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A powerefficient synchronization scheme for the MBOFDMbased UWB systems
EURASIP Journal on Wireless Communications and Networking volume 2015, Article number: 14 (2015)
Abstract
Due to the complexity of multiband orthogonal frequencydivision multiplexing (MBOFDM), achieving low power is still one of the major difficulties in designing the MBOFDMbased ultra wideband (UWB) receiver. This paper targets at providing a very low power synchronization scheme with high performance for the MBOFDM UWB receiver. First, the power consumptions of the individual functions are analyzed based on the overall architecture of the scheme. Then, the individual functions are designed. A timefrequency code (TFC) identification scheme is proposed, which divided the TFC identification process into two steps. A signal detector (SD) is designed for the first step to judge whether preamble symbols are present, and a TFC type detector (TTD) is designed for the second step to identify the TFC type. A packet detector is proposed, in which a coarse detector (CD) and a final detector (FD) are designed for reducing power consumption and improving performance, respectively. Also, a matched filter is proposed to simplify the correlation operation between the received symbol and the preamble symbol. The evaluation results show that the proposed TFC identification scheme and the packet detector have obvious lower computational complexity compared to existing solutions. The results also show the TFC identification scheme has satisfactory performance and that the performance of the packet detector is excellent.
1 Introduction
Highspeed wireless communication through wireless personal area network (WPAN) can bring great convenience to users. Ultra wideband (UWB) technology can offer a solution for the data rate, quality of service (QoS) and power consumption requirements of the nextgeneration highspeed WPAN due to features such as wide bandwidth, low interference from other wireless systems, and low power density. Compared to another UWB technique, impulse radio (IR)UWB, the multiband orthogonal frequencydivision multiplexing (MBOFDM) has higher spectrum efficient and better robustness in dispersive channel and is more suitable for high data rate transmission. Due to these good features, the MBOFDM has been adopted by many groups such as Wireless USB Promoter Group.
The Federal Communications Commission (FCC) has allocated 7,500 MHz (3.1 to 10.6 GHz) spectrum for UWB devices. In MBOFDM UWB, The entire 7,500 MHz spectrum is divided into five band groups and fourteen 528 MHz bands [1]. During data transmission, a timefrequency code (TFC) is used for frequency hopping among the bands that of the same band group [1,2].
Due to high activity, the synchronization block is one of the major power consumption sources in the MBOFDM UWB receiver. To design the synchronization block with low power consumption is still one of the major challenges in the design of the MBOFDM UWB receiver. Also, OFDMbased systems are very sensitive to timing and carrier frequency offset. Synchronization errors cause intersymbol interference (ISI) and intercarrier interference (ICI), which destroy the orthogonality of the OFDM subcarriers and result in severe system performance degradation.
The synchronization design for OFDMbased systems has been researched for years. Many methods have been proposed for normal OFDM systems [37]. Although some experiences of normal OFDM systems can be developed for MBOFDM UWB systems, most methods for normal OFDM systems cannot be used for the MBOFDM UWB system directly due to some special features of MBOFDM UWB such as the frequency hopping and low power spectral density (PSD). By considering the special features of MBOFDM UWB, many effective methods have been proposed specially for MBOFDM UWB systems [818]. A crosscorrelation (CC)based method is proposed in [8] for TFC identification. Some methods [1012,14] are proposed for packet detection. In [11], a signbit CC algorithm is proposed to simplify the operation of the normal CC. The signbit CC algorithm is further simplified in [12]. Some methods [13,15,17] focus on symbol timing. In [15], a method is proposed to distinguish the first significant multipath. Some methods for carrier frequency offset (CFO) estimation are proposed in [14,16,18]. The method in [16] is proposed based on the best linear unbiased estimation (BLUE) principle, which has a high performance. However, these methods all focus on optimizing the implementation complexity or performance of individual functions, and none of them offer optimization on the power consumption of a comprehensive synchronization process.
A comprehensive synchronization design for the MBOFDM UWB system is proposed in [9]. Based on autocorrelation (AC) algorithm, the design in [9] has low implementation complexity and can provide a comprehensive synchronization solution for the design of practical MBOFDM UWB receiver. However, the major consideration of the design in [9] is implementation complexity. Low implementation complexity generally can bring a certain degree of power consumption decrease, but it is not equivalent to low power consumption. For example, the major consideration in a method for low implementation complexity is the hardware cost of the operation. However, in lowpower design, not only the hardware cost of the operation but also the number of the operation to be executed need to be reduced.
With the development of very large scale integrated circuits (VLSI), the hardware cost of electronic products reduces greatly every year. Therefore, in recent years, low power consumption is usually more essential than low implementation complexity for electronic devices, especially the portable devices.
The design in this paper targets at providing a synchronization solution with very low power for the design of practical MBOFDM UWB receivers. From the perspective of the comprehensive synchronization process, the power consumptions of the individual blocks in the proposed overall architecture are analyzed. Based on the result of the analysis, a novel TFC identification scheme with very low power and a novel packet detector with low power and high performance are proposed. In addition, a new matched filter with high noise immunity and low complexity is designed.
Using the same hardware implementation techniques, the algorithmic approach with lower computational complexity usually has lower power consumption. Therefore, in this paper, power consumption majorly refers to computational complexity and a method with low power refers to that the method has low computational complexity, as the major concern of the design is not the detailed circuits but the algorithmic methods.
The key contributions of this paper include the following:

The analysis on the power consumption of the individual blocks from a new perspective. The power consumption of the individual blocks is analyzed from the perspective of the comprehensive synchronization process.

A TFC identification scheme. In the scheme, a signal detector (SD) is designed to judge whether the preamble symbols are present, and a TFC type detector (TTD) is designed to identify TFC type. The TFC identification scheme has very low power consumption with satisfactory performance.

A packet detector. In the packet detector, a coarse detector (CD) is designed to detect symbols with low computational complexity and a final detector (FD) is designed to achieve high performance. The packet detector has low power with high performance.

A matched filter. The matched filter has low computational complexity with high noise immunity. Moreover, the matched filter can roughly maintain the orthogonality between the preamble patterns and can be used to identify the preamble pattern.

Different operations are implemented according to whether the preamble symbols are present or not. In this way, the power consumption of the blocks in the interval that no preamble symbol is present can be greatly reduced.
The rest of this paper is organized as follows. Section 2 gives a brief description of the signal and channel models. In Section 3, the power consumption of the scheme is analyzed. The individual blocks are designed in Section 4. Evaluation and simulation results are shown in Section 5. Section 6 concludes the paper.
2 Signal and channel models
The synchronization in the MBOFDM UWB system is dataaided [2]. Dedicated preambles are used for synchronization. A standard preamble sequence consists of 21 packet synchronization (PS) symbols, 3 frame synchronization (FS) symbols, and 6 channel estimation (CE) symbols. In addition, zeropadded prefix intervals are used instead of cyclic prefix intervals in the symbol. A transmitted symbol is constructed by appending 32 null prefix samples, called zeropadded guard intervals, and five null guard samples to N = 128 length IFFT output samples. A total of N _{ s } = 165 samples are included in a symbol.
The IEEE 802.15.3a Task Group adopted a modified SV model for the UWB channel. The model is cluster based with independent fading and lognormal distribution of rays. The channel impulse response (CIR) can be expressed as
where X is lognormal shading, T _{ l } is the delay of the lth cluster, τ _{ k,l } is the kth ray delay related to the lth cluster, and a _{ k,l } is multipath gain coefficient.
Let s _{ n } is the transmitted signal at the nth time index. The received signal r _{ n } can be expressed as
where l is the length of channel impulse response, n is the time index, θ is the timing offset and v _{ n } is white Gaussian noise with zero mean and variance σ ^{2}.
3 Power consumption analysis
To ensure that the designed individual blocks are compatible with each other and can construct a comprehensive synchronization solution for the design of practical MBOFDM UWB receiver, an overall architecture is proposed in this section. Also, the overall architecture is needed for analyzing the power consumptions of the synchronization blocks in the comprehensive synchronization process because the synchronization process is dependent on the overall architecture of the synchronizer. The block diagram of the proposed overall architecture is shown in Figure 1.
In MBOFDM UWB systems, the data of the user are transmitted in the form of data block. One or several data frames are included in a data block. The structure of the data block is shown in Figure 2. In the figure, A is the interblock interval, B is the interframe interval, C is the time interval that used for packet detection and D is the time interval used for symbol and frame timing. Let N _{ a }, N _{ b }, N _{ c }, and N _{ d } are the numbers of symbols in A, B, C, and D, respectively. In most MBOFDM UWB systems, N _{ a } (usually > N) is much larger than N _{ b } (usually >40), and N _{ b } is much larger than N _{ d } (usually <15) and N _{ c } (usually <6).
In Figure 1, the dashed line is the control line. It means that the individual function is activated by the condition that the previous individual function has been executed successfully. As shown from Figure 1, when signals are received, TFC identification is done firstly, and then the received signals go through the packet detection block. After packet detection is completed successfully, the timing offset and the CFO are estimated. Finally, frame synchronization is done.
Therefore, the TFC identification block needs to detect almost all of the symbols in A because the position of the PS symbols is unknown to it. Another characteristic of TFC identification is that it does not need to be executed for each received frame. For example, because the TFC type has been identified during the first received frame, TFC identification is not executed any more when the second frame of the same data block is received. However, the packet detection block needs to be executed for each received frame to avoid offset accumulation. Therefore, almost all the symbols in B need to be detected by the packet detection block. Thus, a total of about N _{ a } N _{ s } and N _{ b } N _{ s } times detections have been executed by the TFC identification block and the packet detection block, respectively, before PS symbols are present.
Let n ^{^} is the time index at which packet is detected. Symbol timing is to estimate the time offset between n ^{^} and the exact starting boundary of the symbol. The time offset is not large (usually <10 samples). Therefore, only the samples, which are closed to n ^{^}, need to be detected for estimating the time offset. Because N _{ d } < 15, a total of less than 2 N times detections are executed in the whole process of symbol and frame timing offset estimation. For CFO estimation, the CFO estimation block does not need to wait the arrival of PS symbols because PS symbols have been detected by the packet detection block. According to the view in [3], the simplest CFO estimation method can be implemented with one complex multiplication.
Therefore, most of the samples that detected by the synchronizer are used for TFC identification and packet detection. Only optimizing the implementation complexity is not lowpower solution for TFC identification and packet detection. However, because the number of samples used for CFO estimation and timing offset estimation is small, the normal method with low implementation complexity is lowpower design. Therefore, the key for implementing the synchronizer with low power is designing the TFC identification block and the packet detection block with low power consumption.
4 Individual functions
4.1 TFC identification
In MBOFDM UWB systems, the transmitted data are spread on the frequency bands of the same band group, and TFCs are used for frequency hopping. The TFCs in specification [1] are shown in Table 1.
During time interval A (in Figure 2), the most concern of the TFC identification process is not identifying TFC type but judging whether the PS symbol is present or not.
In this paper, we design a SD to judge whether PS symbols are present. In MBOFDM UWB systems, the repeated PS symbols are used for synchronization. The transmitted sample at time index n + dN _{ s } (d is the symbol interval on one frequency band of the used TFC type) is identical to the sample at the nth time index. Thus, at receiver, \( \left{r}_n+{r}_{n+d{N}_s}\right \) is larger than r _{ n } and \( \left{r}_{n+d{N}_s}\right \) with very high probability. However, if noise is transmitted at points n and n + dN _{ s }, r _{ n } and \( {r}_{n+d{N}_s} \) have opposite polarity with high probability (≈1/2). Therefore, \( {\displaystyle \sum_m\left{r}_{n+m}+{r}_{n+d{N}_s+m}\right} \) can be used for detecting the repeated PS symbols.
As shown in Table 1, if d = 6, the transmitted data can be received again at the same frequency band for all TFC types. Let W _{ n } be the accumulated value of \( \left{r}_n+{r}_{n+6{N}_s}\right \) at the nth time index.
By using iterative structure, W _{ n } can be rewritten as
In this way, the major operations for calculating W _{ n } are three real additions. The structure of the SD is shown in Figure 3b.
Once the SD determines that PS symbols are present, a TTD is activated to identify the used TFC type. The CC operation is not sensitive to noise and is suitable for TFC identification in low signaltonoise ratios (SNRs). However, the computational complexity of CC operation is very high. For example, if it is directly used to identify TFC type in a MBOFDM UWB system of specification [1], 4 N complex multiplications need to be executed when a sample is received.
In this paper, the proposed TTD is shown in Figure 3a. To simplify the complexity of CC algorithm, a PSMF is designed. The structure of the PSMF is shown in Figure 3c. In Figure 3c, S _{ r,n } is the signbit of the received sample at the nth time index, S _{ c,k } is the signbit of the kth coefficient of preamble pattern c(c = 1, 2, 3, 4 ) , Q _{ c,k } is an integer and is the rounding of the kth coefficient of preamble pattern c, and G _{ c,n } is the output of the PSMF at the nth time index. First, S _{ r,n + k } compares with S _{ c,k }, and the comparison result I _{ n + k } is given by
After I _{ n + k }(k = 0, 1, … N − 1) are achieved,
Of the coefficients in the preamble patterns in specification [1], some have large absolute value (≈3), and some have the small absolute value (≈0). By assuming c = i and G _{ i,n } is the output of the PSMF at the nth time index, there are three scenarios of the transmitted samples:

(1)
The coefficients of preamble pattern i are transmitted. In this case, the mean probability of I _{ n + k }(k = 0, 1, … N − 1) = 1 is much higher than that of I _{ n + k }(k = 0, 1, … N − 1) = − 1. Moreover, the signs of the coefficients with larger absolute value are more unlikely be changed at the receiver. Therefore, most of I _{ n + k } = − 1 occurs on the coefficients with small absolute value. In this case, G _{ i,n } is close to \( {\displaystyle {\sum}_{k=0}^{N1}\left{Q}_{i,k}\right}. \)

(2)
The noise is transmitted. In this case, the probability of I _{ n + k } = 1 is 1/2. Therefore, for most G _{ i,n } in this case, G _{ i,n } ≈ 0.

(3)
The coefficients of preamble pattern j(j ≠ i) are transmitted. In this case, the mean probability of I _{ n + k }(k = 0, 1, … N − 1) = 1 is approximately 1/2 because the orthogonality between the preamble patterns. In addition, the occurrence of I _{ n + k } = 1 are uncorrelated and independent with the absolute value of the coefficient. Therefore, for most G _{ i,n } in this case, G _{ i,n } ≈ 0.
Therefore, the preamble pattern can be identified by comparing the maximum of {G _{ c,n }c = 1, 2, 3, 4} with a predefined threshold. If G _{ i,n } = max{G _{ c,n }c = 1, 2, 3, 4} and G _{ i,n } is larger than the threshold, preamble pattern i is determined as the preamble pattern used in transmission.
After the preamble pattern has been identified, TFC type can be identified easily. As seen in Table 1, TFC 1 and TFC 2 share the same preamble pattern with TFC 5 and TFC 6, respectively. Of TFC 3 and TFC 4, each occupies one preamble pattern exclusively. Therefore, if TFC 3 or TFC 4 is used, the TFC type can be directly identified by identifying preamble pattern. If TFC 1, 2, 5, or 6 is used, the TFC type can also be identified easily in an additional step according to the hopping order of the TFC type. For example, we assume that preamble pattern 1 is determined as the right preamble pattern at the nth time index. If TFC 5 is used, the signal would be received again at \( {G}_{1,n+2{N}_s} \) on the same frequency band. For TFC 1, the point is \( {G}_{1,n+3{N}_s} \). Therefore, we can compare \( {G}_{1,n+2{N}_s} \) with \( {G}_{1,n+3{N}_s} \). If \( {G}_{1,n+2{N}_s}>{G}_{1,n+3{N}_s}, \) TFC 5 is the TFC type used in transmission, else TFC 1 is used.
4.2 Packet detection
As analyzed in Section 3, for the methods which employ the same operations regardless of whether PS symbols are present or not, most of the power consumption during the packet detection process is cost in interval B. Moreover, because errors occur in the TFC identification process can be corrected easily by the following packet detection process but errors occur in the packet detection process can hardly be corrected by the following processes such as symbol timing, the performance requirement for packet detection is much higher than that for TFC identification.
In this paper, a packet detector, which divides the packet detection process into two steps, is proposed. The first step is to detect symbols with low computational complexity, and the second step targets at obtaining high performance.
The major components of the proposed packet detection scheme are a CD and a FD. The CD is designed to detect the received signals with low computational complexity and low miss detection probability by allowing high false alarm probability. The FD is designed to confirm the PS symbols detected by the CD with both low miss detection probability and low false alarm probability.
The structure of the CD is shown in Figure 4b. In the CD, a CC block, which based on the signbit CC algorithm [14,15], is designed. In the CC block, only the even samples are used for CC operation. The outputs of the CC block are given by
where S _{ r,n + 2k } is the sign of the received sample at time index n + 2 k and S _{ c,2k } are the sign of the 2kth coefficient of preamble pattern c.
However, the noise immunity of the CC will be greatly deteriorated by shorting the length to N/2. Though low false alarm probability is not required for the CD, too high false alarm probability will decrease the effectiveness of it. To improve the noise immunity of the CD, the CC output with dN _{ s } delay can be added to E _{ n }, where dN _{ s } is the intersymbol spacing on a frequency band. In Figure 4b, U _{ n } is given by
When different TFC type is used, d is different. As seen from Table 1,
In the CD, U _{ n } is compared with a predefined threshold. If U _{ n } is larger than the threshold, the PS symbol is detected by the CD.
However, because the threshold for the CD is selected with very small value to ensure low miss detection probability, the false alarm probability of the CD is still much high. This means that many PS symbols detected by the CD are not real PS symbols. The FD is designed to confirm the results of the CD. The structure of the FD is shown in Figure 4a. The FD is designed based on the PSMF, which is designed in the TFC identification scheme.
Because the accumulated value of several outputs of the PSMF has better noise immunity, several outputs of the PSMF are accumulated to compare with a predefined threshold. In Figure 4a, D _{ n } is the accumulated value. As TFC type has been identified and the CD has detected a PS symbol, frequency hopping can be used in the FD. Therefore, D _{ n } is given by
where \( {G}_{n+i{N}_s} \) is the output of the PSMF at time index n + iN _{ s }(i = 0, 1, …, M), N _{ s } is the length of a symbol (also the sample interval between two consecutive symbols) and M is the number of symbols used by the FD (M can be selected according to the performance requirement of the actual system). If D _{ n } is larger than the predefined threshold for the FD, packet is detected.
5 Evaluation
The proposed design is evaluated in this section. The computational complexities of the proposed methods are evaluated. Also, the performances of them are simulated. The parameters used in simulation follow ones’ in specification [1]: for N = 128, N _{ s } = 165, the three carrier frequencies are {3432, 3960, 4488} MHz and the subcarrier spacing f _{ i } = 4.125 MHz.
5.1 TFC identification
The performance of the SD is first estimated. The simulated miss detection probabilities of the SD under different false alarm probabilities are shown in Figure 5. In the simulation, TFC 1 and CM1 are used. To achieve the given false alarm probability, the threshold for each SNR is optimized by searching the suitable value. The miss detection probability of the SD is simulated under the condition that the given false alarm probability is 0.2, 0.45, and 0.7, respectively. When the given false alarm probability is 0.45, the miss detection probabilities are approximately 0.09 and 0.008 with SNR = −8 dB and SNR = −5 dB, respectively, which are satisfactory values for most systems. In addition, the computational complexity of the SD is about 1/16 of that of the TTD (analyzed in the last paragraph of this subsection). Therefore, SD is effective for reducing the computational complexity.
The performance of the TTD is also evaluated. If the PSMF for preamble pattern i is considered, there are three scenarios of the transmitted samples:

(1)
The transmitted samples are the coefficients of preamble pattern i. In this case, let P _{1,λ } = P[G _{ i,n } > λ]. According to (26) in Appendix 1,
$$ {P}_{1,\lambda}\approx 1\varPhi \left(\frac{\frac{N+{2}_{\lambda }}{4}\frac{1}{2} erfc\left(\frac{\overline{m}}{\sqrt{2}{\sigma}^2}\right)N}{\sqrt{\frac{1}{2} erfc\left(\frac{\overline{m}}{\sqrt{2}{\sigma}^2}\right)\left[1\frac{1}{2} erfc\left(\frac{\overline{m}}{\sqrt{2}{\sigma}^2}\right)\right]N}}\right), $$(11)where erfc() is the complementary error function, \( \overline{m} \) is the mean of the absolute values of the preamble coefficients and \( \varPhi (x)={\displaystyle {\int}_{\infty}^x\frac{1}{\sqrt{2\pi }}}{e}^{\frac{1}{2}{t}^2}dt. \)

(2)
The transmitted samples are noise. In this case, let P _{2,λ } = P[G _{ i,n } > λ]. According to (29) in Appendix 1,
$$ {P}_{2,\lambda}\approx 1\varPhi \left(\frac{\frac{N+\lambda }{2}\frac{1}{2}N}{\frac{1}{2}\sqrt{N}}\right) $$(12) 
(3)
The transmitted samples are the coefficients of preamble pattern j(j ≠ i). In this case, let P _{3,λ } = P[G _{ i,n } > λ]. According to (30) in Appendix 1,
$$ {P}_{3,\lambda}\approx 1\varPhi \left(\frac{\frac{N+\lambda }{2}\frac{1}{2}N}{\frac{1}{2}\sqrt{N}}\right) $$(13)
Let P _{ t_e } be the error probability of the TTD and λ is the predefined threshold for the TTD. If TFC 3 or 4 is used in the transmission, the TFC type can be identified directly by identifying the preamble pattern. In the case that TFC 3 or 4 is used, P _{ t_e } can be given by
If TFC 1, 2, 5, or 6 is used, an additional step is needed to identify the TFC after the preamble pattern has been identified. In this case, P _{ t_e } is given by
As to the overall performance of the proposed TFC identification scheme, the overall performance would be deteriorated by taking the miss detection of the SD into consideration, but it would be improved by taking the false alarm of the SD into consideration. By selecting an appropriate false alarm probability for the SD, the influence of the SD to the overall performance can be ignored. Therefore, P _{ t_e } can be seen as the error probability of the proposed TFC identification scheme.
Figure 6 shows the error probability comparison among TFC identification methods. The scheme in [9] and the traditional CC algorithm are included in the comparison. In the simulation, TFC 1 and the CM1 channel are used. The predefined thresholds used in the methods are all optimized by searching the optimal value for each SNR. As shown in Figure 6, the traditional CC algorithm has the best performance, and the performance of the proposed scheme is close to that of the traditional CC algorithm. In actuality, the performance of the traditional CC algorithm is optimal among the TFC identification algorithms. However, the complexity of it is too high to be used in most actual systems.
The comparison on computational complexity among TFC identification methods is shown in Table 2. The scheme in [9] and the traditional CC algorithm are included in the comparison. In the table, D _{ a } is the number of the symbols in interval A. In the proposed scheme, the adders of the PSMF in the TTD can be implemented by 64 3, 32 4, 16 5, and 8 9bit integer adders. The computational complexity of these adders is approximately equal to that of eight real adders. Thus, 128 XNOR gates and eight real adders can be seen as the major components of the PSMF. In the proposed scheme, we assume that the false alarm of SD is 0.45. Therefore, to complete a TFC identification process, about 3N _{ s } D _{ a } real additions, N _{ s } D _{ a } real comparisons, N _{ s } D _{ a } taking absolute value operations (ABS), and N _{ s } D _{ a } XNOR gate operations are consumed by the SD, and about 4 × 8 × 0.45N _{ s } D _{ a } real additions, 4 × 0.45N _{ s } D _{ a } integer comparison, and 4 × 128 × 0.45N _{ s } D _{ a } XNOR gate operations are consumed by the TTD. To simplify the comparison in Table 2, the computational complexity of three integer additions is equal to that of one real addition, three integer comparisons are converted to one real comparison, and eight XNOR gates are converted to one integer comparator. In Table 2, the computational complexities of the real addition and the real comparison are far lower than that of the complex multiplication but are much higher than that of the ABS. As the complex multiplication is not used, the proposed scheme has much lower computational complexity than the other methods in the table.
5.2 Packet detection
The performance of the CD is evaluated firstly. Let P _{ c_m } be the miss detection probability of the CD. According to (35) in Appendix 2, P _{ c_m } is given by
where ā is the mean of the absolute values of the even coefficients and λ _{ c } is the threshold for the CD.
Let P _{ c_f } be the false alarm probability of the CD. According to (36) in Appendix 2, P _{ c_f } is given by
The simulated miss detection probabilities of the CD under different false alarm probabilities are shown in Figure 7. In the simulation, TFC 1 and the CM1 channel are used. The simulated results are achieved by assuming the false alarm probability is 0.1, 0.2, and 0.3, respectively. If 0.2 is the given false alarm probability, the miss detection probabilities are lower than 0.008 and 0.001 with SNR = −8 dB and SNR = −6 dB, respectively. These miss detection results are low enough to satisfy most MBOFDM systems, and the computational complexity of the CD is approximately 1/3 of that of the FD. So, the proposed CD is effective for reducing computational complexity.
The performance of the FD is also evaluated. Let P _{ f_m } be the miss detection probability of the FD and λ _{ f } is the threshold for the FD. According to (38) in Appendix 2, P _{ f_m } is given by
Let P _{ f_f } be the probability of false alarm of the FD. According to (39) in Appendix 2, P _{ f_f } is given by
Let P _{ p_e } be the error probability of the proposed packet detection scheme. P _{ p_e } is given by
By selecting an appropriate false alarm probability for the CD, the influence of the CD on P _{ p_e } can be ignored. Therefore, P _{ p_e } can be simplified as
The simulated error probabilities of packet detection methods are shown in Figures 8 and 9. Except the proposed packet detector, the error probabilities of the traditional AC algorithm, the traditional CC algorithm, and the signbit CCbased method in [12] are also shown in the figures. In the simulation, two channel types, CM1 and CM4, and TFC 1 are used, and the threshold of each method is optimized by searching the optimal value for each SNR. As seen from Figures 8 and 9, the proposed method has the lowest error probability among the methods in the figures.
The comparison on computational complexity among packet detectors is shown in Table 3. The traditional AC algorithm, the traditional CC algorithm, and the method in [12] are included in the comparison. In Table 3, D _{ b } is the number of the symbols in interval B. We assume the false alarm of the CD is 0.2. The CC block in the CD can be implemented with 64 XNOR gates and the adders of 32 2, 16 3, 8 4, and 4 7bit integer adders. The computational complexity of these adders is about equal to that of three real adders. In the packet detection process, the CD approximately consumes 3N _{ s } D _{ b } real additions, 64N _{ s } D _{ b } XNOR gate operations, and N _{ s } D _{ b } integer comparisons, and the FD approximately needs 0.2 × 8N _{ s } D _{ b } real additions, 0.2N _{ s } D _{ b } integer comparisons, and 0.2 × 128N _{ s } D _{ b } XNOR gate operations. To simplify the comparison, the computational complexity of the adders of the CC block in [12] is equal to that of six real adders, and the computational complexity of the integer multipliers in [12] is equal to 1/10 the computational complexity of one complex multiplier. As seen from Table 3, the proposed scheme has much lower computational complexity than the other schemes.
6 Conclusions
In this paper, a powerefficient synchronization design is proposed for MBOFDM UWB systems. Based on the proposed overall architecture, the power consumptions of the individual blocks are analyzed from the perspective of the comprehensive synchronization process. A TFC identification structure with very low power consumption is proposed. A structure with low computational complexity and high performance is proposed for packet detection. In addition, a matched filter, which has low computational complexity and can be used for identifying TFC type, is designed.
Abbreviations
 AC:

autocorrelation
 BLUE:

best linear unbiased estimation
 CC:

crosscorrelation
 CE:

channel estimation
 CD:

coarse detector
 CFO:

carrier frequency offset
 CIR:

channel impulse response
 FCC:

Federal Communications Commission
 FD:

final detector
 FS:

frame synchronization
 ICI:

intercarrier interference
 IR:

impulse radio
 ISI:

intersymbol interference
 MBOFDM:

multiband orthogonal frequencydivision multiplexing
 PS:

packet synchronization
 PSD:

power spectral density
 QoS:

quality of service
 SD:

signal detector
 TFC:

timefrequency code
 TTD:

TFCtype detector
 UWB:

ultra wideband
 VLSI:

very large scale integrated circuits
 WPAN:

wireless personal area network
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Appendices
Appendix 1 The probabilities of G _{ c,n } > λ
If we assume c = i in Equation 6, then the PSMF for preamble pattern i is considered and G _{ i,n } is the output of the PSMF at the nth time index. The probabilities of G _{ i,n } > λ of the three scenarios of the transmitted samples are analyzed as follows:

(1)
The transmitted samples are the coefficients of preamble pattern i. In this case, if s _{ n + k } is transmitted and is received as r _{ n + k } at the receiver, r _{ n + k } follows a Gaussian distribution with mean s _{ n + k }. In (6), the value domain of Q _{ c,k }(k = 0, 1, …, N − 1) is {0, 1, 2, 3}. Because most I _{ n + k } = − 1 occur on the coefficients with small absolute value, the mean of I _{ n + k }Q _{ c,k } (k = 0, 1, …, N − 1; I _{ n + k } = − 1) is approximately equal to − 1/2. Also, usually I _{ n + k } = 1 occurs on the coefficients with large absolute value. The mean of I _{ n + k }Q _{ c,k }(k = 0, 1, …, N − 1; I _{ n + k } = 1) is approximately equal to 3/2. To simplify the analysis, I _{ n + k }Q _{ c,k } is simplified as
$$ {I}_{n+k}\left{Q}_{c,k}\right=\left\{\begin{array}{c}\hfill 1/2\kern2.8em {I}_{n+k}=1\hfill \\ {}\hfill 3/2\kern3.9em {I}_{n+k}=1\hfill \end{array}\right. $$(23)In this case, if the sign of r _{ n + k } is equal to that of s _{ n + k }, I _{ n + k } = 1. Let p _{ k } be the probability of I _{ n + k } = 1, and \( \overline{p} \) is the mean of p _{ k }(k = 0, 1, … N − 1). \( \overline{p} \) can be given by
$$ \overline{p}\approx \frac{1}{2} erfc\left(\frac{\overline{m}}{\sqrt{2}{\sigma}^2}\right) $$(24)where erfc() is the complementary error function, and \( \overline{m} \) is the mean of the absolute values of the preamble coefficients. Because N(=128) is large, according to the central limit theorem, P _{1,λ } is given by
$$ {P}_{1,\lambda}\approx \frac{1}{\sqrt{2\pi }}{\displaystyle {\int}_{\left(\frac{N+2\lambda }{4} Np\right)/\sqrt{Np\left(1p\right)}}^{\infty }{e}^{\frac{1}{2}{t}^2}}dt $$(25)$$ =1\varPhi \left(\frac{\frac{N+2\lambda }{4}\frac{1}{2} erfc\left(\frac{\overline{m}}{\sqrt{2}{\sigma}^2}\right)N}{\sqrt{\frac{1}{2} erfc\left(\frac{\overline{m}}{\sqrt{2}{\sigma}^2}\right)\left[1\frac{1}{2} erfc\left(\frac{\overline{m}}{\sqrt{2}{\sigma}^2}\right)\right]N}}\right) $$(26)where \( (x)={\displaystyle {\int}_{\infty}^x\frac{1}{\sqrt{2\pi }}}{e}^{\frac{1}{2}{t}^2}dt \). In (25), not λ but \( \frac{N+2\lambda }{4} \) is used because not {0, 1} of the Bernoulli distribution but {−1/2, 3/2} is the value domain for I _{ n + k }Q _{ c,k }(k = 0, 1, …, N − 1).

(2)
The transmitted samples are noise. In this case, the received samples are Gaussian noise with zero mean. In this case, to simplify the analysis, I _{ n + k }Q _{ c,k } can be simplified as
$$ {I}_{n+k}\left{Q}_{c,k}\right=\left\{\begin{array}{c}\hfill 1\kern2.3em {I}_{n+k}=1\hfill \\ {}\hfill \kern0.6em 1\kern3.4em {I}_{n+k}=1\hfill \end{array}\right. $$(27)In this case, the probability of I _{ n + k } = 1 is equal to \( \frac{1}{2} \). P _{2,λ } is given by
$$ {P}_{2,\lambda}\approx \frac{1}{\sqrt{2\pi }}{\displaystyle {\int}_{\left(\frac{N+\lambda }{2} Np\right)/\sqrt{Np}\left(1p\right)}^{\infty }{e}^{\frac{1}{2}{t}^2}}dt $$(28)$$ =1\varPhi \left(\frac{\frac{N+\lambda }{2}\frac{1}{2}N}{\frac{1}{2}\sqrt{N}}\right) $$(29) 
(3)
The transmitted samples are coefficients of preamble pattern j(j ≠ i). Because the coefficients of preamble pattern j(j ≠ i) are orthogonal with the coefficients of preamble pattern i, the received samples can be seen as noise for calculating G _{ i,n }. Therefore, P _{3,λ } is given by
$$ {P}_{3,\lambda}\approx 1\varPhi \left(\frac{\frac{N+\lambda }{2}\frac{1}{2}N}{\frac{1}{2}\sqrt{N}}\right) $$(30)
Appendix 2 Probabilities of miss detection and false alarm of the CD and the FD
To calculate P[E _{ n } > λ], there are two scenarios of the transmitted signals to be considered.

(1)
A PS symbol is transmitted. In this case, the received sample r _{ n+k } follows a Gaussian distribution with mean s _{ n + k }. In Equation 7, if r _{ n+2k } and s _{ n+2k } have the same sign, \( {\mathrm{C}}_{\mathrm{r},\mathrm{n}+2\mathrm{k}}{\mathrm{C}}_{\mathrm{c},2\mathrm{k}}^{*}=1, \) else \( {\mathrm{C}}_{\mathrm{r},\mathrm{n}+2\mathrm{k}}{\mathrm{C}}_{\mathrm{c},2\mathrm{k}}^{*}=1 \). In this case, let P _{4,λ } = P[E _{ n } > λ]. The same as the derivation of P _{1,λ } in Appendix 1, P _{4,λ } is given by
$$ {P}_{4,\lambda}\approx 1\varPhi \left(\frac{\frac{N/2+\lambda }{4}\frac{1}{2} erfc\left(\frac{\overline{a}}{\sqrt{2}{\sigma}^2}\right)N/2}{\sqrt{\frac{1}{2} erfc\left(\frac{\overline{a}}{\sqrt{2}{\sigma}^2}\right)\left(\left[1\frac{1}{2} erfc\left(\frac{\overline{a}}{\sqrt{2}{\sigma}^2}\right)\right]N/2\right.}}\right) $$(31)where ā is the mean of the absolute values of the even coefficients. Because N/2 samples are used for calculating E _{ n }, and not {0, 1} of the Bernoulli distribution but {−1, 1} is the value domain for \( {C}_{r,n+2k}{C}_{c,2k}^{*}, \) instead of λ, \( \frac{N/2+\lambda }{2} \) is used in (31).

(2)
Noise is transmitted. In this case, the received samples are Gaussian noise with zero mean. In this case, let P _{5,λ } = P[E _{ n } > λ]. The same as the derivation of P _{2,λ } in Appendix 1, P _{5,λ } is given by
$$ {P}_{5,\lambda}\approx 1\varPhi \left(\frac{\frac{N/2+\lambda }{2}\frac{1}{4}N}{\frac{1}{2}\sqrt{N/2}}\right) $$(32)In the CD, U _{ n } is used to detect the received samples by comparing it with the predefined threshold. We assume that λ is the threshold used for the CD. If PS symbols are present in both segments and U _{ n } < λ, miss detection occurs. As N samples are used for calculating U _{ n }, according to (31), P[U _{ n } > λ] in this case is given by
$$ P\left[{U}_n>\lambda \right]\approx 1\varPhi \left(\frac{\frac{N+\lambda }{2}\frac{1}{2} erfc\left(\frac{\overline{a}}{\sqrt{2}{\sigma}^2}\right)N}{\sqrt{\frac{1}{2} erfc\left(\frac{\overline{a}}{\sqrt{2}{\sigma}^2}\right)\left[1\frac{1}{2} erfc\left(\frac{\overline{a}}{\sqrt{2}{\sigma}^2}\right)\right]N}}\right) $$(33)Therefore, the miss detection probability of the CD is given by
$$ {P}_{c\_m}=1P\left[{U}_n>\lambda \right] $$(34)$$ \approx \varPhi \left(\frac{\frac{N+\lambda }{2}\frac{N}{2} erfc\left(\frac{\overline{a}}{\sqrt{2}{\sigma}^2}\right)}{\sqrt{\frac{N}{2} erfc\left(\frac{\overline{a}}{\sqrt{2}{\sigma}^2}\right)\left[1\frac{1}{2} erfc\left(\frac{\overline{a}}{\sqrt{2}{\sigma}^2}\right)\right]}}\right) $$(35)If U _{ n } > λ and no PS symbol is present in either segment, false alarm occurs. According to (32), in this case,
$$ {P}_{c\_f}=P\left[{U}_n>\lambda \right]\approx 1\varPhi \left(\frac{\frac{N+\lambda }{2}\frac{1}{2}N}{\frac{1}{2}\sqrt{N}}\right) $$(36)The discrete value domain of I _{ n + k }Q _{ c,k } of the PSMF in the FD is simplified as that in Appendix 1, and we assume that M = 2 is selected for the FD and λ is the predefined threshold.
If D _{ n } < λ and PS symbols are present in both segments, miss detection occurs.
$$ {P}_{f\_m}=1P\left[{D}_n>\lambda \right] $$(37)When M = 2, a total of 2 N samples are used for calculating D _{ n }. According to (26) in Appendix 1,
$$ {P}_{f\_m}\approx \varPhi \left(\frac{\frac{N+\lambda }{2} erfc\left(\frac{\overline{m}}{\sqrt{2}{\sigma}^2}\right)N}{\sqrt{erfc\left(\frac{\overline{m}}{\sqrt{2}{\sigma}^2}\right)\left[1\frac{1}{2} erfc\left(\frac{\overline{m}}{\sqrt{2}{\sigma}^2}\right)\right]N}}\right) $$(38)If D _{ n } > λ and no PS symbol is present in either segments, false alarm occurs. According to (29) in Appendix 1,
$$ {P}_{f\_f}\approx 1\varPhi \left(\frac{\frac{2N+\lambda }{2}N}{\frac{1}{2}\sqrt{2N}}\right) $$(39)
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Yin, XW., Tan, HZ. A powerefficient synchronization scheme for the MBOFDMbased UWB systems. J Wireless Com Network 2015, 14 (2015). https://doi.org/10.1186/s1363801502540
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DOI: https://doi.org/10.1186/s1363801502540