Line Spectrum of Signal

Q. Plot the line spectrums of x(t) =12 + 6sin (140πt +30°) - 9cos(80πt-70°).

Solution:

Let’s Analyze and Plot the Line Spectrum of

x(t) = 12 + 6sin(140πt+30⁰) − 9cos(80πt−70⁰)

Step 1: Convert all terms to cosine form

Use the identity: sin(θ) = cos(θ−90⁰)

So,

6sin(140πt+30⁰) = 6cos(140πt−60⁰)

−9cos(80πt−70⁰) = 9cos(80πt+110⁰)

Thus, x(t) = 12 + 6cos(140πt−60⁰) + 9cos(80πt+110⁰)

Step 2: Frequencies, Amplitudes, Phases

Term	Frequency (Hz)	Amplitude	Phase
12 (DC)	0	12	0°
6cos(140πt−600)	f=70	6	-60°
9cos(80πt+1100)	f=40	9	+110°

 We split each cosine into two spectral lines at ±f with half the amplitude.

Now, let's plot the magnitude and phase spectrum.

Magnitude Spectrum: 12 at 0 Hz (DC component) 4.5 at ±40 Hz 3 at ±70 Hz

Phase Spectrum: 0° at 0 Hz ±110° at ±40 Hz ∓60° at ±70 Hz