Angle Differential Modulation Scheme

for Odd-Bit QAM

Syed Safwan Khalid and Shafayat Abrar

Department of Electrical Engineering,COMSATS Institute of Information Technology, Islamabad

{safwan khalid,sabrar}@comsats.edu.pk

Abstract. In this paper, a method for differential encoding of odd-bitquadrature amplitude modulation (QAM) is presented. Differential en-coding would allow QAM to be transmitted over media where coherentdetection is not possible owing to phase ambiguity; it would allow such atransmission without the expense of the overhead incurred in data aidedtransmission. An example 32-QAM constellation is proposed and thealgorithms for differential encoding and decoding of the proposed con-stellation are presented. The analytical performance of the given schemeis also discussed and is verified using simulation results. Results of dif-ferentially encoded detection scheme are also compared with those ofcoherent detection.

Keywords: Odd-bit quadrature amplitude modulation (QAM), crossQAM, differential encoding, coherent detection, phase ambiguity, errorperformance.

1 Introduction

For high speed wireless communication, quadrature amplitude modulation (QAM)system, owing to its increased spectral efficiency, is an attractive alternative tophase shift keying (PSK) systems. Since a wireless channel can introduce exces-sive phase shifts in the received signal, a coherent detection of received QAMsignals is difficult even if the receiver is equipped with a carrier recovery mecha-nism. The received signal would suffer from a phase ambiguity problem i.e. thewhole constellation would get rotated by an unknown multiples of π/2 [1]. Toovercome this problem, an additional overhead of pilot signal or a continuousinsertion of synchronization sequences within the transmitted data is required.In multiple-antenna systems, this overhead for coherent detection becomes evenlarger. This has resulted in a renewed interest in recent years in non-coherent dif-ferentially encoded modulation schemes [2]. The literature regarding differentialcoding for QAM constellations is few in number mostly discussing star 8-QAMor 16-QAM constellations [2,3,4]. In [2,5], methods are proposed to differentiallyencode square QAM constellations; however, these methods are not applicabledirectly to odd-bit cross QAM constellations. In [6], a general differential schemebased on higher-order statistics is discussed but it imposes a significant perfor-mance loss and complexity. More recently in [7], a differential encoding scheme

B.S. Chowdhry et al. (Eds.): IMTIC 2012, CCIS 281, pp. 240–248, 2012.c© Springer-Verlag Berlin Heidelberg 2012

Angle Differential Modulation Scheme for Odd-Bit QAM 241

for 16-QAM and 32-QAM is proposed using look-up tables instead of rule-basedencoding. However, the feasibility and scalability of using look tables for higherorder QAM or memory limited systems requires more investigation.

In this paper, we propose a modification to the differential encoding schemediscussed in [2] to incorporating QAM constellations with odd number of bits persymbol. We propose a modified constellation diagram for 32-QAM and discussthe method for differentially encoding and decoding the proposed constellation.Moreover, the analytical performance analysis and computer simulation resultsare presented for bit error rates (BER) of the proposed scheme under AWGNchannel. The performance of the same constellation diagram for coherent detec-tion is also presented and the results are compared with the performance of theclassical coherently detected cross 32-QAM constellation.

The rest of the paper is organized as follows: in section 2, the proposed 32-QAM constellation is discussed along with its differential encoding and decoding.In section 3, the probability of bit error is analytically obtained. In section 4, thesimulation results are presented along with the analytical curves and conclusionsare drawn.

2 Differential QAM Encoding and Decoding Algorithm

2.1 Encoding

To differentially encode 32-QAM constellation, we have divided the constellationinto a set of eight circles with two overlapping circles in each quadrant as shownin Fig. 1(a). The minimum distance between constellation points is 2b, where b isa positive number. A group of five bits is mapped onto the 32 possible transitionssuch that the first two bits determine the change in the quadrant, the third bitdecides which of the possible two circles to choose in each quadrant and the lasttwo bits determine the position of the one of the four complex symbols on thegiven circle. Let S(i) be the transmitted symbol and S(i − 1) be the previouslytransmitted symbol. Let S(i) be defined as

S(i) = C(i) +D(i) + E(i) (1)

where C(i) is a position vector from the origin to the quadrant center. E(i) is aposition vector from C(i) to the center of a circle. D(i) is a position vector fromthe center of circle to the transmitted symbol on the constellation diagram.

C(i) = R1ejθ1 , R1 = 3b+

√2b (2)

D(i) = R2ejθ2 , R2 = 2b (3)

E(i) = R3ejθ3 , R3 = b (4)

where θ1, θ2 and θ3 belong to the set {π/4, 3π/4, 5π/4, 7π/4}. The differentialencoding rule can be described as three updating formulas

C(i) = C(i− 1)ejΔθ1 (5)

D(i) = D(i − 1)ejΔθ2 (6)

E(i) = E(i − 1)ejΔθ3 (7)

242 S.S. Khalid and S. Abrar

Γ2b

Γ 2b

Γ2blb

lb

lb

lb

Γ2b

Γ 2b

Γ2b

Γ 2b

Γ 2b

Γ2b

Γ 2b

Γ2b

Γ 2b

2b 2b2b2b .b .b

2b

2b

2b

2b

.b.b

a) b)

Fig. 1. Constellations of 32-QAM: a) proposed and b) traditional

where the values of Δθ1 and Δθ2 are determined using Table 1. The value ofΔθ3 is calculated as Δθ3 = Δθ3 −Δθ1, where Δθ3 is calculated using Table 2.

Table 1. Dibit to angle mapping for Δθ1 and Δθ2

00 0

01 π/2

11 π

10 3π/2

Table 2. Single bit to angle mapping for Δθ3

0 0

1 π

Example: Without loss of generality, suppose that the value of b is set equal to1 and the previous transmitted symbol S(i− 1) be 3.8284 + 5.2426j as follows:

S(i− 1) = C(i − 1) +D(i− 1) + E(i− 1) (8)

where C(i − 1) = R1ejπ/4, D(i − 1) = R2e

jπ/4 and E(i − 1) = R3ej3π/4. Let

a segment of data bits 01011 is to be transmitted in the next symbol S(i).Using the encoding tables, we get the values of angles as Δθ1 = π/2, Δθ2 = π

and Δθ3 = 0; hence Δθ3 = 0 − Δθ1 = −π/2. Using the updating formulas asdescribed above we get C(i) = −3.1213 + 3.1213j, D(i) = −1.4142 − 1.4142jand E(i) = 0.7071 + 0.7071j. Adding all three components the value of S(i) iscalculated to be −3.8284 + 2.4142j, refer to Fig. 2.

Angle Differential Modulation Scheme for Odd-Bit QAM 243

s(i)

S(i-1)

11,�����2

0,�������3

01,������1

Fig. 2. Illustration of the encoding algorithm

2.2 Decoding

The received signal X(i) is given as X(i) = S(i)ejφ + N(i), where S(i) is thetransmitted signal, N(i) is the additive white Gaussian noise component and φis the phase ambiguity which could take any value from a set of multiples ofπ/2. The first step in decoding is to compare the received signal X(i) from eachsignal point in the constellation diagram and choose the signal with minimumEuclidian distance as the decoded signal S(i). Now to determine C(i) from S(i)we use the equation

C(i) = R1 [sgn(�(S(i)) + j sgn(�(S(i)))] (9)

where sgn(·) is the signum function, �(z) and �(z) are used to denote real andimaginary components of z, respectively. To decode Δθ1 and hence the first twobits C(i) is correlated with C(i− 1) i.e. multiplied by the complex conjugate ofC(i−1). As long as both C(i) and C(i−1) experience the same phase ambiguityφ, the signal shall be decoded correctly. The value of Δθ1 is detected as

if C(i) C(i− 1)∗ =

⎧

⎪

⎪

⎨

⎪

⎪

⎩

R21 then Δθ1 = 0

jR21 then Δθ1 = π/2

−R21 then Δθ1 = π

−jR21 then Δθ1 = 3π/2

(10)

Now to determine D(i) we use the equation

D(i) = R2[sgn(�(S(i)− C(i))) + j sgn(�(S(i)− C(i)))] (11)

244 S.S. Khalid and S. Abrar

Similar to Δθ1, Δθ2 is calculated from the value of D(i) and D(i− 1) using

if D(i) D(i− 1)∗ =

⎧

⎪

⎪

⎨

⎪

⎪

⎩

R22 then Δθ2 = 0

jR22 then Δθ2 = π/2

−R22 then Δθ2 = π

−jR22 then Δθ2 = 3π/2

(12)

Similarly

E(i) = R3[sgn(�(S(i)− C(i)− D(i))) + j sgn(�(S(i)− C(i)− D(i)))] (13)

Now E(i) is further rotated by an angle Δθ1 to compensate for the angle sub-

tracted during the encoding phase hence E(i) ← E(i)ejΔθ1 . Finally for Δθ3

if E(i) E(i− 1)∗ =

{

R23 then Δθ3 = 0−R2

3 then Δθ3 = π(14)

As long as the phase ambiguity φ is same in the previous and the current detectedsymbol it will not create any errors in decoding since the decoding is based on therelative change of the angle of the previous and the current detected symbol andnot based on their absolute phases. Fig. 3 depicts the architecture of decodingmechanism.

��1

��3

��2

Sgn( )

Sgn( )

Sgn( )

Z-1

Z-1

Z-1

Phase

Detector

Phase

Detector

Phase

Detector

E(i)^

E(i-1)^

D(i)^

D(i-1)^

C(i)^

C(i-1)^

S(i)^

+

+

-

-

Fig. 3. Three stage decoding scheme for 32-QAM

3 Error Performance Analysis

To determine the probability of bit error for the proposed constellation andencoding scheme we utilize the approach as given in [8] and approximate theprobability of bit error as

Pb ≈ 1

bn

M−1∑

i=0

Ni∑

j=1

Px(i)P {εij}nb (15)

Angle Differential Modulation Scheme for Odd-Bit QAM 245

where M is the possible number of signal points transmitted, Ni is the numberof nearest neighbors i.e. no. of symbols at dmin from the ith symbol in theconstellation diagram. Px(i) is the probability of transmission of symbol. P {εij}is the probability of symbol error when symbol ′i′ is erroneously detected assymbol ′j′. nb is the number of bit errors when symbol ′i′ is erroneously detectedas symbol ′j′ and bn is the number of bits per symbol.

P{εij}diff = P(

S(i) is incorrect and S(i− 1) is correct)

+ P(

S(i) is correct and

S(i− 1) is incorrect)

+ P(

S(i) is incorrect and S(i− 1) is incorrect)

= P {εij} (1− P {εij}) + (1− P {εij})P {εij} + P {εij}P {εij}= 2P {εij} (1− P {εij}) + P {εij}2

= 2P {εij} − P {εij}2(16)

Assuming P {εij} is small, the higher order term can be neglected from the aboveexpression and we obtain

P{εij}diff ≈ 2P {εij} (17)

Hence for differential encoding the probability of bit error is approximated as

Pb,diff ≈ 2

bn

M−1∑

i=0

Ni∑

j=1

Px(i)P {εij}nb(i, j) (18)

Using the union bound and replacing P {εij} with Q(dmin/(2α)), we get

Pb,diff ≈ 2

bnQ

(

dmin

2α

)M−1∑

i=0

Ni∑

j=1

Px(i)nb(i, j) (19)

Let nb(i)Δ=

Ni∑

j=1

nb(i, j) and NbΔ=

M−1∑

i=1

Px(i)nb(i) then (19) can be written as

Pb,diff ≈ 2Nb

bnQ

(

dmin

2α

)

(20)

The value of Nb depends upon the particular bit mapping onto the constella-tion symbols. For differential encoding, the bit mapping is redefined after eachsymbol transmission; however, the value of Nb remains unchanged and henceNb can be calculated without loss of generality using any symbol. Let the pre-vious transmitted symbol S(i − 1) be the same as in example of Section 2 i.e.3.8284 + 5.2426j. For this particular choice the bit mapping is shown in Fig. 4.

From Fig. 4, the value of Nb may be calculated. For each symbol in the con-stellation, we compute the number of bit errors that would result if a transmittedsymbol is erroneously detected as one of its nearest neighbor, i.e., the symbolslying at dmin from the transmitted symbol. The average of the total bit errorsgives the value of Nb, as given by

246 S.S. Khalid and S. Abrar

01001

1000111001

00001

01101

1010111101

00101

01011

1001111011

00011

01111

1011111111

00111

01000

1000011000

00000

01100

1010011100

00100

01010

1001011010

00010

01110

1011011110

00110

Fig. 4. Bit mapping for the example given in Section 2

Nb =4

32

[

(3 + 1 + 1) + (3 + 1) + (2 + 2 + 1) + (1 + 1 + 2 + 3)+

(2 + 3 + 2 + 1) + (2 + 1) + (1 + 2) + (2 + 1)]

= 4.75

(21)

The expression for the probability of bit error becomes

Pb,diff ≈ 1.9Q

(

dmin

2α

)

(22)

Now to figure out the probability of bit error in terms of Eb/No, we first notethat dmin is equal to 2b and α =

√

N0/2.

Pb,diff ≈ 1.9Q

(

2b

2α

)

= 1.9Q

(

b√

N0/2

)

(23)

Now to determine the relation between b and Eb, we note that the symbol energyis given by Esym = E[|si|2], where si = ith vector on constellation diagram

Esym =4

32

[

4b2 + 4(b+√2b)2 + 4(b+ 2

√2b)2 + 4(b+ 3

√2b)2

]

(24)

Angle Differential Modulation Scheme for Odd-Bit QAM 247

0 2 4 6 8 10 12 14 16

10−4

10−3

10−2

10−1

100

Eb/N

0 [dB]

Pro

babi

lity

of b

it er

ror

Simulation: Differentiall 32.QAM (proposed)Analysis: Differential 32.QAM (proposed)Simulation: Coherent 32.QAM (proposed)Analysis: Coherent 32.QAM (proposed)Analysis: Coherent 32.QAM (traditional)

Fig. 5. Analytical and simulated BER curves under AWGN for the proposed differentialencoding scheme and its comparison with coherent ones

⇒ b =

√

Esym

2(

8 + 3√2) =

√

5Eb

2(

8 + 3√2) (25)

So the final expression for probability of bit error for differential encodingbecomes

Pb,diff ≈ 1.9Q

(√

Eb

N0

10

2(

8 + 3√2)

)

(26)

To compare our scheme with the traditional cross-shaped 32-QAM coherentmodulation scheme, we use the expression for its approximate probability of biterror as given in [9]:

Pb,cross ≈ 4

5Q

(√

15Eb

31No

)

(27)

4 Results and Conclusion

To verify the performance of the proposed scheme, the simulated as well asthe analytical probability of errors are presented in Fig. 5, where, in legend,“proposed” and “traditional” respectively refer to the constellations as depicted

248 S.S. Khalid and S. Abrar

in Fig. 1(a) and 1(b). It can be observed that the simulation results match withthe analytical performance very well specifically for higher SNR. The probabilityof error for coherent demodulation of 32-QAM for the proposed constellationwith the bit mapping same as that shown in Fig. 3 is also plotted.The analyticalprobability of error for the coherent 32-QAM of the proposed scheme can beapproximated as half the probability of error of the differential encoding schemei.e. Pb,coherent ≈ 0.5Pb,diff . At higher SNR the difference between coherent andnon-coherent detection becomes very small. Finally the performance of a classical32 cross QAM coherent modulation is also plotted for comparison with ourscheme. We observe that our scheme does incur a penalty of slightly greaterthan 1dB when compared with 32 cross QAM coherent detection however ourscheme proposes a method for differential encoding that is simple and scalableto higher order odd bit QAM to enable their transmission in such media wherecoherent detection owing to phase ambiguity is not possible.

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