# SINR is 3-tap zero forcing filter

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)?

• Calculate the signal out/signal in and then calculate interference out/interference in and take the ratio of the two – Dan Boschen Apr 3 at 21:46
• 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. – Marcus Müller Apr 3 at 21:47
• 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 $\frac{\sum |h*s|^2}{\sum |h*i|^2+\sum |h*n|^2}$ – Engineer Apr 3 at 22:09
• 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 – user49544 Apr 3 at 23:16
• 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 – Engineer Apr 4 at 0:22

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.