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For the causal LTI digital filter with impulse response given by $h(n)=0.3 \delta(n)- \delta(n-1)+ 0.38 \delta(n-3)$, sketch the magnitude spectrum of the filter. Using DFT

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written 7.9 years ago by |

A causal system is one that does not depend on future inputs. Thus a causal system cannot respond before the input is applied.

The impulse response satisfy the linear difference equation.

$h(n)=0.3δ(n)-δ(n-1)+ 0.38δ(n-3) \\ ∴ h(n)=0.3δ(n)-δ(n-1)+0δ(n-2)+0.38δ(n-3) \\ h(0)=0.3δ(0)-δ(-1)+0δ(-2)+0.38δ(-3) = 0.3 \\ h(1)=0.3δ(1)-δ(0)+0δ(-1)+0.38δ(-2) = (-1) \\ h(2)=0.3δ(2)-δ(1)+0δ(0)+0.38δ(-1) = 0 \\ h(3)=0.3δ(3)-δ(2)+0δ(1)+0.38δ(0) = 0.38 \\ h(4)=0.3δ(4)-δ(3)+0δ(2)+0.38δ(1) = 0 \\ . \\ . \\ h(n) = \{0.3, -1, 0, 0.38, 0, 0, 0, . . . . \}$

Here, h(n) can be written as, h(n)= { 0.3, -1, 0, 0.38 }

.....(Since DFT of δ(n)δ(n) is always 1)

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