Variance of white noise

[Tom Killwhang]

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
http://www.ru.ac.bd/stat/wp-content/uploads/sites/25/2019/03/504_05_Box_Time-Series-Analysis-Forecasting-and-Control-2015.pdf

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?

[Phil Hobbs]

Post by Tom Killwhang
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
http://www.ru.ac.bd/stat/wp-content/uploads/sites/25/2019/03/504_05_Box_Time-Series-Analysis-Forecasting-and-Control-2015.pdf
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

[Les Cargill]

Post by Phil Hobbs

Post by Tom Killwhang
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
http://www.ru.ac.bd/stat/wp-content/uploads/sites/25/2019/03/504_05_Box_Time-Series-Analysis-Forecasting-and-Control-2015.pdf
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
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

[Phil Hobbs]

Post by Les Cargill

Post by Phil Hobbs

Post by Tom Killwhang
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
http://www.ru.ac.bd/stat/wp-content/uploads/sites/25/2019/03/504_05_Box_Time-Series-Analysis-Forecasting-and-Control-2015.pdf
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
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

[Phil Hobbs]

Post by Phil Hobbs

Post by Les Cargill

Post by Phil Hobbs

Post by Tom Killwhang
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
http://www.ru.ac.bd/stat/wp-content/uploads/sites/25/2019/03/504_05_Box_Time-Series-Analysis-Forecasting-and-Control-2015.pdf
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
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

[Steve Pope]

Post by Les Cargill
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.

(Unless you have unused FFT capacity laying around for free, which
is sometimes the case.)

Steve