Question

A modulating signal m(t) is given by:

(a) m(t) = cos(100πt) + 2 cos(300πt).
(b) m(t) = sin 100 pi t sin(500πt)

In each case:
(i) Sketch the spectrum of m(t).

Answer 1

The spectrum of the first signal, m(t) = cos(100πt) + 2 cos(300πt), consists of two peaks at 50 Hz and 150 Hz, with the latter having twice the amplitude. The spectrum of the second signal, m(t) = sin(100πt) sin(500πt), can be determined using product-to-sum identities, resulting in peaks at 250 Hz and 200 Hz.

Step-by-step explanation:

The student has been asked to sketch the spectrum of a modulating signal m(t) for two different functions. When sketching the spectrum of these signals, we are essentially looking at their frequency components.

For m(t) = cos(100πt) + 2 cos(300πt), there will be two peaks in the spectrum. One peak corresponds to a frequency of 50 Hz (100π radians per second) with a certain amplitude, while the other corresponds to a frequency of 150 Hz (300π radians per second) with twice the amplitude of the previous peak.

For m(t) = sin(100πt) sin(500πt), the signal can be expressed using the product-to-sum trigonometric identities. This results in a spectrum with peaks at the frequencies resulting from the sum and difference of the two original frequencies, which would be at 250 Hz (500π + 100π radians per second) and 200 Hz (500π - 100π radians per second).

Understanding these concepts is essential in the field of signal processing and communications engineering, which is part of a Physics curriculum at a College level.