SER Expressions for Double Differential Modulation

SER Expressions for Double Differential Modulation Manav R. Bhatnagar and Are Hjørungnes UniK – University Graduate Center University of Oslo Instituttveien 25, P. O. Box 70 N-2027 Kjeller, Norway Email: {manav,arehj}@unik.no

Abstract— We present symbol error rate (SER) expressions for the double differential (DD) modulation with MPSK constellations (DDMPSK) over AWGN, Rayleigh, and Ricean fading channels with carrier offsets in a single-input single-output (SISO) system. A closed-form expression is derived for SER in double differential binary phase shift keying (DDBPSK) over AWGN channels with carrier offset. Next, we derive a closedform solution for the SER of DDBPSK and lower and upper bounds over the SER for DDMPSK signals over Rayleigh fading channels with carrier offset. A closed-form SER formulation for DDBPSK over Ricean fading channel with carrier offset is also presented.

I. I NTRODUCTION Double differential modulation [1] is a key differential technique to remove the effect of carrier offset. It differs from single differential modulation in the sense that the decoder uses three consecutive received data samples for decoding of the current symbol. Two consecutive levels of single differential modulations are employed at the transmitter, and the decoding process has double complexity as compared to the single differential demodulation. The literature over SER analysis of single differential modulation is quite rich [2], [3], [4], [8], [9], [12], [13], [15], however, very less is explored about the analytical performance expressions of the double differential modulation. One reason for this is the complicated decoding process. In [5], the bit error rate (BER) of DDBPSK over AWGN channel is derived for a suboptimal decoder, nevertheless, no closed-form BER formulation is provided and a few numerically calculated performance results of DDBPSK are presented. An approximate probability density function of second order phase error for Ricean fading channels is presented in [6] but it also provides no closed-form formulation of BER/SER in DDPSK and presents only a few numerically calculated performance results. BER performance of DDQPSK is evaluated only in faded channel characterized by Gaussian impulsive noise and Doppler effects by [7]. In this paper, we are presenting the following basic results about the double differential modulation for PSK constellations: 1) We derive a closed-form SER expression for This work was supported by the Research Council of Norway project 176773/S10 called OptiMO, which belongs to the VERDIKT program.

DDBPSK over AWGN channel with carrier offset. 2) A closed-form SER expression for DDBPSK over Rayleigh fading channel with carrier offset is found. 3) We have derived lower and upper bounds over SER for DDMPSK over Rayleigh fading channel with carrier offset. 4) We have also derived closed-form SER of DDBPSK over Ricean fading channel with carrier offset. The rest of this paper is organized as follows: In Section II, performance analysis of DDPSK over AWGN channels with carrier offset is given. The performance of DDPSK modulation over Rayleigh fading channels is explained in Section III and over Ricean fading channels in Section IV. In Section V, the analytical results are compared with the experimental results and previously existing results. Section VI contains some conclusions. Abbreviations: Single differential (D), Double differential (DD), Double differential phase shift keying (DDPSK), Double differential binary phase shift keying (DDBPSK), Double differential quadrature phase shift keying (DDQPSK), Double differential M -ary phase shift keying (DDMPSK), symbol error rate (SER), bit error rate (BER), single-input singleoutput (SISO). II. P ERFORMANCE A NALYSIS OF DDPSK OVER AWGN C HANNEL WITH C ARRIER O FFSET Let zn denote the unitary symbols belonging to the MPSK constellation to be transmitted at the time n. In a DD modulation based system shown in Fig. 1(a), z n is encoded into v n in the following way: pn = pn−1 zn , vn = vn−1 pn , n = 1, 2, . . . ,

(1)

where vn is the transmitted signal and |p 0 | = |v0 | = 1. As |zn | = 1 for the MPSK symbols, it follows from (1) that |vn | = |pn | = 1, when n ≥ 0. We consider an AWGN SISO channel with carrier offset described by √ xn = Eejωn vn + en , n = 0, 1, . . . , (2) where xn is the received signal, E is the energy of the transmitted signal, e n is a white circular symmetric Gaussian noise, and ω is an unknown frequency offset due to either a carrier frequency mismatch between the transmitter and

receiver or a Doppler shift caused by the relative motion between the transmitter and receiver [11], [13], [14]. The decoding of z n is performed in the following way: The receiver makes a decision variable as shown in Fig. 1(b)  ∗  ∗ dn = xn x∗n−1 xn−1 x∗n−2 = Xn Xn−1 , (3) where Xn = xn x∗n−1 . It can be shown that √ √ ∗ en + Eejωn vn e∗n−1 Xn = Eej(ω+ΔΦn ) + Ee−jω(n−1) vn−1 +

en e∗n−1

= Ee

j(ω+ΔΦn )

+ ϑn ,

pn

vn

z -1

pn-1

z -1

vn-1 (a)

xn

dn

Xn

(4)

where ΔΦn = arg(vn ) − arg(vn−1 ) and ϑn is the total noise present in Xn . It can be further demonstrated that in the noiseless case the phase of the decision variable (3) is equal to Δ2 Φn =ΔΦn − ΔΦn−1 =arg(vn ) − 2 arg(vn−1 ) + arg(vn−2 ), (5) where Δ2 Φn is the phase to be transmitted at time n. It is shown in [10] that the heuristic decoder [10, Eq. (15)] coincides with the exact maximum likelihood decoder (MLD) [10, Eq. (16)] in the case of DD modulation. Hence, the optimal [10] decoding of DDMPSK is performed as ∗ (6) zn∗ , zˆn = arg max Re Xn Xn−1 zn ∈Ξ

where Ξ is MPSK constellation. A. Analogy between DMPSK and DDMPSK In a single differential modulation based system, p n is obtained from z n in the following way [14]: pn = pn−1 zn , n = 1, 2, . . . , (7) where pn is the transmitted signal and |p 0 | = 1. The differential symbol is transmitted as follows: √ (8) xn = Epn + en , n = 0, 1, . . . . The decoding of z n is performed in the following way: The receiver makes a decision variable (9) dn = xn x∗n−1 , which has the following phase in the noiseless case: ΔΦn = Φn − Φn−1 = arg(zn ) − arg(zn−1 ).

zn

(10)

The optimal MLD decoding [10] is performed  as [11], [14] zˆn = arg max Re xn x∗n−1 zn∗ . (11) zn ∈Ξ

It can be observed that the optimal decoding of double differentially modulated signal is analogous to the single differential modulation and this fact could be verified by comparing (3) and (9), and (6) and (11). Therefore, we can say that the performance of DDMPSK can be characterized by the SER expressions of DMPSK with the signal to noise ratio (SNR) of Xn . This connection is shown in more detail in [1], [11]. Next, we take an approximate assumption that X n and Xn−1 are Gaussian distributed. We make this assumption to avoid the complicacy of the exact probability of error calculation of DD modulation. This assumption is reasonable since at the SNRs of practical interest the higher-order noise term is very small relative to the dominant noise terms [14]

z -1

z -1

( )* x*

n -1

( )* X n*-1

(b)

Fig. 1.

Double differential (a) encoder and (b) decoder.

and we can assume the higher-order noise to be Gaussian distributed. A similar kind of assumption is also made in [14] to find the BER of DPSK over AWGN channels, however, the second order noise term was removed from the analysis. Nevertheless, we are still considering the power of the secondorder noise term in our analysis and we assume that it is Gaussian distributed. We can find the SNR of X n as Es E/N0 = , (12) EN 2 + (E/N0 )−1 where N0 is the variance of noise e n in (2). We use the analogy between the DPSK and DDPSK modulations explained above to find the expression of SER of the DDPSK modulated signals. The SER of DMPSK modulation over AWGN channel is given in [13] as   π  π/2 exp − Es ξ π (ψ) sin M EN M dψ, (13) Ps = π (ψ) 2π ξ −π/2 M π π (ψ) = 1 − cos where ξ M M cos ψ. Using the analogy, we substitute the SNR of Xn from (12)in (13) for finding the SER performance of DDMPSK modulation over AWGN channel with offset   E/N0 π π  π/2 exp − sin M 2+(E/N0 )−1 ξ M (ψ) dψ. (14) Ps = π (ψ) 2π ξM −π/2

B. SER of DDBPSK over AWGN Channel with Carrier Offset For DDBPSK, we put M = 2 in (14), and it can be shown that

E/N0 1 Ps = exp − , (15) 2 2 + (E/N0 )−1 which is a closed-form solution for SER of DDBPSK. If we compare (15) and [13, Eq. (8.85)], it can be shown that DDBPSK requires additional 3 dB to perform similar to the DBPSK at higher SNRs.

III. SER OF DDPSK OVER R AYLEIGH FADING C HANNEL WITH C ARRIER O FFSET

30

In fading channels, the SER of DDMPSK can be found as follows [13], [14]:  ∞ Ps = Ps {γ} pγ (γ) dγ, (16)

25

Exact Approximate

20

0

A. SER of DDBPSK over Rayleigh Fading Channel with Carrier Offset For DDBPSK M = 2, then (18) reduces

into 1 ∞ γ Ps = exp − (19) pγ (γ) dγ. 2 0 2 + γ −1 It is difficult, if not impossible, to find a closed-form expression for (19), therefore, we take the following approximation: 1 γ γ (20) ≈ − . 2 + γ −1 2 4 However, as a cross-check, we have compared the exact and approximate SNRs in Fig. 2 and it is satisfying to see that the approximate SNR follows closely the exact one for γ >5 dB, which is the region of γ values of most practical interest. From (19) and (20), it can be that for shown

DDBPSK 1 1 1 Ps = exp Mγ − , (21) 2 4 2 γ s)−1 is the moment generating function where Mγ (s) = (1−¯ (MGF) for Rayleigh fading. B. Bounds over SER of DDMPSK over Rayleigh Fading Channel with Carrier Offset It can be shown

forπDDMPSK from (18) and (20) that 1 sin M Ps = exp 4 π  

 π/2 π (ψ) exp − 14 g M ξ π (ψ) × Mγ − M dψ, (22) π (ψ) ξM 2 0 π π (ψ) = cos where g M  M cos ψ. It can be observed that  1  − g π (ψ) ≤ 1 , ∀ψ, (23)  4 M  4

N

[dB]

15

10

s

E /E

where γ = δ 2 Es /EN and δ is the instantaneous amplitude of the fading coefficient. P s {γ} is the conditional probability of error given the instantaneous SNR γ of the fading channel. For Rayleigh fading channel, the probability density function (PDF) of the instantaneous SNR given as [14] is

1 γ , (17) pγ (γ) = exp − γ¯ γ¯ where γ¯ is the average SNR. As SER formulations for AWGN channel with carrier offset, derived in Section II, are in the form of exponential function or integral of the exponential function, therefore, P s {γ} can be simply obtained by replacing Es /EN in Ps by γ [13]. The probability of error of DDMPSK over fading channel can be obtained by averaging Ps {γ} over the channel by (16)  as follows:  γ   π π π/2 ∞ exp − sin M 2+γ −1 ξ M (ψ) Ps = pγ (γ) dγdψ. π (ψ) 2π ξM −π/2 0 (18)

5

0

−5

−10

0

5

Fig. 2.

10

15 g [dB]

20

25

30

Comparison of exact and approximate SNRs.

therefore, we can use a first-order Taylor series as a tight π (ψ) approximation for exp − 41 g M as

1 1 π (ψ) π (ψ) . exp − g M ≥ 1 − gM (24) 4 4 Therefore, the SER of DDMPSK can be lower bounded with the help of (24) as

π 1 sin M Ps ≥ exp 4 π

 π/2 π (ψ) 1 − 14 g M ξ π (ψ) × Mγ − M dψ. (25) π (ψ) ξM 2 0 After some manipulations we can find a closed-form solution for the integral of (25), hence, the lower bound on SER of DDMPSK can be written as  

π y exp 14 sin M 3¯ γ − 1 arctanh t Ps ≥ 2πtu 2 u

π 1 + cos M −3 arctanh u , (26) t π 1/2  γ ¯ cos M π −1 ,y= + γ¯2 + 1, and u = where t = cos2 M 2   2 2 π 1/2 2 γ ¯ cos M − γ¯4 − γ¯ − 1 . 4   π (ψ) is a Further, it can be observed that exp − 41 g M monotonically increasing function for 0 ≤ ψ ≤ π/2, with the maximum value of unity. Hence, an upper bound for SER of DDMPSK modulation can be obtained as follows:

ξ π (ψ) M

 M − γ π π/2 2 1 sin M dψ. (27) Ps ≤ exp π (ψ) 4 π ξM 0

0

10

The integral of (27) can be solved and a closed-form upper bound over SER is   of DDMPSK π  y exp 14 sin M γ¯ arctanh t Ps ≤ πtu u π 1 + cos M −2 arctanh u , (28) t where t, y, and u are same as defined in (26).

−1

10

DDBPSK

−2

10

IV. SER OF DDPSK OVER R ICEAN FADING C HANNEL WITH C ARRIER O FFSET For Ricean fading channel, the PDF of the instantaneous SNR γ is given as [14]



1+K (1 + K) γ + K γ¯ pγ (γ) = exp − γ¯ γ¯    K (1 + K) γ , (29) ×I0 2 γ¯ where K is the Ricean factor, which corresponds to the ratio of the power of the line of sight (LOS) component to the average power of the scattered component and I 0 (·) is the zeroth-order modified Bessel function of the first kind. From (18), (20), (29), and also using the relation [15, Eq. (4)], namely

 ∞  √  ν 1 exp (−μx)I0 2 νx dx = exp , (30) μ μ 0 the SER in DDMPSK modulation over Ricean fading channel with carrier offset can be expressed as  

π  π/2(1 + K) exp − 1 g π (ψ) 1 sin M 4 M   Ps = exp γ ¯ π π (ψ) 1 + K + 4 π 0 ξM 2 ξ M (ψ)   π (ψ) K γ¯2 ξ M dψ. (31) × exp − π (ψ) 1 + K + γ¯2 ξ M A. SER of DDBPSK over Ricean Fading Channel with Carrier Offset Putting M = 2 into (31), the SER of DDBPSK over Ricean fading channel with carrier offset can be obtained in the closed-form as   K (2¯ γ − 1) − γ¯2 − 1 1 1+K   exp − . (32) Ps = 2 1 + K + γ¯2 4 1 + K + γ¯2 It can be observed that for K = 0 (32) reduces into the SER expression for Rayleigh fading channel with offset (21) and for K = ∞ it reduces into the SER expression for AWGN channel with offset (apply (20) into (15)). V. A NALYTICAL AND EXPERIMENTAL R ESULTS All the simulations are performed with a random carrier offset, uniformly distributed in the range of [0, π/2]. It is assumed that the offset remains unchanged for at least three transmission periods. The noise is assumed to be circularly Gaussian additive white noise (AWGN) with unity variance. The fading channel is assumed to be complex Gaussian with zero mean (Rayleigh fading) and nonzero mean (Ricean fading), and power of the received signal is varied by changing the variance of the channel.

SER

DBPSK −3

10

−4

10

−5

10

−6

10

0

1

2

3

4

5 6 SNR [dB]

7

8

9

10

11

Fig. 3. Performance of DDBPSK by our analysis (15)  and experimental –◦– in AWGN channel with offset, performance of DDBPSK analytical by [5, Eq. (18)] ∗ and DBPSK analytical [14, Eq. (5.2-69)]  and experimental –◦– in AWGN channel without offset.

A. Performance of DDBPSK over AWGN Channels with Carrier Offset Fig. 3 shows the performance of DDPSK over AWGN channel with the carrier offset. We have compared our analytical expression for SER of DDBPSK (15), with the analytical SER of DBPSK of [14, Eq. (5.2-69)]. It can be seen that DDPSK performs approximately 4 dB poorer than the DPSK at lower SNRs, however, at higher SNRs it performs approximately (slightly more than) 3 dB poorer than DPSK. The new analytical results are verified with the Monte-Carlo simulations in Fig 3. We have also plotted the formulation given in [5, Eq. (18)] for SER in DDBPSK. The decoder used in [5] is a suboptimal decoder, hence, its performance is approximately 4 dB poorer than DBPSK, even in the case of no offset. Moreover, the formulation of SER in [5] also needs the fixed values of carrier offset, whereas, in our case, we consider a random carrier offset with uniform distribution. Therefore, the formulation of [5] cannot predict the performance of DDBPSK in the random offset case. Nevertheless, we have plotted the analytical result of [5] for DDBPSK with no carrier offset. B. Performance of DDPSK over Rayleigh Channel with Carrier Offset Fig. 4 shows the performance of DDPSK over Rayleigh fading channels with carrier offset. We have compared our analytical formulation for SER of DDBPSK in Rayleigh fading channel (21) with the simulations performed under the same conditions. It is satisfying to see that our analytical formulation very closely follows the experimental results. We have also plotted the analytical performance of DBPSK in Rayleigh

1

0

10

10

−1

10 0

10

−2

K=1

10

DD 32−PSK

K=2 −1

10

DDQPSK

−3

10 SER

SER

DD 16−PSK DBPSK

K=5 −4

10

−2

10

DDBPSK K=10

−5

10 −3

10

−6

10

−4

10

−7

0

5

10

15

20

25

30

35

SNR [dB]

Fig. 4. Performance of DDBPSK  analytical and — experimental, DDQPSK, DD 16-PSK, and DD 32-PSK — exact, − ∗ − lower bound, and · · · + · · · upper bound, over Rayleigh fading channel with offset and DBPSK –– analytical over Rayleigh fading channel without offset.

10

0

5

10

15 SNR [dB]

20

25

30

Fig. 5. Analytical  and experimental –◦– performance of DDBPSK over Ricean fading channel with offset.

R EFERENCES fading channel [13, Eq. (8.201)] to compare with our formulation. It can be seen that DDBPSK performs approximately 3 dB poorer to the DBPSK modulation in Rayleigh fading channel with offset. The analytical SER bounds (26) and (28) are also plotted in Fig. 4 for DDQPSK (M = 4), DD 16-PSK (M = 16) and DD 32-PSK (M = 32) modulations. It can be observed that the lower bound is very tight and almost follows the exact SER. C. Performance of DDPSK over Ricean Channel with Carrier Offset Fig. 5 shows the performance of DDBPSK modulation over Ricean channel with carrier offset. We have plotted our closedform analytical formulation of SER (32) for Ricean factor K = 1, K = 2, K = 5, and K = 10. We have also plotted the experimental results under the similar conditions. It can be seen that our analytical results closely follow the experimental results. VI. C ONCLUSIONS We have presented some basic results about the performance analysis of double differential modulation over AWGN, Rayleigh, and Ricean fading channels. Although our analysis is based on some approximations, the analytical results are simple to calculate and follow the experimental results very closely. Our analysis may now pave the way to implement some schemes that are of current interest like cooperative wireless communications with carrier offsets.

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