Log Scale A logarithmic scale uses decades (factors of 10) or octaves (factors of 2) to represent vast ranges of data, such as frequency, on a compact graph. Decades, useful for spanning magnitudes (10, 100, 1000 Hz), represent a 10:1 ratio, while octaves (doubling) are common in audio. • Decade (Log Scale): Represents a power of 10 increase (e.g., 10 Hz to 100 Hz is one decade; 100 Hz 1000 Hz is a second decade). Often used in electronics, such as Bode plots to display frequency response over a large range. • Octave (Log Scale): Represents a doubling of frequency (e.g., 100 Hz to 200 Hz is one octave; 200 Hz to 400 Hz is another). Used in audio and vibration analysis, derived from music.

Motivation for using 20 log10 |H(jω)| In communications it is standard to measure power gain in decibels,

Since power is the square of voltage, the voltage gain is

From now on, we will drop the base of the logarithm; it is understood to be 10.

Bode plots dB vs. frequency to analyze magnitude and phase Bode Plots and Frequency Response: Logarithmic scales are used to display frequency response over several orders of magnitude. The gain is expressed in decibels dB, calculated as 20*log10|a|), and frequency is plotted logarithmically.

Logarithmic plots of H(jω), or Bode diagrams of H(jω), are two graphs: 1. A plot of 20 log10 |H(jω)| versus the frequency in log scale, that is, versus log10 ω 2. The phase angle 6 H(jω) versus log10 ω The standard representation of the logarithmic magnitude of H(jω) is 20 log10 |H(jω)| dB

Octave and decade An octave is a frequency band from ω1 to ω2 such that ω2 / ω1 = 2 There is an increase in decade from ω1 to ω2 when ω2 / ω1 = 10