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].
so, Eb / No is SNR normalized to the spectral efficiency (b).
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.
taken from : http://www.dsprelated.com/showmessage/81112/1.php
Let’s make it more detail :
Eb / No = SNR / b
Eb/No = SNR / (BitRate/BW)
Eb/No(dB) = SNR(dB) – 10*LOG10(BitRate/BW)
Eb/No(dB) = SNR(dB) + 10*LOG10( BW(Hz) / BitRate(Hz) )