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Signals & Systems → Discrete-Time Signals → DFT & FFT
Last updated 5 September 2026
Question Let
X [ k ] = k + 1 , 0 ≤ k ≤ 7 X[k] = k + 1, 0 \leq k \leq 7 X [ k ] = k + 1 , 0 ≤ k ≤ 7 be 8-point DFT of a sequence
x [ n ] x[n] x [ n ] , where
X [ k ] = ∑ n = 0 N − 1 x [ n ] e − j 2 π n k / N X[k] = \sum_{n=0}^{N-1} x[n] e^{-j2\pi nk/N} X [ k ] = ∑ n = 0 N − 1 x [ n ] e − j 2 π nk / N .
The value (correct to two decimal places) of
∑ n = 0 3 x [ 2 n ] \sum_{n=0}^{3} x[2n] ∑ n = 0 3 x [ 2 n ] is
_______ .
Solution Given an 8-point DFT
X [ k ] = k + 1 X[k] = k+1 X [ k ] = k + 1 for
k = 0 , 1 , … , 7 k=0, 1, \dots, 7 k = 0 , 1 , … , 7 .
The DFT is defined as
X [ k ] = ∑ n = 0 N − 1 x [ n ] e − j 2 π n k / N X[k] = \sum_{n=0}^{N-1} x[n] e^{-j2\pi nk/N} X [ k ] = ∑ n = 0 N − 1 x [ n ] e − j 2 π nk / N with
N = 8 N=8 N = 8 .
We need to find the value of
S = ∑ n = 0 3 x [ 2 n ] S = \sum_{n=0}^{3} x[2n] S = ∑ n = 0 3 x [ 2 n ] .
From the definition of the 8-point DFT:
X [ k ] = ∑ n = 0 7 x [ n ] W 8 n k X[k] = \sum_{n=0}^{7} x[n] W_8^{nk} X [ k ] = ∑ n = 0 7 x [ n ] W 8 nk where
W 8 = e − j 2 π / 8 W_8 = e^{-j2\pi/8} W 8 = e − j 2 π /8 .
Specifically, for
k = 0 k=0 k = 0 :
X [ 0 ] = ∑ n = 0 7 x [ n ] = x [ 0 ] + x [ 1 ] + x [ 2 ] + x [ 3 ] + x [ 4 ] + x [ 5 ] + x [ 6 ] + x [ 7 ] X[0] = \sum_{n=0}^{7} x[n] = x[0] + x[1] + x[2] + x[3] + x[4] + x[5] + x[6] + x[7] X [ 0 ] = ∑ n = 0 7 x [ n ] = x [ 0 ] + x [ 1 ] + x [ 2 ] + x [ 3 ] + x [ 4 ] + x [ 5 ] + x [ 6 ] + x [ 7 ] Given
X [ k ] = k + 1 X[k] = k+1 X [ k ] = k + 1 ,
X [ 0 ] = 0 + 1 = 1 X[0] = 0+1 = 1 X [ 0 ] = 0 + 1 = 1 .
For
k = 4 k=4 k = 4 :
X [ 4 ] = ∑ n = 0 7 x [ n ] W 8 4 n = ∑ n = 0 7 x [ n ] e − j π n = ∑ n = 0 7 x [ n ] ( − 1 ) n X[4] = \sum_{n=0}^{7} x[n] W_8^{4n} = \sum_{n=0}^{7} x[n] e^{-j\pi n} = \sum_{n=0}^{7} x[n] (-1)^n X [ 4 ] = ∑ n = 0 7 x [ n ] W 8 4 n = ∑ n = 0 7 x [ n ] e − j π n = ∑ n = 0 7 x [ n ] ( − 1 ) n X [ 4 ] = x [ 0 ] − x [ 1 ] + x [ 2 ] − x [ 3 ] + x [ 4 ] − x [ 5 ] + x [ 6 ] − x [ 7 ] X[4] = x[0] - x[1] + x[2] - x[3] + x[4] - x[5] + x[6] - x[7] X [ 4 ] = x [ 0 ] − x [ 1 ] + x [ 2 ] − x [ 3 ] + x [ 4 ] − x [ 5 ] + x [ 6 ] − x [ 7 ] Given
X [ k ] = k + 1 X[k] = k+1 X [ k ] = k + 1 ,
X [ 4 ] = 4 + 1 = 5 X[4] = 4+1 = 5 X [ 4 ] = 4 + 1 = 5 .
Adding
X [ 0 ] X[0] X [ 0 ] and
X [ 4 ] X[4] X [ 4 ] :
X [ 0 ] + X [ 4 ] = ( x [ 0 ] + x [ 1 ] + x [ 2 ] + x [ 3 ] + x [ 4 ] + x [ 5 ] + x [ 6 ] + x [ 7 ] ) + ( x [ 0 ] − x [ 1 ] + x [ 2 ] − x [ 3 ] + x [ 4 ] − x [ 5 ] + x [ 6 ] − x [ 7 ] ) X[0] + X[4] = (x[0] + x[1] + x[2] + x[3] + x[4] + x[5] + x[6] + x[7]) + (x[0] - x[1] + x[2] - x[3] + x[4] - x[5] + x[6] - x[7]) X [ 0 ] + X [ 4 ] = ( x [ 0 ] + x [ 1 ] + x [ 2 ] + x [ 3 ] + x [ 4 ] + x [ 5 ] + x [ 6 ] + x [ 7 ]) + ( x [ 0 ] − x [ 1 ] + x [ 2 ] − x [ 3 ] + x [ 4 ] − x [ 5 ] + x [ 6 ] − x [ 7 ]) X [ 0 ] + X [ 4 ] = 2 x [ 0 ] + 2 x [ 2 ] + 2 x [ 4 ] + 2 x [ 6 ] = 2 ∑ n = 0 3 x [ 2 n ] X[0] + X[4] = 2x[0] + 2x[2] + 2x[4] + 2x[6] = 2 \sum_{n=0}^{3} x[2n] X [ 0 ] + X [ 4 ] = 2 x [ 0 ] + 2 x [ 2 ] + 2 x [ 4 ] + 2 x [ 6 ] = 2 ∑ n = 0 3 x [ 2 n ] Therefore,
∑ n = 0 3 x [ 2 n ] = X [ 0 ] + X [ 4 ] 2 = 1 + 5 2 = 6 2 = 3 \sum_{n=0}^{3} x[2n] = \frac{X[0] + X[4]}{2} = \frac{1 + 5}{2} = \frac{6}{2} = 3 ∑ n = 0 3 x [ 2 n ] = 2 X [ 0 ] + X [ 4 ] = 2 1 + 5 = 2 6 = 3 .
The value is 3.00.
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Solution Given an 8-point DFT
X [ k ] = k + 1 X[k] = k+1 X [ k ] = k + 1 for
k = 0 , 1 , … , 7 k=0, 1, \dots, 7 k = 0 , 1 , … , 7 .
The DFT is defined as
X [ k ] = ∑ n = 0 N − 1 x [ n ] e − j 2 π n k / N X[k] = \sum_{n=0}^{N-1} x[n] e^{-j2\pi nk/N} X [ k ] = ∑ n = 0 N − 1 x [ n ] e − j 2 π nk / N with
N = 8 N=8 N = 8 .
We need to find the value of
S = ∑ n = 0 3 x [ 2 n ] S = \sum_{n=0}^{3} x[2n] S = ∑ n = 0 3 x [ 2 n ] .
From the definition of the 8-point DFT:
X [ k ] = ∑ n = 0 7 x [ n ] W 8 n k X[k] = \sum_{n=0}^{7} x[n] W_8^{nk} X [ k ] = ∑ n = 0 7 x [ n ] W 8 nk where
W 8 = e − j 2 π / 8 W_8 = e^{-j2\pi/8} W 8 = e − j 2 π /8 .
Specifically, for
k = 0 k=0 k = 0 :
X [ 0 ] = ∑ n = 0 7 x [ n ] = x [ 0 ] + x [ 1 ] + x [ 2 ] + x [ 3 ] + x [ 4 ] + x [ 5 ] + x [ 6 ] + x [ 7 ] X[0] = \sum_{n=0}^{7} x[n] = x[0] + x[1] + x[2] + x[3] + x[4] + x[5] + x[6] + x[7] X [ 0 ] = ∑ n = 0 7 x [ n ] = x [ 0 ] + x [ 1 ] + x [ 2 ] + x [ 3 ] + x [ 4 ] + x [ 5 ] + x [ 6 ] + x [ 7 ] Given
X [ k ] = k + 1 X[k] = k+1 X [ k ] = k + 1 ,
X [ 0 ] = 0 + 1 = 1 X[0] = 0+1 = 1 X [ 0 ] = 0 + 1 = 1 .
For
k = 4 k=4 k = 4 :
X [ 4 ] = ∑ n = 0 7 x [ n ] W 8 4 n = ∑ n = 0 7 x [ n ] e − j π n = ∑ n = 0 7 x [ n ] ( − 1 ) n X[4] = \sum_{n=0}^{7} x[n] W_8^{4n} = \sum_{n=0}^{7} x[n] e^{-j\pi n} = \sum_{n=0}^{7} x[n] (-1)^n X [ 4 ] = ∑ n = 0 7 x [ n ] W 8 4 n = ∑ n = 0 7 x [ n ] e − j π n = ∑ n = 0 7 x [ n ] ( − 1 ) n X [ 4 ] = x [ 0 ] − x [ 1 ] + x [ 2 ] − x [ 3 ] + x [ 4 ] − x [ 5 ] + x [ 6 ] − x [ 7 ] X[4] = x[0] - x[1] + x[2] - x[3] + x[4] - x[5] + x[6] - x[7] X [ 4 ] = x [ 0 ] − x [ 1 ] + x [ 2 ] − x [ 3 ] + x [ 4 ] − x [ 5 ] + x [ 6 ] − x [ 7 ] Given
X [ k ] = k + 1 X[k] = k+1 X [ k ] = k + 1 ,
X [ 4 ] = 4 + 1 = 5 X[4] = 4+1 = 5 X [ 4 ] = 4 + 1 = 5 .
Adding
X [ 0 ] X[0] X [ 0 ] and
X [ 4 ] X[4] X [ 4 ] :
X [ 0 ] + X [ 4 ] = ( x [ 0 ] + x [ 1 ] + x [ 2 ] + x [ 3 ] + x [ 4 ] + x [ 5 ] + x [ 6 ] + x [ 7 ] ) + ( x [ 0 ] − x [ 1 ] + x [ 2 ] − x [ 3 ] + x [ 4 ] − x [ 5 ] + x [ 6 ] − x [ 7 ] ) X[0] + X[4] = (x[0] + x[1] + x[2] + x[3] + x[4] + x[5] + x[6] + x[7]) + (x[0] - x[1] + x[2] - x[3] + x[4] - x[5] + x[6] - x[7]) X [ 0 ] + X [ 4 ] = ( x [ 0 ] + x [ 1 ] + x [ 2 ] + x [ 3 ] + x [ 4 ] + x [ 5 ] + x [ 6 ] + x [ 7 ]) + ( x [ 0 ] − x [ 1 ] + x [ 2 ] − x [ 3 ] + x [ 4 ] − x [ 5 ] + x [ 6 ] − x [ 7 ]) X [ 0 ] + X [ 4 ] = 2 x [ 0 ] + 2 x [ 2 ] + 2 x [ 4 ] + 2 x [ 6 ] = 2 ∑ n = 0 3 x [ 2 n ] X[0] + X[4] = 2x[0] + 2x[2] + 2x[4] + 2x[6] = 2 \sum_{n=0}^{3} x[2n] X [ 0 ] + X [ 4 ] = 2 x [ 0 ] + 2 x [ 2 ] + 2 x [ 4 ] + 2 x [ 6 ] = 2 ∑ n = 0 3 x [ 2 n ] Therefore,
∑ n = 0 3 x [ 2 n ] = X [ 0 ] + X [ 4 ] 2 = 1 + 5 2 = 6 2 = 3 \sum_{n=0}^{3} x[2n] = \frac{X[0] + X[4]}{2} = \frac{1 + 5}{2} = \frac{6}{2} = 3 ∑ n = 0 3 x [ 2 n ] = 2 X [ 0 ] + X [ 4 ] = 2 1 + 5 = 2 6 = 3 .
The value is 3.00.
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