### 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 §G.3 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* (see Appendix G 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'').^{F.7} In general, any dynamic range
can be mapped to any other dynamic range, subject only to noise
limitations.

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Pulse Code Modulation (PCM)

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Specific DB Scales