FPGA Central

Hi all,

Do you what is the theory of SNR at the output of a FM demodulator
(linear) with the SNR at its input ?

I do measure on a real system that I made and I don't know what the
theory says for a know input SNR and a measure ouput SNR
(demodulated)...

Best regards...

[email protected] wrote:
> Hi all,
>
> Do you what is the theory of SNR at the output of a FM demodulator
> (linear) with the SNR at its input ?
>
> I do measure on a real system that I made and I don't know what the
> theory says for a know input SNR and a measure ouput SNR
> (demodulated)...
>
> Best regards...

>From Principles of Communications, by Ziemer and Tranter:

Pt = transmit carrier power A^2/2
W = output LPF bandwidth
Fd = peak deviation
No = input noise density
m = modulating signal

SNRo = 3 * mean(m^2) * (Fd / W)^2 * Pt / (No * W)


Regards,

John

> Pt = transmit carrier power A^2/2
> W = output LPF bandwidth
> Fd = peak deviation
> No = input noise density
> m = modulating signal
>
> SNRo = 3 * mean(m^2) * (Fd / W)^2 * Pt / (No * W)

so if my m signal is a sinus at 250 kHz with 1 V rms then mean(m^2) =
0.5
if my Fd = 250 kHz and W = 250 kHz then (Fd / W)^2 = 1
if Pt / (No * W) = SNR input, at the input at the modulator I simulate
the SNR (measure it)

then SNRo = 3 * 0.5 * 1 * SNRi = 1.5 * SNRi

is it exact ?

[email protected] wrote:
>> Pt = transmit carrier power A^2/2
>> W = output LPF bandwidth
>> Fd = peak deviation
>> No = input noise density
>> m = modulating signal
>>
>> SNRo = 3 * mean(m^2) * (Fd / W)^2 * Pt / (No * W)
>
> so if my m signal is a sinus at 250 kHz with 1 V rms then mean(m^2) =
> 0.5
> if my Fd = 250 kHz and W = 250 kHz then (Fd / W)^2 = 1
> if Pt / (No * W) = SNR input, at the input at the modulator I simulate
> the SNR (measure it)
>
> then SNRo = 3 * 0.5 * 1 * SNRi = 1.5 * SNRi
>
> is it exact ?
>

You should see SNRo > SNRi when the discriminator operates above
threshold (no clicks). This is called FM improvement. I've obtained good
agreement with the theory in the past.

John

John Sampson <[email protected]> writes:

> [email protected] wrote:
>>> Pt = transmit carrier power A^2/2
>>> W = output LPF bandwidth
>>> Fd = peak deviation
>>> No = input noise density
>>> m = modulating signal
>>>
>>> SNRo = 3 * mean(m^2) * (Fd / W)^2 * Pt / (No * W)
>> so if my m signal is a sinus at 250 kHz with 1 V rms then mean(m^2) =
>> 0.5
>> if my Fd = 250 kHz and W = 250 kHz then (Fd / W)^2 = 1
>> if Pt / (No * W) = SNR input, at the input at the modulator I simulate
>> the SNR (measure it)
>> then SNRo = 3 * 0.5 * 1 * SNRi = 1.5 * SNRi
>> is it exact ?
>>
>
> You should see SNRo > SNRi when the discriminator operates above
> threshold (no clicks). This is called FM improvement. I've obtained
> good agreement with the theory in the past.

Hi John,

If the discriminator is just dtheta/dt where theta is the phase of the
analytic baseband signal, then what is the thing you call the
"threshold?" Just curious.
--
% Randy Yates % "Bird, on the wing,
%% Fuquay-Varina, NC % goes floating by
%%% 919-577-9882 % but there's a teardrop in his eye..."
%%%% <[email protected]> % 'One Summer Dream', *Face The Music*, ELO
http://home.earthlink.net/~yatescr

[email protected] wrote:

> Hi all,
>
> Do you what is the theory of SNR at the output of a FM demodulator
> (linear) with the SNR at its input ?
>

FM demodulator just can't be linear since it deals with the phase. Thus
the SNR question is fairly complicated and I don't know of any closed
form equation. It depends on the pdf of the signal. Also, the gaussian
noise at the input results in non-gaussian noise at the output.


> I do measure on a real system that I made and I don't know what the
> theory says for a know input SNR and a measure ouput SNR
> (demodulated)...

Generally, this is a tg - like dependence.

Vladimir Vassilevsky

DSP and Mixed Signal Design Consultant

http://www.abvolt.com

Randy Yates wrote:

> John Sampson <[email protected]> writes:
>
>
>>[email protected] wrote:
>>
>>>>Pt = transmit carrier power A^2/2
>>>>W = output LPF bandwidth
>>>>Fd = peak deviation
>>>>No = input noise density
>>>>m = modulating signal
>>>>
>>>>SNRo = 3 * mean(m^2) * (Fd / W)^2 * Pt / (No * W)
>>>
>>>so if my m signal is a sinus at 250 kHz with 1 V rms then mean(m^2) =
>>>0.5
>>>if my Fd = 250 kHz and W = 250 kHz then (Fd / W)^2 = 1
>>>if Pt / (No * W) = SNR input, at the input at the modulator I simulate
>>>the SNR (measure it)
>>>then SNRo = 3 * 0.5 * 1 * SNRi = 1.5 * SNRi
>>>is it exact ?
>>>
>>
>>You should see SNRo > SNRi when the discriminator operates above
>>threshold (no clicks). This is called FM improvement. I've obtained
>>good agreement with the theory in the past.
>
>
> Hi John,
>
> If the discriminator is just dtheta/dt where theta is the phase of the
> analytic baseband signal, then what is the thing you call the
> "threshold?" Just curious.

Hello Randy,

The approximation quoted by John is valid on the two assumptions:
1. SNR is high. 2. The detector is a dumb discriminator.

We should distinguish the properties of the FM signal itself from the
inefficiency of the particular demodulator.

The discriminator goofs up when the normalized SNR falls below ~10dB.
The amateurs call this a "threshold".

Vladimir Vassilevsky

DSP and Mixed Signal Design Consultant

http://www.abvolt.com

Randy Yates wrote:

> John Sampson <[email protected]> writes:
>
>
>>[email protected] wrote:
>>
>>>>Pt = transmit carrier power A^2/2
>>>>W = output LPF bandwidth
>>>>Fd = peak deviation
>>>>No = input noise density
>>>>m = modulating signal
>>>>
>>>>SNRo = 3 * mean(m^2) * (Fd / W)^2 * Pt / (No * W)
>>>
>>>so if my m signal is a sinus at 250 kHz with 1 V rms then mean(m^2) =
>>>0.5
>>>if my Fd = 250 kHz and W = 250 kHz then (Fd / W)^2 = 1
>>>if Pt / (No * W) = SNR input, at the input at the modulator I simulate
>>>the SNR (measure it)
>>>then SNRo = 3 * 0.5 * 1 * SNRi = 1.5 * SNRi
>>>is it exact ?
>>>
>>
>>You should see SNRo > SNRi when the discriminator operates above
>>threshold (no clicks). This is called FM improvement. I've obtained
>>good agreement with the theory in the past.
>
>
> Hi John,
>
> If the discriminator is just dtheta/dt where theta is the phase of the
> analytic baseband signal, then what is the thing you call the
> "threshold?" Just curious.


Hello Randy,

The approximation quoted by John is valid on the two assumptions:
1. SNR is high. 2. The detector is a simple discriminator.

We should distinguish the properties of the FM signal itself from the
inefficiency of the particular demodulator.

The discriminator fails when the normalized SNR falls below ~10dB. The
popular word for this is a "threshold".

Vladimir Vassilevsky

DSP and Mixed Signal Design Consultant

http://www.abvolt.com

Randy Yates wrote:
> John Sampson <[email protected]> writes:
>
>> [email protected] wrote:
>>>> Pt = transmit carrier power A^2/2
>>>> W = output LPF bandwidth
>>>> Fd = peak deviation
>>>> No = input noise density
>>>> m = modulating signal
>>>>
>>>> SNRo = 3 * mean(m^2) * (Fd / W)^2 * Pt / (No * W)
>>> so if my m signal is a sinus at 250 kHz with 1 V rms then mean(m^2) =
>>> 0.5
>>> if my Fd = 250 kHz and W = 250 kHz then (Fd / W)^2 = 1
>>> if Pt / (No * W) = SNR input, at the input at the modulator I simulate
>>> the SNR (measure it)
>>> then SNRo = 3 * 0.5 * 1 * SNRi = 1.5 * SNRi
>>> is it exact ?
>>>
>> You should see SNRo > SNRi when the discriminator operates above
>> threshold (no clicks). This is called FM improvement. I've obtained
>> good agreement with the theory in the past.
>
> Hi John,
>
> If the discriminator is just dtheta/dt where theta is the phase of the
> analytic baseband signal, then what is the thing you call the
> "threshold?" Just curious.

I'd have to pull out the books (Schwartz?) to provide equations, but
there's an intuitive (to me, anyway) explanation. Wideband FM uses
excess bandwidth to buy better S/N. The receiver's wide front end lets
in more noise than a narrow one would. Below some threshold* input
level, the extra noise from the wide-open front end overwhelms FM's
inherent noise improvement. Above that level, the output S/N improves
rapidly. The effect is that for an unmodulated carrier, the background
noise -- "FM hiss" drops precipitously as the threshold is exceeded.
This phenomenon is known as "quieting", and a typical receiver spec
might be "30 dB quieting at 5 microvolts" at the antenna input.

Jerry
______________________________
* How is that "thresh hold" (the raised sill that keeps the threshed
grain from spilling out)? I read it either "thresh old" or "thres
hold". I guess dropped aitches are common in English.
--
"The rights of the best of men are secured only as the
rights of the vilest and most abhorrent are protected."
- Chief Justice Charles Evans Hughes, 1927
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

Vladimir Vassilevsky wrote:

>
>
> [email protected] wrote:
>
>> Hi all,
>>
>> Do you what is the theory of SNR at the output of a FM demodulator
>> (linear) with the SNR at its input ?
>>
>
> FM demodulator just can't be linear since it deals with the phase. Thus
> the SNR question is fairly complicated and I don't know of any closed
> form equation. It depends on the pdf of the signal. Also, the gaussian
> noise at the input results in non-gaussian noise at the output.
>
Also, if I recall the analysis correctly, you can't find an "optimum" FM
demodulator the way that you can with AM modulation and Gaussian noise.
This means that you have to start by assuming a demodulation scheme,
then analyze it's performance -- but someone may come by tomorrow with a
better demodulator, that'll outperform yours by a few dB.

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Posting from Google? See http://cfaj.freeshell.org/google/

"Applied Control Theory for Embedded Systems" came out in April.
See details at http://www.wescottdesign.com/actfes/actfes.html

Vladimir Vassilevsky wrote:

...

> The discriminator goofs up when the normalized SNR falls below ~10dB.
> The amateurs call this a "threshold".

You think Mischa Schwartz is an amateur?

Jerry
--
"The rights of the best of men are secured only as the
rights of the vilest and most abhorrent are protected."
- Chief Justice Charles Evans Hughes, 1927
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

Jerry Avins wrote:

>
>> The discriminator goofs up when the normalized SNR falls below ~10dB.
>> The amateurs call this a "threshold".
>
>
> You think Mischa Schwartz is an amateur?

I think that we should distinguish the theoretical performance of the FM
from the inefficiency of the particular demodulator. Don't know about
Schwarts, however if someone does not understand the distinction, then
he is an amateur.
Indeed, the "ideal case" performance of the FM can be derived.

Vladimir Vassilevsky

DSP and Mixed Signal Design Consultant

http://www.abvolt.com

Vladimir Vassilevsky wrote:
>
>
> Randy Yates wrote:
>
>> John Sampson <[email protected]> writes:
>>
>>
>>> [email protected] wrote:
>>>
>>>>> Pt = transmit carrier power A^2/2
>>>>> W = output LPF bandwidth
>>>>> Fd = peak deviation
>>>>> No = input noise density
>>>>> m = modulating signal
>>>>>
>>>>> SNRo = 3 * mean(m^2) * (Fd / W)^2 * Pt / (No * W)
>>>>
>>>> so if my m signal is a sinus at 250 kHz with 1 V rms then mean(m^2) =
>>>> 0.5
>>>> if my Fd = 250 kHz and W = 250 kHz then (Fd / W)^2 = 1
>>>> if Pt / (No * W) = SNR input, at the input at the modulator I simulate
>>>> the SNR (measure it)
>>>> then SNRo = 3 * 0.5 * 1 * SNRi = 1.5 * SNRi
>>>> is it exact ?
>>>>
>>>
>>> You should see SNRo > SNRi when the discriminator operates above
>>> threshold (no clicks). This is called FM improvement. I've obtained
>>> good agreement with the theory in the past.
>>
>>
>> Hi John,
>>
>> If the discriminator is just dtheta/dt where theta is the phase of the
>> analytic baseband signal, then what is the thing you call the
>> "threshold?" Just curious.
>
> Hello Randy,
>
> The approximation quoted by John is valid on the two assumptions:
> 1. SNR is high. 2. The detector is a dumb discriminator.
>
> We should distinguish the properties of the FM signal itself from the
> inefficiency of the particular demodulator.
>
> The discriminator goofs up when the normalized SNR falls below ~10dB.
> The amateurs call this a "threshold".
>

So do the professionals.

John

Randy Yates wrote:
> John Sampson <[email protected]> writes:
>
>> [email protected] wrote:
>>>> Pt = transmit carrier power A^2/2
>>>> W = output LPF bandwidth
>>>> Fd = peak deviation
>>>> No = input noise density
>>>> m = modulating signal
>>>>
>>>> SNRo = 3 * mean(m^2) * (Fd / W)^2 * Pt / (No * W)
>>> so if my m signal is a sinus at 250 kHz with 1 V rms then mean(m^2) =
>>> 0.5
>>> if my Fd = 250 kHz and W = 250 kHz then (Fd / W)^2 = 1
>>> if Pt / (No * W) = SNR input, at the input at the modulator I simulate
>>> the SNR (measure it)
>>> then SNRo = 3 * 0.5 * 1 * SNRi = 1.5 * SNRi
>>> is it exact ?
>>>
>> You should see SNRo > SNRi when the discriminator operates above
>> threshold (no clicks). This is called FM improvement. I've obtained
>> good agreement with the theory in the past.
>
> Hi John,
>
> If the discriminator is just dtheta/dt where theta is the phase of the
> analytic baseband signal, then what is the thing you call the
> "threshold?" Just curious.

If you lower the power of the FM signal entering the discriminator, the
"below threshold" region is entered when clicks begin to appear in the
output. This is the point at which the slope of the SNRo vs SNRi curve
changes.

John

Vladimir Vassilevsky wrote:
>
>
> Jerry Avins wrote:
>
>>
>>> The discriminator goofs up when the normalized SNR falls below ~10dB.
>>> The amateurs call this a "threshold".
>>
>>
>> You think Mischa Schwartz is an amateur?
>
> I think that we should distinguish the theoretical performance of the FM
> from the inefficiency of the particular demodulator. Don't know about
> Schwarts, however if someone does not understand the distinction, then
> he is an amateur.
> Indeed, the "ideal case" performance of the FM can be derived.

The notion of threshold has been around for a long time. See, for
example, M. G. Crosby, "Frequency Modulation Noise Characteristics",
Ptoc. IRE, vol. 25, pp 472-514, April 1937.

Jerry
--
"The rights of the best of men are secured only as the
rights of the vilest and most abhorrent are protected."
- Chief Justice Charles Evans Hughes, 1927
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

Vladimir Vassilevsky wrote:
> Jerry Avins wrote:
>
> >
> >> The discriminator goofs up when the normalized SNR falls below ~10dB.
> >> The amateurs call this a "threshold".
> >
> >
> > You think Mischa Schwartz is an amateur?
>
> I think that we should distinguish the theoretical performance of the FM
> from the inefficiency of the particular demodulator. Don't know about
> Schwarts, however if someone does not understand the distinction, then
> he is an amateur.
> Indeed, the "ideal case" performance of the FM can be derived.
>

Is the threshold effect a function of the inefficiency of a particular
demodulator or is the threshold effect an inherent theoretical function
of FM? Does not even the "ideal case" FM demodulator exhibit a
"threshold"?

Yes, I know diffrent demodulators have diffrent thresholds and
"threshold extension" is an interesting topic but I thought even an
ideal FM demodulator would have a threshold.

Mark

The threshold of FM demodulator is for every type of demodulator
implementation. To demodulate an FM signal you need to differentiate the
phase over time.

At low SNR the trajectory of the phase can cross the origin (zero amplitude)
on the wrong side (due to noise). This create a click. The threshold is the
SNR where this phenomena start occurring often.

Here is a simple example. In a sampled signal if the phase is going from 0
to 45 degree in two steps 0, 22.5 and 45 degree. Let assume that the SNR is
very low and the path from 0 to 45 degree cross the origin at the middle
step and it is interpreted as 202.5 degree. This will result in a click
(large amplitude spike at the output of the demodulator).

LM

"Mark" <[email protected]> wrote in message
news:[email protected] oups.com...
>
> Vladimir Vassilevsky wrote:
>> Jerry Avins wrote:
>>
>> >
>> >> The discriminator goofs up when the normalized SNR falls below ~10dB.
>> >> The amateurs call this a "threshold".
>> >
>> >
>> > You think Mischa Schwartz is an amateur?
>>
>> I think that we should distinguish the theoretical performance of the FM
>> from the inefficiency of the particular demodulator. Don't know about
>> Schwarts, however if someone does not understand the distinction, then
>> he is an amateur.
>> Indeed, the "ideal case" performance of the FM can be derived.
>>
>
> Is the threshold effect a function of the inefficiency of a particular
> demodulator or is the threshold effect an inherent theoretical function
> of FM? Does not even the "ideal case" FM demodulator exhibit a
> "threshold"?
>
> Yes, I know diffrent demodulators have diffrent thresholds and
> "threshold extension" is an interesting topic but I thought even an
> ideal FM demodulator would have a threshold.
>
> Mark
>

Vladimir Vassilevsky wrote:

>
>
> Jerry Avins wrote:
>
>>
>>> The discriminator goofs up when the normalized SNR falls below ~10dB.
>>> The amateurs call this a "threshold".
>>
>>
>>
>> You think Mischa Schwartz is an amateur?
>
>
> I think that we should distinguish the theoretical performance of the FM
> from the inefficiency of the particular demodulator. Don't know about
> Schwarts, however if someone does not understand the distinction, then
> he is an amateur.
> Indeed, the "ideal case" performance of the FM can be derived.
>
Is there such a derivation on the web? I'd be interested in reading it
-- IIRC my Van Trees just gives some upper & lower bounds, not a
theoretical best performance.

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Posting from Google? See http://cfaj.freeshell.org/google/

"Applied Control Theory for Embedded Systems" came out in April.
See details at http://www.wescottdesign.com/actfes/actfes.html

Vladimir Vassilevsky wrote:

(snip)

> The approximation quoted by John is valid on the two assumptions:
> 1. SNR is high. 2. The detector is a simple discriminator.

> We should distinguish the properties of the FM signal itself from the
> inefficiency of the particular demodulator.

> The discriminator fails when the normalized SNR falls below ~10dB. The
> popular word for this is a "threshold".

I used to read about TV satellite signals, the big dish ones, not the
small dish ones now popular for home use. If I remember, they broadcast
something like six watts/channel spread over most of the US, and
probably some of the ocean. They depend on what I believe is called
the "FM advantage" to get a reasonable SNR out. Something like 25dB
better than they would otherwise get.

-- glen

Mark wrote:

>>I think that we should distinguish the theoretical performance of the FM
>>from the inefficiency of the particular demodulator.
>
> Is the threshold effect a function of the inefficiency of a particular
> demodulator or is the threshold effect an inherent theoretical function
> of FM? Does not even the "ideal case" FM demodulator exhibit a
> "threshold"?

Good question.

Consider the FM signal with the deviation (-Fd,+Fd) and the modulation
signal bandwidth (0,Fs). Assume the modulation signal already has the
uniform statistics. It is definitely not the case with the audio signal,
however the predictability is the different question and we are not
going to consider it at this time. The noise is additive gaussian.

The ideal FM demodulator is solving the following problem: estimate the
most likely frequency of the carrier in the range of (-Fd,+Fd) based on
the observation of the input on the duration of 1/Fs.

There are two cases here:

a) The power of the noise in the bandwidth (0,Fs) is comparable to the
power of the signal. In this case the frequency estimate error will look
like the erf(input SNR), i.e. exponential type dependence. The noise at
the output is white and uniformly distributed.

b) The power of the noise in the bandwidth (0,Fs) is much smaller then
the power of the signal. The output error is directly proportional to
the input SNR (linear dependence). The noise at the output is white
gaussian.

Where do those cases met?
(-Fd,+Fd)
Here is a coarse estimate: SNR ~ 3 ---------
(0,Fs)

Note: the SNR is defined as the ratio of the RMS input signal to the RMS
noise in the bandwidth (0,Fs).

We can call this SNR value the threshold of the ideal FM demodulator.

I let the others to elaborate on the exact numbers.


>
> Yes, I know diffrent demodulators have diffrent thresholds and
> "threshold extension" is an interesting topic but I thought even an
> ideal FM demodulator would have a threshold.
>
> Mark


Vladimir Vassilevsky

DSP and Mixed Signal Design Consultant

http://www.abvolt.com

L. M. wrote:

> The threshold of FM demodulator is for every type of demodulator
> implementation.

A kind of.

> To demodulate an FM signal you need to differentiate the
> phase over time.

This is not right. The task is to measure the frequency. You don't have
to differentiate the phase to do that.

>
> At low SNR the trajectory of the phase can cross the origin (zero amplitude)
> on the wrong side (due to noise). This create a click. The threshold is the
> SNR where this phenomena start occurring often.

Don't mix the details of the implementation with the fundamental problems.


>
> Here is a simple example. In a sampled signal if the phase is going from 0
> to 45 degree in two steps 0, 22.5 and 45 degree. Let assume that the SNR is
> very low and the path from 0 to 45 degree cross the origin at the middle
> step and it is interpreted as 202.5 degree. This will result in a click
> (large amplitude spike at the output of the demodulator).
>
> LM

Vladimir Vassilevsky

DSP and Mixed Signal Design Consultant

http://www.abvolt.com

Tim Wescott wrote:

> Vladimir Vassilevsky wrote:
>

>> I think that we should distinguish the theoretical performance of the
>> FM from the inefficiency of the particular demodulator.
>> Indeed, the "ideal case" performance of the FM can be derived.

> Is there such a derivation on the web?

For the ideal FM demodulator, it appears to be fairly straightforward.
For the discriminators, it looks much more complex. I am too lazy to
derive the estimates.

I'd be interested in reading it
> -- IIRC my Van Trees just gives some upper & lower bounds, not a
> theoretical best performance.

This has to do with a classical problem of the frequency estimation of
the noisy signal.

Vladimir Vassilevsky

DSP and Mixed Signal Design Consultant

http://www.abvolt.com

glen herrmannsfeldt wrote:

> Vladimir Vassilevsky wrote:
>
>> The approximation quoted by John is valid on the two assumptions:
>> 1. SNR is high. 2. The detector is a simple discriminator.
>
> I used to read about TV satellite signals, the big dish ones, not the
> small dish ones now popular for home use. If I remember, they broadcast
> something like six watts/channel spread over most of the US, and
> probably some of the ocean. They depend on what I believe is called
> the "FM advantage" to get a reasonable SNR out. Something like 25dB
> better than they would otherwise get.

Now about the "FM advantage":

Suppose the FM is operating at the SNR well above the so-called
"treshold". In this case, the absolute error in the frq. estimate is
directly proportional to the SNR. However voltage error at the output of
the demodulator is relative to the deviation. Therefore:

(-Fd,+Fd)
SNR at the output = SNR at the input ---------
(0, Fs)


Consider the FM broadcast: Fd ~ 75kHz, Fs ~ 15kHz, SNRout ~ 60dB.

According to my coarse estimates:

Threshold SNR at the input: ~ 30dB (in the bandwidth of Fs = 15kHz).
FM advantage in SNR: ~20dB if operating at the SNR much higher then the
threshold. The required SNR at the input ~ 40dB.

So, the great ancient engineers definitely knew what they were doing...

Vladimir Vassilevsky

DSP and Mixed Signal Design Consultant

http://www.abvolt.com

"Vladimir Vassilevsky" <[email protected]> wrote in message
news:Bw30h.1287$[email protected] ..
>
>
> Randy Yates wrote:
>
> > John Sampson <[email protected]> writes:
> >
> >
> >>[email protected] wrote:
> >>
> >>>>Pt = transmit carrier power A^2/2
> >>>>W = output LPF bandwidth
> >>>>Fd = peak deviation
> >>>>No = input noise density
> >>>>m = modulating signal
> >>>>
> >>>>SNRo = 3 * mean(m^2) * (Fd / W)^2 * Pt / (No * W)
> >>>
> >>>so if my m signal is a sinus at 250 kHz with 1 V rms then mean(m^2) =
> >>>0.5
> >>>if my Fd = 250 kHz and W = 250 kHz then (Fd / W)^2 = 1
> >>>if Pt / (No * W) = SNR input, at the input at the modulator I simulate
> >>>the SNR (measure it)
> >>>then SNRo = 3 * 0.5 * 1 * SNRi = 1.5 * SNRi
> >>>is it exact ?
> >>>
> >>
> >>You should see SNRo > SNRi when the discriminator operates above
> >>threshold (no clicks). This is called FM improvement. I've obtained
> >>good agreement with the theory in the past.
> >
> >
> > Hi John,
> >
> > If the discriminator is just dtheta/dt where theta is the phase of the
> > analytic baseband signal, then what is the thing you call the
> > "threshold?" Just curious.
>
> Hello Randy,
>
> The approximation quoted by John is valid on the two assumptions:
> 1. SNR is high. 2. The detector is a dumb discriminator.
>
> We should distinguish the properties of the FM signal itself from the
> inefficiency of the particular demodulator.
>
> The discriminator goofs up when the normalized SNR falls below ~10dB.
> The amateurs call this a "threshold".



Not just amateurs! It's called the threshold effect in any text book on FM.
You will see a graph of SNRo vs SNRin for various values of beta (FM
modulation index).


M.




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"Vladimir Vassilevsky" <[email protected]> wrote in message
news:eOo0h.1676$[email protected] ..
>
>
> L. M. wrote:
>
> > The threshold of FM demodulator is for every type of demodulator
> > implementation.
>
> A kind of.
>
> > To demodulate an FM signal you need to differentiate the
> > phase over time.
>
> This is not right. The task is to measure the frequency. You don't have
> to differentiate the phase to do that.
>
> >
You are confusing phase modulation with FM. You need a differentiator of
sorts. Even a PLL does this - it has an integrator (VCO) in its feedback
path.


M.



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