In order to effectively mitigate interference between UWB sensors and other legal networks, from the signal processing aspect, two jobs can be suggested in consideration of the underlay nature of UWB. Firstly, UWB sensors accurately identify available spectrums by monitoring the nearby perspective networks. Then, based on the discovered spectral environment, they adjust the RF emissions to advantageously perform their functions, without interfering other networks [26]. These two functions will be elaborated in Sections 2.1 and 2.2, respectively.
2.1. Spectrum Sensing in UWB Sensors
In cognitive UWB sensors, spectrum sensing is mainly adopted to obtain the current states of other networks. Most traditional sensing schemes assume that whether the authorized users exist maintains independent between two adjacent detection periods, and the probabilities of active state and idle state remain the same. This processing strategy can greatly simplify the sensing algorithms; however, it also results in a suboptimal sensing performance.
Practically, the behavior of PU is close associated to its corresponding wireless service, leading to specific probability features on its working states to some extent. This potential information can be properly explored to improve sensing accuracy. There have existed a few literatures that seek to employ partial probability characteristics to enhance sensing performance [27, 28], including the optimization of spectrum detection scheduling to improve multiple channels utilization. Nevertheless, to the best of our knowledge, extensive investigation on spectrum detection by totally utilizing the state transition information of PU has not been addressed in the literature. The interrelation between sensing gain and the prior information also remains not discussed.
Our main contributions in spectrum detections may lie in that, for the first time, we model the working states of PU as a binary sequence characterized by finite state machine. Then, by fully exploring the potential information carried by PU, we employ MAP to perform optimal spectrum sensing and greatly enhance the detection performance. This processing strategy provides a novel insight into spectrum sensing, and our original revealment of the rough interrelation between the achieved sensing gain and PU's state transition characteristics may substantially benefit future researches in cognitive networks.
2.1.1. General Sensing Strategy
As UWB sensors cannot cause much interference to the authorized networks when using spectrum, they should search unused spectrum before establishing their data links. When specific unoccupied authorized band has been detected, the UWB sensor will send its data during the following time slot. However, since PUs may reclaim their spectrum at any time, UWB users should periodically sense the spectrum to avoid interfering nearby networks. So, we adopt the cycle spectrum sensing mode in this paper. The fixed frame duration F is assumed in which the sensing duration is T and the remaining duration FT is used for data transmission [27]. It is noteworthy that the transmission here means either the data communications or some other dedicated functions, such astarget detection and positioning operations.
Generally, cooperative sensing can alleviate the problem that one single sensor cannot detect the spectrum correctly when there is serious shadow fading [29]. If UWB sensors are taken into consideration, however, single node spectrum sensing is still a reasonable choice. Firstly, in a distributed UWB sensor network with highly dynamic characteristics caused by movement or birthanddeath process of sensor nodes, effective collaboration in spectrum sensing seems hard to be realized. Moreover, the required overhead may create heavy load for UWB network. The cooperative sensing even becomes impractical when the control channels are not available [30]. Additionally, the whole sensing time may become intolerantly long in a cooperative fashion. So, in this paper, we mainly focus on the single node spectrum sensing.
2.1.2. Energy Detection
Given the uncomplicated implementation of ED, it always remains the first choice for spectrum sensing in UWB sensors. Thus, this paper establishes the general sensing model based on ED. Before proceeding, it is necessary to briefly illustrate ED algorithm, which is always formulated as the following two hypotheses:
where is the received signal in UWB sensors, is the nearby network's signal with its variance denoted by , and is the additive white Guassian noise (AWGN). In (1), and denote the hypotheses corresponding to the absence and presence of the primary networks, respectively. In realization, a bandpass filter is usually adopted to extract spectral components of interest. Then, the test statistics is constructed as the observed energy summation within consecutive segments:
where and represent the spectral components of the primary signal and on the interested subband, respectively. Without loss of generality, we assume that is white complex Gaussian noise with zero mean and variance . Then, the test statistics follows a central chisquare distribution with degrees of freedom under H _{
0
}, and a noncentral chisquare distribution with degrees under , that is,
If a decision threshold is properly determined, the false alarm probability can be defined as , and the corresponding detection probability is . Correspondingly, we have
Accordingly, the missed probability is given by [15]. is the generalized Marcum function which is given by
In practice, the probabilities of false alarm and missing detection have different implications for UWB sensors. Generally, low probability of false alarm is necessary to maintain high spectral utilization in UWB systems, since a false alarm would prevent the unused spectral segments from being accessed by UWB users. One the other hand, the probability of missed detection measures the interference of UWB users to PU, which should be limited in the opportunistic spectrum access. The most popular strategy is to determine a threshold to satisfy a false alarm probability, which is based on NeymanPearson criterion [27]. This scheme maximizes the detection probability for a given false alarm probability, for example, , which is suitable in most sensor networks. However, in some other scenes such as UWB radar sensors, the collected data for different vital applications should be transmitted without delay, which puts significant importance on spectral utilization, and the NeymanPearson criterion is not applicable anymore. Therefore, considering spectrum sensing by combining spectral utilization to unused bands and interference to PU, from a much wider sense, we minimize the total error probability in this paper. So the expense of spectrum sensing is given by
2.1.3. State Transition of PU
For most wireless networks, the evolution process of their working states over time can be reasonably abstracted as a finite state machine. Specifically, as is shown in Figure 1, the state transition of PU can be described by a trellis diagram that is similar to NRZI code [31]. If PU is in active state at current moment, then in the next sensing slot, it will stay in with a probability of and enter into sleep state with . Alternatively, if it is current in sleep state , then it will stay in in the next slot with a probability of and change into with . Obviously, we have:
Usually, these transition probabilities vary with time. It is found that the state of authorized networks in each sensing duration corresponds with certain state symbol which constantly changes along the trellis diagram. Specifically, when the state transition occurs at , the following primary state keeps different from that in . As the state transition further extended, can be viewed as a BPSK coding sequence with memory, which is also characterized by a Markov chain. Therefore, the main objective of spectrum detection lies in correctly demodulating this coded sequence . Denoting the two states of PU as binary symbol "0" and "1", the optimum spectrum detection in (6) is equivalent to minimizing the symbol error probability (SER).
The MAP criterion can be properly employed in the successive symbol detection, which is optimum in the sense that it minimizes the detection probability of symbol errors [31]. Suppose that it is desired to detect the state symbol in the k th sensing duration, and let be the observed test statistics, where D is the delay parameter which is always chosen to exceed the signal memory. If we denote the inherent memory of the equivalent sequence by , then we have . On the basis of the already received signal, we compute the posterior probability:
For possible state symbols , we choose with the largest probability. Then, we have
Since the denominator is common for two probabilities, the MAP criterion is equivalent to choose the value of that maximize the numerator of (9). Thus, the criterion for deciding on thek th state symbol is
The solution for (10) recursively begins with the first symbol ; then the following state symbols are sequentially obtained. The amplitude levels are only 2; so the computational complexity is basically acceptable. But, it is noteworthy that there exist fixed D delays in MAP algorithm. Consider that the idle spectrum may be reclaimed again during a short period, this presented detection algorithm withD accessing delays may miss most chances of using idle spectrum. Therefore, this original MAPbased sensing algorithm may not be appropriate to UWB sensors.
Furthermore, most recent research shows that the primary state of specific networks can be modeled as an alternating renewal source, in practice, which can be explored to simplify above MAP detection algorithm. As indicated by investigation in [32], the exponential distribution can be assumed for the probability density functions of the busy state and idle state periods as
where and are the transition rates from busy to idle and from idle to busy, respectively. With the aid of Komogorov Equation [33], we obtain the probability of the idle state remaining unchanged during successive detection periods:
Similarly, the probability of the active state lasting for detection periods is
If we have identified the initial primary state in idle at the moment , which means , then the probability that the primary state always stays in idle after sensing transmission periods can be given by
Accordingly, the probability of the primary state entering into busy in the k th detection period, after staying idle for periods, can be expressed as .
It is noted from the above discussion that PU would stay in present state for certain sensingtransmission periods before jumping into another one. Thus, the key point of tracing a state path is to accurately determine the state transition moment. Furthermore, a careful observation on (14) shows that the state evolution following an exponential distribution has a limited memory, which means that the current state of authorized network is only related to the latest previous state rather than the upcoming ones. Based on these two points above, we may further simplify MAP algorithm in (10). If we have learned that the authorized user enters into state "1" at the moment and stays for periods, the optimal state estimation at the k th moment should meet
2.1.4. Spectrum Detection
In early stage, UWB sensors may take a long duration or employ some other sophisticated sensing algorithms (i.e., cyclostationary feature detection) to get the initial PU state correctly. Without loss of generality, supposing that the initial primary state is , then a prior probability of staying at this state for sensingtransmission cycles is . In order to maximize a posteriori probability detection, (15) requires determining an optimum decision threshold to meet
Here, is the weighting coefficient ranging in , which can be carefully used to adjust relative preference between missed probability and detection probability. In practice, means the cost of spectral efficiency decline because a missed detection is relatively larger than that of interfering the PU, which implies that the nearby networks possess a strong antiinterference ability. In this situation, we may aggressively improve the utilization efficiency of idle spectrums to facilitate data transmissions of urgent UWB applications. On the other hand, implies that we show much favor to the unperturbed communication link of PU compared to the spectrum efficiency. Hence, strict protection to the authorized networks is necessary.
After the optimum decision threshold has been obtained according to (16), the detection probability P _{
0
} in the k th moment can be given by
Similarly, the threshold for initial state 1, , can be also obtained. Then, the detection probability can be written as
Since the state symbol may not appear equally, the overall probability of correct detection is
It is obvious that maximizing gives the minimum sensing expense . Actually, from the information theory aspect, spectrum sensing is equal to detecting a binarycoded sequence that obeys an unsteady state transition, given the reward of detection probability and missed probability. Like in most traditional spectrum sensing algorithms, if two sequential states of authorized users are simply assumed to be independent, the detection probability would be rather limited. However, our MAPbased optimum spectrum sensing is supposed to be much superior to ED, in consideration of entire exploration of the implicit state evolution and corresponding potential coding gain. Compared to matched algorithm and cyclostationary detection, our method only requires a statistic traffic model rather than the specific signal parameters of different networks, which can be conveniently obtained by experiential data or by learning.
2.1.5. Robustness Analysis
When the last state transitions of PU are exactly estimated, the simplified algorithm is equal to MAP algorithm, and it has optimal detection performance. However, from (16), spectrum detection errors in state transition moment will cause error extension in the upcoming periods. To avoid this unfavorable situation, following measures are suggested when PU has entered one state and lasted beyond its mean duration. () We may change the relative ratio between sensing period and transmission period. In an extreme case, the whole cycle is allocated for spectrum detection. () Other advanced sensing algorithms can be employed to assist estimating the exact state transition. () After sensingtransmission periods, usually being among 5–10, we may employ thetruncated sequential detection and retake a long duration to detect the initial state and then repeat this process.
The bad effect on detection probability caused by state transition estimation errors will be discussed in this part, which is instructive if these suggested measures cannot be realized. In the situation with low detection probability, the state transition point can be correctly estimated by adjusting sensing duration. Compared to ED, however, detection gain in this case is quite limited but with a high complexity. So, we mainly analyze the performance decline caused by the state transition estimation errors under high detection probability. Assuming that the average detection probability is , in the worst case, the detection error occurs successively, whose length is about .
Firstly, we focus on state detection errors that occur near the actual state transition point, which has a serious influence on the upcoming detections. For convenience, suppose that the estimation errors take detection cycles in advance the actual transition moment, which is referred to as advanced transition detection error (ATDE). Given that the initial state is 0, accordingly, affected by this error initial state, subsequent optimum thresholds are determined by
The thresholds in following decision are denoted by , and the corresponding detection probability is
Secondly, we investigate the nonideal case that the worst estimation errors occur in the middle of state transition (MPDE); namely, the estimated transition moment is much earlier than the actual state transition. In order to avoid this false state transition, ED is preferred to MAP detection in (15) after the false state estimation has occurred. Then, a counter is adopted for the sustaining periods of this false state. If the sustaining periods are smaller than , we may conclude that the state transition does not actually occur and the upcoming detection still follows the correct state before. If it is larger than , then we judge that the state transition indeed happens, and the current sustaining period of this new transited state is set as . It can be easily found that the detection error is not diffusive in this way; thus the upcoming sensing performance is basically not affected. When is far less than , the detection probability lowbound can be approximated by
where and represent the probabilities of false alarm and missed detection of ED algorithm, respectively.
The analysis above provides the detection performance in the case that the false state estimations cause error diffusion during following detections. However, it is also noteworthy that the successive estimation errors of are almost impossible to happen; so our analysis only provides a loose low bound for sensing performance with state estimation errors under high detection probability.
2.2. Spectrum Sculpting in UWB Sensors
Detecting the presence of other nearby networks in a given primary band is just the first step in operation of a cognitive UWB sensor. In order to best adapt to current spectral environment and minimize mutual interference, UWB sensors should dynamically adjust the RF emissions after probing the current spectrum. Usually, this process covers the physical layer design as well as the upper layer joint optimization. Cross layer optimization is recommended for selecting transmission parameters according to the upper layer quality of service (QoS). Unfortunately, this process is basically computational and also has intolerable delay in practice. In contrast, the UWB waveform designing rarely considers QoS information from the upper layer, but this flexible strategy allows the simple implementation and fast accessing to idle spectrums and hence is much more suitable for distributed UWB sensors.
2.2.1. RF Requirements in CR
Based on an overall consideration of various factors, UWB waveforms design should meet the following requirements.

(1)
Avoid the licensed frequency band flexibly and effectively. It is possible to avoid authorized frequencies based on frequency hopping technique [34]; but in this mechanism, UWB sensors can only use one single free band, resulting in rather low spectrum utilization. In addition, oscillator operating at multiple frequencies is required, which also complicates the hardware implementation. Moreover, the switching time for typical PLL can even reach 1 ms, which may prevent UWB sensors from the timely utilization of idle spectrum. On the other hand, spectrum avoidancebased schemes have a limited spectral attenuation, which can hardly eliminate the accumulated interference from multiple UWB sensors to other networks [21, 22].

(2)
Use the idle spectrum entirely. Generally, more than one free spectrum hole exist, which always isolates a long spectral distance from each other. The traditional methods can use only one free band. Considering the high uncertainty of authorized band, data transmission of UWB nodes is easy to be interrupted by the reclaim of primary band. In order to ensure seamless communications, UWB sensors should utilize multiple idle bands simultaneously in case one PU reoccupies its primary band. TDCS can take advantage of multiple frequency bands. But, the designed waveform has an infinite long tail which inevitably causes ISI and hence undermines its transmission performance. Yet, truncation by windowing will in turn lead to an obvious degradation on spectral efficiency and remarkable outband leakage [25].

(3)
Simplify the upper layer design. Most of traditional spectrum access strategies are based on competitive mechanism or centralized scheduling. On one hand, it has to occupy remarkable bandwidth resource to pass the global control signaling. On the other hand, it also has to take a long time to coordinate transmission of each UWB node, which also inevitably misses most spectrum holes.
In this part, we suggest a novel UWB waveform based on the RBF neural network. The designed signal is highly reconfigurable which can entirely match target spectrum shape after an extremely short switching time. Also, efficient spectral attenuation can be produced to eliminate mutual interference between PU and UWB sensors. After the spectral holes have been identified based on our presented sensing algorithm, UWB sensors can immediately access in the following transmission slot by means of orthogonal waveforms, without waiting for a coordination control. Thus, the upper layer control can be considerably simplified, and the UWB sensors capacity can be enhanced at the same time.
2.2.2. System Architecture
With the excellent capability of the function approximation, RBF neural network can be properly applied to design UWB waveforms given any target spectrum shaping [35]. The main philosophy behind RBF is to adjust a set of basis functions and ultimately match any function in high dimension, at any precision. In fact, as pointed out in [36], the general spectrum forming can be viewed as twodimensional multivariable interpolation problem, which means that given a set containing N different frequency values and another set containingN sampled target spectrum values, the objective is to find a mapping function to meet
where represents the i th discrete frequency value, and is the corresponding sampled spectrum. With the help of the continuity of mapping function, spectrum values other than can be also obtained. If a set of basis functions are properly chosen, then the mapping function can be represented as a linear combination of the basis functions:
In UWB waveform designing, the selection of basis functions is rather different from the traditional sense. Since the basis functions mainly act as the interpolation functions, they are not required to keep orthogonal from each other. When selecting basis functions, it should ensure the localization property firstly, which means . Meanwhile, the basis functions should be central even symmetry, for example, . Besides, they should satisfy the requirement as indicated by Micchelli Theorem [37]. For the convenience of analysis, Gaussian function is served as the basis function in our following analysis; however, other candidate functions meeting above requirements include the raised cosine function and the exponential function [35]:
where and are both adjustable parameters of the transmission functions , which can be employed to modify their center and width, respectively; is the sampling interval in frequency domain.
The implementation architecture of the generalized spectrum shaping network is shown in Figure 2. Based on the spectrum sensing result in Section 2.1, the target output can be firstly determined with a purpose of fully utilizing unoccupied spectrums and also respecting the active primary bands. Then, the parameters of RBF network, including the network weight , the position, and shape of basis functions, are modified adaptively until the error signal between the network output a and the expected output reaches the minimum value. This adjusting process is shown by the solid line in Figure 2. Meanwhile, by introducing the spectrum pruning technique, further slight adjustments will be performed on partial network weigh after convergence, so as to obtain satisfactory spectrum efficiency and also meet specific spectrum constraints. This process corresponding to this feedback process is depicted by the dotted line, which is controlled by a switching circuit.
Each part of signal is discussed in detail as follows.
(1) Transform Function
Transmission functions are mainly used to produce the discrete input sequence (), which corresponds to the basis functions in (25). Supposing that denotes an dimensional vector composed of , then the input matrix can be written as
where represents the network offset. In the block diagram above, the transmission function has to be implemented by the group of filters in practice, and the time bandwidth product BT of these Gaussian filters is determined by in (25). Note that the number of transform functions is always less than N in order to simplify implementations [37].
(2) Target Output
The target output corresponding to the optimal emission spectrum depends on the current spectral environment. If we assume that there are total kinds of active legal systems, then the expected spectrum can be expressed as
where represents the indication gating function corresponding to the i th kind of PU. If the legal network locating at has been detected using the sensing method in Section 2.1, then the associated gating function is enabled:
where represents current working state of the i th PU.
(3) Parameters Updating
During the parameters adjustment process, by adaptively changing the network weights and the transmission parameters and , the mean square error (MSE) between the actual output and the target output will be minimized. And finally, we obtain the UWB signal with optimal spectrum shaping. The MSE can be defined as
where the coefficient 1/2 is just for facilitating the elaboration. The parameters updating of this UWB waveform generator, including the network weights and transmission parameters, can be divided into two phases [37]. The network weights are firstly adjusted using WindrowHoff rule. Then, the transmission parameters and can be modified by resorting to the gradient descent rule. Correspondingly, the partial derivatives of E on transmission para meters can be written as
(4) Implementation
In UWB sensors applications, the duration of the learning algorithm directly determines the accessing delay to idle spectrums. Therefore, we need to optimize the transmission parameters beforehand to further shorten the switching time, also simplifying the implementation of UWB nodes.
In fact, when is big enough, we may let each evenly distributed in frequency axis and employ one single Gaussian filter to generate the basis function . Then, the other basis functions can be obtained by samples cycle shifting operation on , where l represents the shifting factor (). So we can further optimize the single parameter . A reasonable parameter should actually be neither too large to avoid serious ripples in the pass band nor too small in case that the designed waveform has an obvious outband leakage which may interfere PU in adjacent band [35].
Based on the above simplification efforts, the structure of UWB pulse generator can be obtained as is shown in Figure 2. Firstly, an impulse sequences with a period of N is produced, which is then fed into a Gaussian shaping filter whose key parameter, the timebandwidth product BT, is determined by the already optimized . Then the network basis function can be formed. The sampling frequency of is set to , where is the maximum frequency of UWB signals. After that, the network input sequences can be constructed after () samples delay has been performed on . Notice that, here, sample delay is equivalent to cycle shift considering the periodic input impulse sequence. Finally, in the updating stage, the UWB waveform shaper makes adjustment to its weighting vector according to target output that relies on nearby spectrum environment.
When the RBF network is directly applied to UWB sensors, it is usually difficult to meet some given spectral constraints. For example, the designed UWB waveforms under specific spectral masks will have some serious mismatch near the abrupt spectral edges [35]. This would bring in serious negative effects in cognitive scenarios and reduce its applications significantly. Assuming that there is a spectral mismatch during the frequency band , then the corresponding network weight subset is denoted by , where m and n can be obtained by
The main objective of the spectrum pruning is to repair the remarkable local spectral mismatch. This process is also iterative, and the basic idea is to further modify the converged network weights falling into the mismatch range. When is large enough () and the maximum tolerance of the spectral mismatch is denoted by , the spectrum pruning can be summarized into
where is the pruning step. represents the UWB emission spectrum obtained from this shaping network, and is the regulatory spectral constraint. This iterative spectrum pruning process will be continued until all frequency bands have met the given spectral constraints.
2.2.3. Orthogonal Pulse Design
After both the network weights and the transmission parameters have converged to their optimal solutions, the frequency response of the designed signal can best approximate to the expected spectrum. So, we may immediately produce the UWB waveform by IDFT on :
where purelin() is the output function of RBF network [37], denotes the conjugate symmetric spectrum constructed from z which is the representative spectrum of the equivalent lowpass form [31]. The dimensional vector Θ represents the user defined phase response, and the operator ⊗ denotes the vector multiplication between the corresponding two vectors.
As is well known, orthogonal waveforms allow multiple UWB sensors to access the same idle band at the same time and in the same location, without causing serious collision. In a UWB sensor network, therefore, the orthogonal waveform division multiple accessing (WDMA) can also greatly simplify the upper layer protocol design and reduce accessing delay, hence significantly reducing scheduling complexity and improving spectrum efficiency.
For convenience, we assume that the emission power in idle frequency band is only related with the hardware specification of UWB devices. When the idle spectrum is detected, the emitted waveform remains a constant power, denoted by A, in the unused spectrum. For arbitrary two orthogonal waveforms and with their Fourier transform denoted by and , respectively, we have [31]. So, the following relation should be satisfied:
where B is the available frequency bands. It can be found that careful design of the phase response can produce the mutual orthogonal UWB waveforms. One simple scheme is to let the phase response to be
Here, we specify to be a binary sequence, for example, . Then, the designed UWB waveforms will keep orthogonal with each other so long as to ensure orthogonality of discrete sequence . If an appropriate pseudorandom sequence, such as sequenc e, is selected based on the length of sampling lengthN, orthogonal UWB waveforms can be easily derived.
It is noted that from (34) the orthogonality design requires UWB signal remain constant in the whole frequency axis. If the regulatory UWB emission mask is taken into account, such as the FCC emission limits, however, this algorithm has to be further modified. Practically, we may represent the whole spectral line by a combination of constant spectral lines along the frequency axis. Thus, this orthogonality design algorithm is still applicable to each constant spectrum.