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I have a following toy example: received signal: $y[n] = x[n] - x[n-1] + z[n]$ and with $\text{Pr}(x[n]=1)=0.3=\text{Pr}(x[n]=-1)$ and $\text{Pr}(x[n]=0)=0.4$ and $z[n]$ is a Gaussian distribution.

I want to use a 3 tap zero forcing filter as am equalizer. How can I calculate the signal to interference ratio (SINR)?

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  • $\begingroup$ Calculate the signal out/signal in and then calculate interference out/interference in and take the ratio of the two $\endgroup$ – Dan Boschen Apr 3 at 21:46
  • $\begingroup$ I'm sorry, can't read your mind. What system? What interferers? What signal power? Basically: you've described something that is (probably) part of an equalizer. It's absolutely not clear what this has to do with SINR, you'll need to give a lot more background. $\endgroup$ – Marcus Müller Apr 3 at 21:47
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    $\begingroup$ Assuming you want a detailed answer you should add some more details like more information about your signal, interference, noise, and channel. Otherwise, it is just a matter of writing out your composite received signal ($s + i + n$) after it is passed through your filter ($h*s + h*i + h*n$) then your SINR is $\frac{\sum |h*s|^2}{\sum |h*i|^2+\sum |h*n|^2}$ $\endgroup$ – Engineer Apr 3 at 22:09
  • $\begingroup$ Hi thank you for your responses and apologies for providing a not very detailed question. Above I tried to explain my question through numbers, if I could get a walkthrough on how to calculate the filter coefficients and the SINR it would be great $\endgroup$ – user49544 Apr 3 at 23:16
  • $\begingroup$ Something is wrong with your source signal $x[n]$, its probabilities don't add up to one its probably just a typo but you should fix it $\endgroup$ – Engineer Apr 4 at 0:22
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The ZF equalizer, $f[n]$, is a filter (I'm assuming a FIR filter) that tries to force, $f[n]*h[n]$, to be $\delta[n-d]$, where $d$ is the sample delay introduced by the filter (I choose $d=2$ below) and $h[n]$ is the channel impulse response. You can use this convolution equation to solve for the filter weights. I assume that both $f[n]$ and $h[n]$ are zero for all $n < 0$.

For $n=0$ we have: $f[0]h[0]= 0$.

For $n=1$ we have: $f[0]h[1]+f[1]h[0]=1$.

For $n=2$ we have: $f[0]h[2]+f[1]h[1]+f[2]h[0]=0$.

For $n=3$ we have: $f[0]h[3]+f[1]h[2]+f[2]h[1]=0$.

(We go from $n=0$ to $n=3$ because the length of $f[n]*h[n]$ is $4$) We can re-write these equations in a matrix form as $\mathbf{H}\mathbf{f}=\mathbf{d}$, where $\mathbf{d}$ is all zero except for the $d^{th}$ element which is equal to one. From here we can solve for the filter weights as $\mathbf{f}=\mathbf{H}^{+}\mathbf{d}$, where $\mathbf{H}^+$ is the pseudo-inverse of $\mathbf{H}$.

Note: You are given the received signal model in terms of the input signal and the additive noise. From this, you can see the channel impulse response $h[n]=\delta[n]-\delta[n-1]$. Now you are equipped with everything you need to find the filter weights, and be sure at the end to check that $f[n]*h[n]$ does actually look "something like" (there may be small perturbations) an impulse at whatever sample delay that you choose.

To compute the SINR, I am not sure but I am guessing you are considering the $x[n-1]$ term to be the interference. In that case, you can still use the approach from my comment since the signal model is still in the same form.

Code: https://github.com/B-William/DSPSE/blob/master/zfEqualizerScript.m

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