# BER performance of a colored noise

I have generated colored noise from the white gaussian noise using auto-regressive model (by solving Yule-walker equation). The covariance matrix that I used to generate the colored noise is [1 0.2;0.2 1].

It shows that variance(power) of white noise and colored noise are same. So if I don't whiten the colored noise, the BER performance of the white noise and colored noise should be same because power of both noise are same (1 in this case).

But the simulation shows that the BER performance of colored noise is nearly 1.4 dB degraded as compared to the white noise. It seems the covariance term (0.2) is causing this BER degradation.

Can anyone please explain why this happened? I used the matched filter as the original pulse shape in both cases. Does this affect the performance?

I am considering my threshold as: if the matched filter output at signal interval >0 then bit 1 is transmitted else bit 0 is transmitted.

• wow! complex problem. However, this really reads like a giant runaway sentence. Could you use empty lines to mark logical paragraphs, so that it's clearer which deductions base on what? Thank you! – Marcus Müller Jun 12 '18 at 19:24
• Thanks for the response. I have edited the question. Hope the question becomes clear now. – San Jun 12 '18 at 19:38

When the noise is not white, the filter matched to the pulse shape does not maximize SNR anymore. The Wikipedia article on matched filter addresses this situation and shows the new matched filter to be proportional to $R_v^{-1}s$, where $R_v$ is the noise covariance matrix and $s$ is the signal vector. Using the same notation as the Wikipedia article, if you kept the filter matched to the original pulse shape $s$, your signal component would be $\alpha s^{H}s$, your noise component would be $\alpha s^{H} v$, and the SNR would be
$$\frac{(s^{H}s)^2}{s^{H}R_vs} = \frac{s^Hs}{\bar{s}^HR_v\bar{s}}=\frac{E_s}{\bar{s}^HR_v\bar{s}}$$
Here $E_s$ is the energy per symbol and $\bar{s}=\frac{s}{\sqrt{s^Hs}}$ is the signal normalized to unit energy. This just expresses as an equation what we already concluded: that the SNR will depend on the noise correlation (or power spectral density) with respect to the signal.
If you applied a noise whitening filter, then $R_v$ returns to a scaled identity matrix but you have now introduced intersymbol interference.