers TABLE 3.1 Powers in the Denary Number System Thousands 1,000 10 3 Hundreds 100 10 2 Tens 10 10 1 Units 1 10 0 Tenths 1/10 10 21 Hundredths 1/100 10 22 Thousandths 1/1,000 10 23 The terms `hundreds, tenths', etc. are simply powers of the base, in this case 10. The binary system works in exactly the same way, but because it uses 2 as its base, rather than 10, its table is slightly different. TABLE 3.2 Powers in the Binary Number System 32 2 5 16 2 4 8 2 3 4 2 2 2 2 1 1 2 0 1/2 2 21 1/4 2 22 1/8 2 23 1/16 2 24 1/32 2 25 requires one-and-a-half times as much memory to store each word (24/16 5 1 1 / 2 ). As a rule of thumb (ignoring dither), the Dynamic Range (DR) of a digi- tal system is: DR dB 5 6n where n 5 number of bits. Thus, a 16 bit system has a theoretical dynamic range of 6 3 16 5 96 dB and a 24 bit system 144 dB (never achieved because Analogue to Digital Convertors (ADCs) simply aren't that good). We could decide to be more precise by making more numerical measure- ments. Sampling twice as often doubles the memory required. To sum up, a more precise description generates more data, requiring more memory, and this will become significant later. THE FAST FOURIER TRANSFORM (FFT) The reason for converting our analogue signal to a digital signal was to allow mathematical techniques to be applied to the resulting numbers and allow patterns to be seen. (Humans are good at recognising patterns, so any tech- nique that reveals patterns helps understanding.) An oscilloscope allows us to spot patterns that repeat in time, such as a spike that occurs each time a sine wave changes polarity, but a spectrum analyser allows us to spot pat- terns in frequency, perhaps a small spike that indicates fifth harmonic distortion.