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GATE EC 2019 Set 1 — Question 54

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NAT+2 / -0HardDTFT & PropertiesDiscrete-Time SignalsSignals & Systems

Signals & Systems → Discrete-Time Signals → DTFT & Properties

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Let h[n]h[n] be a length-7 discrete-time finite impulse response filter, given byh[0]=4,h[1]=3,h[2]=2,h[3]=1,h[0] = 4, \quad h[1] = 3, \quad h[2] = 2, \quad h[3] = 1,h[−1]=−3,h[−2]=−2,h[−3]=−1,h[-1] = -3, \quad h[-2] = -2, \quad h[-3] = -1,and h[n]h[n] is zero for ∣n∣≥4|n| \ge 4. A length-3 finite impulse response approximation g[n]g[n] of h[n]h[n] has to be obtained such thatE(h,g)=∫−ππ∣H(ejω)−G(ejω)∣2dωE(h, g) = \int_{-\pi}^{\pi} |H(e^{j\omega}) - G(e^{j\omega})|^2 d\omegais minimized, where H(ejω)H(e^{j\omega}) and G(ejω)G(e^{j\omega}) are the discrete-time Fourier transforms of h[n]h[n] and g[n]g[n], respectively. For the filter that minimizes E(h,g)E(h, g), the value of 10g[−1]+g[1]10g[-1] + g[1], rounded off to 2 decimal places, is ________.
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