# Variance of white noise

Started by September 28, 2020
```I was reading Box and Jenkins Time series analysis and noticed that when they calculated power spectrum they had a factor 2 in the numerator - see

equation (3.1.12).

I couldn't figure out where the 2 is coming from but then I wondered if they define noise a different way in stats. Just like when we have sine waves and take an FFT the magnitude is divided by 2 when we show the two sided spectrum, is it fair to do the same with white-noise? I think them may have multiplied it by 2 so that for the full spectrum =pi to +pi it gets halved. We don't seem to do this in engineering do we?
```
```On 2020-09-27 23:38, Tom Killwhang wrote:
> I was reading Box and Jenkins Time series analysis and noticed that when they calculated power spectrum they had a factor 2 in the numerator - see
>
> equation (3.1.12).
>
> I couldn't figure out where the 2 is coming from but then I wondered if they define noise a different way in stats. Just like when we have sine waves and take an FFT the magnitude is divided by 2 when we show the two sided spectrum, is it fair to do the same with white-noise? I think them may have multiplied it by 2 so that for the full spectrum =pi to +pi it gets halved. We don't seem to do this in engineering do we?
>

The analytic signal convention is used almost universally in test
equipment and other areas.  It allows one to use exp(i omega t) instead
of sines and cosines, which saves half the algebra and therefore three
quarters of the blunders. ;)

You form the analytic signal from a real signal by adding +-i times its
Hilbert transform (depending on your sign convention), which  has the
effect of :

1. doubling the positive frequency amplitudes
2. zeroing out the negative frequency ones
3. leaving DC alone.

Normal people of course apply rules 1-3 instead of Hilbert transforming. ;)

The analytic signal convention is responsible for many of those strange
factors of 2 that show up in noise calculations, e.g. the 1-Hz shot
noise density of a current I = e N is

i_N = sqrt(2 e I) = e * sqrt(2N)

rather than e * sqrt(N)

The reason is that a 1-second boxcar has a bandwidth of 0.5 Hz on
account of the negative frequencies being chopped off, so the sqrt(N)
noise is compressed into half the bandwidth.

Cheers

Phil Hobbs

--
Dr Philip C D Hobbs
Principal Consultant
ElectroOptical Innovations LLC / Hobbs ElectroOptics
Optics, Electro-optics, Photonics, Analog Electronics
Briarcliff Manor NY 10510

http://electrooptical.net
http://hobbs-eo.com

```
```Phil Hobbs wrote:
> On 2020-09-27 23:38, Tom Killwhang wrote:
>> I was reading Box and Jenkins Time series analysis and noticed that
>> when they calculated power spectrum they had a factor 2 in the
>> numerator - see
>>
>>
>> equation (3.1.12).
>>
>> I couldn't figure out where the 2 is coming from but then I wondered
>> if they define noise a different way in stats. Just like when we have
>> sine waves and take an FFT the magnitude is divided by 2 when we show
>> the two sided spectrum, is it fair to do the same with white-noise? I
>> think them may have multiplied it by 2 so that for the full spectrum
>> =pi to +pi it gets halved. We don't seem to do this in engineering do we?
>>
>
> The analytic signal convention is used almost universally in test
> equipment and other areas.&nbsp; It allows one to use exp(i omega t) instead
> of sines and cosines, which saves half the algebra and therefore three
> quarters of the blunders. ;)
>
> You form the analytic signal from a real signal by adding +-i times its
> Hilbert transform (depending on your sign convention), which&nbsp; has the
> effect of :
>
> 1. doubling the positive frequency amplitudes
> 2. zeroing out the negative frequency ones
> 3. leaving DC alone.
>
> Normal people of course apply rules 1-3 instead of Hilbert transforming. ;)
>
> The analytic signal convention is responsible for many of those strange
> factors of 2 that show up in noise calculations, e.g. the 1-Hz shot
> noise density of a current I = e N is
>
> i_N = sqrt(2 e I) = e * sqrt(2N)
>
> rather than e * sqrt(N)
>
> The reason is that a 1-second boxcar has a bandwidth of 0.5 Hz on
> account of the negative frequencies being chopped off, so the sqrt(N)
> noise is compressed into half the bandwidth.
>
> Cheers
>
> Phil Hobbs
>

Sp why do so many people treat the Hilbert transform as if it were
equivalent to the analytic signal? You get massive DC with the usual FFT
method of constructing a Hilbert transform.

I will have to try your list, just for giggles. But anything that is
basically "cat signal | s/sin/cos/g " will not be pleasant with respect
to DC. Er, "what is cos(0)? :)

--
Les Cargill
```
```On 2020-10-09 22:01, Les Cargill wrote:
> Phil Hobbs wrote:
>> On 2020-09-27 23:38, Tom Killwhang wrote:
>>> I was reading Box and Jenkins Time series analysis and noticed that
>>> when they calculated power spectrum they had a factor 2 in the
>>> numerator - see
>>>
>>>
>>> equation (3.1.12).
>>>
>>> I couldn't figure out where the 2 is coming from but then I wondered
>>> if they define noise a different way in stats. Just like when we have
>>> sine waves and take an FFT the magnitude is divided by 2 when we show
>>> the two sided spectrum, is it fair to do the same with white-noise? I
>>> think them may have multiplied it by 2 so that for the full spectrum
>>> =pi to +pi it gets halved. We don't seem to do this in engineering do
>>> we?
>>>
>>
>> The analytic signal convention is used almost universally in test
>> equipment and other areas.&nbsp; It allows one to use exp(i omega t)
>> instead of sines and cosines, which saves half the algebra and
>> therefore three quarters of the blunders. ;)
>>
>> You form the analytic signal from a real signal by adding +-i times
>> its Hilbert transform (depending on your sign convention), which&nbsp; has
>> the effect of :
>>
>> 1. doubling the positive frequency amplitudes
>> 2. zeroing out the negative frequency ones
>> 3. leaving DC alone.
>>
>> Normal people of course apply rules 1-3 instead of Hilbert
>> transforming. ;)
>>
>> The analytic signal convention is responsible for many of those
>> strange factors of 2 that show up in noise calculations, e.g. the 1-Hz
>> shot noise density of a current I = e N is
>>
>> i_N = sqrt(2 e I) = e * sqrt(2N)
>>
>> rather than e * sqrt(N)
>>
>> The reason is that a 1-second boxcar has a bandwidth of 0.5 Hz on
>> account of the negative frequencies being chopped off, so the sqrt(N)
>> noise is compressed into half the bandwidth.
>>
>> Cheers
>>
>> Phil Hobbs
>>
>
> Sp why do so many people treat the Hilbert transform as if it were
> equivalent to the analytic signal? You get massive DC with the usual FFT
> method of constructing a Hilbert transform.
>
> I will have to try your list, just for giggles. But anything that is
> basically "cat signal | s/sin/cos/g " will not be pleasant with respect
> to DC. Er, "what is cos(0)? :)
>
> --
> Les Cargill

Well, you can't phase shift DC after all.

(BTW remember to switch back to sines and cosines before doing anything
very nonlinear such as computing the power. )

Cheers

Phil Hobbs

--
Dr Philip C D Hobbs
Principal Consultant
ElectroOptical Innovations LLC / Hobbs ElectroOptics
Optics, Electro-optics, Photonics, Analog Electronics
Briarcliff Manor NY 10510

http://electrooptical.net
http://hobbs-eo.com

```
```On 2020-10-10 13:24, Phil Hobbs wrote:
> On 2020-10-09 22:01, Les Cargill wrote:
>> Phil Hobbs wrote:
>>> On 2020-09-27 23:38, Tom Killwhang wrote:
>>>> I was reading Box and Jenkins Time series analysis and noticed that
>>>> when they calculated power spectrum they had a factor 2 in the
>>>> numerator - see
>>>>
>>>>
>>>> equation (3.1.12).
>>>>
>>>> I couldn't figure out where the 2 is coming from but then I wondered
>>>> if they define noise a different way in stats. Just like when we
>>>> have sine waves and take an FFT the magnitude is divided by 2 when
>>>> we show the two sided spectrum, is it fair to do the same with
>>>> white-noise? I think them may have multiplied it by 2 so that for
>>>> the full spectrum =pi to +pi it gets halved. We don't seem to do
>>>> this in engineering do we?
>>>>
>>>
>>> The analytic signal convention is used almost universally in test
>>> equipment and other areas.&nbsp; It allows one to use exp(i omega t)
>>> instead of sines and cosines, which saves half the algebra and
>>> therefore three quarters of the blunders. ;)
>>>
>>> You form the analytic signal from a real signal by adding +-i times
>>> its Hilbert transform (depending on your sign convention), which&nbsp; has
>>> the effect of :
>>>
>>> 1. doubling the positive frequency amplitudes
>>> 2. zeroing out the negative frequency ones
>>> 3. leaving DC alone.
>>>
>>> Normal people of course apply rules 1-3 instead of Hilbert
>>> transforming. ;)
>>>
>>> The analytic signal convention is responsible for many of those
>>> strange factors of 2 that show up in noise calculations, e.g. the
>>> 1-Hz shot noise density of a current I = e N is
>>>
>>> i_N = sqrt(2 e I) = e * sqrt(2N)
>>>
>>> rather than e * sqrt(N)
>>>
>>> The reason is that a 1-second boxcar has a bandwidth of 0.5 Hz on
>>> account of the negative frequencies being chopped off, so the sqrt(N)
>>> noise is compressed into half the bandwidth.
>>>
>>> Cheers
>>>
>>> Phil Hobbs
>>>
>>
>> Sp why do so many people treat the Hilbert transform as if it were
>> equivalent to the analytic signal? You get massive DC with the usual
>> FFT method of constructing a Hilbert transform.
>>
>> I will have to try your list, just for giggles. But anything that is
>> basically "cat signal | s/sin/cos/g " will not be pleasant with
>> respect to DC. Er, "what is cos(0)? :)
>>
>> --
>> Les Cargill
>
> Well, you can't phase shift DC after all.
>
> (BTW remember to switch back to sines and cosines before doing anything
> very nonlinear such as computing the power. )

I should add that the problem with computing wideband Hilbert transforms
is that the impulse response has an infinite spike at the origin and the
tails also contain infinite energy.  It's okay for reasonably narrowband
signals.

Cheers

Phil Hobbs

--
Dr Philip C D Hobbs
Principal Consultant
ElectroOptical Innovations LLC / Hobbs ElectroOptics
Optics, Electro-optics, Photonics, Analog Electronics
Briarcliff Manor NY 10510

http://electrooptical.net
http://hobbs-eo.com

```
```Les Cargill  <lcargil99@gmail.com> wrote:

>Sp why do so many people treat the Hilbert transform as if it were
>equivalent to the analytic signal? You get massive DC with the usual FFT
>method of constructing a Hilbert transform.

Using a FIR approximation to a Hilbert transform does not have
this problem and (given a typical error budget) is less computation /
silicon than the FFT method.