# Harmonics of DAC + Amp, cumulative?

#### Pdxwayne

##### Major Contributor
Let's say when playing 1Khz tone at 0db, the DAC's second harmonic is at -90db. Let's say an amp is measured to have second harmonic of -90db too.

When playing 1Khz test tone into such DAC and then to such amp, what kind of second harmonic would be outputted to the speakers?

Assuming -140db is dead quite, -90db is 50db of noise. So, would the cumulative second harmonic of the DAC + Amp combo now at 50+50=100db of noise?

Thanks!

#### Speedskater

##### Addicted to Fun and Learning
a] dB's don't add that way.
b] if -140dB is dead quite, maybe -100dB is also dead quite.
c] if the DAC has -90dB total harmonic distortion and the amp has the same. Then the sum might be -87dB but then again it might not.

#### DVDdoug

##### Senior Member
Let's say when playing 1Khz tone at 0db, the DAC's second harmonic is at -90db. Let's say an amp is measured to have second harmonic of -90db too.
+6dB is a doubling of amplitude so worst case (with both "signals" identical in-phase, etc.) you'd have -84dB.

Assuming -140db is dead quite, -90db is 50db of noise.
What? Are trying to calibrate/correlate digital or electrical decibels to acoustic SPL? There is no fixed calibration. The acoustic loudness depends on a LOT of variables, including the volume control setting.

There is a direct correlation... If you "don't touch anything" and everything is linear (if you're driving your amp into distortion, etc.), if you reduce the electrical or digital level by -6dB the acoustic level also drops by 6dB.

True dead silence is -infinity dB. You can get true digital silence (sample-values of zero is -infinity dB) or you can get "acoustical" silence in outer space. 0dB SPL is approximately the quietest sound that can be heard so dB SPL levels are positive.* The digital reference is 0dBFS (zero decibels full scale) which with integer formats is the "highest you can count" with a given number of bits, so digital dB levels are normally negative.

* From what I've read there are anechoic chambers that can be in the negative SPL range and as far as I know that's the only place on earth where it's that quiet.

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OP

#### Pdxwayne

##### Major Contributor
+6dB is a doubling of amplitude so worst case (with both "signals" identical in-phase, etc.) you'd have -84dB.

What? Are trying to calibrate/correlate digital or electrical decibels to acoustic SPL? There is no fixed calibration. The acoustic loudness depends on a LOT of variables, including the volume control setting.

There is a direct correlation... If you "don't touch anything" and everything is linear (if you're driving your amp into distortion, etc.), if you reduce the electrical or digital level by -6dB the acoustic level also drops by 6dB.
Let's say second harmonic of amp is -80 and DAC is -87, how would do I do the math? Thx!

Edit:
Let's say amp is -80 and DAC is -110, what would be the combination result?

OP

#### Pdxwayne

##### Major Contributor
Thanks!
Looks correct to you?

#### flipflop

##### Addicted to Fun and Learning
It's 3 dB for incoherent signals and 6 dB for coherent signals. If both are the second order harmonics of a 1 kHz tone, they're both 2 kHz tones (i.e., coherent).

EDIT: the 3 and 6 dB figures only work if the signals are equal in SPL.

#### abdo123

##### Major Contributor
Forum Donor
Power and voltage are different, @abdo123 posted a calculator based on sound power, not voltage. +3 dB corresponds to doubling of power (1.414 x voltage) and +6 dB corresponds to doubling of voltage (4 x power).

Michael

can you please explain when each type of calculation is valid?

I know that two speakers in mono are 3 dB louder than one speaker, which is why I knew the answer (in sound power) is +3 dB.

but if i doubled my DAC's output from 2V to 4V i would have a 6 dB increase in SPL?

#### mdsimon2

##### Addicted to Fun and Learning
can you please explain when each type of calculation is valid?

I know that two speakers in mono are 3 dB louder than one speaker, which is why I knew the answer (in sound power) is 3 dB.

but if i doubled my DAC's output from 2V to 4V i would have a 6 dB increase in SPL?

Let's say you are feeding DAC output to an amp with 20 dB gain in to a 4 ohm load (makes the math easy).

The amplifier output voltage can be calculated as Vamp = Vdac*10^(G/20) and amplifier power can be calculated as Pamp = Vamp^2 / R

So at 2 V from your DAC.

Vamp = 2*10^(20/20) = 20 V
Pamp = 20^2 / 4 = 100 W

Now at 4 V from your DAC.

Vamp = 4*10^(20/20) = 40 V
Pamp = 40^2 / 4 = 400 W

Increase in voltage in dB can be calculated as dB = 20*log(V2/V1), therefore voltage increase is 20*log(40/20) = 6 dB and power has increased 4 times.

If you run the same calculations for a DAC output voltage of 2.828 V you will see that this results in doubling of power and voltage has increased by 3 dB.

Michael

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#### GaryH

##### Senior Member

The calculator at the bottom of this page on the same site specifically titled "Addition of harmonic distortion attenuation of multiple devices" gives an answer of 3dB (3.01dB to be exact) not 6dB increase in distortion when combining one device at -90dB with another at -90dB.

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#### flipflop

##### Addicted to Fun and Learning
The calculator at the bottom of this page on the same site specifically titled "Addition of harmonic distortion attenuation of multiple devices" gives an answer of 3dB not 6dB increase in distortion when combining one device at -90dB with another at -90dB.
Because that page is about THD.
Even if two devices had identical THD figures, the makeup of the harmonics would be different, thus making the signals incoherent.

#### GaryH

##### Senior Member
Because that page is about THD.
Even if two devices had identical THD figures, the makeup of the harmonics would be different, thus making the signals incoherent.

Will a DAC and amp's signals typically be exactly coherent though? If not the calculator I linked to will better reflect reality even for individual harmonics.

#### AnalogSteph

##### Major Contributor
You basically have to determine the vector sum of all individual harmonics, as represented in magnitude and phase.

At most, two harmonics of identical magnitude might be exactly in phase, at which point you'd get +6 dB, i.e. -84 dB.
They might, however, be exactly 180° out of phase by some stroke of luck, in which case they would cancel entirely.
Real life is going to be somewhere in between. Cancellation tends not to be particularly reliable, affecting one type of harmonic at most (either even or odd), and often just a single one. Statistically, +3 dB probably isn't such a bad bet, but it's a rather crude simplification that made calculations manageable when those were still done on paper.

#### mdsimon2

##### Addicted to Fun and Learning
You basically have to determine the vector sum of all individual harmonics, as represented in magnitude and phase.

At most, two harmonics of identical magnitude might be exactly in phase, at which point you'd get +6 dB, i.e. -84 dB.
They might, however, be exactly 180° out of phase by some stroke of luck, in which case they would cancel entirely.
Real life is going to be somewhere in between. Cancellation tends not to be particularly reliable, affecting one type of harmonic at most (either even or odd), and often just a single one. Statistically, +3 dB probably isn't such a bad bet, but it's a rather crude simplification that made calculations manageable when those were still done on paper.

+1, was just about to the post this!

Michael

#### mdsimon2

##### Addicted to Fun and Learning
Here is something I whipped up quickly that shows the impact of phase offset. This is for two signals of equal magnitude.

As you can see at 0 deg phase offset (correlated) you get +6 dB. At 90 deg phase offset (uncorrelated) you get + 3 dB. Above 120 deg phase offset you start getting cancellation as @AnalogSteph mentioned.

Michael

OP

#### Pdxwayne

##### Major Contributor
Here is something I whipped up quickly that shows the impact of phase offset. This is for two signals of equal magnitude.

View attachment 132283

As you can see at 0 deg phase offset (correlated) you get +6 dB. At 90 deg phase offset (uncorrelated) you get + 3 dB. Above 120 deg phase offset you start getting cancellation as @AnalogSteph mentioned.

Michael
For that +6db number, let's assume also translates to spl of +6db for one speaker. For a stereo setup, assuming each speaker get +6, then both speakers would get a cumulative of +9 or +12?

Thanks!

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