Energy Consumption Analysis for Continuous Phase Modulation in Smart-Grid Internet of Things of beyond 5G

In wireless communication systems, the general format of the physical layer packet structure is shown in Figure 1 and consists of three parts: a pilot code for clock synchronization, a packet header specifying the configuration of transmission parameters, and a data payload carrying the transmitted data.

We assume that the entire packet uses the same modulation and that the symbol error probability (SEP) is determined by the received SNR $\gamma$ and the modulation scheme adopted. We also assume that symbol errors are independently and identically distributed (i.i.d.) and no channel coding is used, i.e., no redundant bits are added (redundant bits increase EC). If a symbol is erroneously detected at the receiver, the entire packet will be retransmitted until all the symbols in the packet are correctly detected at the receiver. The packet error probability (PEP) can be expressed as a function of the SEP, the packet length $L$, and the number of bits per symbol $m=\log_{2}M$, i.e.,

where $M$ denotes the size of the modulation constellation.

Hence, by utilizing the packet length $L$ and the probability that a packet is successfully transmitted, i.e., $1-\mathsf{PEP}$, the average amount of data successfully delivered per transmission duration of a packet is formulated as

Then, upon assuming that the feedback signal indicating whether a retransmission is needed or not is reliably transmitted on the reverse link, the average number of transmissions required for successfully delivering a packet is given by

CPM is an attractive modulation scheme for WSNs underpinning SG-IoT because its carrier phase is modulated in a continuous manner and it is typically implemented as a constant-envelope waveform, i.e., the transmitted carrier power is constant. The phase continuity requires a relatively small percentage of the power to occur outside of the intended band (e.g., low fractional out-of-band power), leading to high spectral efficiency. Meanwhile, the constant envelope yields excellent power/energy efficiency. However, the primary drawback of CPM is the high implementation complexity required for an optimal receiver.

For systems that employ CPM, the transmitted signal at time instant $t$ can be expressed as[19]

where $E$ is the symbol energy, $T$ is the symbol interval, $f_{c}$ is the carrier frequency, and $\phi_{0}$ is an arbitrary constant initial phase shift that can be set to zero without loss of generality when coherent transmission is considered. In addition, $\phi(t,\boldsymbol{I})$ is the time-varying information-carrying phase formulated as

where $\boldsymbol{I}=\{I_{k}|k\in(-\infty,\cdots,-1,0,+1,\cdots,K)\}$ is an infinitely long sequence of uncorrelated $M$-ary data symbols, each having one of the values from the alphabet$\mathcal{A}=\{\pm 1,\pm 3,\cdots,\pm(M-1)\}$ with equal probability $1/M$; $\{h_{k}\}$ is a sequence of modulation indices defined as $h_{k}=2f_{d,k}T$, with $f_{d,k}$ being the peak frequency deviation. When $h_{k}=h$ for all $k$, the modulation index remains fixed for all symbols. When the modulation index changes from one symbol to another, the signal is called multi-$h$ CPM, with $h_{k}$ varying in a cyclic manner. $q(t)$ is some normalized waveform shape that represents the baseband phase response (i.e., phase pulse) and is obtained from the frequency pulse $g(t)$ by

If the duration of $g(t)$ is equal to the symbol interval $T$, namely, $g(t)=0$ for $t>T$, the modulated signal is called full-response CPM. If the duration of $g(t)$ is larger than the symbol interval $T$, namely, $g(t)\neq 0$ for $t>T$, the modulated signal is called partial-response CPM.

Suppose the length of the frequency pulse $g(t)$ in terms of the number of symbol intervals is $N$. Thus, $N=1$ yields full-response CPM. If $g(t)$ is selected as a rectangular pulse, namely,

then for a full-response CPM, we have

It is evident that the performance of CPM is influenced by certain parameters, including but not limited to $M$, $h_{k}$, $N$, and the frequency pulse $g(t)$. Note that by choosing different pulse shapes $g(t)$ and varying $M$, $h_{k}$, and $N$, an infinite variety of CPM signals may be generated, each with its unique characteristics and performance.

For a CPM signal, the error rate performance can be derived based on the maximum-likelihood sequence detection (MLSD) receiver, which is conventionally computed using the Viterbi Algorithm (VA). Specifically, for a given CPM scheme, we have

According to Anderson’s seminal work on digital phase modulation [20], $K_{\textrm{min}}$ denotes the total number of feasible paths that satisfy the constraint of the minimum Euclidean distance $d_{\textrm{min}}$ within the observation interval on the CPM phase grid. The value of $K_{\textrm{min}}$ increases with the modulation order $M$. Both $K_{\textrm{min}}$ and $d_{\textrm{min}}$ depend on critical parameters including $M$, $h_{k}$, $N$, and the pulse shaping function $g(t)$.

In wireless communication systems, a significant portion of energy is dedicated to signal transmission and reception circuits, which are mainly composed of the baseband (BB) digital signal processing unit and the radio frequency (RF) signal processing unit, as shown in Figure LABEL:entire_power_model. To elaborate a little further, the BB signal processing unit mainly includes source coding/decoding, pulse shaping, channel coding/decoding, digital modulation/demodulation, channel estimation, synchronization, and so on. For a wireless sensor, since the data rate requirement is usually low, the BB symbol rate is also low. Meanwhile, typically, no computation-intensive signal processing techniques, such as multi-user detection and iterative decoding, are used in an energy-constrained wireless sensor; hence, the BB power consumption is significantly smaller than the RF circuit power consumption.

A typical model of an RF signal processing unit, also known as an RF chain, is shown in Figure 3 [14, 16, 21, 22, 23]. Specifically, on the transmitter side, the BB signal is first converted to an analog signal by the digital-to-analog converter (DAC). Then, the analog signal is filtered by the low-pass filter and upconverted by the mixer, whose output is then filtered again, amplified by the power amplifier (PA), passed through the duplexer, and finally transmitted to the wireless channel. On the receiver side, the RF signal is sequentially filtered, amplified by the low-noise amplifier (LNA), cleaned by the anti-aliasing filter, downconverted by the mixer, filtered again before passing through the intermediate frequency amplifier (IFA) that has an adjustable gain, and finally converted back to a digital signal by the analog-to-digital converter (ADC). Note that the mixers operate with the aid of the local oscillator (LO) and, among all the RF components, the PA and LNA usually have much higher power consumption than the others.

Accordingly, the total power consumption during transmission can be expressed as

where $P_{\textrm{TBB}}$ and $P_{\underline{\textrm{TRF}}}$ represent the power consumption of the transmitter’s BB processing unit and RF processing unit excluding the PA, respectively, and both of them can be regarded as constant values that are collectively denoted by $P_{\textrm{T0}}$; $P_{\textrm{PA}}(d)$ and $P_{\textrm{T}}(d)$ are defined as the power consumption of the PA and the transmit power, respectively, both of which are functions of the transmission distance $d$ upon assuming adaptive power control and are related via $P_{\textrm{PA}}(d)=\frac{\xi}{\eta}P_{\textrm{T}}(d)$; and $\eta$ and $\xi$ represent the drain efficiency and the peak-to-average power ratio (PAPR) of the PA, respectively [18].

Similarly, the total power consumption of the receiver is expressed as

where $P_{\textrm{RBB}}$ and $P_{\underline{\textrm{RRF}}}$ represent the power consumption of the receiver’s BB processing unit and RF processing unit excluding LNA, respectively, and both of them can be regarded as constant values; $P_{\textrm{LNA}}$ represents the power consumption of the LNA, which is also constant upon assuming that the LNA is appropriately designed and biased, so that necessary sensitivity is provided for reliably receiving, demodulating, and decoding a minimum power signal. Hence, $P_{\textrm{RBB}}$, $P_{\underline{\textrm{RRF}}}$, and $P_{\textrm{LNA}}$ are collectively denoted by the constant $P_{\textrm{R0}}$.

Due to the path-loss, scattering, reflection, and other phenomena in the wireless channel, a certain amount of energy is inevitably lost during the transmission of electromagnetic waves that carry data symbols, thus resulting in transmission energy dissipation. The path-loss depends on several factors, such as the distance between the transmitter and receiver, the frequency of the signal, the type of antennas used, and the environment through which the signal propagates.

When the signal with a transmit power $P_{\textrm{T}}(d)$ is propagated through the wireless channel, the received power $P_{\textrm{R}}(d)$ at the receiver can be formulated as

according to Friis’ transmission equation. This expression characterizes the dependency between the received power with respect to several parameters. Specifically, the constant $A_{0}$ depends on the transmit antenna gain $G_{\textrm{T}}$, the receive antenna gain $G_{\textrm{R}}$, and the carrier wavelength $\lambda$. Additionally, the path-loss exponent is $\alpha$. It is noted that the received power is inversely proportional to the distance $d$ raised to the power of $\alpha$, and the received SNR $\gamma$ is proportional to $1/A_{0}$ divided by $d^{\alpha}$. These relationships collectively describe how the aforementioned parameters influence the received power and the received SNR [18]:

where $W$ represents the transmission bandwidth, $N_{0}$ denotes the power spectral density of the baseband-equivalent additive white Gaussian noise (AWGN), and they are primary determinants of $\gamma$. Furthermore, the noise figure of the RF front-end of the receiver, denoted as $N_{\textrm{f}}$, and any additional noise or interference, represented by the link margin term $M_{\textrm{l}}$, can also impact the received SNR $\gamma$.

Accordingly, by substituting Equation (12) into Equation (13), the relationship between the transmit power $P_{\textrm{T}}(d)$, the communication distance $d$, the received SNR $\gamma$, and other parameters can be quantitatively expressed as

where we have $A=A_{0}N_{0}WN_{\textrm{f}}M_{\textrm{l}}$.

As mentioned in Section II, we assume that each packet transmitted in the forward direction is matched by an error-free feedback packet in the reverse direction in order to guarantee reliable transmission. Both directions of transmissions consume energy. The above transmission process, usually incorporating retransmissions, continues until the entire packet of the forward direction is correctly decoded at the receiver. Additionally, we assume that the sensor node transceiver circuitry works in a multi-mode manner: (1) when there are data to transmit or receive, all circuits of the sensor work in the active mode; (2) when neither transmission nor reception are needed, the circuits of the sensor enter sleep mode by default, which uses the minimum possible power (small enough to be negligible) to ensure that the circuits can be activated when necessary; (3) when the sensor is in the period of switching from sleep mode to active mode, it is in a transient mode that also consumes non-negligible energy. Note that the transient duration from active mode to sleep mode is sufficiently short to be neglected but the start-up process from sleep mode to active mode can be slow.

Therefore, for a single round-trip transmission (forward direction transmission and reverse direction feedback) on a point-to-point communication link, the total EC of the system can be divided into two parts: the EC of the forward direction transmission and the EC of the reverse direction feedback. For the forward direction transmission, the EC is composed of the start-up energy consumption $2E_{\textrm{ST}}$ in transient mode (both the transmitting node and the receiving node may be in sleep mode initially), the transmitter energy consumption $E_{\textrm{Tx}}(d)$ in active mode, and the receiver energy consumption $E_{\textrm{Rx}}$ in active mode. Therefore, the EC of the forward direction transmission is expressed as

where $T_{\textrm{DTA}}$ is the transmission duration for sending a data packet in the forward direction and $T_{\textrm{DRA}}$ is the corresponding duration of signal processing at the receiver of the forward direction.

Additionally, the EC of the reverse direction transmission is expressed as

where $\underline{P}_{\textrm{Tx}}(d)$ is the total power consumption for transmission of a feedback packet in the reverse direction by the receiving node of the forward direction and $T_{\textrm{FTA}}$ is the corresponding time duration. Note that $\underline{P}_{\textrm{Tx}}(d)$ may be different from $P_{\textrm{Tx}}(d)$ of the forward direction because a feedback packet may have a different transmission rate and reliability requirements compared with a data packet. $T_{\textrm{FRA}}$ is the duration of processing the feedback packet at its receiver (i.e., the transmitting node of the forward direction) in active mode.

Based on the above analysis, we obtain the EC per successfully transmitted bit as (17),

where $\underline{P}_{\textrm{T}}(d)$ is the transmit power for the feedback packet transmission and $N_{\textrm{re}}$ is the average number of retransmissions.

From Equation (17), it can be observed that the EC per successfully transmitted bit, $E_{b}$, is predominantly determined by four parameters: the forward-link transmit power $P_{\textrm{T}}(d)$, the reverse-link transmit power $\underline{P}_{\textrm{T}}(d)$, the specific CPM scheme adopted, and the received SNR $\gamma$. From another perspective, we can also say that $E_{b}$ is mainly determined by $P_{\textrm{T}}(d)$, $\underline{P}_{\textrm{T}}(d)$, the achievable $\mathsf{SEP}$, and the modulation efficiency $m$. Upon inspection of Equations (13) and (9), it becomes clear that the $\mathsf{SEP}$ on the forward link can be expressed as

Although a high transmit power is desirable for achieving a low SEP and thus reducing the number of retransmissions $N_{\textrm{re}}$, it may also result in excessively large transmission power for the sensor and increase the EC of each single-direction transmission. Hence, an optimal transmit power (or in turn the received SNR $\gamma$) exists and is yet to be found for minimizing the EC per successfully transmitted bit. Similarly, the relationship between the EC per successfully transmitted bit and other major parameters mentioned above needs to be studied.

In the following numerical simulations, we consider the radio links of a WSN designed for smart grid and operating in the industrial–science–medical (ISM)-oriented 2.4 GHz frequency band. Table II provides a summary of the pertinent simulation parameters, including circuit-related parameter settings as well. Due to the constant envelope characteristic of the CPM signal, a nonlinear PA with a high $\eta$ value is employed, in contrast to the general radio architecture.

First, let us compare the EC performance of different CPM schemes and of other representative modulation schemes, such as OQPSK and 16QAM, by observing how $E_{b}$ varies with the received SNR $\gamma$. As shown in Figure 4, for all modulation schemes, the achievable $E_{b}$ values firstly descend and then increase with the received SNR. This is because in the low-SNR region, the number of retransmissions plays an important role, while the number of retransmissions is reduced to its minimum under a sufficiently high SNR, and then an even higher SNR means unnecessary energy wastage. In addition, for most SNRs, the achievable $E_{b}$ values of REC ($d_{\rm min}^{2}=2.831$, $h=0.75$ and $N=3$), RC ($d_{\rm min}^{2}=8.16596$, $h=0.75$ and $N=3$), and GMSK ($d_{\rm min}^{2}=7.93348$, $h=0.75$ and $N=3$) are lower than those of OQPSK and 16QAM, which verifies the advantages of CPM in terms of energy saving. The reason why a higher order CPM scheme has a lower $E_{b}$ can be explained by referring to Equation (17) as follows: (1) a larger $M$ causes a smaller number of symbols per packet (i.e., the smaller exponent $L/m$) and a smaller $\mathsf{SEP}$ (i.e., the larger base number $1-\mathsf{SEP}$), thus the denominator $L(1-\mathsf{SEP})^{L/m}$ increases with $M$; (2) the variable $T_{\textrm{DTA}}$ in the numerator becomes smaller when $M$ becomes larger, while all the other terms can be regarded as constants. Furthermore, we can see that the lowest values of $E_{b}$ achieved by the four modulation schemes having $M=16$ (i.e., REC, RC, GMSK, and 16QAM) are almost the same, while OQPSK having $M=4$ achieves the largest $E_{b}$ in the high-SNR region and the second-largest $E_{b}$ in the low-SNR region. It is also observed that when the SNR is above a specific threshold, the three different CPM waveforms exhibit the same EC performance.

Figure 5 characterizes the relationship between $E_{b}$ and the communication distance $d$ for OQPSK, 16QAM, and the CPM signals with different pulse shaping functions. We can see that $E_{b}$ increases with $d$ for all the modulation schemes considered. This is because large distance reduces the received SNR value under a given transmit power, thus degrading the SEP performance and increasing the number of retransmissions. We also see that all the three CPM waveforms considered have the same EC performance curves, which are consistently and significantly better than those of OQPSK and 16QAM when $d$ is sufficiently large. Here, we assume $\gamma=8$ dB for the three CPM waveforms and $\gamma=15$ dB for OQPSK and 16QAM. The two SNR values are selected according to the results shown in Figure 4, where the three CPM waveforms achieve their optimal $E_{b}$ at about $\gamma=8$ dB, while OQPSK and 16QAM achieve their near-optimal $E_{b}$ at about $\gamma=15$ dB.

Figure 6 shows how the $E_{b}$ values of the three CPM waveforms vary with the modulation efficiency $m$, while assuming $\gamma=8$ dB and $d=10$ m. It is observed that for each given CPM waveform, the $E_{b}$ value becomes smaller as $m$ increases. Although the RC-based CPM signaling exhibits the highest $E_{b}$ when $m=1,2,3$, all the three CPM waveforms achieve almost the same $E_{b}$ when $m=4$ (i.e., $M=16$). This is because the three CPM waveforms achieve almost the same SEP performance when $m=4$, $\gamma=8$ dB, and $d=10$ m. These observations are consistent with the results shown in Figure 4.

Figure 7 demonstrates the relationship between the average number of transmissions $N_{\textrm{re}}$ required for successfully sending a single packet and the received SNR $\gamma$ over the AWGN channel employing different modulation schemes, including OQPSK, 16QAM, and the three CPM waveforms. Obviously, 16QAM incurs the largest $N_{\textrm{re}}$, while the REC- and GMSK-based CPM schemes require the smallest $N_{\textrm{re}}$, under all the three SNR values of $6$ dB, $8$ dB, and $10$ dB. In addition, the OQPSK scheme requires a smaller and a larger $N_{\textrm{re}}$ than the RC-based CPM scheme under $\gamma=6$ dB and $\gamma=8$ dB, respectively. However, when the received SNR is sufficiently high, e.g., $\gamma=10$ dB, all the modulation schemes, except 16QAM, require only a single transmission on average for successfully sending a packet.