According to Fourier, any complex sound wave can be reconstructed by the addition of sine waves. Based on that, my simple question is this. If it were possible to isolate a single frequency component of a white noise source, would the amplitude of that single frequency component be constant or fluctuating?

From my reading on the properties of white noise, I believe that it is said that statistically the amplitude of every frequency- component making up the white noise is the same, but no prediction as to the instantaneous amplitude of the total white noise can be made. Is this correct? I suppose the indeterminate instantaneous amplitude of the white noise could be attributed to changing amplitude or phase of the components, or both.

Related to this is the following. If I had two independently running white noise sources, would relative amplitude at a particular frequency of a component of the two sources be the same or constantly changing?

Knowing this would be helpful for some testing that I'm doing.

Regards,
Pete

 

Since white noise is basically an infinite series of sinewaves at an infinite number of frequencies, it would thus seem that the amplitude of a single frequency would approach a limit of zero.
If you look at a small bandwidth of frequencies, then you will get a finite voltage.

Why do you want to know that for the secret testing you are doing?

 

Since white noise is basically an infinite series of sinewaves at an infinite number of frequencies, it would thus seem that the amplitude of a single frequency would approach a limit of zero.
If you look at a small bandwidth of frequencies, then you will get a finite voltage.

Why do you want to know that for the secret testing you are doing?

Yes what you are saying I think agrees with what I've read on the web concerning white noise, although I'm not following all that I've read.

Basically I've developed a processing circuit that is supposed to attenuate uncorrelated signals in stereo (recorded music) line level left and right channels. Where there is an identical signal in the two channels, then the processor passes that identical signal with little to no attenuation.

My problem is finding two uncorrelated test signals to put into the two channels that will mimic a recording of musical instruments each in only one or the other channel. What I have been using is two independently operating pink noise generators with each separately outputting into one of the two inputs of the processor. The pink noise generators are low-pass filtering of two separate white noise generators. I'm uncertain as to whether or not this is a correct setup to measure the attenuation that would occur with a stereo music recording. Hence my question about what the make up of white noise is.

Relative phase besides relative amplitude of uncorrelated signals in the two channels greatly affects how much processing attenuation occurs and this is problematic in that it is quite possible that a signal of a certain frequency in one channel might not have a signal of the same frequency in the other channel. This condition makes it impossible to speak of relative phase at all. Another question that I think needs to be answered is which is more likely, the frequency content of the two channels the same, or the frequency content mostly different.

Well, there it is. I hope that this isn't too confusing.

-Pete

 

According to Fourier, any complex sound wave can be reconstructed by the addition of sine waves. Based on that, my simple question is this. If it were possible to isolate a single frequency component of a white noise source, would the amplitude of that single frequency component be constant or fluctuating?

From my reading on the properties of white noise, I believe that it is said that statistically the amplitude of every frequency- component making up the white noise is the same, but no prediction as to the instantaneous amplitude of the total white noise can be made. Is this correct? I suppose the indeterminate instantaneous amplitude of the white noise could be attributed to changing amplitude or phase of the components, or both.

Related to this is the following. If I had two independently running white noise sources, would relative amplitude at a particular frequency of a component of the two sources be the same or constantly changing?

Knowing this would be helpful for some testing that I'm doing.

Regards,
Pete

One thing that is probably tripping up your thinking is the shallow view of your initial claim (which is an easy trap to fall into).

Imagine that I have a signal that is, nominally, a pure sine wave that is turned on and off according to some input. Think of a Morse Code transmission, for instance.

In the time domain, we can visualize this very easily which is that, at any moment in time, the signal is either zero, or the instantaneous value of a sine wave.

But, in the Fourier domain, all notion of time disappears. We have a set (or, in the general case, a continuous spectrum) of frequencies at different amplitudes and phases. Each one of them is a fixed constant from the beginning of time, to the end of the universe. Specifically, this applies to whatever frequency we are turning on and off with our transmitter. In the time domain in which we live, sometimes that frequency is present, and other times it's not. But in the frequency domain, it has constant amplitude and phase that never changes. What is happening is that the collection of all of the frequencies, each at their constant amplitude and phase, combine to either perfectly cancel out during intervals when our transmitter is off, or to perfectly leave just the transmitter's sine wave, at its amplitude and phase, when the transmitter is on. Note that the amplitude and phase of the transmitter's output is going to have no fixed relationship to the amplitude or phase of the Fourier component of the total signal at that frequency. Change the message that is transmitted, without touching the transmitter's sine wave generator, and you change the amplitude and phase of the Fourier component, because it is whatever it needs to be in order to interact with all of the other components in order to construct that particular time-domain waveform.

Moving out of the pristine world of Fourier in which our frequency spectra represents the components of the time-domain signal all at once and for all time, we cannot build such devices. What we can do is build devices that take a time-domain signal over some period of time and give us a representation of how we could approximate the frequency spectra that would produce an approximation of that portion of the time-domain signal (and for which we don't care about what it would claim the time-domain signal looks like outside of that time window). We do this at each time window of interest.

 

Since white noise is basically an infinite series of sinewaves at an infinite number of frequencies, it would thus seem that the amplitude of a single frequency would approach a limit of zero.
If you look at a small bandwidth of frequencies, then you will get a finite voltage.

Why do you want to know that for the secret testing you are doing?

I have two independently running pink noise generators producing two voltage outputs that are uncorrelated. The RMS voltage of the output of the two sources measures the same. Considering the same small bandwidth of frequencies of both sources, is the relative voltage of that bandwidth of frequencies of the two sources the same or fluctuating?

 

I have two independently running pink noise generators producing two voltage outputs that are uncorrelated. The RMS voltage of the output of the two sources measures the same. Considering the same small bandwidth of frequencies of both sources, is the relative voltage of that bandwidth of frequencies of the two sources the same or fluctuating?

If you measure than separately with a small band-width, then they will vary.
The larger the band-width, the less the variation.

 

If you measure than separately with a small band-width, then they will vary.
The larger the band-width, the less the variation.

Not sure what to conclude from that, but thanks.

 

Not sure what to conclude from that, but thanks.

What is not clear about what I said?

 

What is not clear about what I said?

Nothing is unclear about what you said. What is unclear to me is how I might apply what you said to the testing that I want to do. This is lack of understanding on my part.

 

One thing that is probably tripping up your thinking is the shallow view of your initial claim (which is an easy trap to fall into).

Imagine that I have a signal that is, nominally, a pure sine wave that is turned on and off according to some input. Think of a Morse Code transmission, for instance.

In the time domain, we can visualize this very easily which is that, at any moment in time, the signal is either zero, or the instantaneous value of a sine wave.

But, in the Fourier domain, all notion of time disappears. We have a set (or, in the general case, a continuous spectrum) of frequencies at different amplitudes and phases. Each one of them is a fixed constant from the beginning of time, to the end of the universe. Specifically, this applies to whatever frequency we are turning on and off with our transmitter. In the time domain in which we live, sometimes that frequency is present, and other times it's not. But in the frequency domain, it has constant amplitude and phase that never changes. What is happening is that the collection of all of the frequencies, each at their constant amplitude and phase, combine to either perfectly cancel out during intervals when our transmitter is off, or to perfectly leave just the transmitter's sine wave, at its amplitude and phase, when the transmitter is on. Note that the amplitude and phase of the transmitter's output is going to have no fixed relationship to the amplitude or phase of the Fourier component of the total signal at that frequency. Change the message that is transmitted, without touching the transmitter's sine wave generator, and you change the amplitude and phase of the Fourier component, because it is whatever it needs to be in order to interact with all of the other components in order to construct that particular time-domain waveform.

Moving out of the pristine world of Fourier in which our frequency spectra represents the components of the time-domain signal all at once and for all time, we cannot build such devices. What we can do is build devices that take a time-domain signal over some period of time and give us a representation of how we could approximate the frequency spectra that would produce an approximation of that portion of the time-domain signal (and for which we don't care about what it would claim the time-domain signal looks like outside of that time window). We do this at each time window of interest.

John Woodgate, in the article that I'm giving the link to below, makes the following statement:

"Phase is meaningful only for sinusoidal signals, but since any signal can be represented as a set of sinusoidal signals, this is not a big problem".

However I think that I do remember reading in an article on the web stating that the Fourier Transform of white noise is white noise.

https://www.allaboutcircuits.com/projects/designing-quadrature-networks-using-all-pass-filters/

 

I have two independently running pink noise generators producing two voltage outputs that are uncorrelated. The RMS voltage of the output of the two sources measures the same. Considering the same small bandwidth of frequencies of both sources, is the relative voltage of that bandwidth of frequencies of the two sources the same or fluctuating?

You are confuscating amplitude of a sine wave with power in noise.
By definition, white noise is random. Hence it is problematic to define the amplitude of the noise signal.
You can measure power in noise and the amplitude can be expressed in RMS voltage which by definition is an average over time.
By definition, the RMS voltage of a signal is the voltage of a DC signal that has the same power as the signal being measured.

True RMS value has to be measured over infinite time in order to measure all frequencies. By the same token, you need infinite bandwidth. If you narrow the bandwidth, you reduce the sample size and hence the RMS variability increases. If you take this to a limit, a very narrow bandwidth will exhibit a very random RMS voltage.

Here is something to think about.
White noise is a fractal. If you were to zoom in or out at a white noise signal, it would always look the same at any magnification.

 

Nothing is unclear about what you said. What is unclear to me is how I might apply what you said to the testing that I want to do. This is lack of understanding on my part.

How about applying a different (uncorrelated) noise generator to each channel while also summing in a monophonic tone or music signal equally to both channels.
I would then expect that your circuit should largely suppress the noise, while passing through the monophonic signal.
That should tell you how well it suppresses the non-common signal versus the common signal you want.