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Doc. no. ESS140/ext:10, rev. A

Notes on Quadrature Amplitude Modulation

Erik Strom

October 22, 2002

1 Introduction

This document should be viewed as a complement to Sections 7.3.3, 7.5.6, and7.6.5 of Proakis and Salehi’s book [1]. In particular, an error in the book’sequation (7.6.71) is pointed out, and a corrected formula is presented.

2 Quadrature Modulation

A block diagram for a generic inphase quadrature phase modulator (IQ-modulator)is found in Figure 1. We see that the transmitted signal can be viewed as thesum of two PAM processes with different pulse shapes: ψI(t) in the top (in-phase)branch and ψQ(t) in the lower (quadrature) branch, where

ψI =

√2

EggT (t) cos(2πfct)

ψQ =

√2

EggT (t) sin(2πfct),

and where Eg is the energy of the pulse shape gT (t). The pulse shape andthe carrier frequency fc is chosen such that the power spectral density of thetransmitted signal will fit the frequency response of the channel.

As implied by the IQ-modulator block diagram, the signals ψI(t) and ψQ(t)spans a two-dimensional signal space. In fact, it can be shown that signals areapproximately orthonormal if the carrier frequency fc is much larger than thebandwidth of gT (t). The approximation is very good for many practical choicesof fc and gT (t), and we will from now consider ψI(t) and ψQ(t) to be exactly or-thonormal and to make up the standard basis for the IQ signal space. The readershould here be cautioned that some texts use −ψQ(t) as one of the basis func-tions. Hence, we need to be a bit careful when using material on IQ modulationfrom different sources.

1

2 (12) Notes on Quadrature Amplitude Modulation

bits/

symbol

symbol/

vector

PAM

1( )T

g

g tE

PAM

1( )T

g

g tE

2 cos(2 )cf tπ

2 sin(2 )cf tπ

,I ms

,Q ms

transmitted vectorI

Q

s

s

=

s

( )s t

PAM with ( )I tψ

PAM with ( )Q tψ

2-dimensional

constellation

mbits

Figure 1: Generic IQ-modulator

A generic demodulator is shown in Figure 2. We will assume that the channelnoise is white and Gaussian with power spectral density N0/2. It then followsthat the noise vector elements, nI and nQ, are independent Gaussian randomvariables with zero mean and variance N0/2.

Let us consider the transmission of the kth signal vector an M-ary constel-

lation s1, s2, . . . , sM. The transmitted vector is sk =[sI,k sQ,k

]T, and the

transmitted signal is therefore

sk(t) = sI,kψI(t) + sQ,kψQ(t)

= sI,k

√2

EggT (t) cos(2πfct) + sQ,k

√2

EggT (t) sin(2πfct)

If we define Ak and θk as (see Figure 3 for a geometrical interpretation)

Ak = ‖sk‖ =√s2

I,k + s2Q,k,

θk = − tansQ,k

sI,k.

According to the choice of definition of the angle θk, we have that

sI,k = Ak cos(−θk) = Ak cos(θk)

sQ,k = Ak sin(−θk) = −Ak sin(θk),

Doc. no.: ESS140/ext:10, rev.: A, date: October 22, 2002, file: qam-notes.tex

Notes on Quadrature Amplitude Modulation 3 (12)

2 cos(2 )cf tπ

2 sin(2 )cf tπ

Matched filter with respect to ( )I tψ

0

1( )T

g

g T tE

−

0

1( )T

g

g T tE

−

Matched filter with respect to ( )Q tψ

( )s t

( )n t

( )r t

0t nT T= +

Decision

rulem

0t nT T= +

I I

Q Q

s n

s n

= + = +

r s n

Figure 2: Generic IQ-demodulator

,

,

I k

kQ k

s

s

=

s

( )Q tψ

( )I tψ

kθ

,Q ks

,I ks

kA

Figure 3: Geometric interpretation of Ak and θk. Note that θ taken as the anglefrom the vector to the ψI(t) axis, not the other way around. In this figure,θk ≈ −0.12π.

and we can rewrite the transmitted signal as

sk(t) = Ak cos(θk)

√2

Eg

gT (t) cos(2πfct) −Ak sin(θk)

√2

Eg

gT (t) sin(2πfct)

= Ak

√2

Eg

gT (t) cos(2πfct+ θk),

where we have used the trigonometric identity cos(α) cos(β) − sin(α) sin(β) =cos(α+ β).

We conclude that the transmitted signal can be seen as the pulse gT (t) mul-tiplied with a cosine-carrier, where the amplitude and phase of the carrier is



determined by sI,k and sQ,k. Hence, the transmitted information can affect boththe amplitude and phase of the transmitted signal.

We can choose the signal constellation such that the amplitude is the same forall signal alternatives, by placing the signal vectors on a circle in the signal space.This implies that A1 = A2 = · · · = AM , and the transmitted information is thencarried by the phase of the carrier. Phase-shift keying (PSK) is an example ofsuch a modulation scheme.

Conversely, if the signal vectors are placed on a straight line that crosses theorigin in the signal space, then the carrier phase will be the same for all signalalternatives. (Which follows from the fact that sI,k/sQ,k is the same for all k.)Amplitude-shift keying (ASK) is an example of such a modulation scheme.

The general case when both amplitude and phase is allowed to change betweensignal alternatives is called quadrature amplitude modulation (QAM). There ex-ists (of course) many possible QAM constellations, but we will limit the discus-sion here to when the signal points are placed on a regular rectangular grid inthe signal space, see Figure 4 for two examples. Moreover, we will only considerconstellation sizes such that M = 2k, where k is an integer. That is, each symbolrepresents k bits.

( )ψ I t

( )ψ Q t

−A3− A A 3A

A

3A

−A

3− A

16-QAM

( )ψ I t

( )ψ Q t

−A3− A A 3A

A

3A

−A

3− A

8-QAM

Figure 4: Rectangular QAM constellations

As seen from Figure 4, the signal vectors are spaced with the distance 2Aalong the axes. Hence, the minimum distance of the constellation is dmin = 2A.When M is an even square, e.g., when M = 16 = 42, there will be

√M possible

amplitudes for both sI and sQ. In fact,

sI ∈ ±A,±3A, . . . ,±A(√M − 1)

sQ ∈ ±A,±3A, . . . ,±A(√M − 1).



When M is not an even square, e.g., M = 8, the signal points are more spreadalong one axis than the other.

Regardless if M is an even square or not, the maximum likelihood decisionregions will be rectangular-shaped. The regions can be squares (type a), squareswith one open side (type b) or squares with two open sides (type c), see Figure 5.

( )ψ I t

( )ψQ t

−A3− A A 3A

A

3A

−A

3− A

type a type b

type c

type a

type b

type c

Figure 5: ML decision regions for 16-QAM

To compute the symbol error probability, we therefore need only to computethe conditional error probabilities for the three types of decision regions. Con-ditioned on that we send a symbol that has a decision region of type a, theprobability of wrong decision is denoted Pe|a and the probability of correct de-cision is denoted Pc|a. Since the elements on the noise vector, nI and nQ, areindependent Gaussian random variables with zero mean and variance N0/2, wecan express

Pc|a = Pr−A < nI < A,−A < nQ < A= Pr−A < nI < APr−A < nQ < A= [Pr−A < nI < A]2.

From Figure 6, we see that Pr−A < nI < A can be written in terms of theQ-function as

Pr−A < nI < A = Ω2 = 1 − 2Q

(√2A2

N0

).



Hence,

Pe|a = 1 −[1 − 2Q

(√2A2

N0

)]2

= 4Q

(√2A2

N0

)− 4Q2

(√2A2

N0

).

3

0 / 2

AQ

N

Ω =

A

2

00

1( ) exp

In

xf x

NNπ

= −

A−

1

0 / 2

AQ

N

Ω =

2

0

1 2/ 2

AQ

N

Ω = −

x

Figure 6: The probability density function for nI and nQ is a zero mean Gaussiandistribution with variance N0/2. The areas Ω1, Ω2, and Ω3 represents the prob-abilities Ω1 = PrnI < −A, Ω2 = Pr−A < nI < A, and Ω3 = PrnI > A.

Proceeding with the type b region, we note that

Pc|b = PrnI > −A,−A < nQ < A= PrnI > −APr−A < nQ < A= (1 − Ω1)Ω2

=

[1 −Q

(√2A2

N0

)][1 − 2Q

(√2A2

N0

)]

= 1 − 3Q

(√2A2

N0

)+ 2Q2

(√2A2

N0

),

and

Pe|b = 3Q

(√2A2

N0

)− 2Q2

(√2A2

N0

).



Finally,

Pc|c = Pr−A < nI ,−A < nQ= PrnI > −APrnQ > −A= (1 − Ω1)(1 − Ω1)

=

[1 −Q

(√2A2

N0

)]2

= 1 − 2Q

(√2A2

N0

)+Q2

(√2A2

N0

),

Pe|c = 2Q

(√2A2

N0

)−Q2

(√2A2

N0

).

Suppose that all symbols are transmitted with the same probability 1/M . Ifna denotes the number of type a regions in the constellation then na/M is theprobability that a symbol with a type a decision region is transmitted. Hence,we can compute the (average) symbol error probability as

Pe =1

M(naPe|a + nbPe|b + nbPe|b). (1)

We always have nc = 4 (there is one type c region per corner, and there arealways four corners), but na and nb depends on M . For 16-QAM, we have thatna = 4 and nb = 8, and for 8-QAM na = 0 and nb = 4. If M is an even square,we have na = (

√M − 2)2 and nb = 4(

√M − 2).

It is desirable to put the final result in terms of Es/N0, where Es is the averageenergy per symbol. Assuming equally likely transmitted symbols, it is possibleto compute Es for any QAM constellation by just considering the first quadrantin the signal space. For 16-QAM,

Es =1

4[(A2 + A2) + 2(A2 + 9A2) + (9A2 + 9A2)] = 10A2,

and for 8-QAM, we have

Es =1

2[(A2 + A2) + (A2 + 9A2)] = 6A2.

It can be shown (see Section A) that when M is an even square, we have that

Es =A22(M − 1)

3⇒ 2A2

N0

=3Es

(M − 1)N0

. (2)



We can now write the symbol error probability for QAM when M is an evensquare (assuming equally likely transmitted symbols, ML detection, and AWGNchannel with noise power spectral density N0/2) as

Pe =1

M(naPe|a + nbPe|b + ncPe|c)

=1

M

(√M − 2)2

[4Q

(√3Es

(M − 1)N0

)− 4Q2

(√3Es

(M − 1)N0

)]

+ 4(√M − 2)

[3Q

(√3Es

(M − 1)N0

)− 2Q2

(√3Es

(M − 1)N0

)]

+ 4

[2Q

(√3Es

(M − 1)N0

)−Q2

(√3Es

(M − 1)N0

)]

=4

M(M −

√M)Q

(√3Es

(M − 1)N0

)

− 4

M(M − 2

√M + 1)Q2

(√3Es

(M − 1)N0

)(3)

We can form a simple upper bound on the symbol error probability by notingthat Pe|a > Pe|b > Pe|c. Hence, by replacing Pe|b and Pe|c in (1) with Pe|a andsince na + nb + nc = M , we can write

Pe =1

M(naPe|a + nbPe|b + ncPe|c)

<1

M(naPe|a + nbPe|a + ncPe|a)

= Pe|a

= 4Q

(√2A2

N0

)− 4Q2

(√2A2

N0

)(4)

< 4Q

(√2A2

N0

). (5)

The bounds (4) and (5) are valid for any M (assuming equally likely transmittedsymbols, rectangular constellation, ML detection, and AWGN channel with noisepower spectral density N0/2). The bound presented in Proakis and Salehi [1],equation (7.6.71) is not correct1.

1A counterexample can be found, e.g., by computing the exact symbol error probabilityaccording to (1) for the M = 8 constellation in Figure 4. For Es/N0 slightly larger than 9.5 dB,(7.6.71) is actually smaller than the exact symbol error probability. Hence, (7.6.71) is not aupper bound.



Finally, we recall that the minimum distance of a QAM constellation is dmin =2A, and a standard union bound therefore yields

Pe ≤ (M − 1)Q

√d2

min

2N0

= (M − 1)Q

(√4A2

2N0

)

= (M − 1)Q

(√2A2

N0

), (6)

which is not as tight as (4) or (5).For (4), (5), or (6) to be really useful, we usually need to relate A2 to Es. For

a rectangular constellation, there are MI possible values of sI and MQ possible

values of sQ and M = MIMQ, where s =[sI sQ

]T. Due to symmetry, we can

compute Es by only considering the signal points in the first quadrant, which areof the form

sI

sQ

= A

2i− 1

2j − 1

, for i = 1, 2, . . . ,MI/2 and j = 1, 2, . . . ,MQ/2,

Hence, sI = A(2i − 1) and sQ = A(2j − 1) and the energy of the correspondingsignal alternative is

s2I + s2

Q = A2[(2i− 1)2 + (2j − 1)2].

Since there are M/4 signal vectors in the first quadrant, we can compute Es as

Es =4

MA2

MI/2∑i=1

MQ/2∑j=1

(2i− 1)2 + (2j + 1)2.

This can be simplified to (see Section A)

Es =A2

3(M2

I +M2Q − 2). (7)

By combining (4), (5), or (6) with (7), we can form a bound on the symbol errorprobability in terms of Es/N0.

As an example, we see that for the M = 8 constellation in Figure 4, MI = 4and MQ = 2, and

Es =A2

3(16 + 4 − 2) = 6A2,



and the bound (5) evaluates to

Pe < Q

(√2A2

N0

)= 4Q

(√Es

3N0

).

Plots of the exact error probability and the bounds (4), (5), and (6) are found inFigure 7.

0 2 4 6 8 10 12 14 16 18 2010

−6

10−5

10−4

10−3

10−2

10−1

100

Symbol error probability for rectangular 8−QAM

10 log10

Es/N

0

Standard Union BoundBound 1Bound 2Exact

Figure 7: Plots of the exact expression and some upper bounds on the symbolerror probability for rectangular 8-QAM (see Figure 4). Bound 1 and 2 are definedby (4) and (5), respectively. The Standard Union Bound is defined by (6).

3 Summary

In these notes, we have developed a method for computing the exact symbol errorprobability for rectangular QAM (assuming ML detection, AWGN channel, andequally likely transmitted symbols). The general expression, found in (1), canbe further simplified for the case when M is an even square, see (3). A numberof upper bounds on the symbol error probability has also been presented in (4),(5), and (6), and it was noted that (7.6.71) in Proakis and Salehi [1] is in error.



To compute the symbol energy, we can use (7). The result can be combinedwith the symbol error formulas to form yield expressions that depend only onEs/N0.

A Derivation of (2) and (7)

First of all, we recall the following formulas (see, e.g., [2, p. 189])

n∑j=0

j =n(n + 1)

2

n∑j=0

j2 =n(n + 1)(2n+ 1)

6

Hence,

n∑j=1

(2i− 1)2 =

n∑j=1

4i2 − 4i+ 1

= 4n(n+ 1)(2n+ 1)

6− 4

n(n+ 1)

2+ n

=2

3n(n + 1)(2n+ 1) − 2n(n+ 1) + n

=n

3[2(n+ 1)(2n+ 1) − 6(n+ 1) + 3]

=n

3[4n2 + 6n + 2 − 6n− 6 + 3]

=n

3(4n2 − 1).

In particular,

MQ/2∑j=1

(2j + 1)2 =1

3

MQ

2

(4M2

Q

22− 1

)=MQ

6(M2

Q − 1).



The symbol energy is

Es =4

MA2

MI/2∑i=1

MQ/2∑j=1

(2i− 1)2 + (2j + 1)2

=4

MA2

MI/2∑i=1

MQ/2∑

j=1

(2i− 1)2

+

MQ/2∑

j=1

(2j + 1)2

=4

MA2

MI/2∑i=1

[MQ

2(2i− 1)2 +

MQ

6(M2

Q − 1)

]

=4

MA2

MI/2∑

i=1

MQ

2(2i− 1)2

+

MI/2∑

i=1

MQ

6(M2

Q − 1)

=4

MA2

[MQ

2

MI

6(M2

I − 1) +MI

2

MQ

6(M2

Q − 1)

]

=4

MA2MIMQ

12[M2

I +M2Q − 1]

=A2

3[M2

I +M2Q − 2],

where we have used that MIMQ = M . This completes the derivation of (7).The formula holds for any rectangular QAM; in particular, if M is an even

square and MI = MQ =√M , the formula reduces to

Es =A2

3[M2

I +M2Q − 2] =

A2

3[M +M − 2] =

A2

32(M − 1),

and this justifies (2).

B References

[1] John G. Proakis and Masoud Salehi. Communication Systems Engineering.Prentice-Hall, second edition, 2002.

[2] Lennart Rade and Bertil Westergren. Beta, Mathematics Handbook. Stu-dentlitteratur, fourth edition, 1998.


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