Logarithmic Scales

The magnitude of X(f) measured in decibels is defined by

. A 6-dB increase corresponds (approximately) to doubling the magnitude, and a 20-dB increase corresponds to a factor of 10 increase. If

denotes the angle of X(f), it follows from Y(f) = H(f) X(f) that

and

Thus, by measuring magnitudes in dB, we can account for the convolution of X and H by merely adding the dB values and adding the phase values. In addition, graphs that show how the magnitudes and phase angles change with frequency are usually plotted on a logarithmic frequency scale. Then a doubling of frequency (called an octave, from musical terminology) has the same horizontal displacement for all frequencies.

This is not only convenient mathematically, but also fits pretty well with the facts of human hearing. Human hearing is more or less logarithmic, responding to ratios rather than differences. Roughly speaking, 1 dB is about the smallest perceptible change in loudness, no matter what the starting intensity level, and a one-octave frequency change sounds like the same musical interval, no matter what the starting frequency.

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