A finite duration discrete time signal {x}( {n}) is obtained by sampling the continuous time signal, x(t)=\cos (200 n t) at sampling instants, t=\frac{n}{400}, n=0, . . ., 7 . The 8 -point DFT and x[n] is defined as {X}[ {k}]=\sum_{ {n}=0}^{7} \times[ {n}] {e}^{ {- {j} \pi {kn}}/{4}} {k}=0,1 . . ., 7 Which is true?

Correct Answer is : Only X[2] and X[6] are non-zero.

Only {X}[3] and {X}[5] are non-zero

Only X[2] and X[6] are non-zero.

GATE Electronics and Communications (EC) 2020 Solved Papers

Question : 41 of 65

Marks: +1, -0

A finite duration discrete time signal

{x}( {n})

is obtained by sampling the continuous time signal,

x(t)=\cos (200 n t)

at sampling instants,

t=\;{n}/{400}, n=0, . . ., 7 .

The 8 -point DFT and

x[n]

is defined as

{X}[ {k}]=∑↙{ {n}=0}↖{7} ×[ {n}] {e}^{\;{- {j} π {kn}}/{4}}

{k}=0,1 . . ., 7

Which is true?

Solution:

X(t)=\cos (200 ϕ t)

t=\;{n}/{400} n=0,1, . . . 7

x(n)=\cos (200 π × \;{n}/{400})=\cos (\;{π}/{2} ⋅ n)

x[0] \;\; =1

x[1] \;\; =0

x[2] \;\; =-1

x[3] \;\; =0

x[4] \;\; =1

x[5] \;\; =0

x[6] \;\; =-1

x[7] \;\; =0

x[0] \;\; =\{1,0,-1,0,1,0,-1,0\}

X(K)=∑↙{n=0}↖{7} x[n] e^{-j \;{π}/{4} K n}

X(3)=∑↙{n=0}↖{7} x[n] e^{-j \;{π}/{4} 7 n}

(\;{1}/{√{2}}-j \;{1}/{2})^{n} ≠ 0

e^{-j \;{7 π}/{4}}=\cos \;{7 × 180}/{4}-\;{j \sin 7 × 180}/{4}=\;{1}/{√{2}}-j \;{1}/{√{2}}

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