GATE 2019 EC – Question 54

Let $h[n]$ be a length-7 discrete-time finite impulse response filter, given by

$h[0]=4,\ h[1]=3,\ h[2]=2,\ h[3]=1,\ h[-1]=-3,\ h[-2]=-2,\ h[-3]=-1,$

and $h[n]$ is zero for $|n|\ge4$. A length-3 finite impulse response approximation $g[n]$ of $h[n]$ has to be obtained such that

$$E(h,g)=\int_{-\pi}^{\pi}\left|H(e^{j\omega})-G(e^{j\omega})\right|^2d\omega$$

is minimized, where $H(e^{j\omega})$ and $G(e^{j\omega})$ are the discrete-time Fourier transforms of $h[n]$ and $g[n]$, respectively. For the filter that minimizes $E(h,g)$, the value of $10g[-1]+g[1]$, rounded off to 2 decimal places, is ________.