`As mentioned before, a reason to modulate a message signal
is to match the communication's channel frequency. The reason
for this is that information to be send over the channel is often
at a rather low frequency. For example, the sound of music range
in frequency from about 100 to 15,000 Hz. It is very difficult
to send such low-frequency signals over great distance. Low frequency
signals will be interfered by other similar signals. There are
many techniques for modulation. The technique that we will emphasis
on is standard amplitude modulation. There are also many other
modulations including methods using frequency and phase.
`

`In amplitude modulation, the circuit or the modulator combines
the carrier wave (Fig 1) and the message signal (Fig 2) to form
a modulated wave (Fig 3) that is a carrier wave with change in
amplitude. The frequency that I choose for the figures is for
examples only. In reality, the value of the frequency is higher.
`

`Fig 1. Carrier Wave - A cosine wave with amplitude 3 and frequency
100 Hz.`

`Fig 2. Message Signal - A cosine wave with amplitude 3 and
frequency 10 Hz.`

`Fig 3. Modulated Wave `

`The message signal is a cosine wave or a sine wave. In another
word, the message signal is a sinusoidal wave function. This is
the same for a carrier wave except its frequency is very large
when compare to the frequency of the message signal. Let us use
mathematical terms to relate our signals. For the message signal
in the above example, the following mathematical expression represents
it.
`

`Equation 1. m(t) = Am cos (2*pi*fm*t), Am=3, fm=5 Hz, t=time.
`

`Equation 2 represents the carrier wave which is just another
sinusoidal wave function.
`

`Equation 2. C(t) = Ac cos (2*pi*fc*t), Ac=3, fc=100 Hz, t=time.
`

`Equation 3 is the standard form of the modulated wave by the
method of amplitude modulation.
`

`Equation 3. S(t) = Ac [1 + Ka*m(t)] cos (2*pi*fc*t)
`

`The constant Ka is the amplitude sensitivity of the modulator
or the transmitter. The amplitude of equation 3 is called the
envelope of the AM wave. We can represent that as a(t).
`

`Equation 4. a(t) = Ac | 1 + Ka*m(t) |
`

`The p ercentage of modulation
will depend on the absolute value of Ka*m(t). If the absolute
value of Ka*m(t) is less or equal to 1 for all t, then the percentage
of modulation is less than or equal to 100%. However, if the absolute
value of Ka*m(t) is greater than 1 for some t, then the percent
of modulation is in excess of 100% or overmodulation. The following
table summarizes the modulation percentage. For example, assume
that the value of |Ka*m(t)| is 0.7, then the percentage of modulation
is 70%. Therefore, if |Ka*m(t)| is 1.1, then the percentage of
modulation is 110%. `

Case | Percentage of Modulation |

|Ka*m(t)| < 1 or =1 for all t | less than or equal to 100 % |

|Ka*m(t)| >1 for some t | in excess of 100% |

`If the percentage of modulation is less than or equal to 100%,
then the demodulation circuit used to recover the message signal
from the incoming AM wave is greatly simplified. There are two
conditions to be satisfied in order to produce an a(t) or envelope
of the modulated wave to be the same shape as the message signal
m(t). The first condition is that the percentage of modulation
is less than 100% so the envelope or a(t) is not distorted. The
second condition is that the message bandwidth is small compared
to the carrier's frequency. The demodulation circuit for the envelope
is called the envelope detector. It is just basically a low-pass
filter consists of a resistor, a capacitor and a diode. Figure
4 shows a simple circuit diagram of the envelope detector.
`

`Figure 4 Circuit Diagram of Envelope Detector.`

`The above circuit diagram is a low pass filter which just
consists of a capacitor and resistor. The purpose of a low pass
filter is to allow low frequency component of the s(t) or the
modulated wave to pass to the output while the high frequency
component is filter out in the process. The values of the resistor
and the capacitor determine how low the frequency will be able
to appear at the output.`

`Amplitude modulation is still in used today. Some its usage
is AM broadcast radio stations. AM is still used in television
broadcasting as a method for transmission of picture information
while the sound is done `
`by FM.`