Quadrature Amplitude Modulation : Introduction

Ang Man Shun

December 20, 2012

To send multiple signals at one time, one way is to use QAM.

Some trigonometric identities

cos2 θ =1 + cos 2θ

2sin θ cos θ =

sin 2θ

2sin2 θ =

1− cos 2θ

2

1. QAM structure

Consider the y(t) in the transimitter

y(t) = x1(t) cosωct+ x2(t) sinωct

1

y(t) · cosωct = (x1(t) cosωct+ x2(t) sinωct) cosωct = x1(t) cos2 ωct+ x2(t) sinωct cosωct

Use those identities, the signal is

x1(t)

(1 + cos 2ωct

2

)+ x2(t)

sin 2ωct

2=

1

2x1(t) +

1

2cos 2ωct

1

2x2(t) sin 2ωct

After passing the low pass filter

x′1(t) =

1

2x1(t)

Consider the lower signal in the receiver, the signal is

y(t) · sinωct = (x1(t) cosωct+ x2(t) sinωct) sinωct = x1(t) sinωct cosωct+ x2(t) sin2 ωct

= x1(t)sin 2ωct

2+ x2(t)

1− cos 2ωct

2=

x2(t)

2+ x1(t)

sin 2ωct

2− x2(t)

cos 2ωct

2

So after passing the LPF

x′2(t) =

x2(t)

2

i.e. x′1(t) =

x1(t)

2

x′2(t) =

x2(t)

2Improved QAM

This time the output of the transmitter is

y(t) =√2 [x1(t) cosωct+ x2(t) sinωct]

The signals in receiver are now{Upper y(t)

√2 cosωct

Lower y(t)√2 sinωct

=

{x1(t)2 cos

2 ωct+ x2(t)2 sinωct cosωct

x1(t)2 cosωct sinωct+ x2(t)2 sin2 ωct

=

{x1(t) + x1(t) cos 2ωct+ x2(t) sin 2ωct

x2(t)− x2(t) cos 2ωct+ x1(t) sin 2ωct

2

Consider the upper signal in the receiver, the signal is

After passing low pass filter

x′1(t) = x1(t) x′

2(t) = x2(t)

Thus, this simple QAM is make use of the property of sin-cos function to achieve the objective :sending 2 signal using one channel.

−END−

x1(t) +