2012 7th International ICST Conference on Communications and Networking in China (CHINACOM)
A Generalized FSK-Based PHY Layer Design for Wireless Sensor Networks Pavel Loskot College of Engineering Swansea University, Swansea, United Kingdom, SA2 8PP E-mail:
[email protected]
Abstract—A generalized FSK modulation employing FSK, PPM and APSK modulation alphabets is considered. The modulation combines a traditional APSK alphabet extended with a zero-symbol and the multiple-access. Analysis of the power spectral density (PSD) reveals that the bandwidth is significantly expanded whenever the transmission is periodically turned on and off. The overall probability of bit error and the overall data rate for multiple parallel generalized FSK transmissions are compared in single-user and multiuser scenarios. A rate-one repetition diversity scheme for real-valued transmitted signals is proposed to overcome the time and frequency correlations of communication channels between nodes. The overall energy consumption is shown to be reduced with the FSK modulation order, and by using multiple parallel FSK transmissions.
I. I NTRODUCTION The environment monitoring is emerging as one of the most important applications of wireless sensor networks (WSNs). The data rates in such applications are typically small, however, the nodes in the WSNs are battery powered and expected to report the measurements over long periods of time. The frequency shift keying (FSK) can readily trade-off the energy efficiency with the transmission bandwidth [1], and it can be combined with other modulations such as onoff keying (OOK) [2], [3], and QAM [4], [5]. Many design aspects of FSK have been considered in the literature. The peak transmission power and duty-cycle of OOK-FSK are investigated in [2] and [3]. Binary quantization of the received FSK signals is considered in [1]. The diversity reception of the coherent and non-coherent FSK is analyzed in [6] and [7]. The adaptive QAM-FSK is proposed in [4], analyzed in [5], and compared with the adaptive QAM in [8]. The rate-adaptive orthogonal multiplexing of orthogonal modulation signals is considered in [9]. The orthogonal and non-orthogonal OOKFSK realized in the spatial domain is described in [10]. The information bearing modulation is typically created at the physical layer (PHY), however, it can be further combined with the upper layer protocol entities such as the multiple access and the routing decisions. We assume that the PHY of the WSN defines fixed time-frequency (TF) bins. These TFbins corresponding to the TF-spreading are exploited by the FSK-PPM (pulse position modulation) signaling, and are also used to obtain orthogonal multiple-access for the WSN nodes. The FSK-PPM is further combined with the amplitude-phase shift keying (APSK) which is referred to as a generalized FSK. For implementation simplicity, the demodulation of the gener-
alized FSK is obtained sub-optimally in two stages. However, the periodic on-off keying of the modulated signals negatively affect their PSD as will be shown in the next section. We consider multiple parallel generalized FSK transmissions for the single-user and multiuser scenarios corresponding to the uplink and the downlink transmissions in the WSN, respectively. Since the WSN nodes for environment monitoring experience almost static channels with large coherence time and bandwidth, we propose a rate-one repetition diversity scheme. Lastly, we show that increasing the data rate is, in general, beneficial to save the energy. The rest of this paper is organized as follows. Section II describes the modulation and demodulation of the generalized FSK, and its PSD is derived. The performance of the generalized FSK and the repetition diversity signaling including evaluation of the energy consumption is investigated in Section III. Conclusions are given in Section IV. II. G ENERALIZED FSK M ODULATION Assume that there are 𝑁 sensor nodes with the on-board transceivers distributed in a geographical area. The sensors are communicating the environment measurements to a base station in a star-topology network. The base station acts as a network coordinator and a data fusion center. The uplink and downlink communications assume orthogonal time-frequency division multiple access (OTFDMA). Each node is assigned 𝑀12 = 2𝑚12 = 𝑀1 𝑀2 TF-bins across 𝑀1 = 2𝑚1 frequency sub-channels and 𝑀2 = 2𝑚2 time slots. The modulated waveform is a sum of the real-valued sine and cosine pulses of duration 𝑇 having the complex-valued amplitudes, and is transmitted in a given TF-bin. Thus, assuming complex-valued envelopes in the baseband, the transmitted signal is written as [11], ∞ ∑ ℎ(d𝑘 ; 𝑡 − 𝑘𝑇 𝑀2 ) (1) 𝑥(𝑡) = 𝐴 𝑘=−∞
where the modulation pulse, ℎ(d𝑘 ; 𝑡) = 𝑝(d′𝑘 ; 𝑡)𝜂(𝑑T,𝑘 ; 𝑡), and, √ 2 ′ cos(𝜔 𝑑F,𝑘 𝑡 + 𝜙𝑑F,𝑘 ) + 𝑝(d𝑘 ; 𝑡) = 𝑑C,𝑘 √𝑇 2 sin(𝜔 𝑑F,𝑘 𝑡 + 𝜙𝑑F,𝑘 ) 𝑑S,𝑘 𝑇 { 1 𝑡 ∈ ((𝑑T,𝑘 − 1)𝑇, 𝑑T,𝑘 𝑇 ) 𝜂(𝑑T,𝑘 ; 𝑡) = 0 otherwise.
362
978-1-4673-2699-5/12/$31.00 © 2012 IEEE
The information carrying modulation symbols corresponding to the FSK, PPM, and APSK modulation [12], respectively, have the modulation alphabets, 𝑑F,𝑘 𝑑T,𝑘 𝑑C,𝑘 , 𝑑S,𝑘
∈
{1, 2, . . . , 𝑀1 }
∈ ∈
{1, 2, . . . , 𝑀2 } {±1, ±3, . . . , ±(𝑀0 − 1)} + j{±1, ±3, . . . , ±(𝑀0 − 1)}
The coherent demodulator in Fig. 1(b) assumes time and frequency synchronization as well as knowledge of the initial phases 𝜙𝑑F,𝑘 . The demodulator continuously monitors the TFbins by first correlating the received signal 𝑦(𝑡) with the locally generated sine and cosine waveforms of the frequencies 𝜔𝑖 . The correlation values 𝐶𝑖,𝑗 and 𝑆𝑖,𝑗 , 𝑗 = 1, 2, . . . , 𝑀2 , are combined to obtain the decision variables, 𝐷𝑖,𝑗,1 𝐷𝑖,𝑗,2
d′𝑘
= where j is the imaginary unit, log2 𝑀0 = 𝑚0 , and {𝑑C,𝑘 , 𝑑S,𝑘 , 𝑑F,𝑘 }. The modulation symbols are assumed to be independent and equally probable. The initial phases 𝜙𝑑F,𝑘 are set arbitrarily, and 𝜔 = 2𝜋/𝑇 is the frequency separation of the sub-channels, i.e., the so-called modulation index is unity [12]. The scaling factor 𝐴 in (1) is used to set the average energy per transmitted bit to unity, i.e., [11] √ 3 log2 𝑀 𝐴= 2(𝑀02 − 1) where 𝑀 = 𝑀12 𝑀02 is the modulation order of the generalized FSK. The modulator structure employing a frequency synthesis with the TF-multiplexing is shown in Fig. 1(a). It is useful to consider special cases of the modulated signal (1). In particular, if 𝑑S,𝑘 = j𝑑C,𝑘 = j𝑑𝑘 , then the waveform 𝑝(d′𝑘 ; 𝑡) can be rewritten as [11, (4-3-27)], √ 2 j(𝜔 𝑑F,𝑘 𝑡+𝜙𝑑F,𝑘 ) ′ e 𝑝(d𝑘 ; 𝑡) = 𝑑𝑘 𝑇 and the modulated signal (1) corresponds to the hybrid QAMFSK modulation [4]. For 𝑑C,𝑘 ∈ {±1} and 𝑑S,𝑘 = 0, we have a bi-orthogonal FSK modulation (BFSK). The hybrid OOKFSK modulation assumes the data symbol alphabet 𝑑C,𝑘 ∈ {0, 1} and 𝑑S,𝑘 = 0 [1], [2]. The FSK modulation is obtained for 𝑀2 = 1, and the PPM corresponds to 𝑀1 = 1; thus, the case 𝑀1 > 1 and 𝑀2 > 1 is a combination of the FSK and the PPM and is referred to as the generalized FSK.
so that,
where the impulse response of the multipath channel 𝑔(𝑡; 𝜏 ) = ∑ 𝐿 𝑙=1 𝑔𝑙 (𝑡)𝛿(𝜏 −𝜏𝑙 ), ⊗ is the convolution, 𝑤(𝑡) is a zero-mean additive [white Gaussian noise (AWGN) with the variance ] 2 = E ∣𝑤(𝑡)∣2 being equal to the noise double-sided PSD 𝜎𝑤 𝑁0 , and E[⋅] and ∣ ⋅ ∣ denotes expectation and the absolute value, respectively. Provided that the maximum delay spread 𝜏𝐿 of the wireless channel is much smaller than the modulation pulse duration 𝑇 , the transmitted signal at the 𝑖-th subchannel undergoes a frequency non-selective fading given by the discrete-time Fourier transform (DTFT) [13],
𝑙=1
where 𝜔𝑖 = 2𝜋𝑑F,𝑖 /𝑇 , 𝑖 = 1, 2, . . . , 𝑀1 .
{
𝐷𝑖,𝑗,1
=
𝐷𝑖,𝑗,2
=
𝐺𝑖,𝑗 𝑑C,𝑘 + 𝑤C,𝑘 𝑑F,𝑘 = 𝑖, 𝑑T,𝑘 = 𝑗 otherwise 𝑤C,𝑘 { 𝐺𝑖,𝑗 𝑑S,𝑘 + 𝑤S,𝑘 𝑑F,𝑘 = 𝑖, 𝑑T,𝑘 = 𝑗 otherwise. 𝑤S,𝑘
𝑖=1,...,𝑀1 𝑗=1,...,𝑀2
Consequently, the estimates of the transmitted data symbols 𝑑F,𝑘 and 𝑑T,𝑘 are given as, 𝑑ˆF,𝑘 = ˆ𝚤 and 𝑑ˆT,𝑘 = ˆ, respectively. Finally, assuming perfect knowledge of the channel fadings2 𝐺𝑖,𝑗 at the receiver, the decisions about the transmitted symbols 𝑑C,𝑘 and 𝑑S,𝑘 are obtained as, 𝑑ˆC,𝑘
=
𝑑ˆS,𝑘
=
argmin
ˆ 𝑘,1 − 𝐺ˆ𝚤,ˆ 𝑑C ∣ ∣𝐷
argmin
ˆ 𝑘,2 − 𝐺ˆ𝚤,ˆ 𝑑S ∣ ∣𝐷
𝑑C ={±1,...,±(𝑀0 −1)} 𝑑S ={±1,...,±(𝑀0 −1)}
ˆ 𝑘,1 = 𝐷ˆ𝚤,ˆ,1 and 𝐷 ˆ 𝑘,2 = 𝐷ˆ𝚤,ˆ,2 . where 𝐷
𝑦(𝑡) = 𝑔(𝑡; 𝜏 ) ⊗ 𝑥(𝑡) + 𝑤(𝑡)
𝑔𝑙 (𝑡) e−j𝜔𝑖 𝜏𝑙
𝐶𝑖,𝑗 sin 𝜙𝑖 + 𝑆𝑖,𝑗 cos 𝜙𝑖
{ˆ𝚤,ˆ} = argmax ∣𝐷𝑖,𝑗,1 ∣2 + ∣𝐷𝑖,𝑗,2 ∣2 .
The received signal in the baseband can be written as,
𝐺𝑖 (𝑡) =
𝐶𝑖,𝑗 cos 𝜙𝑖 − 𝑆𝑖,𝑗 sin 𝜙𝑖
=
The channel fading coefficients 𝐺𝑖,𝑗 = 𝐺𝑖 ((𝑗 + 𝑘)𝑇 ) assume that the modulation pulse duration 𝑇 is much smaller than the fading coherence time 𝑇𝑐 = 9/(16𝜋𝑓𝐷 ) where 𝑓𝐷 is the signal Doppler spread which is proportional to the relative speed of the transmitter and the receiver antennas [13]. The zero-mean ]AWGN [ samples ] 𝑤C,𝑘 and 𝑤S,𝑘 have the variance, [ E ∣𝑤C,𝑘 ∣2 = E ∣𝑤S,𝑘 ∣2 = 𝑁0 , and we can show that these AWGN samples are mutually uncorrelated. The TF-bin containing the transmitted signal is decided using the maximum energy detection1 , i.e.,
A. Demodulator
𝐿 ∑
=
B. PSD of Transmitted Signal It is clear that the periodic on-and-off transmissions of the modulated signal adversely affects its PSD. In general, the PSD of a modulated signal 𝑥(𝑡) can be evaluated as [14], 𝑆𝑥 (𝑓 ) = ℱ𝜏 {AvE[𝑥(𝑡 + 𝜏 )𝑥∗ (𝑡)]} where ℱ𝜏 {⋅} is the Fourier transform with respect to variable 𝜏 , (⋅)∗ is the complex conjugate, and AvE[⋅] = Av[E[⋅]] denotes the time-averaging, i.e., Av[⋅] = ∫ 𝑈/2 lim𝑈 →∞ 𝑈1 −𝑈/2 (⋅)d𝑢, and expectation is taken over all 1 Such non-coherent selection does not require knowledge of the channel fading 𝐺𝑖,𝑗 , and in the literature, is also referred to as a S+N selection. 2 The channel estimates are obtained by probing all the sub-channels with continuous pilot waveforms.
363
2 T
cos φ1
cos(ω1 t)
D1, j,1
C1, j
f1 , φ1
sin fM1 , φM1
cos
T-F MUX
S1, j
cos fk , φk sin
sin φ1
T
dC,k
cos
hk (t)y(t)
D1, j,2
T 2 T
2 T
Dˆ k,1
sin(ω1 t)
cos φ1
cos(ωM1 t)
T 2 T
SM1, j
DM1, j,2
sin(ωM1 t)
cos φM1
dT,k dF,k (a) Figure 1.
(b) The generalized FSK modulator and the corresponding coherent demodulator.
0
] 𝐴2 [ E ∣𝐻(d; 𝑓 )∣2 = 𝑇 𝑀1 ∣ ej2𝜋𝑓 𝑇 − 1∣2 ∑ (𝑖 − 1)2 + (𝑓 𝑇 )2 = const× 𝑀1 𝜋 2 (4(𝑖 − 1)4 + (𝑓 𝑇 )4 ) 𝑖=1
-10
𝑆𝑥 (𝑓 𝑇 )
random modulation symbols d𝑘 . After some manipulations, assuming that E[𝑑C,𝑘 ] = E[𝑑S,𝑘 ] = 0, and that the sequence of symbols d𝑘 is stationary, we obtain, 𝑆𝑥 (𝑓 )
Dˆ k,2
sin φM1
T
DM1, j,1
CM1, j
dS,k
sin
SELECT
cos φM1
G-FSK QAM S-FSK
-20
where 𝐻(d; 𝑓 ) = ℱ𝑡 {ℎ(d; 𝑡)} is the modulation pulse in the frequency domain. The PSD of the modulated signal 𝑥(𝑡) corresponding to the generalized FSK modulation (GFSK), of the QAM modulation with rectangular (non-return to zero, NRZ) pulse shaping, and of the Sunde’s FSK (SFSK) [12] are compared in Fig. 2. We observe that the G-FSK modulation experiences significant expansion of the bandwidth comparable to the bandwidth required for the transmission of the QAM modulation with the rectangular modulation pulses. Thus, the PSD of the G-FSK modulation decreases with the square of the frequency whereas the PSD of the SFSK modulation having a continuous phase decreases with the fourth-power of the frequency. The PSD of the G-FSK could be improved if other than the rectangular window 𝜂(𝑑T,𝑘 ; 𝑡) is used in the modulation signal (1).
(MGF) of the squared fading amplitude ∣𝐺1,1 ∣2 is [11, (2-1117)],
III. P ERFORMANCE A NALYSIS
Φ∣𝐺1,1 ∣2 (𝑢) = (1 − 2𝜎 2 𝑢)−𝑛R /2 e 1−2𝜎2 𝑢 .
Recall that the modulation symbols 𝑑C,𝑘 and 𝑑S,𝑘 are normalized, so that the modulation signal (1) has the unit average energy per bit. Without loss of generality, we assume that the modulated signal was transmitted in the first TF-bin having the indices 𝑑F = 𝑑T = 1. We also assume that the fading magnitudes ∣𝐺𝑖,𝑗 ∣ are generalized Ricean distributed ( )2 where 𝑠2 is the nonwith the Ricean factor 𝐾R = 𝜎𝑠 centrality parameter, and 𝜎 2 is the variance of the diffuse component [11, (2-1-143)]. The moment generating function
Furthermore, we denote as[ Γ the average energy of the fading ] amplitude, i.e., let Γ = E ∣𝐺1,1 ∣2 = (𝑛R + 2𝐾R )𝜎 2 . Consider first the case of 𝑀 -ary FSK modulation with the symbols being the parameters 𝑑C,𝑘 = 1 and 𝑑S,𝑘 = 0, i.e., 𝑀 = 𝑀12 . The decision variables 𝐷𝑖,𝑗,1 then correspond to the correlator outputs given by eq. (5-4-32) in [11], i.e., { 𝐺1,1 + 𝑤 𝑖 = 𝑗 = 1 𝐷𝑖,𝑗,1 = 𝑤 otherwise.
-30
-40
-50 0
2
Figure 2.
4
𝑓 𝑇 [−]
6
8
10
Comparison of the PSDs of three modulation schemes.
2𝜎 2 𝑢𝐾R
364
Conditioned on 𝐺1,1 , the probability of bit error is calculated as [11, (5-4-48)], ˜ 12 ) − 𝑛𝑚 ( 𝑀∑ 12 −1 ∣𝐺1,1 ∣2 𝛾𝑆 𝑛+1 ˜ 𝑛 ˜ +1 𝑀12 − 1 e (−1) 𝑃b (𝐺1,1, ) = 𝑎𝑚12 𝑛 ˜ 𝑛 ˜+1 𝑛 ˜ =1 ( ) 𝑚12 −1 where 𝑎𝑚12 = 22𝑚12 −1 , 𝑎𝑏 is the binomial coefficient, and 𝛾𝑆 = 𝑁0−1 is the signal-to-noise ratio (SNR) per transmitted modulation symbol. One can readily use the MGF method to obtain the average probability of bit error over the Ricean fading channels, i.e., ( ) 𝑛 ˜ 𝑚12 )Φ ( 𝑀∑ 12 −1 2 Γ𝛾𝑆 ∣𝐺 ∣ 1,1 𝑛 ˜ +1 𝑀 − 1 12 . (−1)𝑛˜ +1 𝑃¯b (Γ𝛾𝑆 ) = 𝑎𝑚12 𝑛 ˜+1 𝑛 ˜
0
10
16-FSK: 4b, 16 bins 8-FSK: 3b, 8 bins 8-BFSK: 4b, 8 bins 2x(4-FSK): 4b, 8 bins 4x(4-FSK): 8b, 16 bins
-1
10
-2
𝑃b,tot
10
-3
10
-4
10
𝑛 ˜ =1
Consider now the case of 𝑀 -ary bi-orthogonal FSK (BFSK), i.e., 𝑑C,𝑘 ∈ {±1}, 𝑑S,𝑘 = 0, and 𝑀 = 2𝑀12 = 2𝑚 . Assuming the transmission over a AWGN channel without fading, i.e., 𝐺1,1, = 1, the probability of bit error is calculated as, ) ( (√ ) 0.5 + 𝑚12 𝑃b +𝑄 𝛾𝑆 /𝑚 𝑃c 𝑃b (𝛾𝑆 /𝑚) = (1 − 𝑃c ) 1 + 𝑚12 ∫∞ 2 where the Q-function, 𝑄(𝑢) = 𝑢 √12𝜋 e−𝑢 /2 d𝑢, the probability of correctly selecting the first TF-bin3 [11, (5-2-34)], ∫∞ ( ( ) ( ))𝑀12 −1 √ √ 𝑄 −𝑣 − 2𝛾𝑆 − 𝑄 𝑣 + 2𝛾𝑆 𝑃c = × √ − 2𝛾𝑆
2 1 × √ e−𝑣 /2 d𝑣 2𝜋 must be calculated numerically, and the probability of bit error due to the incorrect TF-bin decision is,
2𝑚12 −1 (1 − 𝑃c ). 2𝑚12 − 1 In general, a node that is allocated 𝑀12 TF-bins for the transmission can increase its transmission rate by employing multiple modulation schemes in parallel. In particular, let the 𝑀12 TF-bins be divided into 𝑁 non-overlapping4 parts. Each ∑𝑁 (𝑛) (𝑛) TF-part contains the 𝑀12 TF-bins, so that 𝑛=1 𝑀12 = (𝑛) (𝑛) 𝑀12 . For 𝑀 (𝑛) = 𝑀02 𝑀12 = 2𝑚 , the overall transmission rate (in bits per channel use) is, ∑𝑁 𝑚(𝑛) log2 𝑀 𝑚 ≥ = . 𝑅tot = 𝑛=1 𝑀12 𝑀12 𝑀12 However, such rate increase also causes the probability of bit error to increase since the overall transmission power is shared among the 𝑁 modulated signals as well as the probability of bit error for 𝑀 -ary FSK, in general, increases with the modulation order 𝑀 [11], i.e., ∑𝑁 (𝑛) (𝑛) 𝑛=1 𝑃b 𝑚 𝑃b,tot = ∑ 𝑁 (𝑛) 𝑛=1 𝑚 𝑃b =
that (5-2-34) in [11] is missing the factor, (2𝜋)−1/2 . order to maintain the modulation orthogonality
3 Note 4 in
-5
10
0
2
4
6
8 10 𝛾𝑆 /𝑚 [dB]
12
14
16
18
Figure 3. Comparison of the bit error probabilities for several FSK modulations.
(𝑛)
where 𝑃b is the corresponding probability of bit error of the 𝑛-th modulated signal. As a design example, the probability of bit error 𝑃b,tot for 𝑁 = 1 and 𝑁 = 2 modulation signals using 16 and 8 TF-bins, respectively, is shown in Fig. 3. For example, 8-BFSK modulation with coherent detection has the rate 4 bits per channel use whereas the same rate is also achieved by 𝑁 = 2 4-FSK modulations with noncoherent detection and the SNR penalty of 1 dB. Hence, in the downlink, it is beneficial if the base station employs a single FSK modulation across all the available TF-bins. Each node then demodulates all the transmitted data symbols, however, it only recovers the pertaining data bits. Similar design considerations can be used for the multiuser systems with 𝑁 nodes. However, in this case, the transmission power does not have to be shared among the 𝑁 nodes, and the overall (sum-) rate and the probability of bit error, respectively, are defined as, 𝑅tot
=
𝑁 ∑ log2 𝑀 (𝑛) 𝑛=1
𝑃b,tot
=
(𝑛)
𝑀12
𝑁 1 ∑ (𝑛) 𝑃 . 𝑁 𝑛=1 b (𝑛)
For instance, if the number of TF-bins per node 𝑀12 is (𝑛) halved, the data rate per node is also halved with 𝑀12 -ary FSK, however, the number of nodes in the system can be doubled. A. Diversity In the WSN scenario for the environment monitoring, the sensor nodes are likely to be stationary. The channel models labeled as SUI-{1 − 6} for such scenarios were developed in [15]. Assuming that the WSN is deployed in the first
365
-1
0
10
10
16-FSK, DO=1 8-FSK, DO=1 4-FSK, DO=1 16-FSK, DO=2 8-FSK, DO=2 4-FSK, DO=2
Γ=15dB -2
10
-1
-3
Γ=25
10
𝑃b
𝑃b,tot
10
-2
10 -4
10
-5
10
0
16-FSK 2x(8-FSK) 4x(4-FSK) 2
4
-3
6
8 𝐾
10
12
14
10
16
0
5
10
15 𝛾𝑆 /𝑚 [dB]
20
25
30
Figure 4. The bit error probabilities when 𝐾 TF-bins experience more benign fading than the remaining (16 − 𝐾) TF-bins.
Figure 5. The bit error probabilities of 𝑀 -ary FSK with the diversity transmission of order 1 and order 2, respectively.
3 MHz of the first 802.11 channel5 [16], we can allocate 100 × 16 subcarriers to 𝑁 = 100 nodes with the subcarrier spacing (1/𝑇 ) = 1.875 kHz. In this case, the fading coherence time is between 71 ms for the SUI-5 to 716 ms for the SUI-2 and the SUI-4. The coherence bandwidth is 800, 350, and 50 kHz for the SUI channels 4, 5 and 6, respectively, and greater than 3 MHz for the SUI channels 1, 2 and 3. Hence and importantly, with the single transmitter and receiver antennas, we can allocate the TF-bins to the nodes in order to obtain at least some diversity in the time or the frequency domains. For instance, consider the non-coherent FSK transmission for 𝑁 = 1 node using 𝑀12 = 16 TF-bins, 𝑁 = 2 nodes each using 𝑀12 = 8 TF-bins, and 𝑁 = 4 nodes each using 𝑀12 = 4 TF-bins. Fig. 4 shows the system bit error probabilities assuming that 𝐾 ≥ 0 out of 16 TF-bins experience the Ricean fading with the parameters 𝑛R = 4 and 𝐾R = 0 while the remaining (16 − 𝐾) TF-bins experience the Ricean fading with the parameters 𝑛R = 2 and 𝐾R = 0. We observe that, especially for larger values of the fading power Γ, the bit error probability can be reduced significantly if only at most one sub-channel experiences less benign fading. In general, provided that the TF-bins can be allocated to experience two independent fadings, say, 𝐺1 (𝑡) and 𝐺2 (𝑡), and, provided that the data symbols 𝑑C,𝑘 and 𝑑S,𝑘 are realvalued (i.e., the baseband modulated signal 𝑥(𝑡) is realvalued), we can construct the modulated signal,
independent channels to the receiver, so that the corresponding received signals can be written as,
1 𝑥(𝑡)) 𝑧(𝑡) = √ (𝑥(𝑡) + j˜ 2 having twice the data rate than the original 𝑥(𝑡), [ 2 signal ] [ 2 how] ever, having the same variance, i.e., E 𝑥 (𝑡) = E 𝑥 ˜ (𝑡) = ] [ E ∣𝑧(𝑡)∣2 . The modulated signal 𝑧(𝑡) is transmitted over two 5 This
bandwidth is not used by any other 802.11 channels.
𝑦1 (𝑡) 𝑦2 (𝑡)
=
𝐺1 (𝑡)𝑧(𝑡) + 𝑤1 (𝑡)
=
𝐺2 (𝑡)𝑧(𝑡) + 𝑤2 (𝑡)
where 𝑤1 (𝑡) and 𝑤2 (𝑡) are the zero-mean AWGNs of equal variances. Assuming perfect knowledge of the fadings 𝐺1 (𝑡) and 𝐺2 (𝑡) at the receiver, the received signals 𝑦1 (𝑡) and 𝑦2 (𝑡) are coherently combined, and the receiver estimates the transmitted signals 𝑥(𝑡) and 𝑥 ˜(𝑡) as, √ ∣𝐺1 (𝑡)∣2 + ∣𝐺2 (𝑡)∣2 𝑥(𝑡) + 𝑤1′ (𝑡) 𝑥 ˆ(𝑡) = 2 √ ∣𝐺1 (𝑡)∣2 + ∣𝐺2 (𝑡)∣2 𝑥 ˜(𝑡) + 𝑤2′ (𝑡) 𝑥 ˜ˆ(𝑡) = 2 where 𝑤1′ (𝑡) and 𝑤2′ (𝑡) are again the zero-mean AWGNs of equal variances. On the other hand, if the signals 𝑥(𝑡) and 𝑥 ˜(𝑡) are transmitted independently, they are coherently estimated in the receiver as, 𝑥 ˆ(𝑡) = ∣𝐺1 (𝑡)∣ 𝑥(𝑡) + 𝑤1′ (𝑡) ˆ˜(𝑡) = ∣𝐺2 (𝑡)∣ 𝑥 ˜(𝑡) + 𝑤2′ (𝑡). 𝑥 Hence, both transmission methods have the equal data rate, however, the first method achieves the second-order diversity. As an example, we compare the bit error probabilities of 𝑀 -ary FSK modulations assuming that 𝐺1 (𝑡) and 𝐺2 (𝑡) are independent and generalized Ricean distributed with the parameters 𝑛R = 2 and 𝐾R = 0. The corresponding bit error probabilities are shown in Fig. 5. We observe that the achievable diversity order (DO) is equal to 𝑛R /2 which can be also shown mathematically.
366
IV. C ONCLUSIONS
0 -2 -4 ˆb − 𝜂ˆ [dB] 𝐸
-6 -8
-10 -12 N=1 N=2 N=4 N=8
-14 -16 -18 0
2
4
log2 𝑀
6
8
10
Figure 6. The normalized energy per transmitted bit versus the number of bits per channel use.
The generalized FSK modulation exploiting the TF-bins and the FSK, PPM and APSK modulation alphabets was investigated. The corresponding modulator and demodulator structure was described. The PSD of the generalized FSK was evaluated to confirm that the periodic on-and-off switching of the transmitted signal causes significant expansion of the required bandwidth. The overall probability of bit error and the overall data rate were evaluated for the case of multiple parallel transmissions of 𝑀 -ary FSK and BFSK modulations in single-user as well as multiuser scenarios. Since the communication channels experienced by the WSN nodes in the environment monitoring applications have large coherence time as well as large coherence bandwidth, a simple rate-one repetition diversity analogy of the Alamouti scheme for realvalued transmitted signals was proposed. Finally, it was shown that the overall energy consumption per transmitted bit can be reduced by increasing the number of TF-bins, or by dividing the TF-bins into parts to allow the parallel FSK transmissions. R EFERENCES
B. Energy Consumption In general, in order to transmit 𝐵 bits at the rate 𝑅 = log2 𝑀 bits per channel use, the channel must be used 𝑆 = 𝐵/𝑅 = 𝐵/ log2 𝑀 times. If we split the channel into 𝑁 subchannels, we can increase the data rate to 𝑅 = 𝑁 log2 (𝑀/𝑁 ) bits per channel use as discussed in the previous sub-section. The overall energy consumed at the transmitter can be expressed as, 𝐸tot = (𝑃RF + 𝑃OVH )𝑆 where 𝑃RF = 𝐶BER 𝑅 is the RF radiated power that is proportional to the data rate with the constant 𝐶BER being a function of the target bit error probability, and 𝑃OVH is the so-called overhead power that is constant and independent of the data rate 𝑅. Furthermore, we have that, 𝐶BER = 𝜂ˆ𝑃OVH , where, typically, 𝜂ˆ ≤ 30%. Then, the normalized energy ˆb is evaluated as, consumption per bit 𝐸 1 ˆb = 𝐸tot 1 = 𝜂ˆ + . 𝐸 𝐵 𝑃OVH 𝑁 (log2 𝑀 − log2 𝑁 ) ˆb as a function of the data rate Fig. 6 shows the energy 𝐸 𝑅 = log2 𝑀 for the different number of sub-channels 𝑁 . Using Fig. 6, we can conclude that, in general, the communication systems are more energy efficient if their data rate is increased, for example, by increasing the number of sub-channels 𝑁 which is readily achieved for systems employing the generalized FSK modulations. For example, the energy consumed per transmitted bit is reduced by 3 dB if 𝑀12 = 24 = 16 TF-bins are divided into 𝑁 = 4 sub-channels (𝑛) of 𝑀12 = 4 TF-bins each, or when the number of TF-bins is increased to 𝑀12 = 29 = 512 TF-bins. Furthermore, the larger 𝑅, the smaller 𝑆 which, in turn, reduces the energy consumption at the receiver.
[1] M. Gursoy, “The impact of hard-decision detection on the energy efficiency of phase and frequency modulation,” IEEE Trans. Wireless Commun., vol. 8, no. 9, pp. 4644–4655, Sep. 2009. [2] ——, “Error rate analysis for peaky signaling over fading channels,” IEEE Trans. Commun., vol. 57, no. 9, pp. 2546–2550, Sep. 2009. [3] M. Gursoy and Q. Wang, “Diversity analysis of peaky FSK signaling in fading channels,” IEEE Trans. Wireless Commun., vol. 8, no. 8, pp. 3896–3902, Aug. 2009. [4] F. F. Digham and M.-S. Alouini, “Adaptive hybrid M-FSK M-QAM modulation,” in Proc. PACRIM, vol. 2, 2003, pp. 964–967. [5] ——, “Variable-rate noncoherent M-FSK modulation for power limited systems over nakagami-fading channels,” IEEE Trans. Wireless Commun., vol. 3, no. 4, pp. 1295–1304, Jul. 2004. [6] A. Chandra, C. Bose, and M. Bose, “Performance of non-coherent MFSK with selection and switched diversity over Hoyt fading channels,” Wireless Pers. Comm., pp. 1–21, Nov. 2011. [7] A. Chandra, S. R. Poram, and C. Bose, “SEP calculations for coherent M-ary FSK in different fading channels with MRC diversity,” Int. J. Commun. Syst., vol. 24, no. 2, pp. 202–224, Feb. 2011. [8] F. F. Digham and M.-S. Alouini, “Spectrally efficient hybrid FSK/QAM with optimum bit and power loading,” in Proc. ICC, vol. 11, 2006, pp. 5022–5027. [9] W.-L. Lin and C.-D. Chung, “Adaptive differentially coherent orthogonally multiplexed orthogonal phase modulation over flat fading channels,” IEEE Trans. Wireless Commun., vol. 8, no. 3, pp. 1062–1066, Mar. 2009. [10] M. D. Renzo, H. Haas, and P. M. Grant, “Spatial modulation for multiple-antenna wireless systems: a survey,” IEEE Commun. Mag., vol. 49, no. 12, pp. 182–191, Dec. 2011. [11] J. G. Proakis, C. M. Rader, F. Ling, C. L. Nikias, M. Moonen, and I. K. Proudler, Algorithms for Statistical Signal Processing. Prentice-Hall, 2002. [12] F. Xiong, Digital Modulation Techniques. Artech House, 2000. [13] T. Rappaport, Wireless Communications: Principles and Practice, 2nd ed. Prentice Hall, 2001. [14] J. Sykora, Theory of Digital Communications, 2nd ed. (in Czech), CTU, Prague, 2002. [15] V. Erceg, K. S. Hari, M. Smith, D. Baum, K. Sheikh, C. Tappenden, J. Costa, C. Bushue, A. Sarajedini, R. Schwartz, D. Branlund, T. Kaitz, and D. Trinkwon, “Channel models for fixed wireless applications,” 2001, iEEE 802.16 Broadband Wireless Access, IEEE 802.16.3c01/29r4. [16] IEEE Std 802.11-2007, “Part 11: Wireless MAC and PHY specifications,” Jun. 2007, IEEE Standard.
367
