Eb/No and SNR

Hi, could someone explain me the difference and the relationship between
SNR ratio and Eb/No ratio ?
thanks a lot !

"dude23" <pf_morlat@yahoo.fr> writes:

> Hi, could someone explain me the difference and the relationship between
> SNR ratio and Eb/No ratio ?
> thanks a lot !

Energy per bit (Eb) is defined as the ratio of signal power to
bit-rate,

  Eb = Ps / B [joules / bit],

where Ps is in [watts = joules / sec] and B is in [bits / sec]. The
bit-rate can be expressed in terms of the spectral efficiency b
[bits / sec / Hz] and bandwidth W [Hz] required for the signal as 

  B = b * W [bits / sec].

Therefore

  Eb = Ps / (b * W) [joules / bit].

Now note that if the noise is flat with constant power spectral
density No [watts / Hz = joules], then the noise power Pn in bandwidth
W [Hz = 1 / sec] is

  Pn = No * W [joules / sec = watts].

Thus the signal-to-noise ratio SNR is 

  SNR = Ps / Pn 
      = Ps / (No * W),

and Eb / No is related to SNR as

  Eb / No = Ps / (b * W * No)
          = SNR / b [1 / bit].

See [lessandmesserschmitt3] for a good coverage of this.

To state it more heuristically, Eb/No is a sort of normalized way to
look at SNR in which the SNR is normalized to the bit rate. For
example, you wouldn't want to consider two signals that have the same
SNR to be equivalent (in terms of energy efficiency) if one was twice
the bit rate of the other.

--Randy

@BOOK{leeandmesserschmitt3,
  title = "{Digital Communication}",
  author = "{John R. Barry and Edwared A. Lee and David G. Messerschmitt}",
  publisher = "Springer",
  edition = "third",
  year = "2004"}

-- 
%  Randy Yates                  % "My Shangri-la has gone away, fading like 
%% Fuquay-Varina, NC            %  the Beatles on 'Hey Jude'" 
%%% 919-577-9882                %  
%%%% <yates@ieee.org>           % 'Shangri-La', *A New World Record*, ELO
http://home.earthlink.net/~yatescr

Randy Yates <yates@ieee.org> writes:

> See [lessandmesserschmitt3] for a good coverage of this.

Sorry - that should have been [leeandmesserschmitt3].

> @BOOK{leeandmesserschmitt3,
>   title = "{Digital Communication}",
>   author = "{John R. Barry and Edwared A. Lee and David G. Messerschmitt}",
>   publisher = "Springer",
>   edition = "third",
>   year = "2004"}

--Randy

-- 
%  Randy Yates                  % "She tells me that she likes me very much,
%% Fuquay-Varina, NC            %     but when I try to touch, she makes it
%%% 919-577-9882                %                            all too clear."
%%%% <yates@ieee.org>           %        'Yours Truly, 2095', *Time*, ELO  
http://home.earthlink.net/~yatescr

After pondering this awhile, I think I made couple of wrong statements
and need to correct myself. Everything I initially wrote prior to the
point below was correct.

Randy Yates <yates@ieee.org> writes:
> [...]
> To state it more heuristically, Eb/No is a sort of normalized way to
> look at SNR in which the SNR is normalized to the bit rate. 

No. Eb / No is SNR normalized to the spectral efficiency (b). That's 
pretty much what the mathematical relationship stated directly:

  Eb / No = SNR / b.

> For example, you wouldn't want to consider two signals that have the
> same SNR to be equivalent (in terms of energy efficiency) if one was
> twice the bit rate of the other.

Just ignore the above statement completely. 

If we're concerned with energy efficiency, then we have to look no
further than Eb (energy per bit) alone. 

If we're concerned with spectral efficieny, then we have to look
no further than b (spectral efficiency) alone. 

Eb / No is a significant ratio because the performance (i.e., bit-error
rate) of many common digital comm signals is a function of this ratio.

To see why we wouldn't use SNR in performance computations, consider
comparing two signals S1 and S2 with the same bit rate B and energy
per bit Eb but with varying spectral efficiencies b1 = 1 bit / sec / Hz 
and b2 = 2 bit / sec / Hz:

  Ps = Eb * B
  W = B / b
  Pn = No * W
  SNR = Ps / Pn

  Qty         Signal 1      Signal 2         Units
  ---         --------      --------         -----
  Ps          Eb * B        Eb * B           [watts]
  W           B             B / 2            [Hz]
  Pn          No * B        No * B / 2       [watts]
  SNR         Eb / No       2 * Eb / No      [none]
  Eb / No     Eb / No       Eb / No          [1 / bit]

The point is that the spectral efficiency caused signal 2 to
have a higher SNR even though the transmitted power was the
same due to the fact that the noise bandwidth is half and
thus the noise power is half. However, this would not have
created any better performance in terms of bit-error rate
since its Eb / No is the same as signal 1.

I hope this is correct and hasn't confused the heck out of
anyone. 
-- 
%  Randy Yates                  % "Midnight, on the water... 
%% Fuquay-Varina, NC            %  I saw...  the ocean's daughter." 
%%% 919-577-9882                % 'Can't Get It Out Of My Head' 
%%%% <yates@ieee.org>           % *El Dorado*, Electric Light Orchestra
http://home.earthlink.net/~yatescr

ok, I understand better
thanks a lot


>After pondering this awhile, I think I made couple of wrong statements
>and need to correct myself. Everything I initially wrote prior to the
>point below was correct.
>
>Randy Yates <yates@ieee.org> writes:
>> [...]
>> To state it more heuristically, Eb/No is a sort of normalized way to
>> look at SNR in which the SNR is normalized to the bit rate. 
>
>No. Eb / No is SNR normalized to the spectral efficiency (b). That's 
>pretty much what the mathematical relationship stated directly:
>
>  Eb / No = SNR / b.
>
>> For example, you wouldn't want to consider two signals that have the
>> same SNR to be equivalent (in terms of energy efficiency) if one was
>> twice the bit rate of the other.
>
>Just ignore the above statement completely. 
>
>If we're concerned with energy efficiency, then we have to look no
>further than Eb (energy per bit) alone. 
>
>If we're concerned with spectral efficieny, then we have to look
>no further than b (spectral efficiency) alone. 
>
>Eb / No is a significant ratio because the performance (i.e., bit-error
>rate) of many common digital comm signals is a function of this ratio.
>
>To see why we wouldn't use SNR in performance computations, consider
>comparing two signals S1 and S2 with the same bit rate B and energy
>per bit Eb but with varying spectral efficiencies b1 = 1 bit / sec / Hz 
>and b2 = 2 bit / sec / Hz:
>
>  Ps = Eb * B
>  W = B / b
>  Pn = No * W
>  SNR = Ps / Pn
>
>  Qty         Signal 1      Signal 2         Units
>  ---         --------      --------         -----
>  Ps          Eb * B        Eb * B           [watts]
>  W           B             B / 2            [Hz]
>  Pn          No * B        No * B / 2       [watts]
>  SNR         Eb / No       2 * Eb / No      [none]
>  Eb / No     Eb / No       Eb / No          [1 / bit]
>
>The point is that the spectral efficiency caused signal 2 to
>have a higher SNR even though the transmitted power was the
>same due to the fact that the noise bandwidth is half and
>thus the noise power is half. However, this would not have
>created any better performance in terms of bit-error rate
>since its Eb / No is the same as signal 1.
>
>I hope this is correct and hasn't confused the heck out of
>anyone. 
>-- 
>%  Randy Yates                  % "Midnight, on the water... 
>%% Fuquay-Varina, NC            %  I saw...  the ocean's daughter." 
>%%% 919-577-9882                % 'Can't Get It Out Of My Head' 
>%%%% <yates@ieee.org>           % *El Dorado*, Electric Light Orchestra
>http://home.earthlink.net/~yatescr
>

>and Eb / No is related to SNR as
>
>  Eb / No = Ps / (b * W * No)
>          = SNR / b [1 / bit].

First off, your explanation is really helpful.I have a query regarding the
units of above expression. I understand that expression on LHS (left hand
side) has units of 1/bit. (as Eb is Energy per bit).

Now for RHS expression, You have written units as "[1 / bit]". 
I am unable to understand it because "b" being spectral efficiency has
units bits/s/Hz, so how come you have written only 1/bit.

/s/Hz units are absent. Is this because /s/Hz cancel out each other.

Please help me out.



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a.s. <104988@DSPRelated> wrote:

[ attribution lost wrote]

>>and Eb / No is related to SNR as
>>
>>  Eb / No = Ps / (b * W * No)
>>          = SNR / b [1 / bit].

>First off, your explanation is really helpful.I have a query regarding the
>units of above expression. I understand that expression on LHS (left hand
>side) has units of 1/bit. (as Eb is Energy per bit).

The above sort of formula is the closest you can get to relating
Eb/No to SNR, however there are situations where applying this
formula directly might be misleading.

Ignoring that detail and looking at the units:

     Eb/No  =  (energy per bit ) / (noise power per Hz)

     Signal power = (energy per bit) * (bits per Hz)

     Noise power = (noise power per Hz) *  (BW in Hz)

therefore

     SNR =  Eb * (bits per Hz) / ( (noise power per Hz) (BW in Hz)

         = (Eb/No) * (bits per Hz) / BW

Since "bits per hertz" and "BW" (bandwidth) both have units of
"frequency", SNR is dimensionless as is Eb/No.

Hope this helps.

Steve

[...snip...]

> [Hz = 1 / sec] 
>

[...snip...]

>
>--Randy
>

I realize the original post is from eight years ago, but I disagree with
your units for Hz being 1/sec, it should be cycles per second.

From Wikipedia:

"The hertz (symbol Hz) is the unit of frequency in the International
System of Units (SI) and is defined as one cycle per second."

This makes more sense in calculations like the following:  

Consider a tone at 1000 Hz.

CD Quality = 44,100 Samples per second

44,100 Samples per second / 1000 cycles per second = 44.1 Samples per
cycle

Speed of sound in air 1130 feet per second

1130 feet per second / 1000 cycles per second = 1.13 feet per cycle

Ced
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Cedron <103185@DSPRelated> wrote:

>I realize the original post is from eight years ago, 

I was wondering about that.

> but I disagree with your units for Hz being 1/sec, it should be cycles 
> per second.

Hz denotes cycles per second, but dimensionally it is 1 / time.
A "cycle" is dimensionless, as is a "radian", a "symbol", a
"sample" or a "baud".

To possibly tighten up the language used in previous posts the word
"unit" should probably be replaced by "has dimensions of".

Steve

>Cedron <103185@DSPRelated> wrote:
>
>>I realize the original post is from eight years ago, 
>
>I was wondering about that.
>
>> but I disagree with your units for Hz being 1/sec, it should be cycles

>> per second.
>
>Hz denotes cycles per second, but dimensionally it is 1 / time.
>A "cycle" is dimensionless, as is a "radian", a "symbol", a
>"sample" or a "baud".
>
>To possibly tighten up the language used in previous posts the word
>"unit" should probably be replaced by "has dimensions of".
>
>Steve

I found this tutorial on the topic:

https://www.physics.uoguelph.ca/tutorials/dimanaly/

So, I would say Randy was mostly carrying units all along, not doing a
dimensional analysis.  

In short, the units for Hertz, a measure of frequency, should be
cycles/second and the dimension should be Time^(-1) (not 1/sec).  Well,
that's how I read it.

Ced
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