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Quadrature Amplitude Modulation (QAM)

Module by: Fabio Ussher. E-mail the author

Quadrature Amplitude Modulation (QAM)

All signal communications must adhere to frequency restrictions so that they can be received without interference. This gives rise to the notion of carrier modulation, where a baseband signal is moved to an unoccupied section of the frequency domain before transmission. This is also known as frequency modulation. This also simultaneously addresses the issue that low-frequency signals suffer greatly from attenuation during transmission through a medium.

In order to transmit digital information, symbols are needed to represent the bits. The simplest set is known as BPSK, which consists of just two symbols; one represents 0, while the other represents 1. The baud rate in this case is only one; more complicated methods are necessary if we wish to improve upon this. While there are many types of modulation of varying complexity, we will focus on one of the popular methods known as 16-Quadrature Amplitude Modulation (16-QAM).

16-QAM utilizes both amplitude and phase alterations in conjunction with frequency modulation in a way that allows each symbol to represent four bits rather than just one. This increase in baud rate comes at the cost of design complexity and cost. The transmitter must send two signals simultaneously; in order to do this in a way that the signals can be separated by the receiver, the two signals must be orthogonal to each other. This is implemented via the frequency modulation, except one signal is modulated by a cosine and the other a sine. Thus, the output s(t) can be defined as

s ( t ) = I ( t ) cos ( 2πf c t ) Q ( t ) sin ( 2πf c t ) s ( t ) = I ( t ) cos ( 2πf c t ) Q ( t ) sin ( 2πf c t ) size 12{s \( t \) =I \( t \) "cos" \( 2πf rSub { size 8{c} } t \) - Q \( t \) "sin" \( 2πf rSub { size 8{c} } t \) } {}
(1)

The first signal is known as the in-phase component, while the other is known as the quadrature component. The fact that

cos ( ωt ) sin ( ωt ) = 0 2πk cos ( ωt ) sin ( ωt ) dt = 0, k Z cos ( ωt ) sin ( ωt ) = 0 2πk cos ( ωt ) sin ( ωt ) dt = 0, k Z size 12{"cos" \( ωt \) cdot "sin" \( ωt \) = Int cSub { size 8{0} } cSup { size 8{2πk} } {"cos" \( ωt \) "sin" \( ωt \) * ital "dt"} =0,k in Z} {}

implies the signals' orthogonality. Multiplication by these sinusoids, via properties of the Fourier Transform, centers the frequency representation of the signal around plus and minus fc rather than at baseband.

Figure 1
Figure 1 (graphics1.jpg)

Figure 2
Figure 2 (graphics2.jpg)

The various amplitudes paired with the two phases can be succinctly represented by a constellation map as shown below.

Figure 3
Figure 3 (graphics3.jpg)

Each point corresponds to a particular pair of amplitudes of the two signals. To combat the effects of noise, the points of the constellation are placed as far away from each other as possible so avoid misinterpretation. Many constellation configurations can be used; ours is described below:

Table 1
Bits I(t) Q(t)
0001 1 1
0010 3 1
0011 1 3
0100 3 3
0101 1 -1
0110 1 -3
0111 3 -1
1000 3 -3
1001 -1 1
1010 -1 3
1011 -3 1
1101 -3 3
1110 -1 -1
1111 -3 -1
1110 -1 -3
1111 -3 -3

In order to correctly interpret the data from r(t), the received signal s(t) with the addition of white noise after it passes through the channel, the receiver must recover I(t) and Q(t). I(t) is obtained by modulating s(t) by a cosine of identical frequency and phase as the original modulation, while Q(t) is obtained in the same way but with a sine instead. A low-pass filter will then yield the original signal, as the following equations illustrate:

r ( t ) = I ( t ) cos ( ωt ) + Q ( t ) sin ( ωt ) r ( t ) = I ( t ) cos ( ωt ) + Q ( t ) sin ( ωt ) size 12{r \( t \) =I \( t \) "cos" \( ωt \) +Q \( t \) "sin" \( ωt \) } {}
(2)
I rcvd ( t ) = LPF [ r ( t ) cos ( ωt ) ] I rcvd ( t ) = LPF [ I ( t ) cos 2 ( ωt ) + Q ( t ) sin ( ωt ) cos ( ωt ) ] I rcvd ( t ) = LPF [ 1 2 I ( t ) ( 1 + cos ( 2ωt ) ) + 1 2 Q ( t ) sin ( 2ωt ) ] I rcvd ( t ) = I ( t ) 2 I rcvd ( t ) = LPF [ r ( t ) cos ( ωt ) ] I rcvd ( t ) = LPF [ I ( t ) cos 2 ( ωt ) + Q ( t ) sin ( ωt ) cos ( ωt ) ] I rcvd ( t ) = LPF [ 1 2 I ( t ) ( 1 + cos ( 2ωt ) ) + 1 2 Q ( t ) sin ( 2ωt ) ] I rcvd ( t ) = I ( t ) 2 alignl { stack { size 12{I rSub { size 8{ ital "rcvd"} } \( t \) = ital "LPF" \[ r \( t \) "cos" \( ωt \) \] } {} # I rSub { size 8{ ital "rcvd"} } \( t \) = ital "LPF" \[ I \( t \) "cos" rSup { size 8{2} } \( ωt \) +Q \( t \) "sin" \( ωt \) "cos" \( ωt \) \] {} # I rSub { size 8{ ital "rcvd"} } \( t \) = ital "LPF" \[ { {1} over {2} } I \( t \) \( 1+"cos" \( 2ωt \) \) + { {1} over {2} } Q \( t \) "sin" \( 2ωt \) \] {} # I rSub { size 8{ ital "rcvd"} } \( t \) = { {I \( t \) } over {2} } {} } } {}
(3)

The low-pass filter removes the components of frequency 2ω, leaving only a baseband signal. A similar approach shows that indeed

Q rcvd ( t ) = LPF [ r ( t ) sin ( ωt ) ] = Q ( t ) 2 Q rcvd ( t ) = LPF [ r ( t ) sin ( ωt ) ] = Q ( t ) 2 size 12{Q rSub { size 8{ ital "rcvd"} } \( t \) = ital "LPF" \[ r \( t \) "sin" \( ωt \) \] = { {Q \( t \) } over {2} } } {}
(4)

Thus, both signals I(t) and Q(t) can successfully be recovered at the receiver. Below is the block diagram implementation of a transmitter using 16 QAM:

Figure 4
Figure 4 (graphics4.jpg)

Below is the block diagram implementation of a receiver using 16 QAM:

Figure 5
Figure 5 (graphics5.jpg)

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