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Even and odd part from stem plot

Short answer

From x[n]=1 (n=−4...0) and 2 (n=1...4): even part xe[n]=1.5 for 1≤|n|≤4, xe[0]=1. Odd part xo[n]=−0.5 for −4≤n≤−1, 0 at n=0, +0.5 for 1≤n≤4, using xe=(x[n]+x[−n])/2, xo=(x[n]−x[−n])/2.

Find the odd and even part of the following signal. [Figure: stem plot x[n] for n = −4 to 4; x[n] = 1 for n = −4, −3, −2, −1, 0 and x[n] = 2 for n = 1, 2, 3, 4.]

The stem plot gives x[n] = 1 for n = −4 … 0 and x[n] = 2 for n = 1 … 4.

Use xe[n] = ½{x[n] + x[−n]} and xo[n] = ½{x[n] − x[−n]}.

nx[n]x[−n]xe[n]xo[n]
−4 … −1121.5−0.5
01110
1 … 4211.5+0.5

Even part xe[n]

 xe[n]
 1.5 |  o  o  o  o     o  o  o  o
 1.0 |              o
     +--+--+--+--+--+--+--+--+--+-- n
       -4 -3 -2 -1  0  1  2  3  4

xe[n] = 1.5 for 1 ≤ |n| ≤ 4, and xe[0] = 1. It is symmetric about n = 0.

Odd part xo[n]

 xo[n]
 +0.5 |                 o  o  o  o
    0 +--+--+--+--+--o--+--+--+--+-- n
 -0.5 |  o  o  o  o
        -4 -3 -2 -1  0  1  2  3  4

xo[n] = −0.5 for n = −4 … −1, xo[0] = 0, and xo[n] = +0.5 for n = 1 … 4. It is antisymmetric.

Check: xe[n] + xo[n] = 1 for n ≤ 0 and 2 for n ≥ 1, which equals x[n].

✓ Answer checked by Claude Opus 5.5 (high effort)

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