Attenuation:

   dB = 10 log10(Pr/Ps)

  loss of power during transmission

 we transmit 10W and recieve 9W, what's the attenuation in decibels?

 10 * log_10 (9W/10W)      <- log base 10.
  or 
 10 * log (9/10) / log(10)   <- any log base (often e).
 = -0.45757

brush up on logs: https://www.youtube.com/watch?v=sULa9Lc4pck



Nyquist Bandwidth

 C = 2B log2 M

Bandwidth is 8kHz, number of votage levels is 4, what's the max capacity?

  C = 2 * 8000 * log(4)/log(2)  <- any log
  C = 32000.00000   
   or
  C = 32kbps


if capacitry is 32kbps, and we're using 4 votage levels, what's the bandwidth?

  .... 8kHz


Shannon Capacity

 --signal to noise ratio in decibels

 SNRdB = 10 log10  signal power / noise power


C = B log2 (1 + SNR)

we're transmitting at 31W, noise on the wire estimated to be 1W, 
bandwidth is 8kHz, what's the max error free capacity?

SNR = 31/1 = 31

C = 8000 * log(1 + 31) / log(2)
 C = 40000 = 40kbps

------

we're transmitting at 40kbps, 2% of the bits arrive flipped.
what's the error free capacity of this channel?

assume we have a wire that indicates when an error occurs [e.g. 40kbps, a 1 for an error and 0 for no-error]. what's the entropy of this signal?

H = -(0.02*log(0.02)/log(2) + (1 - 0.02)*log(1 - 0.02)/log(2) )
H = 0.14144

every symbol (1 or 0) of this "error channel" is actually 0.14144 bits of information.

we're transmitting 40kbps of these errors: 40000 * 0.14144. 
 5657.6bps of errors...

 total capacity is 40kbps, errors are 5.657kbps, leaving:
   40000 - 5657 = 34343 bits/second for error-free-capacity.

~ 34.3kbps

max length of LAN: http://theparticle.com/cs/bc/net/ether.pdf
MaxLength = (2 × 108) × (51.2 × 10−6/2) = 5120m