Question $1$

$x(widehat())=e^{-j4\text{w}\text{i}\text{d}\text{e}\text{h}\text{a}\text{t}()}+2e^{-j5\text{w}\text{i}\text{d}\text{e}\text{h}\text{a}\text{t}()}+4e^{-j6\text{h}\text{a}\text{t}()}$

The block diagram of a system and the DTFT of the input $x[n]$ are given above. Answer the following parts with reference to this system and input.

Question $1$ Part $1$

Which expression below would best represent $x[n]?$

A$)4[n-1]+5[n-2]+6[n-4]$

B$)3[n+4]-7[n+6]$

C$)[n-4]+2[n-5]+4[n-6]$

D$)-[n-2]-2[n-3]-4[n-4]$

Which expression below would the DTFT of the $3-$point causa age filter $H(widehat())?$ A$)e^{-\text{j}\text{w}\text{i}\text{d}\text{e}\text{h}\text{a}\text{t}()}+\frac{1}{2}{e}^{-j2\text{w}\text{i}\text{d}\text{e}\text{h}\text{a}\text{t}()}+\frac{1}{3}{e}^{-j3\text{w}\text{i}\text{d}\text{e}\text{h}\text{a}\text{t}()}$

B$)\frac{1}{3}{e}^{-j4\text{w}\text{i}\text{d}\text{e}\text{h}\text{a}\text{t}()}+\frac{1}{3}{e}^{-j5\text{w}\text{i}\text{d}\text{e}\text{h}\text{a}\text{t}()}+\frac{1}{3}{e}^{-j6\text{w}\text{i}\text{d}\text{e}\text{h}\text{a}\text{t}()}$

C$)\frac{1}{3}+\frac{1}{3}{e}^{-\text{j}\text{w}\text{i}\text{d}\text{e}\text{h}\text{a}\text{t}()}+\frac{1}{3}{e}^{-j2\text{w}\text{i}\text{d}\text{e}\text{h}\text{a}\text{t}()}$

D$)\frac{1}{3}+\frac{1}{3}{e}^{-\text{j}\text{w}\text{i}\text{d}\text{e}\text{h}\text{a}\text{t}()}$

Question $1$ Part $4$

What would be the values of the output $y[n]$ for $n=0,1,$dots,$9?$

$)\{0,\frac{9}{2},0,-5,\frac{3}{8},-1,-\frac{7}{4},0,0,3\}$

B$)\{\frac{1}{3},1,0,0,\frac{7}{3},2,\frac{4}{3},0,0,0\}$

$\{0,\frac{4}{3},2,\frac{7}{3},1,\frac{1}{3},0,0,0,0\}$

D$)\{0,0,0,0,\frac{1}{3},1,\frac{7}{3},2,\frac{4}{3},0\}$