# Frequency response of filter <-> signal mix

This might be a weird question but here's the setup: I have a few biquads that filter a signal $$x[n]$$ and output the filtered signal $$y[n]$$. I can calculate the frequency response of those biquads with the z-transform. Now I mix $$x[n]$$ and the filtered signal $$y[n]$$ such that the mixed signal $$y'[n]= a \cdot x[n] + (1 - a) \cdot y[n]$$ where $$0\leq a\leq 1$$. How would I go about calculating the frequency response of $$y'[n]$$?

The Z transform is linear. So if $$Y(z) = H(z) \cdot X(z)$$ then $$Y^{'}(z) = a \cdot X(z) + (1-a) \cdot Y(z) = a \cdot X(z) + (1-a) \cdot H(z) \cdot X(z) = (a + (1-a) \cdot H(z)) \cdot X(z)$$
Hence $$H'(z) = a + (1-a) \cdot H(z)$$