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[mathcad] Modeling White Noise

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[mathcad] Modeling White Noise
Author: Bill E Dumke    Posted: Wed, 2 Apr 2008 10:14:28 -0500
I need a model for bandwidth limited white noise. I have Mathcad 14 but would prefer to use Mathcad 11 for this. Thanks, Bill Bill Dumke, RF Engineer Nsight Telservices dba Cellcom 450 Security Blvd Green Bay, WI 54313 Office: 920 617-7311 e-mail: "bill.dumke"

RE: [mathcad] Modeling White Noise
Author: Neil Harris    Posted: Thu, 3 Apr 2008 10:41:34 +0100
Bill, The simplest way is to generate a pair of vectors of length 2^N/2 + 1; one has the required amplitude response, and the other is a random phase generated by using the function phi := 2*pi*rnd(1). These are combined into a complex vector amplitude * exp(1j x phi) The required noise is the simply the inverse FFT of this complex vector. Noise = IFF(complex vector) Apply appropriate scaling to get the right amplitude. Regards, Neil From: Bill E Dumke "mailto:Bill.Dumke" Sent: 02 April 2008 16:14 To: "mathcad" Subject: [mathcad] Modeling White Noise I need a model for bandwidth limited white noise. I have Mathcad 14 but would prefer to use Mathcad 11 for this. Thanks, Bill Bill Dumke, RF Engineer Nsight Telservices dba Cellcom 450 Security Blvd Green Bay, WI 54313 Office: 920 617-7311 e-mail: "bill.dumke"

Re: [mathcad] Modeling White Noise
Author: Bill Dumke    Posted: Thu, 03 Apr 2008 06:49:19 -0500
Thanks Neil, But doesn't the amplitude vary randomly in amplitude as well? Bill Neil Harris wrote: > Bill, > > The simplest way is to generate a pair of vectors of length 2^N/2 + 1; > one has the required amplitude response, and the other is a random phase > generated by using the function phi := 2*pi*rnd(1). > > These are combined into a complex vector amplitude * exp(1j x phi) > > The required noise is the simply the inverse FFT of this complex vector. > > Noise = IFF(complex vector) > > Apply appropriate scaling to get the right amplitude. > > Regards, > Neil > > > From: Bill E Dumke "mailto:Bill.Dumke" > Sent: 02 April 2008 16:14 > To: "mathcad" > Subject: [mathcad] Modeling White Noise > > I need a model for bandwidth limited white noise. I have Mathcad 14 but > would prefer to use Mathcad 11 for this. > > Thanks, Bill > > Bill Dumke, RF Engineer > Nsight Telservices dba Cellcom > 450 Security Blvd > Green Bay, WI 54313 > > Office: 920 617-7311 > e-mail: "bill.dumke" > > Contributions: "mathcad" > Hosted by: Adept Scientific http://www.adeptscience.com > List Archive: http://lists.adeptscience.co.uk/ > Would you like this to come to a different email address? > Simply leave the mailing list (see below) and re-join by > sending a blank email from the new address to: > "mailto:join-mathcad" or > You are currently subscribed to mathcad as: "n.harris" > To unsubscribe send a blank email to > "leave-52344-494237H" > > Contributions: "mathcad" > Hosted by: Adept Scientific http://www.adeptscience.com > List Archive: http://lists.adeptscience.co.uk/ > Would you like this to come to a different email address? > Simply leave the mailing list (see below) and re-join by > sending a blank email from the new address to: > "mailto:join-mathcad" or > You are currently subscribed to mathcad as: "billd" > To unsubscribe send a blank email to "leave-52400-494029G" > > > > >

RE: [mathcad] Modeling White Noise
Author: Neil Harris    Posted: Thu, 3 Apr 2008 13:36:57 +0100
Bill, The definition of white noise is that it has a (long-term) power spectrum of unity - i.e. the same as an impulse. It is only the phase response that differs between the two. In the time domain, the noise signal is "random" due to its random phase. Neil From: Bill Dumke "mailto:billd" Sent: 03 April 2008 12:49 To: "mathcad" Subject: Re: [mathcad] Modeling White Noise Thanks Neil, But doesn't the amplitude vary randomly in amplitude as well? Bill Neil Harris wrote: > Bill, > > The simplest way is to generate a pair of vectors of length 2^N/2 + 1; > one has the required amplitude response, and the other is a random phase > generated by using the function phi := 2*pi*rnd(1). > > These are combined into a complex vector amplitude * exp(1j x phi) > > The required noise is the simply the inverse FFT of this complex vector. > > Noise = IFF(complex vector) > > Apply appropriate scaling to get the right amplitude. > > Regards, > Neil > > > From: Bill E Dumke "mailto:Bill.Dumke" > Sent: 02 April 2008 16:14 > To: "mathcad" > Subject: [mathcad] Modeling White Noise > > I need a model for bandwidth limited white noise. I have Mathcad 14 but > would prefer to use Mathcad 11 for this. > > Thanks, Bill > > Bill Dumke, RF Engineer > Nsight Telservices dba Cellcom > 450 Security Blvd > Green Bay, WI 54313 > > Office: 920 617-7311 > e-mail: "bill.dumke" > > Contributions: "mathcad" > Hosted by: Adept Scientific http://www.adeptscience.com > List Archive: http://lists.adeptscience.co.uk/ > Would you like this to come to a different email address? > Simply leave the mailing list (see below) and re-join by > sending a blank email from the new address to: > "mailto:join-mathcad" or > You are currently subscribed to mathcad as: "n.harris" > To unsubscribe send a blank email to > "leave-52344-494237H" > > Contributions: "mathcad" > Hosted by: Adept Scientific http://www.adeptscience.com > List Archive: http://lists.adeptscience.co.uk/ > Would you like this to come to a different email address? > Simply leave the mailing list (see below) and re-join by > sending a blank email from the new address to: > "mailto:join-mathcad" or > You are currently subscribed to mathcad as: "billd" > To unsubscribe send a blank email to "leave-52400-494029G" > > > > >

RE: [mathcad] Modeling White Noise
Author: Bill E Dumke    Posted: Thu, 3 Apr 2008 08:37:24 -0500
Neil, This is very confusing to me. I was searching around for info on white noise and discovered this paper: http://www.home.agilent.com/upload/cmc_upload/All/Exp65.pdf The paper confirms what you have said. Only the phase is random. I am in a debate with a PhD friend of mine, and he claims "Noise is a random process but in conductors (including resistors) it is an one-dimensional random process". I created a Mathcad document to match the procedure given in the Agilent document. And true enough, the vector N is one dimensional. (Since the white noise vector contains only real numbers, it would seem to me that adding white noise to a carrier would only affect the amplitude of the carrier, not the phase.) Because of this he claims that since white noise is orthogonal to phase noise, white noise will not affect the signal to noise performance of an FM receiver. (But that is not what I know about FM receivers. FM is limited by noise figure just as AM and SSB receivers are limited by noise figure, the only difference being that FM gives better signal to noise ratio at the output of the FM demodulator for high input signal to noise ratios. And I seem to remember that it gives worse performance below the limiting threshold.) So I am confused. Also I wonder why the "bandlimited White" noise shown at the bottom of the page doesn't show any amplitude variation across the spectrum. Every time I have looked at the noise coming from a filter with a spectrum analyzer or power meter it varies in amplitude as well. Trying to understand this, Bill Bill Dumke, RF Engineer Nsight Telservices dba Cellcom 450 Security Blvd Green Bay, WI 54313 Office: 920 617-7311 e-mail: "bill.dumke" From: Neil Harris "mailto:N.Harris" Sent: Thursday, April 03, 2008 7:37 AM To: "mathcad" Subject: RE: [mathcad] Modeling White Noise Bill, The definition of white noise is that it has a (long-term) power spectrum of unity - i.e. the same as an impulse. It is only the phase response that differs between the two. In the time domain, the noise signal is "random" due to its random phase. Neil From: Bill Dumke "mailto:billd" Sent: 03 April 2008 12:49 To: "mathcad" Subject: Re: [mathcad] Modeling White Noise Thanks Neil, But doesn't the amplitude vary randomly in amplitude as well? Bill Neil Harris wrote: > Bill, > > The simplest way is to generate a pair of vectors of length 2^N/2 + 1; > one has the required amplitude response, and the other is a random phase > generated by using the function phi := 2*pi*rnd(1). > > These are combined into a complex vector amplitude * exp(1j x phi) > > The required noise is the simply the inverse FFT of this complex vector. > > Noise = IFF(complex vector) > > Apply appropriate scaling to get the right amplitude. > > Regards, > Neil > > > From: Bill E Dumke "mailto:Bill.Dumke" > Sent: 02 April 2008 16:14 > To: "mathcad" > Subject: [mathcad] Modeling White Noise > > I need a model for bandwidth limited white noise. I have Mathcad 14 but > would prefer to use Mathcad 11 for this. > > Thanks, Bill > > Bill Dumke, RF Engineer > Nsight Telservices dba Cellcom > 450 Security Blvd > Green Bay, WI 54313 > > Office: 920 617-7311 > e-mail: "bill.dumke" > > Contributions: "mathcad" > Hosted by: Adept Scientific http://www.adeptscience.com > List Archive: http://lists.adeptscience.co.uk/ > Would you like this to come to a different email address? > Simply leave the mailing list (see below) and re-join by sending a > blank email from the new address to: > "mailto:join-mathcad" or > You are currently subscribed to mathcad as: "n.harris" > To unsubscribe send a blank email to > "leave-52344-494237H" > > Contributions: "mathcad" > Hosted by: Adept Scientific http://www.adeptscience.com > List Archive: http://lists.adeptscience.co.uk/ > Would you like this to come to a different email address? > Simply leave the mailing list (see below) and re-join by sending a > blank email from the new address to: > "mailto:join-mathcad" or > You are currently subscribed to mathcad as: "billd" > To unsubscribe send a blank email to "leave-52400-494029G" > > > > >

RE: [mathcad] Modeling White Noise
Author: Neil Harris    Posted: Thu, 3 Apr 2008 18:23:38 +0100
Bill, Firstly, one never measures for an infinite length of time (!), so there are fluctuations in "real-world" measurements. Secondly, if you use the IFFT function, the noise (time-domain) will be real-only. Finally, if you want a band-limited noise, use the required power spectrum for the square of the amplitude; for example Butterworth, nth order low pass |H(f)|^2 = fc^2n/(fc^2n + f^2n) Butterworth, nth order high pass |H(f)|^2 = f^2n/(fc^2n + f^2n) I hope this helps a bit? Neil From: Bill E Dumke "mailto:Bill.Dumke" Sent: 03 April 2008 14:37 To: "mathcad" Subject: RE: [mathcad] Modeling White Noise Neil, This is very confusing to me. I was searching around for info on white noise and discovered this paper: http://www.home.agilent.com/upload/cmc_upload/All/Exp65.pdf The paper confirms what you have said. Only the phase is random. I am in a debate with a PhD friend of mine, and he claims "Noise is a random process but in conductors (including resistors) it is an one-dimensional random process". I created a Mathcad document to match the procedure given in the Agilent document. And true enough, the vector N is one dimensional. (Since the white noise vector contains only real numbers, it would seem to me that adding white noise to a carrier would only affect the amplitude of the carrier, not the phase.) Because of this he claims that since white noise is orthogonal to phase noise, white noise will not affect the signal to noise performance of an FM receiver. (But that is not what I know about FM receivers. FM is limited by noise figure just as AM and SSB receivers are limited by noise figure, the only difference being that FM gives better signal to noise ratio at the output of the FM demodulator for high input signal to noise ratios. And I seem to remember that it gives worse performance below the limiting threshold.) So I am confused. Also I wonder why the "bandlimited White" noise shown at the bottom of the page doesn't show any amplitude variation across the spectrum. Every time I have looked at the noise coming from a filter with a spectrum analyzer or power meter it varies in amplitude as well. Trying to understand this, Bill Bill Dumke, RF Engineer Nsight Telservices dba Cellcom 450 Security Blvd Green Bay, WI 54313 Office: 920 617-7311 e-mail: "bill.dumke" From: Neil Harris "mailto:N.Harris" Sent: Thursday, April 03, 2008 7:37 AM To: "mathcad" Subject: RE: [mathcad] Modeling White Noise Bill, The definition of white noise is that it has a (long-term) power spectrum of unity - i.e. the same as an impulse. It is only the phase response that differs between the two. In the time domain, the noise signal is "random" due to its random phase. Neil From: Bill Dumke "mailto:billd" Sent: 03 April 2008 12:49 To: "mathcad" Subject: Re: [mathcad] Modeling White Noise Thanks Neil, But doesn't the amplitude vary randomly in amplitude as well? Bill Neil Harris wrote: > Bill, > > The simplest way is to generate a pair of vectors of length 2^N/2 + 1; > one has the required amplitude response, and the other is a random phase > generated by using the function phi := 2*pi*rnd(1). > > These are combined into a complex vector amplitude * exp(1j x phi) > > The required noise is the simply the inverse FFT of this complex vector. > > Noise = IFF(complex vector) > > Apply appropriate scaling to get the right amplitude. > > Regards, > Neil > > > From: Bill E Dumke "mailto:Bill.Dumke" > Sent: 02 April 2008 16:14 > To: "mathcad" > Subject: [mathcad] Modeling White Noise > > I need a model for bandwidth limited white noise. I have Mathcad 14 but > would prefer to use Mathcad 11 for this. > > Thanks, Bill > > Bill Dumke, RF Engineer > Nsight Telservices dba Cellcom > 450 Security Blvd > Green Bay, WI 54313 > > Office: 920 617-7311 > e-mail: "bill.dumke" > > Contributions: "mathcad" > Hosted by: Adept Scientific http://www.adeptscience.com > List Archive: http://lists.adeptscience.co.uk/ > Would you like this to come to a different email address? > Simply leave the mailing list (see below) and re-join by sending a > blank email from the new address to: > "mailto:join-mathcad" or > You are currently subscribed to mathcad as: "n.harris" > To unsubscribe send a blank email to > "leave-52344-494237H" > > Contributions: "mathcad" > Hosted by: Adept Scientific http://www.adeptscience.com > List Archive: http://lists.adeptscience.co.uk/ > Would you like this to come to a different email address? > Simply leave the mailing list (see below) and re-join by sending a > blank email from the new address to: > "mailto:join-mathcad" or > You are currently subscribed to mathcad as: "billd" > To unsubscribe send a blank email to "leave-52400-494029G" > > > > >

Re: [mathcad] Modeling White Noise
Author: Eden Mei    Posted: Thu, 3 Apr 2008 11:36:05 -0700
One thing you need to separately distinguish is the amplitude of the power spectral density vs. the amplitude of the real noise. The "white noise spectrum" is essentially the power spectral density function, equal amplitude over frequency. The actual noise behaves like a series of random numbers. So, adding white noise to an actual signal is indeed adding a series of random numbers. The phase is built into the random signal. It's the variability of the phase that allows the signal to have varying amplitude in time, since some frequencies add, while others subtract, precisely because of the varying phase. TTFN, Eden

Re: [mathcad] Modeling White Noise
Author: Eden Mei    Posted: Thu, 3 Apr 2008 11:38:32 -0700
Adder: The white noise is orthogonal to PM phase noise, since the random noise generally does not affect the phase of the FM signal, unless the SNR is REALLY bad. The phase of the noise has nothing to do with the phase noise of the FM signal. TTFN, Eden

RE: [mathcad] Modeling White Noise
Author: Philip Oakley    Posted: Fri, 4 Apr 2008 00:11:40 +0100
Bill, The PhD guy hasn't done enough thinking. The difference between: * white noise (who's frequency response (spectra density) is infinitely wide & infinitely low) and * an impulse response (who's frequency response (spectra density) is infinitely wide & infinitely low) is only that the impulse has all the frequencies co-align to be at zero phase at the same instant, so they are all cos(0) [so peak up to an infinite spike for zero time]. For the white noise, the phases of all those (infinite) frequencies never co-align (so they can't create that infinite peak) rather they produce a random sum. The other thing he does wrong is to miss the tricks that happens in the Fourier Transform. The transform of real data is symmetric in positive and negative frequencies, just so that we have exactly enough spare data values for those phase values. The amplitude is orthogonal to the phase. In fact it has to be. But that is the ampliude of the 'lasts forever' sinewave. When you truncate the sine wave (by only taking a limited time length) you will 'never' manage to get the start and end point to be exactly at the 2N*pi points of that sinewave (remember there are an infinite number of them). So the freqreqency gets adjusted by the FFT to be at FFT's sample points. As the others have said, if you define a nice amplitude spectrum and then use random phase, you can use the IFFT to get the random sequence of any spectral (white/pink/..) distribution you need. Philip From: Bill E Dumke "mailto:Bill.Dumke" Sent: 03 April 2008 2:37 PM To: "mathcad" Subject: RE: [mathcad] Modeling White Noise Neil, This is very confusing to me. I was searching around for info on white noise and discovered this paper: http://www.home.agilent.com/upload/cmc_upload/All/Exp65.pdf The paper confirms what you have said. Only the phase is random. I am in a debate with a PhD friend of mine, and he claims "Noise is a random process but in conductors (including resistors) it is an one-dimensional random process". I created a Mathcad document to match the procedure given in the Agilent document. And true enough, the vector N is one dimensional. (Since the white noise vector contains only real numbers, it would seem to me that adding white noise to a carrier would only affect the amplitude of the carrier, not the phase.) Because of this he claims that since white noise is orthogonal to phase noise, white noise will not affect the signal to noise performance of an FM receiver. (But that is not what I know about FM receivers. FM is limited by noise figure just as AM and SSB receivers are limited by noise figure, the only difference being that FM gives better signal to noise ratio at the output of the FM demodulator for high input signal to noise ratios. And I seem to remember that it gives worse performance below the limiting threshold.) So I am confused. Also I wonder why the "bandlimited White" noise shown at the bottom of the page doesn't show any amplitude variation across the spectrum. Every time I have looked at the noise coming from a filter with a spectrum analyzer or power meter it varies in amplitude as well. Trying to understand this, Bill Bill Dumke, RF Engineer Nsight Telservices dba Cellcom 450 Security Blvd Green Bay, WI 54313 Office: 920 617-7311 e-mail: "bill.dumke" From: Neil Harris "mailto:N.Harris" Sent: Thursday, April 03, 2008 7:37 AM To: "mathcad" Subject: RE: [mathcad] Modeling White Noise Bill, The definition of white noise is that it has a (long-term) power spectrum of unity - i.e. the same as an impulse. It is only the phase response that differs between the two. In the time domain, the noise signal is "random" due to its random phase. Neil From: Bill Dumke "mailto:billd" Sent: 03 April 2008 12:49 To: "mathcad" Subject: Re: [mathcad] Modeling White Noise Thanks Neil, But doesn't the amplitude vary randomly in amplitude as well? Bill Neil Harris wrote: > Bill, > > The simplest way is to generate a pair of vectors of length 2^N/2 + 1; > one has the required amplitude response, and the other is a random phase > generated by using the function phi := 2*pi*rnd(1). > > These are combined into a complex vector amplitude * exp(1j x phi) > > The required noise is the simply the inverse FFT of this complex vector. > > Noise = IFF(complex vector) > > Apply appropriate scaling to get the right amplitude. > > Regards, > Neil > > > From: Bill E Dumke "mailto:Bill.Dumke" > Sent: 02 April 2008 16:14 > To: "mathcad" > Subject: [mathcad] Modeling White Noise > > I need a model for bandwidth limited white noise. I have Mathcad 14 but > would prefer to use Mathcad 11 for this. > > Thanks, Bill > > Bill Dumke, RF Engineer > Nsight Telservices dba Cellcom > 450 Security Blvd > Green Bay, WI 54313 > > Office: 920 617-7311 > e-mail: "bill.dumke" > > Contributions: "mathcad" > Hosted by: Adept Scientific http://www.adeptscience.com > List Archive: http://lists.adeptscience.co.uk/ > Would you like this to come to a different email address? > Simply leave the mailing list (see below) and re-join by sending a > blank email from the new address to: > "mailto:join-mathcad" or > You are currently subscribed to mathcad as: "n.harris" > To unsubscribe send a blank email to > "leave-52344-494237H" > > Contributions: "mathcad" > Hosted by: Adept Scientific http://www.adeptscience.com > List Archive: http://lists.adeptscience.co.uk/ > Would you like this to come to a different email address? > Simply leave the mailing list (see below) and re-join by sending a > blank email from the new address to: > "mailto:join-mathcad" or > You are currently subscribed to mathcad as: "billd" > To unsubscribe send a blank email to "leave-52400-494029G" > > > > >

Re: [mathcad] Modeling White Noise
Author: Bill Dumke    Posted: Thu, 03 Apr 2008 18:33:47 -0500
Eden, In your first e-mail on this subject when you used the term random numbers you are actually speaking of complex random numbers. Is this correct? If it is, I can understand that. In the second e-mail, I think what you are trying to say is that for a good signal to noise ratio into an FM receiver (above limiting threshold) the phase noise in the white noise is insignificant. But for a poor signal to noise ratio the phase noise in the white noise is quite significant and the output signal to noise ratio from the FM demodulator is poor. Is this correct? I don't understand why you say white noise is orthogonal to an FM signal when random phase variations are what make white noise in the first place. Bill Eden Mei wrote: > Adder: > > The white noise is orthogonal to PM phase noise, since the random > noise generally does not affect the phase of the FM signal, unless the > SNR is REALLY bad. The phase of the noise has nothing to do with the > phase noise of the FM signal. > > TTFN, > Eden > > Contributions: "mathcad" > Hosted by: Adept Scientific http://www.adeptscience.com > List Archive: http://lists.adeptscience.co.uk/ > Would you like this to come to a different email address? > Simply leave the mailing list (see below) and re-join by > sending a blank email from the new address to: > "mailto:join-mathcad" or > You are currently subscribed to mathcad as: "billd" > To unsubscribe send a blank email to > "leave-52980-494029G"

Re: [mathcad] Modeling White Noise
Author: Bill Dumke    Posted: Thu, 03 Apr 2008 20:33:53 -0500
Eden, In your first e-mail on this subject when you used the term random numbers you are actually speaking of complex random numbers. Is this correct? If it is, I can understand that. In the second e-mail, I think what you are trying to say is that for a good signal to noise ratio into an FM receiver (above limiting threshold) the phase noise in the white noise is insignificant. But for a poor signal to noise ratio the phase noise in the white noise is quite significant and the output signal to noise ratio from the FM demodulator is poor. Is this correct? I don't understand why you say white noise is orthogonal to an FM signal when random phase variations are what make white noise in the first place. Bill Eden Mei wrote: > Adder: > > The white noise is orthogonal to PM phase noise, since the random > noise generally does not affect the phase of the FM signal, unless the > SNR is REALLY bad. The phase of the noise has nothing to do with the > phase noise of the FM signal. > > TTFN, > Eden > > Contributions: "mathcad" > Hosted by: Adept Scientific http://www.adeptscience.com > List Archive: http://lists.adeptscience.co.uk/ > Would you like this to come to a different email address? > Simply leave the mailing list (see below) and re-join by > sending a blank email from the new address to: > "mailto:join-mathcad" or > You are currently subscribed to mathcad as: "billd" > To unsubscribe send a blank email to > "leave-52980-494029G"

RE: [mathcad] Modeling White Noise
Author: Bill E Dumke    Posted: Fri, 4 Apr 2008 09:45:01 -0500
Phil, Thank you very much for helping out. Let me see if I understand this correctly. The random phase noise used in the model for white noise also produces amplitude noise as well because of interactions between all the frequency components with random phases. If the IFFT function is used in the model, it only produces single valued real numbers for the result. When the ICFFT function is used it produces complex numbers for the result, and the complex result is what we see in the real world. It has both phase and amplitude noise (caused by the phase noise) in it. Noise figure is the result of a decrease of signal to noise ratio between the input and output of a device, such as an amplifier or AM receiver. It is usually caused by thermal noise at the input of the device, (but can also be caused by other types of noise related to the components used in the device.) Thermal noise and white noise are equivalent. As a result of the amplitude noise in thermal noise, the noise figure of an AM receiver will degrade the signal to noise ratio of a received AM signal. As a result of the phase noise in thermal noise, the noise figure of an FM receiver with a limiter (to eliminate AM noise) will degrade the signal to noise ratio of an FM signal. (In this case, the noise figure would have to be measured before the limiter and FM demodulator.) If the signal at the input of an FM receiver, is near the thermal noise level, the output signal to noise ratio will be poor. The improvement in output signal to noise ratio seen in an FM receiver with a limiter at a high input signal level relative to the thermal moise is still limited by the phase noise in the thermal noise. And of course, measured phase noise from a local oscillator will raise the noise floor of an FM receiver, if it is strong enough. Please correct me if I am wrong. Bill Bill Dumke, RF Engineer Nsight Telservices dba Cellcom 450 Security Blvd Green Bay, WI 54313 Office: 920 617-7311 e-mail: "bill.dumke" From: Philip Oakley "mailto:philipoakley" Sent: Thursday, April 03, 2008 6:12 PM To: "mathcad" Subject: RE: [mathcad] Modeling White Noise Bill, The PhD guy hasn't done enough thinking. The difference between: * white noise (who's frequency response (spectra density) is infinitely wide & infinitely low) and * an impulse response (who's frequency response (spectra density) is infinitely wide & infinitely low) is only that the impulse has all the frequencies co-align to be at zero phase at the same instant, so they are all cos(0) [so peak up to an infinite spike for zero time]. For the white noise, the phases of all those (infinite) frequencies never co-align (so they can't create that infinite peak) rather they produce a random sum. The other thing he does wrong is to miss the tricks that happens in the Fourier Transform. The transform of real data is symmetric in positive and negative frequencies, just so that we have exactly enough spare data values for those phase values. The amplitude is orthogonal to the phase. In fact it has to be. But that is the ampliude of the 'lasts forever' sinewave. When you truncate the sine wave (by only taking a limited time length) you will 'never' manage to get the start and end point to be exactly at the 2N*pi points of that sinewave (remember there are an infinite number of them). So the freqreqency gets adjusted by the FFT to be at FFT's sample points. As the others have said, if you define a nice amplitude spectrum and then use random phase, you can use the IFFT to get the random sequence of any spectral (white/pink/..) distribution you need. Philip From: Bill E Dumke "mailto:Bill.Dumke" Sent: 03 April 2008 2:37 PM To: "mathcad" Subject: RE: [mathcad] Modeling White Noise Neil, This is very confusing to me. I was searching around for info on white noise and discovered this paper: http://www.home.agilent.com/upload/cmc_upload/All/Exp65.pdf The paper confirms what you have said. Only the phase is random. I am in a debate with a PhD friend of mine, and he claims "Noise is a random process but in conductors (including resistors) it is an one-dimensional random process". I created a Mathcad document to match the procedure given in the Agilent document. And true enough, the vector N is one dimensional. (Since the white noise vector contains only real numbers, it would seem to me that adding white noise to a carrier would only affect the amplitude of the carrier, not the phase.) Because of this he claims that since white noise is orthogonal to phase noise, white noise will not affect the signal to noise performance of an FM receiver. (But that is not what I know about FM receivers. FM is limited by noise figure just as AM and SSB receivers are limited by noise figure, the only difference being that FM gives better signal to noise ratio at the output of the FM demodulator for high input signal to noise ratios. And I seem to remember that it gives worse performance below the limiting threshold.) So I am confused. Also I wonder why the "bandlimited White" noise shown at the bottom of the page doesn't show any amplitude variation across the spectrum. Every time I have looked at the noise coming from a filter with a spectrum analyzer or power meter it varies in amplitude as well. Trying to understand this, Bill Bill Dumke, RF Engineer Nsight Telservices dba Cellcom 450 Security Blvd Green Bay, WI 54313 Office: 920 617-7311 e-mail: "bill.dumke" From: Neil Harris "mailto:N.Harris" Sent: Thursday, April 03, 2008 7:37 AM To: "mathcad" Subject: RE: [mathcad] Modeling White Noise Bill, The definition of white noise is that it has a (long-term) power spectrum of unity - i.e. the same as an impulse. It is only the phase response that differs between the two. In the time domain, the noise signal is "random" due to its random phase. Neil From: Bill Dumke "mailto:billd" Sent: 03 April 2008 12:49 To: "mathcad" Subject: Re: [mathcad] Modeling White Noise Thanks Neil, But doesn't the amplitude vary randomly in amplitude as well? Bill Neil Harris wrote: > Bill, > > The simplest way is to generate a pair of vectors of length 2^N/2 + 1; > one has the required amplitude response, and the other is a random phase > generated by using the function phi := 2*pi*rnd(1). > > These are combined into a complex vector amplitude * exp(1j x phi) > > The required noise is the simply the inverse FFT of this complex vector. > > Noise = IFF(complex vector) > > Apply appropriate scaling to get the right amplitude. > > Regards, > Neil > > > From: Bill E Dumke "mailto:Bill.Dumke" > Sent: 02 April 2008 16:14 > To: "mathcad" > Subject: [mathcad] Modeling White Noise > > I need a model for bandwidth limited white noise. I have Mathcad 14 but > would prefer to use Mathcad 11 for this. > > Thanks, Bill > > Bill Dumke, RF Engineer > Nsight Telservices dba Cellcom > 450 Security Blvd > Green Bay, WI 54313 > > Office: 920 617-7311 > e-mail: "bill.dumke" > > Contributions: "mathcad" > Hosted by: Adept Scientific http://www.adeptscience.com > List Archive: http://lists.adeptscience.co.uk/ > Would you like this to come to a different email address? > Simply leave the mailing list (see below) and re-join by sending a > blank email from the new address to: > "mailto:join-mathcad" or > You are currently subscribed to mathcad as: "n.harris" > To unsubscribe send a blank email to > "leave-52344-494237H" > > Contributions: "mathcad" > Hosted by: Adept Scientific http://www.adeptscience.com > List Archive: http://lists.adeptscience.co.uk/ > Would you like this to come to a different email address? > Simply leave the mailing list (see below) and re-join by sending a > blank email from the new address to: > "mailto:join-mathcad" or > You are currently subscribed to mathcad as: "billd" > To unsubscribe send a blank email to "leave-52400-494029G" > > > > >

Re: [mathcad] Modeling White Noise
Author: Chris Whitford    Posted: Fri, 04 Apr 2008 10:39:28 +0100
This line of argument seems to lead to the wrong conclusions. If one takes a finite set of data, sampled independently from some distribution (i.e. white noise), and takes the discrete Fourier transform, the sine and cosine terms will be independent and from the central limit theorem will have a Gaussian distribution. Converting from the components to amplitude and phase, the phase is random with a uniform distribution and the amplitude has a distribution f(v)dv = exp(-v^2/2*a^2)*v*dv/a^2, where a is the variance. As more samples are taken, the samples are more closely spaced in frequency, but have the same distribution. Smoothing the spectrum changes the distribution and in the limit, as the number of points approaches infinity, the spectrum becomes flat, with zero variance. Smoothing in the frequency domain is equivalent to multiplying by a window function in the time domain. If the smoothing function is sin(f)/f the window function is a box-car and the effect is to truncate the data set. The snag in this is that computing the amplitude is a non-linear function (v = sqrt(x^2 + y^2)) so smoothing the amplitude is not the same as smoothing the components. The amplitude and phase are not Cartesian, they form a set of curvilinear co-ordinates. The frequency components are orthogonal, amplitude and phase are only so locally. I think what one should do, is generate a set of random data with a Gaussian distribution, take the Fourier transform, apply a filter to this with the required frequency response, and transform back to the time domain. The sample rate should be high enough to avoid aliasing (that is, the filter amplitude is small at the Nyquist frequency). In FM, if the noise is small, the in-phase component modulates the carrier amplitude and this is rejected by the demodulator. The quadrature component modulates the carrier phase. As frequency is the derivative of the phase, the high frequency noise components have larger derivatives and add more FM noise (which is why pre-emphasis is used in FM, to attenuate the HF noise after demodulation). Chris At 20:33 03/04/2008 -0500, you wrote: >Eden, > >In your first e-mail on this subject when you used the term random numbers >you are actually speaking of complex random numbers. Is this correct? If >it is, I can understand that. > >In the second e-mail, I think what you are trying to say is that for a >good signal to noise ratio into an FM receiver (above limiting threshold) >the phase noise in the white noise is insignificant. But for a poor >signal to noise ratio the phase noise in the white noise is quite >significant and the output signal to noise ratio from the FM demodulator >is poor. Is this correct? > >I don't understand why you say white noise is orthogonal to an FM signal >when random phase variations are what make white noise in the first place. > >Bill > >Eden Mei wrote: >>Adder: >> >>The white noise is orthogonal to PM phase noise, since the random noise >>generally does not affect the phase of the FM signal, unless the SNR is >>REALLY bad. The phase of the noise has nothing to do with the phase >>noise of the FM signal. >> >>TTFN, >>Eden >> >>Contributions: "mathcad" >>Hosted by: Adept Scientific http://www.adeptscience.com >>List Archive: http://lists.adeptscience.co.uk/ >>Would you like this to come to a different email address? >>Simply leave the mailing list (see below) and re-join by >>sending a blank email from the new address to: >>mailto: or >>You are currently subscribed to mathcad as: "billd" >>To unsubscribe send a blank email to >> > > >Contributions: "mathcad" > Hosted by: Adept Scientific http://www.adeptscience.com > List Archive: http://lists.adeptscience.co.uk/ >Would you like this to come to a different email address? >Simply leave the mailing list (see below) and re-join by >sending a blank email from the new address to: >mailto: or >You are currently subscribed to mathcad as: "chw" >To unsubscribe send a blank email to > + Chris Whitford + Research Fellow, University of Leicester, Space Research Centre, + Physics and Astronomy Department, University Road, LEICESTER LE1 7RH, UK + Tel: +44 116 252 3496, Fax: +44 116 252 2464 + email: "chw" http://www.star.le.ac.uk/

RE: [mathcad] Modeling White Noise
Author: Philip Oakley    Posted: Fri, 4 Apr 2008 19:36:46 +0100
Bill, After re-checking the email, the following caution applies. People will use the same phrases and terms for different things, especially in special case pleadings. Modulation is one of them (Frequency or amplitude or phase modulation - FM, AM, PM - the latter is rare). The particular feature of modulation is that the carrier frequency is very different to the modulation frequency, so one can talk about random effects on either of the two components and assume that the reader/listener knows which is being talked about in th context!! -- A generic white random noise sequence is a series of independent samples from a gaussian distribution. It can be in as many dimensions as is needed. Thankfully only one in this case. They are real numbers. To make complex numbers just join 2 dimensions together a+ib. The act of choosing a sample rate sets the (Nyquist) bandwith and you also have the choice about aliasing. Yes: Thermal noise and white noise are equivalent. -- "the complex result is what we see in the real world. It has both phase and amplitude noise (caused by the phase noise) in it." This one is "Wrong", in the sense that actual voltages can not be imaginary. However the caveat applies. When we look at the two signal mixing (modulating), then we can represent the result in a variety of standard (mathematical) forms with a pretence of complex mathematics. Think of 3 phase electricity. They try to represent the voltages with complex numbers & vectors and such like, but we all know they are just plain voltages. In fact, think of a direct drive clock as a phase meter - when the clock shows fast then the phase (frequency) has gone ahead of the expected phase/frequency. And an oscilloscope will show all the 3-ph voltages as real - not a complex in sight! -- The first thing to clarify is what 'thing' is being talked about with all the various terms. I have attached a very old mathcad sheet that quickly looks at FM & PM. You probably need to stretch the graph and vary some of the number to make the effects show. -- have a look at wikipedia http://en.wikipedia.org/wiki/Frequency_modulation and http://en.wikipedia.org/wiki/Phase_modulation. They are a start but don't really do full justice to your problem. If you are recieving very low signal levels, so that within the FM tuner bandwith you have a lot of thermal (white / gaussian) noise, then that noise can be represented as either: A / B / C / D etc. (as said above there are lots of choices ;-) each with slightly different simplifying assumptions. Philip From: Bill E Dumke "mailto:Bill.Dumke" Sent: 04 April 2008 3:45 PM To: "mathcad" Subject: RE: [mathcad] Modeling White Noise Phil, Thank you very much for helping out. Let me see if I understand this correctly. The random phase noise used in the model for white noise also produces amplitude noise as well because of interactions between all the frequency components with random phases. If the IFFT function is used in the model, it only produces single valued real numbers for the result. When the ICFFT function is used it produces complex numbers for the result, and the complex result is what we see in the real world. It has both phase and amplitude noise (caused by the phase noise) in it. Noise figure is the result of a decrease of signal to noise ratio between the input and output of a device, such as an amplifier or AM receiver. It is usually caused by thermal noise at the input of the device, (but can also be caused by other types of noise related to the components used in the device.) Thermal noise and white noise are equivalent. As a result of the amplitude noise in thermal noise, the noise figure of an AM receiver will degrade the signal to noise ratio of a received AM signal. As a result of the phase noise in thermal noise, the noise figure of an FM receiver with a limiter (to eliminate AM noise) will degrade the signal to noise ratio of an FM signal. (In this case, the noise figure would have to be measured before the limiter and FM demodulator.) If the signal at the input of an FM receiver, is near the thermal noise level, the output signal to noise ratio will be poor. The improvement in output signal to noise ratio seen in an FM receiver with a limiter at a high input signal level relative to the thermal moise is still limited by the phase noise in the thermal noise. And of course, measured phase noise from a local oscillator will raise the noise floor of an FM receiver, if it is strong enough. Please correct me if I am wrong. Bill Bill Dumke, RF Engineer Nsight Telservices dba Cellcom 450 Security Blvd Green Bay, WI 54313 Office: 920 617-7311 e-mail: "bill.dumke" From: Philip Oakley "mailto:philipoakley" Sent: Thursday, April 03, 2008 6:12 PM To: "mathcad" Subject: RE: [mathcad] Modeling White Noise Bill, The PhD guy hasn't done enough thinking. The difference between: * white noise (who's frequency response (spectra density) is infinitely wide & infinitely low) and * an impulse response (who's frequency response (spectra density) is infinitely wide & infinitely low) is only that the impulse has all the frequencies co-align to be at zero phase at the same instant, so they are all cos(0) [so peak up to an infinite spike for zero time]. For the white noise, the phases of all those (infinite) frequencies never co-align (so they can't create that infinite peak) rather they produce a random sum. The other thing he does wrong is to miss the tricks that happens in the Fourier Transform. The transform of real data is symmetric in positive and negative frequencies, just so that we have exactly enough spare data values for those phase values. The amplitude is orthogonal to the phase. In fact it has to be. But that is the ampliude of the 'lasts forever' sinewave. When you truncate the sine wave (by only taking a limited time length) you will 'never' manage to get the start and end point to be exactly at the 2N*pi points of that sinewave (remember there are an infinite number of them). So the freqreqency gets adjusted by the FFT to be at FFT's sample points. As the others have said, if you define a nice amplitude spectrum and then use random phase, you can use the IFFT to get the random sequence of any spectral (white/pink/..) distribution you need. Philip From: Bill E Dumke "mailto:Bill.Dumke" Sent: 03 April 2008 2:37 PM To: "mathcad" Subject: RE: [mathcad] Modeling White Noise Neil, This is very confusing to me. I was searching around for info on white noise and discovered this paper: http://www.home.agilent.com/upload/cmc_upload/All/Exp65.pdf The paper confirms what you have said. Only the phase is random. I am in a debate with a PhD friend of mine, and he claims "Noise is a random process but in conductors (including resistors) it is an one-dimensional random process". I created a Mathcad document to match the procedure given in the Agilent document. And true enough, the vector N is one dimensional. (Since the white noise vector contains only real numbers, it would seem to me that adding white noise to a carrier would only affect the amplitude of the carrier, not the phase.) Because of this he claims that since white noise is orthogonal to phase noise, white noise will not affect the signal to noise performance of an FM receiver. (But that is not what I know about FM receivers. FM is limited by noise figure just as AM and SSB receivers are limited by noise figure, the only difference being that FM gives better signal to noise ratio at the output of the FM demodulator for high input signal to noise ratios. And I seem to remember that it gives worse performance below the limiting threshold.) So I am confused. Also I wonder why the "bandlimited White" noise shown at the bottom of the page doesn't show any amplitude variation across the spectrum. Every time I have looked at the noise coming from a filter with a spectrum analyzer or power meter it varies in amplitude as well. Trying to understand this, Bill Bill Dumke, RF Engineer Nsight Telservices dba Cellcom 450 Security Blvd Green Bay, WI 54313 Office: 920 617-7311 e-mail: "bill.dumke" From: Neil Harris "mailto:N.Harris" Sent: Thursday, April 03, 2008 7:37 AM To: "mathcad" Subject: RE: [mathcad] Modeling White Noise Bill, The definition of white noise is that it has a (long-term) power spectrum of unity - i.e. the same as an impulse. It is only the phase response that differs between the two. In the time domain, the noise signal is "random" due to its random phase. Neil From: Bill Dumke "mailto:billd" Sent: 03 April 2008 12:49 To: "mathcad" Subject: Re: [mathcad] Modeling White Noise Thanks Neil, But doesn't the amplitude vary randomly in amplitude as well? Bill Neil Harris wrote: > Bill, > > The simplest way is to generate a pair of vectors of length 2^N/2 + 1; > one has the required amplitude response, and the other is a random phase > generated by using the function phi := 2*pi*rnd(1). > > These are combined into a complex vector amplitude * exp(1j x phi) > > The required noise is the simply the inverse FFT of this complex vector. > > Noise = IFF(complex vector) > > Apply appropriate scaling to get the right amplitude. > > Regards, > Neil > > > From: Bill E Dumke "mailto:Bill.Dumke" > Sent: 02 April 2008 16:14 > To: "mathcad" > Subject: [mathcad] Modeling White Noise > > I need a model for bandwidth limited white noise. I have Mathcad 14 but > would prefer to use Mathcad 11 for this. > > Thanks, Bill > > Bill Dumke, RF Engineer > Nsight Telservices dba Cellcom > 450 Security Blvd > Green Bay, WI 54313 > > Office: 920 617-7311 > e-mail: "bill.dumke" > > Contributions: "mathcad" > Hosted by: Adept Scientific http://www.adeptscience.com > List Archive: http://lists.adeptscience.co.uk/ > Would you like this to come to a different email address? > Simply leave the mailing list (see below) and re-join by sending a > blank email from the new address to: > "mailto:join-mathcad" or > You are currently subscribed to mathcad as: "n.harris" > To unsubscribe send a blank email to > "leave-52344-494237H" > > Contributions: "mathcad" > Hosted by: Adept Scientific http://www.adeptscience.com > List Archive: http://lists.adeptscience.co.uk/ > Would you like this to come to a different email address? > Simply leave the mailing list (see below) and re-join by sending a > blank email from the new address to: > "mailto:join-mathcad" or > You are currently subscribed to mathcad as: "billd" > To unsubscribe send a blank email to "leave-52400-494029G" > > > > >
Attachments:  freq_mod.mcd

Re: [mathcad] Modeling White Noise
Author: Eden Mei    Posted: Fri, 4 Apr 2008 20:38:30 -0700
You're mixing the concept of phase of the white noise with phase noise of the FM signal. An FM signal is modulating its frequency to encode the audio signal. This has nothing to do with the phase of the white noise. In the ideal sense, the FM signal is immune to amplitude noise, because the amplitude of the FM signal contains no information pertaining to the encoded signal. The white noise only affects the amplitude of the FM signal, not its phase, hence, white noise is orthogonal to FM phase noise, because there is almost no interaction between the two. The limiting case would where the white noise so corrupts the FM signal that its phase cannot be accurately determined at all. In that case, the white noise will screw up the FM signal. Otherwise, the FM decoder can decode the phase and reconstruct the audio signal. TTFN, Eden

RE: [mathcad] Modeling White Noise
Author: Bill E Dumke    Posted: Thu, 3 Apr 2008 13:13:26 -0500
Thanks, Neil. I believe I understand now. Bill Bill Dumke, RF Engineer Nsight Telservices dba Cellcom 450 Security Blvd Green Bay, WI 54313 Office: 920 617-7311 e-mail: "bill.dumke" From: Neil Harris "mailto:N.Harris" Sent: Thursday, April 03, 2008 12:24 PM To: "mathcad" Subject: RE: [mathcad] Modeling White Noise Bill, Firstly, one never measures for an infinite length of time (!), so there are fluctuations in "real-world" measurements. Secondly, if you use the IFFT function, the noise (time-domain) will be real-only. Finally, if you want a band-limited noise, use the required power spectrum for the square of the amplitude; for example Butterworth, nth order low pass |H(f)|^2 = fc^2n/(fc^2n + f^2n) Butterworth, nth order high pass |H(f)|^2 = f^2n/(fc^2n + f^2n) I hope this helps a bit? Neil From: Bill E Dumke "mailto:Bill.Dumke" Sent: 03 April 2008 14:37 To: "mathcad" Subject: RE: [mathcad] Modeling White Noise Neil, This is very confusing to me. I was searching around for info on white noise and discovered this paper: http://www.home.agilent.com/upload/cmc_upload/All/Exp65.pdf The paper confirms what you have said. Only the phase is random. I am in a debate with a PhD friend of mine, and he claims "Noise is a random process but in conductors (including resistors) it is an one-dimensional random process". I created a Mathcad document to match the procedure given in the Agilent document. And true enough, the vector N is one dimensional. (Since the white noise vector contains only real numbers, it would seem to me that adding white noise to a carrier would only affect the amplitude of the carrier, not the phase.) Because of this he claims that since white noise is orthogonal to phase noise, white noise will not affect the signal to noise performance of an FM receiver. (But that is not what I know about FM receivers. FM is limited by noise figure just as AM and SSB receivers are limited by noise figure, the only difference being that FM gives better signal to noise ratio at the output of the FM demodulator for high input signal to noise ratios. And I seem to remember that it gives worse performance below the limiting threshold.) So I am confused. Also I wonder why the "bandlimited White" noise shown at the bottom of the page doesn't show any amplitude variation across the spectrum. Every time I have looked at the noise coming from a filter with a spectrum analyzer or power meter it varies in amplitude as well. Trying to understand this, Bill Bill Dumke, RF Engineer Nsight Telservices dba Cellcom 450 Security Blvd Green Bay, WI 54313 Office: 920 617-7311 e-mail: "bill.dumke" From: Neil Harris "mailto:N.Harris" Sent: Thursday, April 03, 2008 7:37 AM To: "mathcad" Subject: RE: [mathcad] Modeling White Noise Bill, The definition of white noise is that it has a (long-term) power spectrum of unity - i.e. the same as an impulse. It is only the phase response that differs between the two. In the time domain, the noise signal is "random" due to its random phase. Neil From: Bill Dumke "mailto:billd" Sent: 03 April 2008 12:49 To: "mathcad" Subject: Re: [mathcad] Modeling White Noise Thanks Neil, But doesn't the amplitude vary randomly in amplitude as well? Bill Neil Harris wrote: > Bill, > > The simplest way is to generate a pair of vectors of length 2^N/2 + 1; > one has the required amplitude response, and the other is a random phase > generated by using the function phi := 2*pi*rnd(1). > > These are combined into a complex vector amplitude * exp(1j x phi) > > The required noise is the simply the inverse FFT of this complex vector. > > Noise = IFF(complex vector) > > Apply appropriate scaling to get the right amplitude. > > Regards, > Neil > > > From: Bill E Dumke "mailto:Bill.Dumke" > Sent: 02 April 2008 16:14 > To: "mathcad" > Subject: [mathcad] Modeling White Noise > > I need a model for bandwidth limited white noise. I have Mathcad 14 but > would prefer to use Mathcad 11 for this. > > Thanks, Bill > > Bill Dumke, RF Engineer > Nsight Telservices dba Cellcom > 450 Security Blvd > Green Bay, WI 54313 > > Office: 920 617-7311 > e-mail: "bill.dumke" > > Contributions: "mathcad" > Hosted by: Adept Scientific http://www.adeptscience.com > List Archive: http://lists.adeptscience.co.uk/ > Would you like this to come to a different email address? > Simply leave the mailing list (see below) and re-join by sending a > blank email from the new address to: > "mailto:join-mathcad" or > You are currently subscribed to mathcad as: "n.harris" > To unsubscribe send a blank email to > "leave-52344-494237H" > > Contributions: "mathcad" > Hosted by: Adept Scientific http://www.adeptscience.com > List Archive: http://lists.adeptscience.co.uk/ > Would you like this to come to a different email address? > Simply leave the mailing list (see below) and re-join by sending a > blank email from the new address to: > "mailto:join-mathcad" or > You are currently subscribed to mathcad as: "billd" > To unsubscribe send a blank email to "leave-52400-494029G" > > > > >

RE: [mathcad] Modeling White Noise
Author: Bill E Dumke    Posted: Fri, 4 Apr 2008 14:15:12 -0500
Phil, Thanks for the reply and the file, as well as the simple analogies. The major concern I have is when you say that the white noise is real. That appears to be the heart of the controversy. To me that would say it is orthogonal to phase or frequency noise, so that white noise would not affect the signal to noise ratio of a receiver, which I know is wrong. White noise in the real world certainly does affect FM receiver operation. (If it did not, we could totally ignore noise figure in an FM receiver.) Am I correct? In the context of working with an FM receiver you seem to be saying that phase noise should also be considered, but using real numbers in a separate calculation. Is this correct? You say the white noise model is real only. So is there an equivalent model for "white noise phase noise"? It would have to vary only between -pi and pi. But I thought that was taken care of in the original model of white noise, and the complex transform is what would show it. I am trying to understand this. Bill Bill Dumke, RF Engineer Nsight Telservices dba Cellcom 450 Security Blvd Green Bay, WI 54313 Office: 920 617-7311 e-mail: "bill.dumke" From: Philip Oakley "mailto:philipoakley" Sent: Friday, April 04, 2008 1:37 PM To: "mathcad" Cc: Bill E Dumke Subject: RE: [mathcad] Modeling White Noise Bill, After re-checking the email, the following caution applies. People will use the same phrases and terms for different things, especially in special case pleadings. Modulation is one of them (Frequency or amplitude or phase modulation - FM, AM, PM - the latter is rare). The particular feature of modulation is that the carrier frequency is very different to the modulation frequency, so one can talk about random effects on either of the two components and assume that the reader/listener knows which is being talked about in th context!! -- A generic white random noise sequence is a series of independent samples from a gaussian distribution. It can be in as many dimensions as is needed. Thankfully only one in this case. They are real numbers. To make complex numbers just join 2 dimensions together a+ib. The act of choosing a sample rate sets the (Nyquist) bandwith and you also have the choice about aliasing. Yes: Thermal noise and white noise are equivalent. -- "the complex result is what we see in the real world. It has both phase and amplitude noise (caused by the phase noise) in it." This one is "Wrong", in the sense that actual voltages can not be imaginary. However the caveat applies. When we look at the two signal mixing (modulating), then we can represent the result in a variety of standard (mathematical) forms with a pretence of complex mathematics. Think of 3 phase electricity. They try to represent the voltages with complex numbers & vectors and such like, but we all know they are just plain voltages. In fact, think of a direct drive clock as a phase meter - when the clock shows fast then the phase (frequency) has gone ahead of the expected phase/frequency. And an oscilloscope will show all the 3-ph voltages as real - not a complex in sight! -- The first thing to clarify is what 'thing' is being talked about with all the various terms. I have attached a very old mathcad sheet that quickly looks at FM & PM. You probably need to stretch the graph and vary some of the number to make the effects show. -- have a look at wikipedia http://en.wikipedia.org/wiki/Frequency_modulation and http://en.wikipedia.org/wiki/Phase_modulation. They are a start but don't really do full justice to your problem. If you are recieving very low signal levels, so that within the FM tuner bandwith you have a lot of thermal (white / gaussian) noise, then that noise can be represented as either: A / B / C / D etc. (as said above there are lots of choices ;-) each with slightly different simplifying assumptions. Philip From: Bill E Dumke "mailto:Bill.Dumke" Sent: 04 April 2008 3:45 PM To: "mathcad" Subject: RE: [mathcad] Modeling White Noise Phil, Thank you very much for helping out. Let me see if I understand this correctly. The random phase noise used in the model for white noise also produces amplitude noise as well because of interactions between all the frequency components with random phases. If the IFFT function is used in the model, it only produces single valued real numbers for the result. When the ICFFT function is used it produces complex numbers for the result, and the complex result is what we see in the real world. It has both phase and amplitude noise (caused by the phase noise) in it. Noise figure is the result of a decrease of signal to noise ratio between the input and output of a device, such as an amplifier or AM receiver. It is usually caused by thermal noise at the input of the device, (but can also be caused by other types of noise related to the components used in the device.) Thermal noise and white noise are equivalent. As a result of the amplitude noise in thermal noise, the noise figure of an AM receiver will degrade the signal to noise ratio of a received AM signal. As a result of the phase noise in thermal noise, the noise figure of an FM receiver with a limiter (to eliminate AM noise) will degrade the signal to noise ratio of an FM signal. (In this case, the noise figure would have to be measured before the limiter and FM demodulator.) If the signal at the input of an FM receiver, is near the thermal noise level, the output signal to noise ratio will be poor. The improvement in output signal to noise ratio seen in an FM receiver with a limiter at a high input signal level relative to the thermal moise is still limited by the phase noise in the thermal noise. And of course, measured phase noise from a local oscillator will raise the noise floor of an FM receiver, if it is strong enough. Please correct me if I am wrong. Bill Bill Dumke, RF Engineer Nsight Telservices dba Cellcom 450 Security Blvd Green Bay, WI 54313 Office: 920 617-7311 e-mail: "bill.dumke" From: Philip Oakley "mailto:philipoakley" Sent: Thursday, April 03, 2008 6:12 PM To: "mathcad" Subject: RE: [mathcad] Modeling White Noise Bill, The PhD guy hasn't done enough thinking. The difference between: * white noise (who's frequency response (spectra density) is infinitely wide & infinitely low) and * an impulse response (who's frequency response (spectra density) is infinitely wide & infinitely low) is only that the impulse has all the frequencies co-align to be at zero phase at the same instant, so they are all cos(0) [so peak up to an infinite spike for zero time]. For the white noise, the phases of all those (infinite) frequencies never co-align (so they can't create that infinite peak) rather they produce a random sum. The other thing he does wrong is to miss the tricks that happens in the Fourier Transform. The transform of real data is symmetric in positive and negative frequencies, just so that we have exactly enough spare data values for those phase values. The amplitude is orthogonal to the phase. In fact it has to be. But that is the ampliude of the 'lasts forever' sinewave. When you truncate the sine wave (by only taking a limited time length) you will 'never' manage to get the start and end point to be exactly at the 2N*pi points of that sinewave (remember there are an infinite number of them). So the freqreqency gets adjusted by the FFT to be at FFT's sample points. As the others have said, if you define a nice amplitude spectrum and then use random phase, you can use the IFFT to get the random sequence of any spectral (white/pink/..) distribution you need. Philip From: Bill E Dumke "mailto:Bill.Dumke" Sent: 03 April 2008 2:37 PM To: "mathcad" Subject: RE: [mathcad] Modeling White Noise Neil, This is very confusing to me. I was searching around for info on white noise and discovered this paper: http://www.home.agilent.com/upload/cmc_upload/All/Exp65.pdf The paper confirms what you have said. Only the phase is random. I am in a debate with a PhD friend of mine, and he claims "Noise is a random process but in conductors (including resistors) it is an one-dimensional random process". I created a Mathcad document to match the procedure given in the Agilent document. And true enough, the vector N is one dimensional. (Since the white noise vector contains only real numbers, it would seem to me that adding white noise to a carrier would only affect the amplitude of the carrier, not the phase.) Because of this he claims that since white noise is orthogonal to phase noise, white noise will not affect the signal to noise performance of an FM receiver. (But that is not what I know about FM receivers. FM is limited by noise figure just as AM and SSB receivers are limited by noise figure, the only difference being that FM gives better signal to noise ratio at the output of the FM demodulator for high input signal to noise ratios. And I seem to remember that it gives worse performance below the limiting threshold.) So I am confused. Also I wonder why the "bandlimited White" noise shown at the bottom of the page doesn't show any amplitude variation across the spectrum. Every time I have looked at the noise coming from a filter with a spectrum analyzer or power meter it varies in amplitude as well. Trying to understand this, Bill Bill Dumke, RF Engineer Nsight Telservices dba Cellcom 450 Security Blvd Green Bay, WI 54313 Office: 920 617-7311 e-mail: "bill.dumke" From: Neil Harris "mailto:N.Harris" Sent: Thursday, April 03, 2008 7:37 AM To: "mathcad" Subject: RE: [mathcad] Modeling White Noise Bill, The definition of white noise is that it has a (long-term) power spectrum of unity - i.e. the same as an impulse. It is only the phase response that differs between the two. In the time domain, the noise signal is "random" due to its random phase. Neil From: Bill Dumke "mailto:billd" Sent: 03 April 2008 12:49 To: "mathcad" Subject: Re: [mathcad] Modeling White Noise Thanks Neil, But doesn't the amplitude vary randomly in amplitude as well? Bill Neil Harris wrote: > Bill, > > The simplest way is to generate a pair of vectors of length 2^N/2 + 1; > one has the required amplitude response, and the other is a random phase > generated by using the function phi := 2*pi*rnd(1). > > These are combined into a complex vector amplitude * exp(1j x phi) > > The required noise is the simply the inverse FFT of this complex vector. > > Noise = IFF(complex vector) > > Apply appropriate scaling to get the right amplitude. > > Regards, > Neil > > > From: Bill E Dumke "mailto:Bill.Dumke" > Sent: 02 April 2008 16:14 > To: "mathcad" > Subject: [mathcad] Modeling White Noise > > I need a model for bandwidth limited white noise. I have Mathcad 14 but > would prefer to use Mathcad 11 for this. > > Thanks, Bill > > Bill Dumke, RF Engineer > Nsight Telservices dba Cellcom > 450 Security Blvd > Green Bay, WI 54313 > > Office: 920 617-7311 > e-mail: "bill.dumke" > > Contributions: "mathcad" > Hosted by: Adept Scientific http://www.adeptscience.com > List Archive: http://lists.adeptscience.co.uk/ > Would you like this to come to a different email address? > Simply leave the mailing list (see below) and re-join by sending a > blank email from the new address to: > "mailto:join-mathcad" or > You are currently subscribed to mathcad as: "n.harris" > To unsubscribe send a blank email to > "leave-52344-494237H" > > Contributions: "mathcad" > Hosted by: Adept Scientific http://www.adeptscience.com > List Archive: http://lists.adeptscience.co.uk/ > Would you like this to come to a different email address? > Simply leave the mailing list (see below) and re-join by sending a > blank email from the new address to: > "mailto:join-mathcad" or > You are currently subscribed to mathcad as: "billd" > To unsubscribe send a blank email to "leave-52400-494029G" > > > > >

Re: [mathcad] Modeling White Noise
Author: Tom Gutman    Posted: Sun, 6 Apr 2008 00:44:11 -0700
Aren't phase and frequency modulation almost the same thing? With phase modulation by a signal being the same as frequency modulation by the derivative of the signal? And frequency modulation by a signal being the same as phase modulation by the integral of the signal? And aren't differentiation and integration just frequency dependent scaling, just multiplying or dividing the amplitude of each frequency component by the frequency? So that phase modulation is just frequency modulation with a boost to the higher frequencies? As to the phase/amplitude issue, isn't phase really an artificial concept? The only real reality being the amplitude at each point in time? Phase becomes something that is inferred when a signal has enough structure so that one can identify some sort of repeating unit. So while the "phase" of white noise has no direct effect, and the noise does not directly affect the phase, the distortion in the signal due to the noise reduces the accuracy with which a phase can be recovered. And such reduction in accuracy occurs long before the point where frequency and phase become completely unrecognizable. There is a large measure of immunity to noise using FM, as there is no directly sensitivity to the noise itself, only to the distortions in the carrier introduced by the noise, but it is not absolute. AFAIK the only modulation schemes that have the attributes you propose, a perfect reproduction up to the point where no reconstruction at all is possible, are digital rather than analog schemes. Tom Gutman From: "Philip Oakley" "philipoakley" To: "mathcad" Cc: "Bill.Dumke" Sent: Friday, April 4, 2008 11:36 AM Subject: RE: [mathcad] Modeling White Noise > Bill, > > After re-checking the email, the following caution applies. People > will use > the same phrases and terms for different things, especially in > special case > pleadings. Modulation is one of them (Frequency or amplitude or > phase > modulation - FM, AM, PM - the latter is rare). > > The particular feature of modulation is that the carrier frequency > is very > different to the modulation frequency, so one can talk about random > effects > on either of the two components and assume that the reader/listener > knows > which is being talked about in th context!! > > -- > > A generic white random noise sequence is a series of independent > samples > from a gaussian distribution. It can be in as many dimensions as is > needed. > Thankfully only one in this case. They are real numbers. To make > complex > numbers just join 2 dimensions together a+ib. The act of choosing a > sample > rate sets the (Nyquist) bandwith and you also have the choice about > aliasing. > > Yes: Thermal noise and white noise are equivalent. > > -- > "the complex result is what we see in the real world. It has both > phase and > amplitude noise (caused by the phase noise) in it." > This one is "Wrong", in the sense that actual voltages can not be > imaginary. > However the caveat applies. When we look at the two signal mixing > (modulating), then we can represent the result in a variety of > standard > (mathematical) forms with a pretence of complex mathematics. > Think of 3 phase electricity. They try to represent the voltages > with > complex numbers & vectors and such like, but we all know they are > just plain > voltages. In fact, think of a direct drive clock as a phase meter - > when the > clock shows fast then the phase (frequency) has gone ahead of the > expected > phase/frequency. And an oscilloscope will show all the 3-ph voltages > as > real - not a complex in sight! > > -- > > The first thing to clarify is what 'thing' is being talked about > with all > the various terms. > > > I have attached a very old mathcad sheet that quickly looks at FM & > PM. You > probably need to stretch the graph and vary some of the number to > make the > effects show. > > -- > > have a look at wikipedia > http://en.wikipedia.org/wiki/Frequency_modulation > and http://en.wikipedia.org/wiki/Phase_modulation. They are a start > but > don't really do full justice to your problem. > > > If you are recieving very low signal levels, so that within the FM > tuner > bandwith you have a lot of thermal (white / gaussian) noise, then > that noise > can be represented as either: A / B / C / D etc. (as said above > there are > lots of choices ;-) each with slightly different simplifying > assumptions. > > > Philip > > From: Bill E Dumke "mailto:Bill.Dumke" > Sent: 04 April 2008 3:45 PM > To: "mathcad" > Subject: RE: [mathcad] Modeling White Noise > > > Phil, > > Thank you very much for helping out. Let me see if I understand > this > correctly. > > The random phase noise used in the model for white noise also > produces > amplitude noise as well because of interactions between all the > frequency components with random phases. > > If the IFFT function is used in the model, it only produces single > valued real numbers for the result. When the ICFFT function is used > it > produces complex numbers for the result, and the complex result is > what > we see in the real world. It has both phase and amplitude noise > (caused > by the phase noise) in it. > > Noise figure is the result of a decrease of signal to noise ratio > between the input and output of a device, such as an amplifier or AM > receiver. It is usually caused by thermal noise at the input of the > device, (but can also be caused by other types of noise related to > the > components used in the device.) > > Thermal noise and white noise are equivalent. > > As a result of the amplitude noise in thermal noise, the noise > figure of > an AM receiver will degrade the signal to noise ratio of a received > AM > signal. > > As a result of the phase noise in thermal noise, the noise figure of > an > FM receiver with a limiter (to eliminate AM noise) will degrade the > signal to noise ratio of an FM signal. (In this case, the noise > figure > would have to be measured before the limiter and FM demodulator.) > > If the signal at the input of an FM receiver, is near the thermal > noise > level, the output signal to noise ratio will be poor. > > The improvement in output signal to noise ratio seen in an FM > receiver > with a limiter at a high input signal level relative to the thermal > moise is still limited by the phase noise in the thermal noise. > > And of course, measured phase noise from a local oscillator will > raise > the noise floor of an FM receiver, if it is strong enough. > > Please correct me if I am wrong. > > Bill > > > > > > > > Bill Dumke, RF Engineer > Nsight Telservices dba Cellcom > 450 Security Blvd > Green Bay, WI 54313 > > Office: 920 617-7311 > e-mail: "bill.dumke" > From: Philip Oakley "mailto:philipoakley" > Sent: Thursday, April 03, 2008 6:12 PM > To: "mathcad" > Subject: RE: [mathcad] Modeling White Noise > > Bill, > The PhD guy hasn't done enough thinking. > > The difference between: > * white noise (who's frequency response (spectra density) is > infinitely > wide & infinitely low) and > * an impulse response (who's frequency response (spectra density) is > infinitely wide & infinitely low) is only that the impulse has all > the > frequencies co-align to be at zero phase at the same instant, so > they > are all cos(0) [so peak up to an infinite spike for zero time]. For > the > white noise, the phases of all those > (infinite) frequencies never co-align (so they can't create that > infinite > peak) rather they produce a random sum. > > The other thing he does wrong is to miss the tricks that happens in > the > Fourier Transform. > The transform of real data is symmetric in p