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A question on Shannons Formula Simplification

  1. Aug 12, 2011 #1
    Hello all, i encountered a question which i coudnt really solve, was hoping someone could help me. Its regarding Shannons Formula for finding capacity of a channel.

    So it goes like this:

    C = B log (base2 ) [1 + SNR]

    KEY:
    C = capacity
    B = bandwidth
    SNR = SIgnal Noise Ratio

    when the SNR (Signal Noise Ratio) is high, the above formula is not very reliable and we can use this formula instead:

    *Also it states that if the SNR is high we can ignore the 1.

    C = B * [SNR(subscript db)/3]

    Question: How do we get the second formula after simplifying the first?


    -------------------------------------------------------------------------------------

    This is what i have tried till now.

    SNR(subscript db) = 10 log (base 10) SNR

    We make SNR subject of formula :

    10^(SNRdb/10) = SNR

    Therefore substituting SNR in the text with the SNR (subscript dB)gives you

    *We ignore the 1

    C = B * log (base 2) [ 10 ^ (SNRdb / 10)]

    Now im given to understand that to find log base 2 of a number we can do this:


    C = [B * log (base 10) [ 10 ^ (SNRdb / 10)]] / [log (base 10) 2]

    ....

    I am really confuse how to simplify after that.. i dont even know if im right in the first place -.-

    Any help would be appreciated.

    Thanks




    How do we go to this simplified formula from the one given above??
     
  2. jcsd
  3. Aug 13, 2011 #2

    dynamicsolo

    User Avatar
    Homework Helper

    It looks like you basically have this, if not in a form you are recognizing: you use the "change-of-base" formula for logarithms to write

    log(2) SNR = [ log(10) SNR ] / [ log(10) 2 ] .

    Now from your formula 10^(SNRdb/10) = SNR , we have

    SNRdb/10 = log(10) SNR .

    Upon substituting this into the earlier equation, we get

    log(2) SNR = [ SNRdb/10 ] / [ log(10) 2 ] = [ SNRdb ] / [ 10 log(10) 2 ]

    The common log of 2 is 0.3010... , so 10 log(10) 2 is real close to 3 .

    We can now put these pieces together to get, for SNR >> 1 ,

    C = B log(2) [1 + SNR] ≈ B log(2) SNR ≈ B * [ SNRdb ] / 3 .
     
  4. Aug 14, 2011 #3
    thanks for the reply :) i better understand my mistake.. i forgot to make it into the value 0.3 ..... silly me :|

    thanks anyway :D
     
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