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 QAM signal. Using a solar system analogy, the proposed angle
proposed to solve phase ambiguity problem in non-data-aided differential-QAM (ADQAM) scheme can be easily
continuous transmission system with square QAM constellation. comprehended, and the decoding scheme can also be
Starting from the 16-ADQAM case, we derive differential efficiently done. Under the assumptions of perfect amplitude
encoding and decoding schemes in terms of two differential
estimation, both theoretical performance analysis and
angles, and use a solar system analogy for explanation. The
16-ADQAM system incurs only about 0.5-dB performance computer simulation show that the ADQAM scheme imposes
degradation as compared to the coherent 16-QAM system under only slight performance degradation as compared to coherent
AWGN channel. Generalization to flat fading channel and QAM system, under both the AWGN channel and Rayleigh
higher-level ADQAM are straightforward. For educational fading channel.
purpose, a demonstrative 16-ADQAM system is realized in terms To demonstrate how the proposed differential scheme can
of an audio-band software-defined-radio approach, and be applied to realistic continuous transmission scenario, a
experimental result shows the feasibility of real-time speech 16-ADQAM system is then implemented in terms of an
transmission. audio-band software-defined-radio (AB-SDR) approach [6].
Being an instructional system, the platform needs only two
Keywords ⎯ Differential coding scheme, square QAM, personal computers with sound card support and Matlab
continuous transmission system, phase ambiguity, Rayleigh
software. Experimental results show that the whole system,
fading, performance analysis.
including all necessary synchronization algorithms, can be
designed, tested, and fine tuned in a very flexible way.
1. Introduction The paper is organized as follows. In Section II, the
M-ary quadrature amplitude modulation (QAM) is a widely ADQAM coding and decoding schemes are presented. In
adopted digital modulation technique for its good bandwidth Section III, the BER performance analysis of the proposed
and power efficiency. Herein, consider the problem of ADQAM scheme and the performance simulation result is
applying the square QAM to continuous speech transmission conducted. In Section IV, the AB-SDR design of the
without using preamble or unique word for data-aided receiver 16-ADQAM system is described, and experimental results are
synchronization. In such a case, even the receiver is equipped also presented. Finally, conclusions are made in Section V.
with non-data-aided carrier recovery loop, it will still suffer
from a phase ambiguity problem [1, Sec. 5.3], meaning that 2.ADQAM Encoding and Decoding Schemes
the signal constellation is rotated by an unknown integer
multiple of π/2. To tackle this problem, differential coding is A. ADQAM Encoding Scheme
often used. For example, the differential QPSK (DQPSK)
scheme uses a mapping between input dibit and four possible Since QPSK is a special case of M-QAM for M=4, we
differential angle Δθ ∈{0. π/2, π, 3π/2}. However, very few similarly want to generalize the DQPSK structure to the
literatures can be found about differential coding for the proposed ADQAM scheme by using K/2 differential angles,
square QAM system, although a differential star 16-QAM where K=log2M denotes the number of bits per ADQAM
scheme had been adopted by the CCITT v.29 9600 bps modem symbol. In the following, the coding and decoding processes
[2], and its performance analysis can also be found in [3-4]. In are explained by taking the 16-ADQAM constellation as an
[5], Gini and Giannakis proposed a general differential scheme example.
based on higher-order statistics, but it imposes a significant For 16-ADQAM, a group of four bits is mapped into one of
performance loss for square QAM and the differential coding the 24 possible transitions between two consecutive complex
scheme is quite complicated. symbols S(i-1) and S(i). The transition can be expressed in
In this paper, we propose a new angle differential terms of two differential angles {Δθ1, Δθ2}, where Δθ1 is
encoding/decoding scheme to solve the phase ambiguity determined by the first dibit of the QAM symbol, and Δθ2 is
problem for square QAM system which is already equipped determined by the second dibit. In Table 1, the gray-coded
with a non-data-aided Costas loop and amplitude estimator for
mapping between dibit (b1,b2) and corresponding differential around the Earth by the second differential angle Δθ2(i).
angleΔθ is listed. A simple example is given to illustrate the above
16-ADQAM encoding scheme. Let a segment of data bits is [ 0
TABLE 1. Dibit to Differential Angle Mapping. 1 0 1 1 1 1 0 ] , which correspond to two 16-ADQAM
b0 b1 θ symbols. Using the mapping in Table I, the following
0 0 0 differential angles are obtained:
0 1 π/2
π π 3π
1 1 π Δθ1(1) = ; Δθ2 (1) = ; Δθ1(2) = π ; Δθ2 (2) =
1 0 3π/2 2 2 2
And resulting the 16-ADQAM symbol S (2) =
Referring to Fig. 1, Let the complex symbol S(i) be C (2) + D (2) = 3 - j which the transition is illustrated in Fig.2.
represented as the summation of a quadrant center C(i), and a For higher level ADQAM case, generalization of the above
displacement D(i) : differential coding scheme is straightforward. Fig 3 shows the
S (i ) = C (i ) + D(i) (1) constellation for the 64-ADQAM case, where K=log2(64)=6
bits/symbol. In such a scheme, the 64-ADQAM symbol will
Then the differential 16-QAM encoding rule can be described be decomposed into three components which are rotated by
as two recursive updating formulae: three differential angles {Δθ1, Δθ2, Δθ3}, respectively.
C (i ) = C (i − 1)e j Δθ1 (i ) (2)

D (i ) = D (i − 1)e jΔθ2 (i ) (3)

Without loss of generality, the initial symbol S(0) is set by
π π
letting C (0) = Re j 4 and D(0) = re j 4 , where R = 2 2 denotes
the distance between the origin and the quadrant center, and
r = 2 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
2 ( M -1)
is Es ,av = = 10 . Note that the above differential
3 Fig. 2 Illustrative examples of ADQAM symbol transition.
encoding scheme is also applicable to DVB-T hierarchical
QAM constellation [6].
Δθ 3

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

A Solar System analogy can be used to explain the above Fig. 3 The 64-ADQAM constellation and its three-stage encoding scheme.
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 B. ADQAM Decoding under Phase Ambiguity
Sun, and the 16 constellation points correspond to possible
locations of the Moon revolving around the Earth. With the If the transmitted signal s(t) undergoes a frequency flat
above analogy, the transition from S(i-1) to S(i) can be simply channel, the received equivalent low-pass signal x(t) can be
viewed as two relative revolving movements between the Sun, written as
Earth, and Moon. First, the Earth is rotated around the Sun by
x (t ) = α s (t ) e + n (t ) ,
j (ωo t +θ )
the first differential angle Δθ1(i), and then the Moon is rotated (4)
where α denotes the complex channel attenuation, ωo and higher-level M-ray ADQAM, similar procedure with K/2
θ denote the carrier frequency and phase offsets, and n(t) is the decoding stages can be applied. Besides, for M=4, the above
AWGN noise. For the proposed ADQAM system, two differential decoding scheme degenerates to the simplest
assumptions are made at the receiver side; (1) The received DQPSK case, which has been adopted by the conventional
square QAM constellation has been correctly oriented and Barker-code WLAN and CCK WLAN [7].
aligned with the I/Q axes by a non-data-aided (NDA) recursive Cˆ p ( i )
Costas loop [1, Sec. 5.3.8]. (2) The received constellation has Phase
X(i) sgn(.) Δθˆ1 ( i )
detector
been adjusted to correct voltage level by automatic gain Cˆ *p ( i − 1)

control (AGC), which can converge to a gain reciprocal to the Z-1

absolute value of the channel attenuation. Then the decision Dˆ p ( i ) Phase

sgn(.) Δθˆ2 ( i )
variable at the slicer input can be written as Dˆ *p ( i − 1)
detector

X ( i ) = S ( i ) e jφ + N ( i ) (5)
Z-1

where φ∈ {0, π/2, π, 3π/2} denotes the unknown phase Fig. 4 The two-stage decoding scheme of 16-ADQAM receiver.
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 3. Analysis of ADQAM Error Performance
X ( i ) = C ( i ) e jφ + D ( i ) e jφ + N ( i ) Based on the received signal model (4) and the proposed
(6) ADQAM scheme, we should analyze the receiver
= C p ( i ) + Dp (i ) + N (i ) performance below. First, let the minimum distance of the
16-QAM constellation be denoted as dmin. Note that the error
where the subscript p denotes the rotation caused by the phase
probability in deciding the quadrant center Cp(i) is governed
ambiguity φ. Since the rotated quadrant center can be easily
by the nearest neighbour union bound (NNUB) at sufficiently
decided as : high SNR. Thus, counting the number of line segments with
( ) (
Cˆ p ( i ) = R × ⎡sgn real ⎣⎡ X ( i ) ⎦⎤ + j sgn imag ⎣⎡ X ( i ) ⎦⎤ ⎤
⎣ ⎦ ) (7)
length dmin/2 from the constellation points in the same
quadrant to the quadrant boundary gives the value Np = 4 for
where real[x] and imag[x] denote the real and imaginary parts the 16-ADQAM. For the AWGN channel with α=1, the
of a complex number x, respectively, and sgn(x) is the signum distribution of N(i) is N(0, N0/2), and then the error
probabilities for Cp(i) can be closely approximated as
function. From Cˆ p (i ) and Cˆ p (i − 1) , the first differential angle
1
Δθ1(i) and the corresponding dibit can be detected according Pe ( C p ( i ) ) = Pr ⎡⎣Cˆ p ( i ) ≠ C p ( i ) ⎤⎦ ≈ × N p × p = p , (11)
to the following rule: 4

⎧R2 , say Δθ1 ( i ) = 0 where it is assumed that the four signal points in the quadrant
⎪ 2 are equally likely, and
⎪ jR , say Δθ1 ( i ) = π / 2
( )
*
If Cˆ p ( i ) × Cˆ p ( i − 1) =⎨ 2 (8) ⎛ d ⎞
⎪− R , say Δθ1 ( i ) = π 1
p = erfc ⎜ min ⎟ (12)
⎪ − jR 2 , 2 ⎜2 N ⎟
⎩ say Δθ1 ( i ) = 3π / 2 ⎝ 0 ⎠
is the tail probability in (dmin/2, ∞) of the noise distribution.
Once Cˆ p ( i ) and Δθ1(i) have been detected, the second
Since the detection of the first differential angle Δθ1 ( i )
displacement vector Dp(i) can be detected likewise. By letting
depends on two subsequent quadrant centers, we have the
( )
Dˆ p ( i ) = r × ⎡⎢sgn real ⎡⎣ X ( i ) − Cˆ p ( i ) ⎤⎦ +
⎣ approximate error probability for Δθ1 ( i ) as
j sgn ( imag ⎡⎣ X ( i ) − Cˆ ( i ) ⎤⎦ ) ⎤⎥
(9)
p
⎦ Pe ( Δθ1 ( i ) ) ≈ 2 Pe ( C p ( i ) ) ≈ 2 p .(13)

the second differential angle Δθ2(i) can also be detected as Next, let’s examine the second-stage detection for Dp(i).
From the total probability theorem, we have
⎧r 2 , say Δθ 2 ( i ) = 0
⎪ 2
say Δθ 2 ( i ) = π / 2
If Dˆ p ( i ) × ( Dˆ p ( i − 1) )
⎪ jr , Pe ( D p ( i ) ) = Pr ⎡⎣ Dˆ p ( i ) ≠ D p ( i ) ⎤⎦
*
=⎨ 2 (10)
⎪−r , say Δθ 2 ( i ) = π
⎪ − jr 2 ,
⎩ say Δθ 2 ( i ) = 3π / 2
( ) (
= Pe D p ( i ) | Cˆ p (i) is correct P Cˆ p (i ) is correct + )
(14)
P (De ) (
p ( i ) | C p (i ) is incorrect P C p (i ) is incorrect
ˆ ˆ )
Therefore, regardless of the phase ambiguity φ, the ADQAM (
≈ 2p (1 − p ) + Pe Dp ( i ) | Cˆ p (i ) is incorrect p )
symbol can be correctly detected. The block diagram of the
above two-stage decoding scheme is shown in Fig.4. For where (
Pe D p ( i ) | Cˆ p (i ) is incorrect ) denotes the error
propagation effect due to the first-stage error. The above we can use integration technique [8] to average (23) over the
equation can be simplified by letting p<<1 and PDF of γ, resulting in a closed-form BER as:
( )
Pe D p ( i ) | Cˆ p (i ) is incorrect ≈1. Then ∞
Pf ,16− DQAM = ∫ Pe,16− DQAM ( γ ) f ( γ ) d γ
0
3 ⎛ d ⎞
Pe ( D p ( i ) ) ≈ 2 p + p = erfc ⎜ min ⎟ .(15) ⎛ γ ⎞ ⎛ γ ⎛
1+ γ
⎞⎞ (24)
2 ⎜2 N ⎟ ⎜1 1 10 ⎟ − 3 ⎜ 1 − 1 10 tan −1 ⎜ 10 ⎟ ⎟
⎝ 0 ⎠ = 2⎜ −
γ ⎟ ⎜ γ ⎜ γ ⎟⎟
⎜ 2 2 1 + 10 ⎟ ⎜ 4 π 1 + 10 ⎜ 10 ⎟⎠ ⎟⎠
⎝ ⎠ ⎝ ⎝
Thus we have
For higher-level constellation with M>16, the M-ary ADQAM
Pe ( Δθ 2 ( i ) ) ≈ 2 Pe ( Dp ( i ) ) ≈ 6 p .(16)
symbol consists of K/2 differential angles Δθ1 " ΔθK 2 , and a { }
Finally, the average BER of the 16-ADQAM can be found
K/2-stage decoder is used in the receiver. Its BER under
as AWGN channel can also be approximated by NNUB, and the
1 ⎡ result is
Pe,16− DQAM =
log 2 16 ⎣
( )( )
1 − 1 − Pe ( Δθ1 ( i ) ) 1 − Pe ( Δθ 2 ( i ) ) ⎤
⎦
1
= ⎡⎣ Pe ( Δθ1 ( i ) ) + Pe ( Δθ 2 ( i ) ) − Pe ( Δθ1 ( i ) ) Pe ( Δθ 2 ( i ) ) ⎤⎦
(17) Pe, M − DQAM =
1
log 2 M
⎡ K /2
⎣ n =1
( ( ˆ ))
⎤
⎢1 − ∏ 1 − Pe Δθ n ( i ) ⎥
⎦
4 (25)
= 2 p − 3 p2 ≈
1
log 2 M ( ( (
⎡1 − (1 − 2 p ) K 2 −1 1 − 2 p K + 1
⎢⎣ 2 )) )⎤⎥⎦ ≈ 2 p
For M-ary QAM constellation, the average bit energy Eb is
related to the minimum distance dmin as follows: 1 ⎛ K E ⎞
where p = erfc ⎜⎜ b
⎟.
⎟
Es ,av 2 ( M − 1) 2 2 ⎝ E s , av 4 N 0 ⎠
Eb = = d min (18)
log 2 M 3 × log 2 M To verify the performance of the proposed ADQAM coding
and decoding scheme, Fig.5 plots both the simulated and
Hence, for M=16, we have Eb = 2.5d , and the final 2
min
theoretical BER curves of the 16-ADQAM system. Under
approximate BER expression in terms of Eb/N0 is both the AWGN and flat Rayleigh fading channels, the
⎛ Eb ⎞ theoretical approximate BER curve and the simulated curve
Pe,16 − DQAM ≈ 2 p = erfc ⎜⎜ ⎟⎟ (19) match very well. Moreover, the proposed 16-ADQAM system
⎝ 10 N 0 ⎠ degrades by about 0.6 dB at BER=10-3 under the AWGN
channel, as compared to its coherent counterpart. Fig. 6 shows
On the other hand, a coherent 16-QAM system has the
the BER performance comparison between coherent QAM
approximate BER
and ADQAM system under AWGN channel for M = 4, 16, 64,
3 ⎛ Eb ⎞ 256, and 1024. It is seen that the performance loss increases
Pe ,16−QAM ≈ erfc ⎜
8 ⎜ 10 N ⎟⎟ (20) with M, but is still less than 1dB at BER=10-3 for M=1024.
⎝ 0 ⎠

Therefore the ADQAM-to-QAM BER ratio for M=16 is 0

10

Pe ,16− DQAM 8
= ≈ 2.667 . (21) -1
10
Pe ,16−QAM 3

Next, let us consider the BER analysis under Rayleigh -2

10
fading channel. In such a case, the fading coefficient α is a
zero-mean complex Gaussian random variable. Hence, the -3
BER

10
α 2 Eb
instantaneous SNR γ = has an exponential probability
N0 -4
10
density function as follows
1 -5

f (γ ) =
10
e−γ γ , γ ≥0 (22)
γ
Eb -6
10
where γ = E ⎡⎣α ⎤⎦
2
denotes the average SNR per bit. Hence, 0 5 10 15 20 25 30
N0 Eb/N0 (dB)

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

⎛1
2 Fig. 5 The theoretical and simulated BER curves of the 16-ADQAM
⎛ γ ⎞⎞ ⎛ 1 ⎛ γ ⎞⎞
Pe ,16 − DQAM ( γ ) = 2 ⎜ erfc ⎜⎜ ⎟⎟ ⎟⎟ + 3 ⎜⎜ erfc ⎜⎜ ⎟⎟ ⎟⎟ (23) system under AWGN and flat Rayleigh channels, with the coherent
⎜2
⎝ ⎝ 10 ⎠ ⎠ ⎝ 2 ⎝ 10 ⎠ ⎠ 16-QAM system as a reference.

To find the average error probability under Rayleigh fading,

 0
At the transmitter, the input voice signal is converted into
10
M-QAM
the source bit stream at a bit rate of 44.1 kbps by using the
-1
10
M-ADQAM adaptive delta modulation (ADM). Then the bit stream is
converted to 16-ADQAM passband signal at a carrier
-2
10 frequency of 10 kHz. At the receiver, continuous reception of
bit stream is done with the help of two non-data-aided (NDA)
-3
10
synchronizers: (1) Gardner’s symbol timing recovery loop [1,
BER

-4
Sec. 7.5], and (2) a recursive 2nd-order digital Costas loop [1,
10
Sec. 5.3.8] for carrier frequency synchronization. The primary
-5
10 system parameters settings are listed in Table 2.
M=4 M=16 M=64 M=256 M=1024

-6
10 B. Experimental Results
-7
10
5 10 15 20 25 30 A test speech signal was transmitted by the 16-ADQAM
Eb/N0, dB
system. We deliberately set a carrier frequency offset of fo = 10
Fig. 6 The BER performance comparison between coherent QAM Hz. We have observed that both synchronizers can lock the
and ADQAM system under AWGN . for M = 4, 16, 64, 256, and 1024 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
4. An Audio-Band SDR Realization of the output after the Costas loop, where the de-rotated constellation
16-ADQAM System is now aligned with the signal space axes. Although an
An SDR is able to provide very flexible multi-mode, unknown phase ambiguity still remains, the proposed
multifunction, and multi-band operation. In this paper, we ADQAM decoding scheme can successfully solve the problem
deliberately use a much lower carrier frequency in audio band and detect all the source bits correctly.
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 Fig.8 The 16-ADQAM symbol constellation after
(a) symbol synchronizer , and
PC-1 SPK and the PC-2 LINE IN jackets. Oscilloscope can (b) carrier synchronizer.
also be used to monitor the audio-band ADQAM signal during
transmission.

PC-1 Sound Card 44.1kHz PC-2

16-ADQAM Matlab Tx-Program 16-ADQAM Matlab Rx-Program

4 Adaptive- ⊗
16-ADQAM A A +
Delta- LINE IN
Encoder cos Halfband T/2 SRRC
4 Modulation / /
Σ 2
D D sin filter filter
SRRC ⊗ 3.5mm +

+
audio cable
⊗ j T/2-Interpolation
cos D D Timing Recovery
Σ Σ
sin / /
SPK OUT L Adaptive-
+ A A P Delta- 16-ADQAM Recursive Digital
AWGN
SRRC ⊗ F Demodulation Decoder Costas Loop

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

 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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