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Magnitude and phase spectra of a sum of sinusoids

Short answer

Convert: sin(12t+20°)=cos(12t-70°), -cos(16t-60°)=cos(16t+120°). Signal =5+cos(12t-70°)+cos(16t+120°)+cos(20t+40°). Magnitudes: DC=5, amplitude 1 at ω=12,16,20 rad/s; phases -70°, 120°, 40°. Plot vertical lines at those ω. Two-sided gives lines of 0.5 at ±12, ±16, ±20 with phases odd-symmetric.

Plot the magnitude and phase spectra of X(t) = 5 + sin(12t + 20°) - cos(16t - 60°) + cos(20t + 40°).

X(t) = 5 + sin(12t + 20°) - cos(16t - 60°) + cos(20t + 40°)

Step 1: convert to positive cosines

  • sin(12t + 20°) = cos(12t + 20° - 90°) = cos(12t - 70°)
  • -cos(16t - 60°) = cos(16t - 60° + 180°) = cos(16t + 120°)
  • cos(20t + 40°) is already in the right form.

Step 2: table (ω in rad/s; f = ω/2π)

ω (rad/s)f (Hz)AmplitudePhase
0 (DC)050°
121.911-70°
162.551+120°
203.181+40°

Single-sided plots

Magnitude                    Phase (deg)
5 |*                      120 |            *
  |                        40 |                 *
1 |   |    |    |           0 +--+---+----+----+--> w
  +---+----+----+--> w          0  12   16   20
  0  12   16   20            -70|   *

Two-sided plots

Lines of height 0.5 at ω = ±12, ±16, ±20 and 5 at ω = 0, with phases odd-symmetric: -70°, +120°, +40° at positive ω and +70°, -120°, -40° at negative ω.

The magnitude spectrum is even and the phase spectrum is odd for a real signal.

✓ Answer checked by Claude Opus 5.5 (high effort)

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