Even and odd parts of a signal

Derivation

Any signal can be written as the sum of an even part and an odd part: x(t) = xₑ(t) + xₒ(t), where xₑ(-t) = xₑ(t) and xₒ(-t) = -xₒ(t).

Replace t by -t: x(-t) = xₑ(t) - xₒ(t).

Adding the two equations: x(t) + x(-t) = 2xₑ(t) Subtracting: x(t) - x(-t) = 2xₒ(t)

xₑ(t) = ½ [x(t) + x(-t)]

xₒ(t) = ½ [x(t) - x(-t)]

Numerical

x(t) = 0.5(t + 1) = 0.5t + 0.5 for -1 ≤ t ≤ 1 (zero outside).

x(-t) = 0.5(-t + 1) = 0.5 - 0.5t, for -1 ≤ t ≤ 1.

xₑ(t) = ½ [(0.5t + 0.5) + (0.5 - 0.5t)] = 0.5, for -1 ≤ t ≤ 1

xₒ(t) = ½ [(0.5t + 0.5) - (0.5 - 0.5t)] = 0.5t, for -1 ≤ t ≤ 1

Both are zero outside this interval.

Check: xₑ + xₒ = 0.5 + 0.5t = x(t).

x(t): ramp from 0 at t = -1 to 1 at t = 1
  1 |              /
0.5 |        /
  0 +--/-----+-----+--> t
      -1     0     1

xe(t): rectangle of height 0.5 on -1 <= t <= 1
0.5 |  +-----------+
  0 +--+-----+-----+--> t
      -1     0     1

xo(t): line from -0.5 at t = -1 to 0.5 at t = 1
 0.5 |              /
   0 +--------/--------> t
-0.5 |  /
       -1     0     1