NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University
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Dynamic Range
The dynamic range of a signal processing system can be defined as the maximum dB level sustainable without overflow (or other distortion) minus the dB level of the ''noise floor''.
Similarly, the dynamic range of a signal can be defined as its maximum decibel level minus its average ''noise level'' in dB. For digital signals, the limiting noise is ideally quantization noise.
Quantization noise is generally modeled as a uniform random variable between plus and minus half the least significant bit (since rounding to the nearest representable sample value is normally used). If
denotes the quantization interval, then the maximum quantization-error magnitude is
, and its variance (''noise power'') is
(see Appendix A for a derivation of this value).
The rms level of the quantization noise is therefore
, or about 60% of the maximum error.
The number system and number of bits chosen to represent signal samples determines their available dynamic range. Signal processing operations such as digital filtering may use the same number system as the input signal, or they may use extra bits in the computations, yielding an increased ''internal dynamic range''.
Since the threshold of hearing is near 0 dB SPL, and since the ''threshold of pain'' is often defined as 120 dB SPL, we may say that the dynamic range of human hearing is approximately 120 dB.
The dynamic range of magnetic tape is approximately 55 dB. To increase the dynamic range available for analog recording on magnetic tape, companding is often used. ''Dolby A'' adds approximately 10 dB to the dynamic range that will fit on magnetic tape (by compressing the signal dynamic range by 10 dB), while DBX adds 30 dB (at the cost of more ''transient distortion'').4.6 In general, any dynamic range can be mapped to any other dynamic range, subject only to noise limitations.