# Highpass filter applied to noise

Suppose the input signal is represented as $$y=x+w$$ where $x$ is the signal to be estimated and $w$ is AWGN. I design a high pass filter $H$. What does $$H^{T}Hw$$ represent? Will it be just AWGN but attenuated or will it AWGN completely distorted. We can also assume that $\sigma_w = 0.2$

• $H^\mathrm{T}$ is the adjoint filter and $H^\mathrm{T}Hw$ is simply applying it to the output of the first filter. Multiply the two spectrums of $h$ and its adjoint to see what the combined effect of the two will be. Commented Oct 9, 2013 at 23:13
• It appears that you refering to some literature but you fail to mention what is sigma_w means here. Further Transpose(H).H is a constant (Sum of Squares of filter coefficients). Then Transpose(H).H.w will be a scaled signal.
– Ram
Commented Oct 11, 2013 at 4:12
• $\sigma_w=0.2$ implies the standard deviation of the noise. And btw $H^THw$ is not a scaled signal. Nikita's answer below seems right.
– AAP
Commented Oct 12, 2013 at 4:25