My only experience with this is building and designing guitar amps, which often have 80dB of gain or more, a.k.a., a pain in the ass amount of gain to deal with. It's not something on par with, say, radio astronomy, but it's still a lot of gain to deal with.
Usually the main source of noise will be a 120Hz or 100Hz buzz, but with humbucking pickups and careful orientation of the guitar you can mostly eliminate that. The next source of noise will be a low-level white noise (sounds like a hiss), which is from the amplifier, and consists of a mixture of Johnson noise and shot noise.
In older amps you may hear a louder hiss/crackle which is from old carbon comp resistors, which is an inferior type of resistor that produces additional noise through a different mechanism.
If you're trying to record your guitar directly through a digital interface, you may run into clipping issues and have to enable the pad (a built-in attenuator). Unfortunately, my experience is that the pad often introduces an unacceptable amount of noise, and I believe that it's just plain Johnson noise from a resistive divider.
The experienced and mysterious audio engineer "NwAvGuy" [0] praised the virtue of using two gain stages and moving the volume control away from the first input to reduce Johnson noise in audio amplifier designs [1]. It's a good example of how the basic principle applies both to mundane audio and cutting-edge science: the system noise is dominated by the first amplifier stage. Adding some noise before the first stage significantly degrades signal-to-noise ratio, but adding the same noise after the first stage is often acceptable since the signal is much stronger now. To reduce noise, you move the noise-generating resistor away in an audio amp, or cryogenically cool the resistor in a radio telescope front-end.
> One of the big claims for many audiophile op amps is lower noise. The chip manufactures make a big deal about it and audiophiles, not surprisingly, have jumped on the bandwagon. But, in reality, it’s often the Johnson Noise that limits the noise performance of a headphone amp, not the op amps. Johnson Noise is, literally, self generated noise that’s present in any resistor. The larger the resistor value, the more noise you get. Many DIY headphone amp designs have the volume control at the input to the gain stage. And it’s, at the lowest, usually 10,000 ohms. By comparison the O2 has 274 ohms in series with the input. That’s a huge difference in Johnson Noise. The way volume controls work, the noise is typically worst at half volume where you have 5000 ohms in series with the source and 5000 ohms to ground. So, at typical volume settings, you get a fair amount of Johnson Noise from the volume control that’s amplified by whatever gain your amp has. That noise typically exceeds the op amp’s internal noise. If you put the volume control after the gain stage its Johnson Noise is no longer amplified. And, as a bonus, the volume control at lower settings now attenuates noise from the gain stage. For more, see O2 Circuit Description and Circuit Design.
> To put these numbers in perspective, referenced to the old 400 mV they’re –105.3 dBr and –108.2 dBr. On the exact same test, at half volume, the Mini3 had nearly 11 dB more noise and measured –94.5 and –97.5 dB. Noise of –113 dB below 1 volt is under 3 microvolts.
I was messing about with contact microphones last year and very much akin area to guitar amps as high-impedance, so very much the same issues.
If you run on batteries you will find it works best, as with anything mains, you will want a good ground.
What I did find was that if you use peizo's back to back you can effect a balanced signal and that in itself helps immensely in eliminating much of the noise. You can also use a contact material sandwich in-between the piezo discs and effect how it works tone wise as well as become more zoned in the pick-up area.
But impedance matching is, as with guitars, very much key for pre-amps.
As for input levels and clipping - the rise of 32bit float has made a huge difference and means you can not worry about mic input levels at the ADC stage as much and normalise everything in post, sorting the levels out then without any fear of clipping at all.
Though those just unbalanced input designs, alas I'm not aware of any balanced contact mic's on the market - but can easily make them yourself using the above approach.
I was surprised to find that after replacing most of the op-amps in a ADA MP-1 pre-amp, most of that orientation-sensitive remaining buzz that you still get with humbucking pickups was seriously reduced.
When I'm playing at a low volume, I can just mute the strings and put the guitar on a stand to get it to be quiet. On a high-gain program using the tube board and all.
The reason for some of the buzz is that the circuits are amplifying common mode. The op-amps are operated in feedback meaning that the - and + inputs are at nearly the same voltage. However, the incoming common mode noise moves that entire voltage; and it's possible for the common movement of +/- to itself be amplified.
So that is to say, suppose you have this representative single-ended stage:
The guitar cable's shield is connected to GND. Now suppose that GND is oscillating at 60 Hz due to the cable shield picking up EMI. (The cable shield is a big area of copper bathed in noisy electric fields, with nothing shielding it, and is galvanically connected to your amp!)
This means that the (+) node of the OP amp is seeing this fluctutation, and due to the feedback the (-) node is following.
An ideal op-amp will not amplify any voltage offset that equally affects (+) and (-). But real op-amps do. The degree to which they do not is the CMRR (common mode rejection ratio) which is a data sheet parameter that is better in some parts than others.
Shot noise was originally attributed to electrons hitting the anodes of vacuum tubes, but transistors turned out to make the same kind of noise so it applies to them too.
And it is current-dependent so sometimes you can be quieter at idle by biasing lower.
For resistor noise, nothing wrong with a megohm referencing input to ground since there's not any significant current flowing there.
But in a tube preamp the typical 100K anode resistor will conduct a bit and can get hot. These should be carefully auditioned. Plus higher wattage rating parts give less noise in this service than they put in most commercial amps.
If you increase gain using something like 150K, 220K, or even 330K there will be less current (through the resistor and the tube) but the increased amplification factor will equally multiply any noise which occurs before that gain stage.
Then with power you've got the ability to broadcast audio, naturally over extremely short distances (compared to radio frequencies which hopefully you are filtering out) but those are the distances inside your chassis where different parts of the wiring layout can interact beyond a cetain point as broadcast and receiving antennae, and either provide negative or positive feedback, stabilizing or destabilizing respectively parts of the circuit whether you intended it to happen or not.
On top of the expected acoustic mechanical feedback from the speaker at high volume, which reverses polarity based on distance, you've also got magnetic feedback. Once a very high-power output transformer goes wild it can reach out a lot further and touch your pickups directly from a few feet away.
At this point the noise at idle is usually as loud as as ten pounds of bacon frying and I hate that.
So get out the soldering iron and fix it so you can only tell it's on if you put your ear close to the speaker, when it's actually set loud enough to play with a heavy drummer.
While not exactly for guitar, Phil’s Lab on youtube recently posted a low noise headphone amplifier design that's both over-kill in some ways and interesting and cheap to make.
> In fact, a remarkably common response to a diagnosis of resistor noise is to seek a source of "good" resistors, with "good" being defined as without thermal noise. This is impossible.
It's impossible to make a totally noiseless resistor, but it's also important to understand that all resistors are not created equal.
Most resistors have noise levels that are orders of magnitude above the Johnson limit. Potentiometers are especially bad.
If you want "good" resistors for noise-critical applications, I recommend metal thin-film resistors. They hardly cost any extra anyway.
Also, in cases where resistors are used to set DC signals such as offsets and biases, you can add capacitors to filter the heck out of those lines to decrease their noise contribution.
Electrical engineer here. Thermal noise is the same for all resistors with a given R, regardless of their method of manufacture. You cannot have thermal noise either less or more than this, so it's not a Johnson "limit" but rather a definite value. You are correct that there are other sources of noise such as 1/f noise. But more importantly, the manner in which noise manifests in the end result has to do with the circuit as a whole.
For noise critical applications you should do a noise analysis of the circuit as a whole rather than make ad hoc selections of components.
I recommend measuring the noise. Systems guru Phil Hobbs said that you should know where every dB of noise comes from in your design. Of course a dB could be a little or a lot in your application, but the point is that you should perform a noise budget and then test your assumptions.
It's not necessarily easy, but recommended if possible. I was doing it with DIY equipment, so I don't claim traceable results.
In one case, I literally measured the noise of some resistors, and within the parameters of what I cared about, I found no measurable difference between metal film and carbon film. I was passing no DC current through the resistor. Some sources of "excess noise" are proportionate to DC current and can be corrected by appropriate filtering.
Fun fact: for the most demanding RF applications, namely, radio astronomy, the front-end low-noise amplifiers are indeed cooled to cryogenic temperature by liquid nitrogen. Here's how it's done at NASA for the Deep Space Network [0]. It's a long paper, see Chapter 4 Cryogenic Refrigeration Systems, PDF page 179 (text page 159). Also, nice photos in page 183 and 188.
The reverse problem is interesting. Consider Voyager 1- its high gain antenna is pointed essentially directly at the sun, a powerful wide-band noise source. How does it detect anything from earth? The DSN has to outshine the sun within the tiny S-band window that Voyager listens to: 20 kW and 62 dB antenna gain.
Since the amplifier has both voltage and current input noise sources, there is an optimum source impedance that provides a minimum noise figure, and it’s not an impedance match. This is called a minimum noise match, along with associated noise contours where noise is traded off with impedance match.
Also, the noise from an antenna is dependent on it’s efficiency and what it’s pointing at. Even if it’s input impedance is 50 Ohms, it can generate far less noise than the equivalent resistor.
The opening paragraph of Goodstein's "States of Matter":
"Ludwig Boltzmann, who spent much of his life studying statistical mechanics, died in 1906, by his own hand. Paul Ehrenfest, carrying on the work, died similarly in 1933. Now it is our turn to study statistical mechanics."
One of my first jobs, which I got while I was still an undergrad (in the mid-80s), was designing amplifiers for fiber-optic sensors. I pretty much had no clue what I was doing so I just started futzing around with op-amps and realized very quickly that my signal-to-noise-ratio was much higher than was acceptable. I figured there was some hardware design trick that they hadn't taught me in my EE curriculum, but one day I decided to do the math on resistor noise and discovered that that was in fact my limiting factor and the only way were were going to get it to work was to either cool the first-stage resistor or to use a ridiculously high value because the gain goes up linearly with the resistor value but the noise only increases with the square root. We ended up with a ten gigaohm resistor, which was just enough to get the S/N ration we needed to make it work.
Depending on the voltage across a resistor like that, you may calculate less than one electron passing through the resisitor per second.
Without ceramic or teflon standoffs, the circuit board can often conduct better than the resistor, plus dust can also accumulate on the outside of the resistor and conduct better eventually, which is why they are often encased in glass, so they can be effectively cleaned during a maintenance cycle.
I usually put a single 470 ohm resistor in line with the gate of a discrete jfet in common collector mode as the first gain stage in my projects. Once you boost up the signal voltage it’s way easier to maintain a good signal to noise ratio.
The resistor is there to prevent the jfet from being burned out by over voltage on the gate, which is very sensitive to static electricity. But, I can easily hear the difference if I put a 10k resistor there instead. It’s really important to get that first gain stage really, really quiet, a discrete jfet has a better noise floor than an op amp or a regular transistor.
I was watching an interview with Tom Christiansen (he owns Neurochrome, a company that makes very high-end DIY amplifier designs/kits, with THDs of 0.0001%).
He mentioned something about how resistor noise can actually track with the low frequency portions of the audio signal due to the resistor literally heating up and cooling down as the current through it varies. I thought that was interesting. I knew that noise was proportional to heat, but I didn't realize the temperature could vary that quickly, but I guess it makes sense when you're dealing with tiny parts. There are probably also localized hot spots that have less thermal mass than the entire resistor as a whole.
Chapter 8 of The Art of Electronics, 3rd Edition is a great resource on electrical noise. The first section is almost exclusively about resistors and noise budgeting. Fun read. I should get back to it at some point.
Lots of great tricks on bootstrapped filters, capacitive multipliers, precision transistor noise measurements with some outboard circuits fed into a spectrum analyzer.
It is now partially incorrect, too. Now we can no longer change Boltzmann's constant not because he's dead (indeed, until recently, it was a measurable quantity), but because k_B is now defined as a part of the redefinition of the SI unit system in 2018.
I read it more like a whine "don't blame our opamps, they better than resistors" :-) But it is something I've become much more familiar with building RF frontends for software defined radios. You want as much gain as you can get to pull in weak signals but keeping the whole thing noise quiet is really really hard.
Excellent article! I feel like I understand so much more about analog signals than I did before.
It seems very obvious now, that if you want to have a high signal to noise ratio, you should get as much signal as possible, and keep your signal voltage as high as possible as long as possible before amplification.
This fundamental resistor noise is something that I'm probably going to start seeing everywhere when I look at any analog signals, and will have to take into account when designing things.
It is surreal that this article showed up now. I am going back and forth currently with our electric utility company. They upgraded capacitor pack on the pole by my house. And these new ones are generating so much of this kind of low buzz sound, it is unbearable. I was researching into what generates this noise, and found this article very timely. Only solution is to move these, as nothing else can be done about this sound. Which is turning into quite a project.
It is quite rare for capacitor banks to hum. Much more likely that you're hearing the magnetics (transformers), which indeed have reason to be where they are. Good luck.
I don't know how this one made it, but if there's an Analog Dialogue article on the front page of HN weekly for the next five years it will still barely scratch the surface of all the treasure that's buried in that archive.
To the best of my knowledge time-independent articles don't need a date stamp. I never see one on Wikipedia articles even if the last edit was years ago.
There's a strange relationship between Resistance, Noise (audible and above the auditory spectrum, such as RF), and the concept of Impedance...
I'll put a wager that future (or perhaps even current!) scientists are able to engineer complex waveforms such that the complex waveform effectively negates the resistance/noise/impedance -- effectively turning the resistor into a conductor -- but only for that specific complex waveform -- which very possibly would change over time...
Also, future (and perhaps current!) scientists should be able to use an electrical signal of known characteristics -- to determine exactly what the complex impedance of the resistor/resistance element/impedance element -- in a circuit is, exactly...
In other words, given one of the above things (complex waveform, complex impedance) -- derive what the other one is, from it...
klodolph|4 years ago
Usually the main source of noise will be a 120Hz or 100Hz buzz, but with humbucking pickups and careful orientation of the guitar you can mostly eliminate that. The next source of noise will be a low-level white noise (sounds like a hiss), which is from the amplifier, and consists of a mixture of Johnson noise and shot noise.
In older amps you may hear a louder hiss/crackle which is from old carbon comp resistors, which is an inferior type of resistor that produces additional noise through a different mechanism.
If you're trying to record your guitar directly through a digital interface, you may run into clipping issues and have to enable the pad (a built-in attenuator). Unfortunately, my experience is that the pad often introduces an unacceptable amount of noise, and I believe that it's just plain Johnson noise from a resistive divider.
bcaa7f3a8bbc|4 years ago
> One of the big claims for many audiophile op amps is lower noise. The chip manufactures make a big deal about it and audiophiles, not surprisingly, have jumped on the bandwagon. But, in reality, it’s often the Johnson Noise that limits the noise performance of a headphone amp, not the op amps. Johnson Noise is, literally, self generated noise that’s present in any resistor. The larger the resistor value, the more noise you get. Many DIY headphone amp designs have the volume control at the input to the gain stage. And it’s, at the lowest, usually 10,000 ohms. By comparison the O2 has 274 ohms in series with the input. That’s a huge difference in Johnson Noise. The way volume controls work, the noise is typically worst at half volume where you have 5000 ohms in series with the source and 5000 ohms to ground. So, at typical volume settings, you get a fair amount of Johnson Noise from the volume control that’s amplified by whatever gain your amp has. That noise typically exceeds the op amp’s internal noise. If you put the volume control after the gain stage its Johnson Noise is no longer amplified. And, as a bonus, the volume control at lower settings now attenuates noise from the gain stage. For more, see O2 Circuit Description and Circuit Design.
> To put these numbers in perspective, referenced to the old 400 mV they’re –105.3 dBr and –108.2 dBr. On the exact same test, at half volume, the Mini3 had nearly 11 dB more noise and measured –94.5 and –97.5 dB. Noise of –113 dB below 1 volt is under 3 microvolts.
[0] https://spectrum.ieee.org/tech-history/silicon-revolution/nw...
[1] https://nwavguy.blogspot.com/2011/07/o2-headphone-amp.html
Zenst|4 years ago
If you run on batteries you will find it works best, as with anything mains, you will want a good ground.
What I did find was that if you use peizo's back to back you can effect a balanced signal and that in itself helps immensely in eliminating much of the noise. You can also use a contact material sandwich in-between the piezo discs and effect how it works tone wise as well as become more zoned in the pick-up area.
But impedance matching is, as with guitars, very much key for pre-amps.
As for input levels and clipping - the rise of 32bit float has made a huge difference and means you can not worry about mic input levels at the ADC stage as much and normalise everything in post, sorting the levels out then without any fear of clipping at all.
Some nice low-noise preamp designs to check out here: http://www.richardmudhar.com/piezo-contact-microphone-hi-z-a...
Though those just unbalanced input designs, alas I'm not aware of any balanced contact mic's on the market - but can easily make them yourself using the above approach.
kazinator|4 years ago
When I'm playing at a low volume, I can just mute the strings and put the guitar on a stand to get it to be quiet. On a high-gain program using the tube board and all.
The reason for some of the buzz is that the circuits are amplifying common mode. The op-amps are operated in feedback meaning that the - and + inputs are at nearly the same voltage. However, the incoming common mode noise moves that entire voltage; and it's possible for the common movement of +/- to itself be amplified.
So that is to say, suppose you have this representative single-ended stage:
The guitar cable's shield is connected to GND. Now suppose that GND is oscillating at 60 Hz due to the cable shield picking up EMI. (The cable shield is a big area of copper bathed in noisy electric fields, with nothing shielding it, and is galvanically connected to your amp!)This means that the (+) node of the OP amp is seeing this fluctutation, and due to the feedback the (-) node is following.
An ideal op-amp will not amplify any voltage offset that equally affects (+) and (-). But real op-amps do. The degree to which they do not is the CMRR (common mode rejection ratio) which is a data sheet parameter that is better in some parts than others.
fuzzfactor|4 years ago
Shot noise was originally attributed to electrons hitting the anodes of vacuum tubes, but transistors turned out to make the same kind of noise so it applies to them too.
And it is current-dependent so sometimes you can be quieter at idle by biasing lower.
For resistor noise, nothing wrong with a megohm referencing input to ground since there's not any significant current flowing there.
But in a tube preamp the typical 100K anode resistor will conduct a bit and can get hot. These should be carefully auditioned. Plus higher wattage rating parts give less noise in this service than they put in most commercial amps.
If you increase gain using something like 150K, 220K, or even 330K there will be less current (through the resistor and the tube) but the increased amplification factor will equally multiply any noise which occurs before that gain stage.
Then with power you've got the ability to broadcast audio, naturally over extremely short distances (compared to radio frequencies which hopefully you are filtering out) but those are the distances inside your chassis where different parts of the wiring layout can interact beyond a cetain point as broadcast and receiving antennae, and either provide negative or positive feedback, stabilizing or destabilizing respectively parts of the circuit whether you intended it to happen or not.
On top of the expected acoustic mechanical feedback from the speaker at high volume, which reverses polarity based on distance, you've also got magnetic feedback. Once a very high-power output transformer goes wild it can reach out a lot further and touch your pickups directly from a few feet away.
At this point the noise at idle is usually as loud as as ten pounds of bacon frying and I hate that.
So get out the soldering iron and fix it so you can only tell it's on if you put your ear close to the speaker, when it's actually set loud enough to play with a heavy drummer.
Hearing protection required beyond this point.
Renaud|4 years ago
https://www.youtube.com/watch?v=Z2GUoi63pJs
floxy|4 years ago
Any reason for guitar pads couldn't use a capacitive divider in place of a resistive divider? (I have no idea what a guitar pad is)
jbay808|4 years ago
It's impossible to make a totally noiseless resistor, but it's also important to understand that all resistors are not created equal.
Most resistors have noise levels that are orders of magnitude above the Johnson limit. Potentiometers are especially bad.
If you want "good" resistors for noise-critical applications, I recommend metal thin-film resistors. They hardly cost any extra anyway.
Also, in cases where resistors are used to set DC signals such as offsets and biases, you can add capacitors to filter the heck out of those lines to decrease their noise contribution.
xenocyon|4 years ago
For noise critical applications you should do a noise analysis of the circuit as a whole rather than make ad hoc selections of components.
analog31|4 years ago
It's not necessarily easy, but recommended if possible. I was doing it with DIY equipment, so I don't claim traceable results.
In one case, I literally measured the noise of some resistors, and within the parameters of what I cared about, I found no measurable difference between metal film and carbon film. I was passing no DC current through the resistor. Some sources of "excess noise" are proportionate to DC current and can be corrected by appropriate filtering.
wellthereitis|4 years ago
prpl|4 years ago
bcaa7f3a8bbc|4 years ago
[0] https://descanso.jpl.nasa.gov/monograph/series10/Reid_DESCAN...
jhallenworld|4 years ago
https://descanso.jpl.nasa.gov/DPSummary/Descanso4--Voyager_e...
https://ipnpr.jpl.nasa.gov/progress_report/42-175/175E.pdf
exmadscientist|4 years ago
Fortunately for the rest of us, that meant they had to make some measurements, which they were kind enough to share: https://dcc.ligo.org/LIGO-T0900200/public
madengr|4 years ago
Also, the noise from an antenna is dependent on it’s efficiency and what it’s pointing at. Even if it’s input impedance is 50 Ohms, it can generate far less noise than the equivalent resistor.
cryptonector|4 years ago
Worth the read just for that punch line.
pfdietz|4 years ago
"Ludwig Boltzmann, who spent much of his life studying statistical mechanics, died in 1906, by his own hand. Paul Ehrenfest, carrying on the work, died similarly in 1933. Now it is our turn to study statistical mechanics."
mgleason_3|4 years ago
lisper|4 years ago
fuzzfactor|4 years ago
Depending on the voltage across a resistor like that, you may calculate less than one electron passing through the resisitor per second.
Without ceramic or teflon standoffs, the circuit board can often conduct better than the resistor, plus dust can also accumulate on the outside of the resistor and conduct better eventually, which is why they are often encased in glass, so they can be effectively cleaned during a maintenance cycle.
analog31|4 years ago
https://patents.google.com/patent/US4744105A
carapace|4 years ago
The trick is to twist it into a Möbius strip!
http://www.rexresearch.com/davis/davis.htm
fallingfrog|4 years ago
The resistor is there to prevent the jfet from being burned out by over voltage on the gate, which is very sensitive to static electricity. But, I can easily hear the difference if I put a 10k resistor there instead. It’s really important to get that first gain stage really, really quiet, a discrete jfet has a better noise floor than an op amp or a regular transistor.
klodolph|4 years ago
This used to be true, but you can get really good low-noise op amps these days.
Unklejoe|4 years ago
He mentioned something about how resistor noise can actually track with the low frequency portions of the audio signal due to the resistor literally heating up and cooling down as the current through it varies. I thought that was interesting. I knew that noise was proportional to heat, but I didn't realize the temperature could vary that quickly, but I guess it makes sense when you're dealing with tiny parts. There are probably also localized hot spots that have less thermal mass than the entire resistor as a whole.
The interview is posted in this thread: https://www.audiosciencereview.com/forum/index.php?threads/t...
cushychicken|4 years ago
Lots of great tricks on bootstrapped filters, capacitive multipliers, precision transistor noise measurements with some outboard circuits fed into a spectrum analyzer.
mattkrause|4 years ago
ISL|4 years ago
The value is 1.380649×10^{−23} J/K, exactly.
buescher|4 years ago
OnlyOneCannolo|4 years ago
https://www.analog.com/media/en/selection-guides/Equation_Pu...
ChuckMcM|4 years ago
pontifier|4 years ago
It seems very obvious now, that if you want to have a high signal to noise ratio, you should get as much signal as possible, and keep your signal voltage as high as possible as long as possible before amplification.
This fundamental resistor noise is something that I'm probably going to start seeing everywhere when I look at any analog signals, and will have to take into account when designing things.
ng55QPSK|4 years ago
xbar|4 years ago
Footkerchief|4 years ago
unknown|4 years ago
[deleted]
dekhn|4 years ago
sunjain|4 years ago
exmadscientist|4 years ago
jeffrallen|4 years ago
kazinator|4 years ago
unknown|4 years ago
[deleted]
failwhaleshark|4 years ago
My HP 32SII made terrible resistor noises. Bzzzzzzz like some sort of bad tinnitus. I could also hear it on an HP 48G when I placed my ear up to it.
Wouldn't it be possible to use SMT resistors and pot them in silastic to quiet them down?
bsmith0|4 years ago
unknown|4 years ago
[deleted]
zihotki|4 years ago
jeffbee|4 years ago
Causality1|4 years ago
FriendWithMoon|4 years ago
peter_d_sherman|4 years ago
I'll put a wager that future (or perhaps even current!) scientists are able to engineer complex waveforms such that the complex waveform effectively negates the resistance/noise/impedance -- effectively turning the resistor into a conductor -- but only for that specific complex waveform -- which very possibly would change over time...
Also, future (and perhaps current!) scientists should be able to use an electrical signal of known characteristics -- to determine exactly what the complex impedance of the resistor/resistance element/impedance element -- in a circuit is, exactly...
In other words, given one of the above things (complex waveform, complex impedance) -- derive what the other one is, from it...
DiabloD3|4 years ago
h2odragon|4 years ago