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

# Energy or Power?

Started by ●June 22, 2004

Reply by ●June 22, 20042004-06-22

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 > > TomPower 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

Reply by ●June 22, 20042004-06-22

"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 > > TomThe 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

Reply by ●June 22, 20042004-06-22

> 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.

Reply by ●June 22, 20042004-06-22

"Rune Allnor" <allnor@tele.ntnu.no> wrote in message news:f56893ae.0406220201.fd2df2b@posting.google.com...> "Tom" <somebody@knowherex.netgx> wrote in messagenews:<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 signalis> > random yet its variance is also the average power. Can a random signalhave> > 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. > > RuneI 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

Reply by ●June 23, 20042004-06-23

"Tom" <somebody@knowherex.netgx> wrote in message news:<1087953908.807055@radsrv1.tranzpeer.net>...> "Rune Allnor" <allnor@tele.ntnu.no> wrote in message > news:f56893ae.0406220201.fd2df2b@posting.google.com... > > "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 have seen many papers on voice activity detectors that use the energy of > speech as a measure. I assume this should really be power? > > TomCould 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

Reply by ●June 25, 20042004-06-25

This is a multi-part message in MIME format. --------------050009000503010302070600 Content-Type: text/plain; charset=us-ascii; format=flowed Content-Transfer-Encoding: 7bit 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 >>news:f56893ae.0406220201.fd2df2b@posting.google.com... >> >> >>>"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 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 --------------050009000503010302070600 Content-Type: text/html; charset=us-ascii Content-Transfer-Encoding: 7bit <!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"> <html> <head> <meta content="text/html;charset=ISO-8859-1" http-equiv="Content-Type"> <title></title> </head> <body bgcolor="#ffffff" text="#000000"> Rune Allnor wrote: <blockquote cite="midf56893ae.0406222202.3ebda828@posting.google.com" type="cite"> <pre wrap="">"Tom" <a class="moz-txt-link-rfc2396E" href="mailto:somebody@knowherex.netgx"><somebody@knowherex.netgx></a> wrote in message news:<a class="moz-txt-link-rfc2396E" href="mailto:1087953908.807055@radsrv1.tranzpeer.net"><1087953908.807055@radsrv1.tranzpeer.net></a>... </pre> <blockquote type="cite"> <pre wrap="">"Rune Allnor" <a class="moz-txt-link-rfc2396E" href="mailto:allnor@tele.ntnu.no"><allnor@tele.ntnu.no></a> wrote in message <a class="moz-txt-link-freetext" href="news:f56893ae.0406220201.fd2df2b@posting.google.com">news:f56893ae.0406220201.fd2df2b@posting.google.com</a>... </pre> <blockquote type="cite"> <pre wrap="">"Tom" <a class="moz-txt-link-rfc2396E" href="mailto:somebody@knowherex.netgx"><somebody@knowherex.netgx></a> wrote in message </pre> </blockquote> <pre wrap=""> news:<a class="moz-txt-link-rfc2396E" href="mailto:1087878297.439424@radsrv1.tranzpeer.net"><1087878297.439424@radsrv1.tranzpeer.net></a>... </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 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> </html> --------------050009000503010302070600--