# Energy or Power?

Started by June 22, 2004
```I read in the books that a periodic signal can have power but a random
signal has energy. I am a bit confused by this as a white noise signal is
random yet its variance is also the average power. Can a random signal have
power or am I confused...By power here I mean instantaneous power V.I
and not its integral which defines average power.

thanks

Tom

```
```On Tue, 22 Jun 2004 16:24:58 +1200, Tom wrote:

> I read in the books that a periodic signal can have power but a random
> signal has energy. I am a bit confused by this as a white noise signal is
> random yet its variance is also the average power. Can a random signal have
> power or am I confused...By power here I mean instantaneous power V.I
> and not its integral which defines average power.
>
> thanks
>
> Tom

Power is the first derivative of work respect the time: P(t) = dW(t)/dt
but it does mean that W(t) must be continuous and it's first derivative
too respect to time. In a random wave this is not the case due to its
randomness, in fact energy is proportional to the squared signal module.
So it is pretty impossible to define power in this case. In fact you
was talking about "average" power: Delta(P) = Delta(W) / Delta(t) this is
correct unless you want to describe the evolution of power in time, thus
P(t) = lim(delta(t) -> 0) (W(t+delta(t)) - W(t)) / delta(t) = P(t) that
is power.

Angelo
```
```"Tom" <somebody@knowherex.netgx> wrote in message news:<1087878297.439424@radsrv1.tranzpeer.net>...
> I read in the books that a periodic signal can have power but a random
> signal has energy. I am a bit confused by this as a white noise signal is
> random yet its variance is also the average power. Can a random signal have
> power or am I confused...By power here I mean instantaneous power V.I
> and not its integral which defines average power.
>
> thanks
>
> Tom

The formal definitions are:

inf
Energy:    E = integral |x(t)|^2 dt                       [1]
-inf

1      T2
Power:     P = ----- integral  |x(t)|^2 dt                [2]
T2-T1    T1

The formal requirements for the Fourier analysis to work, is that the
signals are "absolutely integrable".

The problem is, some signals, like sin(t) and cos(t), have infinite
energy if integrated over infinite domains according to [1] above.
However, if integrated over a period, according to [2] above, they
remain finite.

Stationary random signals can, if integrated over infinite time,
yield infinite energy according to [1] above. So they must be
integrated according to [2] to get meaningful results. So speaking
of "noise energy" appears to be a somewhat dodgy terminology.

Now, given an observation window of length T, the noise observed
has an energy E_N given by

E_N = T*P_N                                               [3]

where P_N is the noise power. I guess one way of speaking of
noise energy is to regard [3] as some sort of "per sample mess-up",
when T is the signal sampling period.

Rune
```
```> I read in the books that a periodic signal can have power but a random
> signal has energy. I am a bit confused by this as a white noise signal is
> random yet its variance is also the average power. Can a random signal have
> power or am I confused...By power here I mean instantaneous power V.I
> and not its integral which defines average power.

Truly-white noise contains the same power at all frequencies up to
infinity... i.e. it has a uniform power spectral density. The total
power in the  noise is therefore infinite (the area under the psd
curve). Actual systems impose bandwidth restrictions which ensure
non-infinite power. The reason a sinusoid is finite power is because
its psd is a spike at the relevant frequency, and the area under this
spike is proportional to the square of the amplitude of the sinusoid.
```
```"Rune Allnor" <allnor@tele.ntnu.no> wrote in message
> "Tom" <somebody@knowherex.netgx> wrote in message
> > I read in the books that a periodic signal can have power but a random
> > signal has energy. I am a bit confused by this as a white noise signal
is
> > random yet its variance is also the average power. Can a random signal
have
> > power or am I confused...By power here I mean instantaneous power V.I
> > and not its integral which defines average power.
> >
> > thanks
> >
> > Tom
>
> The formal definitions are:
>
>                      inf
>    Energy:    E = integral |x(t)|^2 dt                       [1]
>                     -inf
>
>                     1      T2
>    Power:     P = ----- integral  |x(t)|^2 dt                [2]
>                   T2-T1    T1
>
> The formal requirements for the Fourier analysis to work, is that the
> signals are "absolutely integrable".
>
> The problem is, some signals, like sin(t) and cos(t), have infinite
> energy if integrated over infinite domains according to [1] above.
> However, if integrated over a period, according to [2] above, they
> remain finite.
>
> Stationary random signals can, if integrated over infinite time,
> yield infinite energy according to [1] above. So they must be
> integrated according to [2] to get meaningful results. So speaking
> of "noise energy" appears to be a somewhat dodgy terminology.
>
> Now, given an observation window of length T, the noise observed
> has an energy E_N given by
>
>    E_N = T*P_N                                               [3]
>
> where P_N is the noise power. I guess one way of speaking of
> noise energy is to regard [3] as some sort of "per sample mess-up",
> when T is the signal sampling period.
>
> Rune

I have seen many papers on voice activity detectors that use the energy of
speech as a measure. I assume this should really be power?

Tom

```
```"Tom" <somebody@knowherex.netgx> wrote in message news:<1087953908.807055@radsrv1.tranzpeer.net>...
> "Rune Allnor" <allnor@tele.ntnu.no> wrote in message
> > "Tom" <somebody@knowherex.netgx> wrote in message
> > > I read in the books that a periodic signal can have power but a random
> > > signal has energy. I am a bit confused by this as a white noise signal
>  is
> > > random yet its variance is also the average power. Can a random signal
>  have
> > > power or am I confused...By power here I mean instantaneous power V.I
> > > and not its integral which defines average power.
> > >
> > > thanks
> > >
> > > Tom
> >
> > The formal definitions are:
> >
> >                      inf
> >    Energy:    E = integral |x(t)|^2 dt                       [1]
> >                     -inf
> >
> >                     1      T2
> >    Power:     P = ----- integral  |x(t)|^2 dt                [2]
> >                   T2-T1    T1
> >
> > The formal requirements for the Fourier analysis to work, is that the
> > signals are "absolutely integrable".
> >
> > The problem is, some signals, like sin(t) and cos(t), have infinite
> > energy if integrated over infinite domains according to [1] above.
> > However, if integrated over a period, according to [2] above, they
> > remain finite.
> >
> > Stationary random signals can, if integrated over infinite time,
> > yield infinite energy according to [1] above. So they must be
> > integrated according to [2] to get meaningful results. So speaking
> > of "noise energy" appears to be a somewhat dodgy terminology.
> >
> > Now, given an observation window of length T, the noise observed
> > has an energy E_N given by
> >
> >    E_N = T*P_N                                               [3]
> >
> > where P_N is the noise power. I guess one way of speaking of
> > noise energy is to regard [3] as some sort of "per sample mess-up",
> > when T is the signal sampling period.
> >
> > Rune
>
> I have seen many papers on voice activity detectors that use the energy of
> speech as a measure. I assume this should really be power?
>
> Tom

Could be, formally speaking. However, speech processing (what little
I know of it, anyway) is based on segmenting the speech signal into
framelengths and do the processing on a frame-by-frame basis. In
that case E=T*P where T is the frame length, and the two terms are
for all practical purposes interchangable.

Rune
```
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Rune Allnor wrote:

>"Tom" <somebody@knowherex.netgx> wrote in message news:<1087953908.807055@radsrv1.tranzpeer.net>...
>
>
>>"Rune Allnor" <allnor@tele.ntnu.no> wrote in message
>>
>>
>>>"Tom" <somebody@knowherex.netgx> wrote in message
>>>
>>>
>>
>>
>>>>I read in the books that a periodic signal can have power but a random
>>>>signal has energy. I am a bit confused by this as a white noise signal
>>>>
>>>>
>> is
>>
>>
>>>>random yet its variance is also the average power. Can a random signal
>>>>
>>>>
>> have
>>
>>
>>>>power or am I confused...By power here I mean instantaneous power V.I
>>>>and not its integral which defines average power.
>>>>
>>>>thanks
>>>>
>>>>Tom
>>>>
>>>>
>>>The formal definitions are:
>>>
>>>                     inf
>>>   Energy:    E = integral |x(t)|^2 dt                       [1]
>>>                    -inf
>>>
>>>                    1      T2
>>>   Power:     P = ----- integral  |x(t)|^2 dt                [2]
>>>                  T2-T1    T1
>>>
>>>The formal requirements for the Fourier analysis to work, is that the
>>>signals are "absolutely integrable".
>>>
>>>The problem is, some signals, like sin(t) and cos(t), have infinite
>>>energy if integrated over infinite domains according to [1] above.
>>>However, if integrated over a period, according to [2] above, they
>>>remain finite.
>>>
>>>Stationary random signals can, if integrated over infinite time,
>>>yield infinite energy according to [1] above. So they must be
>>>integrated according to [2] to get meaningful results. So speaking
>>>of "noise energy" appears to be a somewhat dodgy terminology.
>>>
>>>Now, given an observation window of length T, the noise observed
>>>has an energy E_N given by
>>>
>>>   E_N = T*P_N                                               [3]
>>>
>>>where P_N is the noise power. I guess one way of speaking of
>>>noise energy is to regard [3] as some sort of "per sample mess-up",
>>>when T is the signal sampling period.
>>>
>>>Rune
>>>
>>>
>>I have seen many papers on voice activity detectors that use the energy of
>>speech as a measure. I assume this should really be power?
>>
>>Tom
>>
>>
>
>Could be, formally speaking. However, speech processing (what little
>I know of it, anyway) is based on segmenting the speech signal into
>framelengths and do the processing on a frame-by-frame basis. In
>that case E=T*P where T is the frame length, and the two terms are
>for all practical purposes interchangable.
>
>Rune
>
>
Yep. Most speech processing (compression, voice recognition, speaker
recognition, synthesis, etc.) segments the audio into blocks. These are
generally fixed length blocks of  the order of 20 to 30ms duration. The
voice energy per block is what is generally used for things like voice
activity detection, so it really is based on energy, and not power. Of
course, the two have a rather intimate relationship. :-)

Regards,
Steve

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Rune Allnor wrote:
type="cite">
</pre>
<blockquote type="cite">
<pre wrap="">"Rune Allnor" <a class="moz-txt-link-rfc2396E" href="mailto:allnor@tele.ntnu.no">&lt;allnor@tele.ntnu.no&gt;</a> wrote in message
</pre>
<blockquote type="cite">
<pre wrap="">"Tom" <a class="moz-txt-link-rfc2396E" href="mailto:somebody@knowherex.netgx">&lt;somebody@knowherex.netgx&gt;</a> wrote in message
</pre>
</blockquote>
</pre>
<blockquote type="cite">
<blockquote type="cite">
<pre wrap="">I read in the books that a periodic signal can have power but a random
signal has energy. I am a bit confused by this as a white noise signal
</pre>
</blockquote>
</blockquote>
<pre wrap=""> is
</pre>
<blockquote type="cite">
<blockquote type="cite">
<pre wrap="">random yet its variance is also the average power. Can a random signal
</pre>
</blockquote>
</blockquote>
<pre wrap=""> have
</pre>
<blockquote type="cite">
<blockquote type="cite">
<pre wrap="">power or am I confused...By power here I mean instantaneous power V.I
and not its integral which defines average power.

thanks

Tom
</pre>
</blockquote>
<pre wrap="">The formal definitions are:

inf
Energy:    E = integral |x(t)|^2 dt                       [1]
-inf

1      T2
Power:     P = ----- integral  |x(t)|^2 dt                [2]
T2-T1    T1

The formal requirements for the Fourier analysis to work, is that the
signals are "absolutely integrable".

The problem is, some signals, like sin(t) and cos(t), have infinite
energy if integrated over infinite domains according to [1] above.
However, if integrated over a period, according to [2] above, they
remain finite.

Stationary random signals can, if integrated over infinite time,
yield infinite energy according to [1] above. So they must be
integrated according to [2] to get meaningful results. So speaking
of "noise energy" appears to be a somewhat dodgy terminology.

Now, given an observation window of length T, the noise observed
has an energy E_N given by

E_N = T*P_N                                               [3]

where P_N is the noise power. I guess one way of speaking of
noise energy is to regard [3] as some sort of "per sample mess-up",
when T is the signal sampling period.

Rune
</pre>
</blockquote>
<pre wrap="">I have seen many papers on voice activity detectors that use the energy of
speech as a measure. I assume this should really be power?

Tom
</pre>
</blockquote>
<pre wrap=""><!---->
Could be, formally speaking. However, speech processing (what little
I know of it, anyway) is based on segmenting the speech signal into
framelengths and do the processing on a frame-by-frame basis. In
that case E=T*P where T is the frame length, and the two terms are
for all practical purposes interchangable.

Rune
</pre>
</blockquote>
Yep. Most speech processing (compression, voice recognition, speaker
recognition, synthesis, etc.) segments the audio into blocks. These are
generally fixed length blocks of&nbsp; the order of 20 to 30ms duration. The
voice energy per block is what is generally used for things like voice
activity detection, so it really is based on energy, and not power. Of
course, the two have a rather intimate relationship. :-)<br>
<br>
Regards,<br>
Steve<br>
<br>
</body>
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