# SNR for channel after equalization

Suppose I have a channel described by $h[n]$ or $H(z)$ over which I send a simple PAM2 signal. After the channel AWGN is added.

• What is the best possible SNR with equalization?

As suggested in the comments I could assume a zero forcing or MMSE equalizer and e.g. for the former arrive at:

$$\mathrm{SNR_{out}} = \mathrm{SNR_{in}} \frac{1}{\frac{2}{N} \sum_{k=0}^{N/2+1} \frac{1}{|H[k]|^2} }$$

which matches well with my simulations. However, what I am looking for is the best theoretic bound, not assuming any particular equalization technique. For this derivation, it is assumed that I split the channel into small pieces of frequency bands (over which the channel is assumed constant and hence no equalization required) and equate the channel capacity to the channel capacity of an ideal AWGN channel. This results in:

$$10\log_{10}(1+\mathrm{SNR_{out}}) = \frac{2}{N} \sum_{k=0}^{N/2-1} 10\log_{10}(1 + \mathrm{SNR}_k) \approx \mathrm{SNR_{out,dB}}$$

and $\mathrm{SNR}_k = |H[k]|^2/\sigma^2/(N/2)$.

However, something is odd with this: Assuming a perfect channel, an input signal of unit power and $\sigma=0.001$, the SNR would be 60 dB. Executing the formula above for any lowpass filter (e.g. $N=1000$), the SNR is better than 60 dB because the low frequency SNRs are so high. For example, for low $k$, $|H[k]|\approx 1$ and hence $\mathrm{SNR}_k=1/\sigma^2/(N/2)=90\,\mathrm{dB}$. Even if they degrade at higher frequencies, the average is still more than 60 dB.

• I think the keywords should be "zero forcing" and "MMSE equalizers". – AlexTP Jun 30 '17 at 7:19
• Variance is equal to the average power only when the random variable is zero mean. Make sure you are not calculating the power incorrectly. See the answer I just gave to another question. – msm Jun 30 '17 at 19:18
• AlexTP: Thanks I can confirm results with ZF or MMSE equalizer formulas. But I am looking for a theoretic bound independent of the equalizer. I think the approach is based on Shannon channel capacity with variable SNR. I updated my question to make it more clear. – divB Jul 5 '17 at 18:26