# Why is frequency not measured in db in bode's plot?

While drawing bode's graph paper where Y axis is gain given in db while the X axis is frequency given in absolute value but scale used is log. I think the reason for log scale on X axis is to increase frequency range over which plot can be drawn. But can't same advantage be achieved if frequency is represented in db and scale used is linear for both X as well as Y axis.

• The advantage of using dB is that multiplication turns into addition when converting to dB. Multiplying gains of various elements in a system is a common operation. However, multiplying frequencies is not. What is the advantage of converting frequency to dB then? – AJN Dec 2 '20 at 14:24
• @AJN Any musician deals with multiplying frequencies; it's fundamental (heh) to the notion of a scale. However a scale already has a logarithmic base; dB is just less convenient than octave (or semitone) – Brian Drummond Dec 2 '20 at 14:38
• You can take the log of frequency and plot it on a linear scale, or plot the absolute frequency on a log scale. Roundabouts and swings, really. – Chu Dec 2 '20 at 14:39
• @BrianDrummond As a musician, composer, producer, and audio engineer, I never multiply frequencies unless I use a pitch shifter like a Digitech Whammy Pedal. But of course I'm constantly considering different types of gain and attenuation for signal levels. From a musical point of view, a pitch is a perception of frequency, and music theory is concerned with pitches, not frequencies. Even when using different tunings and temperaments, we think about frequencies a bit, but creating the correct pitch sense often requires generating the "wrong" frequencies. – Todd Wilcox Dec 3 '20 at 5:52

Frequency is not a measure of signal amplitude or power hence, talking about "this or that" frequency in decibels makes no sense. If somebody says "increase the frequency by 3 dB"; it is meaningless because it isn't a signal or power quantity. If someone else says "increase the signal by 3 dB", we know that means the power is doubled (or the signal is increased by $$\\sqrt2\$$). To apply that concept to frequency is pointless and meaningless.

There is nothing intrinsic to "frequency" that makes it relatable to power or root-power (as in voltage, current or other signal levels) hence, to use it so, makes no sense. After all, we use the base 10 logarithm of a power ratio multiplied by ten to convert it to decibels and, we use the log of a voltage ratio multiplied by twenty to produce an answer in dB. Which formula would make sense for frequency?

$$20\cdot \log_{10}(\text{voltage ratio}) = 10\cdot \log_{10}(\text{power ratio})$$

The number "20" is used because volts are squared to make power hence, the logarithm of a squared voltage ratio is multiplied by "20" to make it numerically equivalent decibel power. The base factor of "10" puts power into "deci" units.

Does adding a decibel frequency ratio to another decibel frequency ratio have any actual meaning that is useful? On the other hand, with gain stages you can make a lot of sense of adding the gains in decibels to produce an overall gain value in dB. Does that make sense for frequency; no.

• Describing frequency as a power of ten may not be meaningful, but describing it as a power of two would be. If log2(f1) is one greater than log2(f2), then they represent pitches that are a musical octave apart. If there were a prefix that meant (1/12), describing frequencies in units of 1/12 of a musical octave would for some purposes be even more useful. – supercat Dec 3 '20 at 21:52
• That doesn’t fit well with the word “deci”. If deci were a twelfth then sure, it would fit with a musical semitone but, what would be its usefulness when we already have sharp or flat versions of the letters a to g that inform us of the correct pitch to play. Again, I make the reminder that frequency has no implicit relationship with power or square root of power; that’s why we use decibels or bels. – Andy aka Dec 3 '20 at 22:37
• The existing term "decibel" refers to log10(x)/10. My point was that it would be useful if there were a term for a log2(x)/12 increase in frequency--perhaps something like ddo for "duodecioctave", though that would represent a curious mix of Latin fractional prefix and Italian musical terminology--not that the term "decibel" would be suitable for such a purpose. – supercat Dec 3 '20 at 23:16
• It’s called a semitone. Well that’s what I think you refer to @supercat – Andy aka Dec 3 '20 at 23:46
• [bonking head]Yeah, I guess if one is describing things purely in relative terms that could work, and I suppose as an absolute measure, one could kinda sorta "MIDI note number" to refer to the number of semitones above 8.175Hz.though fractional values would seem a little odd with that phrase. – supercat Dec 3 '20 at 23:52

We could start using dBHz and see how it feels. My guess is it won't make life any simpler or help understanding.

NOTE however : logarithmic frequency scale is already in use, and has been for a while ... a few millennia of precedence for the octave (base 2 logarithm) will make a decade-based system difficult to sell. Octaves are in reasonably common use in audio electronics thanks to their prior use in music.

Furthermore the "Bel" in decibel refers to loudness and not pitch, so its transfer to power ratios is quite logical, applying it to frequency is less so.

• Agreed, and if you look at almost any Bode plot you can already see that frequency is plotted logarithmically, even if the numbers are unscaled. – Justin Dec 2 '20 at 14:37

While drawing bode's graph paper where Y axis is gain given in db while the X axis is frequency given in absolute value but scale used is log.

So we are plotting log response against log frequency.

I think the reason for log scale on X axis is to increase frequency range over which plot can be drawn.

It has that effect, but that's more of a byproduct. The real reason we plot frequency on a log scale is that many things are proportional to the ratios of frequencies, like the upper and lower responses of a bandpass filter, or the stopbands of filters. A straight line on a log-log plot like Bode's means that we have a power ratio between frequency and response, the slope of the line giving us the exponent.

But can't same advantage be achieved if frequency is represented in db and scale used is linear for both X as well as Y axis.

We could theoretically do that, it's trivial in a plotting program to plot in either way, but that's not the way humans think and work. We use the two axes in different ways. We continue to use dBs with the gain response because it's easy to do and makes sense to most people. We think of the edge of the passband as 0.1dB or 1dB down, because we are interested in gain relative to a level. We want to know when the stopband has got down to -40dB. We tend not to use frequency like that.

That's all it is. We continue using these graph conventions because they make sense to the humans that use them, and there's no real reason to change.

It's worth noticing why synthesisers and counters read out in linear units, and signal sources and analysers read level in dBs. These instruments aren't forcing us to use those units. We make the instruments like that because those are the units we find most convenient to use.

Gain is relative, frequency is absolute. A dB with no reference can refer to a gain because a dB with no reference is relative. Absolute levels require a reference value to have units in dB, examples include dBm, dBu, dBFS, dB-SPL. Since gain is basically dimensionless, it can be measured in dB. But frequency has a dimension - the Hz.

Another way to look at this is why do we measure power, voltage, sound pressure, and other wave "levels" in dB tied to a reference instead of with a simple unit? Well one good reason is if we have any kind of amplifier or attenuator with a dimensionless gain or attenuation factor in dB, we can simply add or subtract the dB of gain or attenuation of the device from the absolute input level to get the absolute output level.

Also note that levels are frequently and easily changed by all sorts of signal processing devices, and frequencies are changed far less often. It could be that using dB of "shift" as a measurement for the action of a frequency shifting device could be useful, but it could also be confusing, since we've essentially standardized that referenced dB means a level or intensity, and dimensionless dB means gain or attenuation. Adding other kinds of dB into the mix without a good reason doesn't seem helpful.

Measuring frequency in dB would just create confusion. With amplitude the absolute values are not really important -- it's the relative values that one is concerned with in the vast majority of cases.

On the other hand, with frequency it's rare to be concerned with relative values. If you are designing an RC filter to control bandwidth, you need the absolute value of the cutoff frequency. If you are designing an amplifier, you need the absolute lower and upper bandpass numbers, not the relative numbers.

• That's not really true. Digital filters for example are just about always designed in terms of normalized frequency (relative to FS) and it's not unheard of elsewhere either as filter types, strategies and shapes are relative in their behavior. – Chris Stratton Dec 3 '20 at 4:50
• Note that it's not amplitude that plotted in dB in the context of the question, it's gain, which is dimensionless. – Todd Wilcox Dec 3 '20 at 5:41

There are ratio-based terms which describe frequency changes by powers of two ("octaves", referring to the fact that the eighth note of common musical scales will have a frequency twice that of the first), or by one twelfth of that ("semitone", referring to the fact that while most musical intervals in a scale are 1/6 of an octave, there are two smaller intervals that are 1/12 octave). The use of the former term is common when describing some filter or frequency-drop-off characteristics (e.g. 6dB/octave). The term decibel is used to describe 1/10 of a power-of-ten change in power, or 1/20 of a power-of-ten change in voltage or current. Depending upon which scaling factor seemed more appropriate for frequency, an octave would either be 3.01dB or 6.02dB, and a semitone would be 0.251dB or 0.502dB.