Performance Analysis of an Angle Differential-QAM Scheme for Resolving Phase Ambiguity

Jeng-Kuang Hwang†, and Yu-Lun Chiu

Communication Signal Processing Laboratory Institute of Communication Engineering

Yuan-Ze University, 32026, Taiwan E-mail†: eejhwang@saturn.yzu.edu.tw

Abstract ⎯ An angle differential-QAM (ADQAM) scheme is proposed to solve phase ambiguity problem in non-data-aided continuous transmission system with square QAM constellation. Starting from the 16-ADQAM case, we derive differential encoding and decoding schemes in terms of two differential angles, and use a solar system analogy for explanation. The 16-ADQAM system incurs only about 0.5-dB performance degradation as compared to the coherent 16-QAM system under AWGN channel. Generalization to flat fading channel and higher-level ADQAM are straightforward. For educational purpose, a demonstrative 16-ADQAM system is realized in terms of an audio-band software-defined-radio approach, and experimental result shows the feasibility of real-time speech transmission. Keywords ⎯ Differential coding scheme, square QAM, continuous transmission system, phase ambiguity, Rayleigh fading, performance analysis.

1. Introduction M-ary quadrature amplitude modulation (QAM) is a widely

adopted digital modulation technique for its good bandwidth and power efficiency. Herein, consider the problem of applying the square QAM to continuous speech transmission without using preamble or unique word for data-aided receiver synchronization. In such a case, even the receiver is equipped with non-data-aided carrier recovery loop, it will still suffer from a phase ambiguity problem [1, Sec. 5.3], meaning that the signal constellation is rotated by an unknown integer multiple of π/2. To tackle this problem, differential coding is often used. For example, the differential QPSK (DQPSK) scheme uses a mapping between input dibit and four possible differential angle Δθ ∈{0. π/2, π, 3π/2}. However, very few literatures can be found about differential coding for the square QAM system, although a differential star 16-QAM scheme had been adopted by the CCITT v.29 9600 bps modem [2], and its performance analysis can also be found in [3-4]. In [5], Gini and Giannakis proposed a general differential scheme based on higher-order statistics, but it imposes a significant performance loss for square QAM and the differential coding scheme is quite complicated.

In this paper, we propose a new angle differential encoding/decoding scheme to solve the phase ambiguity problem for square QAM system which is already equipped with a non-data-aided Costas loop and amplitude estimator for

QAM signal. Using a solar system analogy, the proposed angle differential-QAM (ADQAM) scheme can be easily comprehended, and the decoding scheme can also be efficiently done. Under the assumptions of perfect amplitude estimation, both theoretical performance analysis and computer simulation show that the ADQAM scheme imposes only slight performance degradation as compared to coherent QAM system, under both the AWGN channel and Rayleigh fading channel.

To demonstrate how the proposed differential scheme can be applied to realistic continuous transmission scenario, a 16-ADQAM system is then implemented in terms of an audio-band software-defined-radio (AB-SDR) approach [6]. Being an instructional system, the platform needs only two personal computers with sound card support and Matlab software. Experimental results show that the whole system, including all necessary synchronization algorithms, can be designed, tested, and fine tuned in a very flexible way.

The paper is organized as follows. In Section II, the ADQAM coding and decoding schemes are presented. In Section III, the BER performance analysis of the proposed ADQAM scheme and the performance simulation result is conducted. In Section IV, the AB-SDR design of the 16-ADQAM system is described, and experimental results are also presented. Finally, conclusions are made in Section V.

2.ADQAM Encoding and Decoding Schemes

A. ADQAM Encoding Scheme

Since QPSK is a special case of M-QAM for M=4, we similarly want to generalize the DQPSK structure to the proposed ADQAM scheme by using K/2 differential angles, where K=log2M denotes the number of bits per ADQAM symbol. In the following, the coding and decoding processes are explained by taking the 16-ADQAM constellation as an example.

For 16-ADQAM, a group of four bits is mapped into one of the 24 possible transitions between two consecutive complex symbols S(i-1) and S(i). The transition can be expressed in terms of two differential angles {Δθ1, Δθ2}, where Δθ1 is determined by the first dibit of the QAM symbol, and Δθ2 is determined by the second dibit. In Table 1, the gray-coded

mapping between dibit (b1,b2) and corresponding differential angleΔθ is listed.

TABLE 1. Dibit to Differential Angle Mapping.

b0 b1 θ 0 0 0 0 1 π/2 1 1 π 1 0 3π/2

Referring to Fig. 1, Let the complex symbol S(i) be

represented as the summation of a quadrant center C(i), and a displacement D(i) :

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

Then the differential 16-QAM encoding rule can be described as two recursive updating formulae:

( ) ( ) ( )11 j iC i C i e θΔ= − (2)

( ) ( ) ( )21 j iD i D i e θΔ= − (3)

Without loss of generality, the initial symbol S(0) is set by letting 4(0) jC Re

π= and 4(0) jD re

π= , where 2 2R = denotes

the distance between the origin and the quadrant center, and 2r = is the distance between the quadrant center and the

constellation point. Hence, S(0) = 3+j3, and all the subsequent symbols S(i) can be expressed as ( ) ( ) ( )S i a i jb i= + , where a,b ∈ {± 1, ± 3}. The averaged symbol energy

is ( ),

2 -1 10

3s av

ME = = . Note that the above differential

encoding scheme is also applicable to DVB-T hierarchical QAM constellation [6].

Fig. 1 The 16-ADQAM constellation and its Solar System analogy.

A Solar System analogy can be used to explain the above encoding scheme. Referring to Fig. 1, let the origin represents the Sun which is stationary. Then the four quadrant centers denote the possible positions of the Earth revolving around the Sun, and the 16 constellation points correspond to possible locations of the Moon revolving around the Earth. With the above analogy, the transition from S(i-1) to S(i) can be simply viewed as two relative revolving movements between the Sun, Earth, and Moon. First, the Earth is rotated around the Sun by the first differential angle Δθ1(i), and then the Moon is rotated

around the Earth by the second differential angle Δθ2(i). A simple example is given to illustrate the above

16-ADQAM encoding scheme. Let a segment of data bits is [ 0 1 0 1 1 1 1 0 ] , which correspond to two 16-ADQAM symbols. Using the mapping in Table I, the following differential angles are obtained:

1 2(1) ; (1)2 2π πθ θΔ = Δ = ; 1 2

3(2) ; (2)2πθ π θΔ = Δ =

And resulting the 16-ADQAM symbol ( )2S = ( ) ( )2 2 3 - C D j+ = which the transition is illustrated in Fig.2.

For higher level ADQAM case, generalization of the above differential coding scheme is straightforward. Fig 3 shows the constellation for the 64-ADQAM case, where K=log2(64)=6 bits/symbol. In such a scheme, the 64-ADQAM symbol will be decomposed into three components which are rotated by three differential angles {Δθ1, Δθ2, Δθ3}, respectively.

Fig. 2 Illustrative examples of ADQAM symbol transition.

3θΔ

Fig. 3 The 64-ADQAM constellation and its three-stage encoding scheme.

B. ADQAM Decoding under Phase Ambiguity

If the transmitted signal s(t) undergoes a frequency flat channel, the received equivalent low-pass signal x(t) can be written as

( ) ( ) ( ) ( )oj tx t s t e n tω θα += + , (4)

where α denotes the complex channel attenuation, oω and θ denote the carrier frequency and phase offsets, and n(t) is the AWGN noise. For the proposed ADQAM system, two assumptions are made at the receiver side; (1) The received square QAM constellation has been correctly oriented and aligned with the I/Q axes by a non-data-aided (NDA) recursive Costas loop [1, Sec. 5.3.8]. (2) The received constellation has been adjusted to correct voltage level by automatic gain control (AGC), which can converge to a gain reciprocal to the absolute value of the channel attenuation. Then the decision variable at the slicer input can be written as

( ) ( ) ( )jX i S i e N iφ= + (5)

where φ∈ {0, π/2, π, 3π/2} denotes the unknown phase ambiguity, and N(i) denotes the noise sample. A two-stage differential decoding is then proposed for the 16-ADQAM system to detect the correct bit sequence. Substituting (2) and (3) into (4), we have

( ) ( ) ( ) ( )

( ) ( ) ( )

j j

p p

X i C i e D i e N i

C i D i N i

φ φ= + +

= + + (6)

where the subscript p denotes the rotation caused by the phase ambiguity φ. Since the rotated quadrant center can be easily decided as :

( ) ( )( ) ( )( )ˆ sgn real sgn imagpC i R X i j X i⎡ ⎤⎡ ⎤ ⎡ ⎤= × +⎣ ⎦ ⎣ ⎦⎣ ⎦ (7)

where real[x] and imag[x] denote the real and imaginary parts of a complex number x, respectively, and sgn(x) is the signum function. From ˆ ˆ( ) and ( 1)p pC i C i − , the first differential angle Δθ1(i) and the corresponding dibit can be detected according to the following rule:

If ( ) ( )( )( )( )( )( )

21

2* 1

21

21

say 0 ,say / 2,ˆ ˆ 1say ,

, say 3 / 2

p p

iRijR

C i C iiR

jR i

θ

θ π

θ π

θ π

⎧ Δ =⎪

Δ =⎪× − = ⎨

Δ =−⎪⎪− Δ =⎩

(8)

Once ( )ˆpC i and Δθ1(i) have been detected, the second

displacement vector Dp(i) can be detected likewise. By letting

( ) ( ) ( )( )

( ) ( )( )ˆˆ sgn

ˆ sgn

p p

p

D i r real X i C i

j imag X i C i

⎡ ⎡ ⎤= × − +⎢ ⎣ ⎦⎣⎤⎡ ⎤− ⎥⎣ ⎦ ⎦

(9)

the second differential angle Δθ2(i) can also be detected as

If ( ) ( )( )( )( )( )( )

22

2* 2

22

22

say 0 ,say / 2, ˆ ˆ 1

say , , say 3 / 2

p p

irijr

D i D iir

jr i

θ

θ π

θ π

θ π

⎧ Δ =⎪

Δ =⎪× − = ⎨

Δ =−⎪⎪− Δ =⎩

(10)

Therefore, regardless of the phase ambiguity φ, the ADQAM symbol can be correctly detected. The block diagram of the above two-stage decoding scheme is shown in Fig.4. For

higher-level M-ray ADQAM, similar procedure with K/2 decoding stages can be applied. Besides, for M=4, the above differential decoding scheme degenerates to the simplest DQPSK case, which has been adopted by the conventional Barker-code WLAN and CCK WLAN [7].

X(i)( )ˆ

pC i

( )*ˆ 1pC i −

( )*ˆ 1pD i −

( )ˆpD i

( )1̂ iθΔ

( )2̂ iθΔ

sgn(.)

Z-1

Phase detector

sgn(.)

Z-1

Phase detector

Fig. 4 The two-stage decoding scheme of 16-ADQAM receiver.

3. Analysis of ADQAM Error Performance

Based on the received signal model (4) and the proposed ADQAM scheme, we should analyze the receiver performance below. First, let the minimum distance of the 16-QAM constellation be denoted as dmin. Note that the error probability in deciding the quadrant center Cp(i) is governed by the nearest neighbour union bound (NNUB) at sufficiently high SNR. Thus, counting the number of line segments with length dmin/2 from the constellation points in the same quadrant to the quadrant boundary gives the value Np = 4 for the 16-ADQAM. For the AWGN channel with α=1, the distribution of N(i) is N(0, N0/2), and then the error probabilities for Cp(i) can be closely approximated as

( )( ) ( ) ( ) 1ˆPr4e p p p pP C i C i C i N p p⎡ ⎤= ≠ ≈ × × =⎣ ⎦ , (11)

where it is assumed that the four signal points in the quadrant are equally likely, and

min

0

12 2

dp erfcN

⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠ (12)

is the tail probability in (dmin/2, ∞) of the noise distribution. Since the detection of the first differential angle ( )1 iθΔ depends on two subsequent quadrant centers, we have the approximate error probability for ( )1 iθΔ as

( )( ) ( )( )1 2 2e e pP i P C i pθΔ ≈ ≈ .(13)

Next, let’s examine the second-stage detection for Dp(i). From the total probability theorem, we have

( )( ) ( ) ( )

( )( ) ( )( )( ) ( )

( ) ( )( )

ˆPr

ˆ ˆ | ( ) is correct ( ) is correct

ˆ ˆ | ( ) is incorrect ( ) is incorrect

ˆ 2 1 | ( ) is incorrect

e p p p

e p p p

e p p p

e p p

P D i D i D i

P D i C i P C i

P D i C i P C i

p p P D i C i p

⎡ ⎤= ≠⎣ ⎦

= +

≈ − +

(14)

where ( )( )ˆ| ( ) is incorrecte p pP D i C i denotes the error

propagation effect due to the first-stage error. The above equation can be simplified by letting p<<1 and

( )( )ˆ| ( ) is incorrecte p pP D i C i ≈1. Then

( )( ) min

0

322 2e p

dP D i p p erfcN

⎛ ⎞≈ + = ⎜ ⎟⎜ ⎟

⎝ ⎠ .(15)

Thus we have

( )( ) ( )( )2 2 6e e pP i P D i pθΔ ≈ ≈ .(16)

Finally, the average BER of the 16-ADQAM can be found as

( )( )( ) ( )( )( )

( )( ) ( )( ) ( )( ) ( )( )

,16 1 22

1 2 1 2

2

1 1 1 1log 161 4

2 3

e DQAM e e

e e e e

P P i P i

P i P i P i P i

p p

θ θ

θ θ θ θ

−⎡ ⎤= − − Δ − Δ⎣ ⎦

⎡ ⎤= Δ + Δ − Δ Δ⎣ ⎦

= −

(17)

For M-ary QAM constellation, the average bit energy Eb is related to the minimum distance dmin as follows:

( ), 2

min2 2

2 1log 3 log

s avb

E ME d

M M−

= =×

(18)

Hence, for M=16, we have 2min2.5bE d= , and the final

approximate BER expression in terms of Eb/N0 is

,160

210

be DQAM

EP p erfcN−

⎛ ⎞≈ = ⎜ ⎟⎜ ⎟

⎝ ⎠ (19)

On the other hand, a coherent 16-QAM system has the approximate BER

,160

38 10

be QAM

EP erfcN−

⎛ ⎞≈ ⎜ ⎟⎜ ⎟

⎝ ⎠ (20)

Therefore the ADQAM-to-QAM BER ratio for M=16 is

,16

,16

8 2.6673

e DQAM

e QAM

PP

−

−

= ≈ . (21)

Next, let us consider the BER analysis under Rayleigh fading channel. In such a case, the fading coefficient α is a zero-mean complex Gaussian random variable. Hence, the

instantaneous SNR 2

0

bEN

αγ = has an exponential probability

density function as follows

( ) 1 , 0f e γ γγ γγ

−= ≥ (22)

where 2

0

bEEN

γ α⎡ ⎤= ⎣ ⎦ denotes the average SNR per bit. Hence,

from (17), the instantaneous BER is also a function of γ:

( )2

,161 12 32 10 2 10e DQAMP erfc erfcγ γγ−

⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠

(23)

To find the average error probability under Rayleigh fading,

we can use integration technique [8] to average (23) over the PDF of γ, resulting in a closed-form BER as:

( ) ( ),16 ,160

111 1 1 110 10 102 3 tan

2 2 41 110 10 10

f DQAM e DQAMP P f dγ γ γ

γ γ γ

γ γ γπ

∞

− −

−

=

⎛ ⎞⎛ ⎞ ⎛ ⎞+⎜ ⎟⎜ ⎟ ⎜ ⎟= − − −⎜ ⎟⎜ ⎟ ⎜ ⎟+ +⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

∫

(24)

For higher-level constellation with M>16, the M-ary ADQAM symbol consists of K/2 differential angles{ }1 2

Kθ θΔ Δ , and a K/2-stage decoder is used in the receiver. Its BER under AWGN channel can also be approximated by NNUB, and the result is

( )( )( )( ) ( )( )( )

/ 2

,12

12

2

1 ˆ1 1log

1 1 1 2 1 2 1 22log

K

e M DQAM e nn

K

P P iM

Kp p pM

θ−=

−

⎡ ⎤= − − Δ⎢ ⎥

⎣ ⎦

⎡ ⎤≈ − − − + ≈⎢ ⎥⎣ ⎦

∏ (25)

where , 0

12 4

b

s av

EKp erfcE N

⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠.

To verify the performance of the proposed ADQAM coding and decoding scheme, Fig.5 plots both the simulated and theoretical BER curves of the 16-ADQAM system. Under both the AWGN and flat Rayleigh fading channels, the theoretical approximate BER curve and the simulated curve match very well. Moreover, the proposed 16-ADQAM system degrades by about 0.6 dB at BER=10-3 under the AWGN channel, as compared to its coherent counterpart. Fig. 6 shows the BER performance comparison between coherent QAM and ADQAM system under AWGN channel for M = 4, 16, 64, 256, and 1024. It is seen that the performance loss increases with M, but is still less than 1dB at BER=10-3 for M=1024.

0 5 10 15 20 25 3010-6

10-5

10-4

10-3

10-2

10-1

100

Eb/N0 (dB)

BE

R

Fig. 5 The theoretical and simulated BER curves of the 16-ADQAM system under AWGN and flat Rayleigh channels, with the coherent

16-QAM system as a reference.

5 10 15 20 25 3010

-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Eb/N0, dB

BE

R

M-QAMM-ADQAM

M=4 M=16 M=64 M=256 M=1024

.

4. An Audio-Band SDR Realization of the

16-ADQAM System An SDR is able to provide very flexible multi-mode,

multifunction, and multi-band operation. In this paper, we deliberately use a much lower carrier frequency in audio band for educational and budget consideration.

A. The AB-SDR Approach and Block Diagram

As is shown in Fig.7, the audio-band SDR (AB-SDR) platform needs only two personal computers (PCs) with soundcard support and the Matlab software. Specifically, PC-1 is for running the transmitter (TX) program, including the speech coding, 16-ADQAM mapping, and I/Q modulation. Then the soundcard SPK output is used for sending out the modulated ADQAM signal. On the other hand, PC-2 is for running the complete 16-ADQAM real-time receiver (RX) program which acquires its input signal from the PC-2 LINE IN jacket. Besides, a 3.5 mm audio wire is used to connect the PC-1 SPK and the PC-2 LINE IN jackets. Oscilloscope can also be used to monitor the audio-band ADQAM signal during transmission.

At the transmitter, the input voice signal is converted into the source bit stream at a bit rate of 44.1 kbps by using the adaptive delta modulation (ADM). Then the bit stream is converted to 16-ADQAM passband signal at a carrier frequency of 10 kHz. At the receiver, continuous reception of bit stream is done with the help of two non-data-aided (NDA) synchronizers: (1) Gardner’s symbol timing recovery loop [1, Sec. 7.5], and (2) a recursive 2nd-order digital Costas loop [1, Sec. 5.3.8] for carrier frequency synchronization. The primary system parameters settings are listed in Table 2.

B. Experimental Results

A test speech signal was transmitted by the 16-ADQAM system. We deliberately set a carrier frequency offset of fo = 10 Hz. We have observed that both synchronizers can lock the signal quickly. Fig. 8(a) shows the signal constellation after the symbol synchronizer, where the carrier frequency offset gives rise to constellation rotation. Finally, Fig. 8(b) shows the output after the Costas loop, where the de-rotated constellation is now aligned with the signal space axes. Although an unknown phase ambiguity still remains, the proposed ADQAM decoding scheme can successfully solve the problem and detect all the source bits correctly.

Fig.8 The 16-ADQAM symbol constellation after (a) symbol synchronizer , and

(b) carrier synchronizer.

3.5mmaudio cable

16-ADQAMEncoder

Adaptive-Delta-

Modulation

4

SRRC

4

SRRC

A/D

D/A

Σ

AWGN

⊗+

+

cos sin

⊗

Σ

A/D

D/A Adaptive-

Delta-Demodulation

Recursive DigitalCostas Loop

16-ADQAMDecoder

⊗+

+

cos sin

⊗

Σ

j

Halfbandfilter

T/2 SRRC filter

T/2-Interpolation Timing Recovery

2

LPF

SPK OUT

LINE IN

PC-116-ADQAM Matlab Tx-Program

PC-216-ADQAM Matlab Rx-Program

Sound Card 44.1kHz

Fig. 7 Block diagram of the 16-ADQAM AB-SDR transceiver.

Fig. 6 The BER performance comparison between coherent QAM and ADQAM system under AWGN for M = 4, 16, 64, 256, and 1024

5. Conclusions

In this paper, we have proposed an angle differential-QAM (ADQAM) system to solve the phase ambiguity problem for real-time continuous transmission system without resorting to any training sequence. It is shown that the differential coding/decoding scheme is very systematic and costs only a little extra computational load. As for the BER performance, the proposed ADQAM system just incurs a little performance degradation, as compared to the coherent square-QAM system. An instructive AB-SDR implementation of the 16-ADQAM system is also presented, which can serve as a cost-effective algorithm verification and prototyping workbench.

TABLE 2. Parameters of the AB-SDR 16-ADQAM transciver 1. Transmitter Parameters Symbol rate Rs= Rb/4=11.025 ksps Over sampling rate OVR=4 TX D/A sampling rate fst = OVR*Rs = 44.1 ksps TX carrier frequency fc= 10 kHz SRRC filter roll-off factor α=0.5 2. Receiver Parameters RX A/D passband sampling rate fsr = 44.1 ksps RX carrier frequency offset fo = 10 Hz Matched filter and decimation ratio 0.5-SRRC with 2:1 decimation

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