# Modulated Sinusoids

## Introduction

Modulated signals are composed of a carrier signal which is modified in some way based on another signal, generally called the baseband signal. The baseband signal refers to the original range of a signal before it is modulated to a different frequency range. Two primary modulation approaches are amplitude modulation and frequency modulation.

In amplitude modulation, the amplitude of the carrier signal is changed based on the baseband signal. Amplitude modulated (or AM) radio signals use the baseband signal corresponding to the audio signal which is to be broadcast. The carrier signal is a sinusoid whose frequency corresponds to the frequency band allocated by the FCC (Federal Communications Commission) to the radio station. When the amplitude of the carrier signal is modulated by the baseband signal, the baseband signal is “shifted” so that it is centered around the carrier signal's frequency. We tune to the carrier signal frequency in order to acquire the signal; the signal is demodulated in order to obtain the desired audio signal.

In frequency modulation, the frequency of the carrier signal is changed based on the amplitude of the baseband signal. The idea is similar to the idea of swept sinusoids, which are also signals whose frequency varies. In frequency modulation (abbreviated FM), if the amplitude of the carrier frequency is high, the frequency of the modulated frequency is high, and vice-versa. This approach is used, for example, by FM radio stations.

Both amplitude modulated and frequency modulated signals are discussed in more detail below. The mathematics associated with the signals is not emphasized; we will instead rely on qualitative descriptions of the signals.

## Amplitude Modulation

We will consider only the case of a pure sinusoidal carrier signal which is amplitude modulated by a pure sinusoidal baseband signal. The mathematical representations of these signals are:

(1.)

${f_c}(t) = {A_c}\cos ({\omega _c}t)$

(2.)

${f_c}(t) = {A_c}\cos ({\omega _c}t)$

Where fc(t) is the carrier signal (with amplitude Ac and radian frequency ωc) and f b(t) is the baseband signal (with amplitude Ab and radian frequency ωb). For simplicity, we neglect the phase of these signals—it is not really important to our discussion and will merely complicate matters. Examples of carrier and baseband signals are shown in Fig. 1(a) and 1(b), respectively.

The amplitude modulated signal is the product of the carrier signal and the baseband signal, such that:

(3.)

${f_{AM}}(t) = {f_b}(t) \times {f_c}(t)$

The amplitude modulated signal created from the product of the signals shown in Figs. 1(a) and 1(b) is shown in Fig. 1(c).

## Frequency Modulation

The mathematics associated with frequency modulation is a bit tedious, so we will provide a simpler discussion of the basic concepts involved. In general, the basis of frequency modulation is similar to the swept sinusoids we examined in Experiment 2 of this project. The overall idea is that the frequency of the carrier signal is varied according to the amplitude of the baseband signal—when the amplitude of the baseband signal is high, the frequency of the FM signal is high, and when the amplitude of the baseband signal is low, the frequency of the FM signal is low. The frequency modulated signal created from the carrier signal shown in Fig. 1(a) and the baseband signal shown in Fig. 1(b) is shown in Fig. 1(c).