APPARATUS AND METHOD FOR ENCODING A MESSAGE HAVING A TARGET PROBABILITY DISTRIBUTION OF CODE SYMBOLS TECHNICAL FIELD

In general, the present invention relates to encoding and decoding in communication systems. More specifically, the present invention relates to apparatuses and methods for encoding and decoding messages based on a polar transform.

BACKGROUND

In order to achieve the capacity of a transmission channel in communication systems, the channel input symbols should have a certain probability distribution. For example, in the case of an additive white Gaussian noise (AWGN) channel, a Gaussian distribution is required in order to achieve the capacity of the channel. However, in many practical communication systems, uniformly distributed channel input symbols are used, thereby causing a gap to the capacity. This loss is also called the shaping loss and can be up to 1 .53 dB on AWGN channels, in case uniformly distributed quadrature amplitude modulation (QAM) symbols are used.

The shaping loss can become significant especially with high order modulation. A common approach for transmission with high order modulation is the so-called bit-interleaved coded modulation (BICM), wherein the message is first encoded by a channel encoder to a code word which is interleaved, and then mapped to channel input symbols via a symbol mapper. In many communication systems, binary channel codes are used, such that the code words are binary vectors. In general, the distribution of ones and zeros in a code word is uniform. This causes the channel input symbols to have a uniform distribution too. In the so-called non-uniform constellations (NUC) approach, the possibility of using a symbol mapper with non-uniform constellations is suggested. In this approach, the output of the channel encoder is mapped to symbols that do not have a regular structure like QAM symbols, but have an optimized structure which can help reducing the shaping loss. This approach does not have any constraints on the used channel code and is also called geometric shaping. However, in this approach the standard QAM symbol mapper and QAM demapper has to be replaced by a more complex mapper/demapper, due to the non-regular constellation structure. The so-called probabilistic amplitude shaping (PAS) approach proposes the use of a shaping encoder prior to using the channel encoder, which converts the uniformly distributed input message to a non-uniformly distributed sequence. Then, this sequence is encoded by a systematic channel encoder (i.e., the output of the channel encoder contains the input of the encoder as a sub-vector), which is then fed to a QAM symbol mapper. At the receiver, a QAM demapper can be used. After decoding the channel code, the shaping decoder processes the output of the channel code in order to retrieve the message. Although this approach can remove almost all the shaping loss, it still requires a shaping encoder and a shaping decoder, thus, increasing the complexity and latency at the transmitter and the receiver. The information is retrieved after being processed by two decoders in serial, which can be suboptimal compared to a joint decoder. Moreover, this approach poses a constraint on the used channel code, since the channel encoder has to be a systematic encoder, which may not be favorable.

Polar codes (see, E. Arikan, "Channel polarization: A method for constructing capacity- achieving codes for symmetric binary-input memoryless channels," IEEE Trans. Inf. Theory, vol. 55, no. 7, pp. 3051 -3073, Jul. 2009) are recently developed forward error correction schemes that can achieve the capacity of binary input discrete memoryless channels and which can be used for the channel coding scheme. However, their performance with high order modulation (such as with BICM) is often poor compared to other modern coding schemes. Moreover, in general, polar code words have P _c (l) = 0.5, i.e., the probabilities of having ones and zeros in the code words are equal.

In the prior art, there are proposals (see the work by Mondelli et al., "How to achieve the capacity of asymmetric channels", Communication, Control, and Computing (Allerton), 2014 52nd Annual Allerton Conference on. IEEE, 2014) to shape the output distribution of a polar code word, such that P _c (l) is not equal to 0.5. This is achieved by allowing some

dependencies between the transmitted information bits. Basically, instead of transmitting k information bits, one transmits k— s information bits, and generates s bits depending on the rest. Those s bits are selected in a way, such that the resulting code word has a certain P _c (l). These bits can also be called the shaping bits. In the BICM case, it is possible to combine BICM with polar codes (using a single polar code combined with a high order modulation). This is compatible with many existing standards, and easy to implement. However, the performance of BICM with polar codes is poor compared to other alternatives.

In the multi-level coding (MLC) approach, the channel input symbols are considered as a combination of different bit levels. For each bit level, a different channel code is used. At the receiver, the demapping and decoding are done successively, i.e., it is started by demapping and decoding the first bit-level. After the first bit-level is processed, the output is used to demap and decode the second bit-level, and continues until all the bit levels are processed. It has been shown that the performance of polar codes combined with MLC can be better than conventional BICM. However, although a good performance can be achieved, a separate channel code for each bit-level needs to be designed, and several channel demapping and decoding processes should be run, which can increase the complexity and the processing latency. Thus, there is a need for improved communication apparatuses and methods for encoding and decoding messages based on polar codes allowing to reduce the gap to the capacity of the transmission channel.

SUMMARY

It is an object of the invention to provide for improved communication apparatuses and method for encoding and decoding messages based on polar codes allowing to reduce the gap to the capacity of the transmission channel. The foregoing and other objects are achieved by the subject matter of the independent claims. Further implementation forms are apparent from the dependent claims, the description and the figures.

According to a first aspect, the invention relates to a communication apparatus for encoding a message to be transmitted over a communication channel, wherein the communication apparatus comprises: a precoder configured to generate on the basis of the message and a polar transform a pre-coded message, and a channel encoder configured to encode the pre- coded message based on the polar transform into a code word, wherein the code word comprises a plurality of bits which have a probability distribution, wherein the precoder is configured to generate the pre-coded message such that the channel encoder generates the plurality of bits of the code word with the probability distribution matching a target probability distribution. rresponding to the

Matching the probability distribution to a target probability distribution can comprise that the precoder generates the bits and/or symbols based on the target probability distribution. Further, this can comprise that the bits and/or symbols generated by the precoder (or the channel encoder that processes the output of the precoder) have a probability distribution that is substantially equal to the target distribution, in particular that the probability distribution has substantially the same mean and/or variance as the target distribution. For example, the distance between the probability distribution of the plurality of bits of the code word and the target probability distribution can be given by the Kullback-Leibler distance.

Thus, the communication apparatus according to the first aspect can generate code words having a non-uniform probability distribution of the plurality of bits. For example, the probability for 1 to occur in a code word can be different from 0.5. Thus, an improved communication apparatus is provided, since its performance is significantly improved compared to conventional BICM with polar codes and compared to multi-level coding with polar codes.

In a possible implementation form of the communication apparatus according to the first aspect, the precoder is a systematic precoder, i.e., the pre-coded message contains the message.

This provides the advantage that, at the receiver side, which is configured to decode the message to be transmitted, the inversion of the precoding operation is not required, allowing for a less complex receiver.

In a further possible implementation form of the communication apparatus according to the first aspect, the communication apparatus further comprises an interleaver configured to interleave the plurality of bits of the code word and/or a modulator, in particular a symbol mapper configured to map the code word to one or more symbols for transmission over the communication channel. Thus, an improved communication apparatus is provided, since, for example, the use of an interleaver makes forward error correction more robust with respect to burst errors. In a further possible implementation form of the communication apparatus according to the first aspect, the polar transform is configured to take as an input a vector of frozen bits /, wherein the precoder is configured to generate the pre-coded message on the basis of the message and the vector of frozen bits /.

In a further possible implementation form of the communication apparatus according to the first aspect, the precoder is configured to generate the pre-coded message using one or more polar decoders, in particular successive cancellation (SC) or successive cancellation list (SCL) decoders.

Thus, an improved communication apparatus is provided, since, for example, using a SC decoder provides the advantage of having a less complex communication apparatus, while using a SCL decoder provides the advantage of having a communication apparatus with a good performance.

In a further possible implementation form of the communication apparatus according to the first aspect, the precoder is configured to take as an input a vector v comprising a vector of information bits e representing the message and the vector of frozen bits /, wherein the precoder is configured to decompose the vector v into m subvectors v _t .

In a further possible implementation form of the communication apparatus according to the first aspect, the precoder comprises m sub-precoders, wherein each sub-precoder is configured to generate a respective subvector d _t of a vector d representing the pre-coded message.

In a further possible implementation form of the communication apparatus according to the first aspect, the precoder is configured to generate the vector d representing the pre-coded message on the basis of the respective subvectors d _t generated by the respective sub- precoders.

In a further possible implementation form of the communication apparatus according to the first aspect, each sub-precoder comprises a polar decoder, in particular a successive cancellation decoder or a successive cancellation list decoder.

In a further possible implementation form of the communication apparatus according to the first aspect, each subvector v _t of the vector v comprises a vector e _t , which is a subvector of the vector of information bits e, and a vector /; , which is a subvector of the vector of frozen bits /, and wherein each sub-precoder is configured to generate a respective subvector d _t of the vector d representing the pre-coded message on the basis of the respective subvector of information bits e _t and the respective subvector of frozen bits /; .

In a further possible implementation form of the communication apparatus according to the first aspect, the target probability distribution comprises a plurality of target probabilities ; and wherein each sub-precoder is configured to generate a respective auxiliary channel decoder input vector y'i on the basis of the respective target probability ;.

This provides the advantage of generating code words that contain different sub-vectors with different probabilities, so that after the symbol mapper channel input symbols with desired distribution can be obtained. Moreover, this allows to reduce the gap to the capacity of the transmission channel.

In a further possible implementation form of the communication apparatus according to the first aspect, each sub-precoder is further configured to generate a subvector of auxiliary frozen bits /'; on the basis of the respective subvector of information bits e _t and the respective subvector of frozen bits /; .

In a further possible implementation form of the communication apparatus according to the first aspect, each sub-precoder comprises a polar decoder, in particular a successive cancellation decoder or a successive cancellation list decoder, configured to generate a respective shaping bit vector s; on the basis of the respective auxiliary channel decoder input vector y'i and the respective subvector of auxiliary frozen bits /';, wherein each sub- precoder is configured to generate the respective subvector d _t of the vector d representing the pre-coded message on the basis of the respective information bits subvector e _t and the respective shaping bit vector s; . In a further possible implementation form of the communication apparatus according to the first aspect, each sub-precoder is configured to generate the respective subvector d _t of the vector d representing the pre-coded message on the basis of the respective information bits subvector e _t and the respective shaping bit vector s; by concatenating the respective shaping bit vector s; with the respective information bits subvector e _t . According to a second aspect, the invention relates to a method for encoding a message to be transmitted over a communication channel, wherein the method comprises: generating on the basis of the message and a polar transform a pre-coded message, and

encoding the pre-coded message based on the polar transform into a code word, wherein the code word comprises a plurality of bits which have a probability distribution, wherein the pre-coded message is generated in such a way that the plurality of bits of the code word are generated with the probability distribution matching a target probability distribution.

According to a third aspect, the invention relates to a computer program comprising program code for performing the method of the second aspect, when executed on a computer or a processor.

According to a fourth aspect, the invention relates to a communication apparatus for decoding a message received over a communication channel, wherein the communication apparatus comprises: a channel decoder configured to generate an estimate of a vector d containing a vector of information bits e and a vector of shaping bits s by decoding the message on the basis of a polar transform on the basis of a vector of frozen bits /.

In a possible implementation form of the communication apparatus according to the fourth aspect, the channel decoder is configured to discard the vector of shaping bits s.

In a further implementation form of the communication apparatus according to the fourth aspect, the channel decoder is configured to generate a sequence s' on the basis of the estimate of the vector d and to output an error message, in case the sequence s' is not equal to the vector of shaping bits s.

In a further implementation form of the communication apparatus according to the fourth aspect, the channel decoder comprises a list decoder configured to select a respective code word from a list of code words by selecting the code word, for which the sequence s' is equal to the vector of shaping bits s.

The invention can be implemented in hardware and/or software.

BRIEF DESCRIPTION OF THE DRAWINGS

Further embodiments of the invention will be described with respect to the following figures, wherein: Fig. 1 shows a schematic diagram illustrating a communication system comprising a communication apparatus for encoding a message according to an embodiment, and a communication apparatus for decoding the message according to an embodiment;

Fig. 2 shows a schematic diagram illustrating the channel encoder of a communication apparatus for encoding a message according to an embodiment;

Fig. 3 shows a schematic diagram illustrating a communication apparatus for decoding a message according to an embodiment;

Fig. 4 shows a schematic diagram illustrating a polar transform used by a communication apparatus for encoding a message according to an embodiment; Fig. 5 shows a schematic diagram illustrating a communication apparatus for encoding a message according to an embodiment;

Fig. 6 shows a schematic diagram illustrating a polar transform used by a communication apparatus for encoding a message according to an embodiment;

Fig. 7 shows a schematic diagram illustrating a precoder of a communication apparatus for encoding a message according to an embodiment;

Fig. 8 shows a schematic diagram illustrating a sub-precoder of a precoder of a

communication apparatus for encoding a message according to an embodiment; and

Fig. 9 shows a schematic diagram of a method for encoding a message according to an embodiment. In the various figures, identical reference signs will be used for identical or at least functionally equivalent features.

DETAILED DESCRIPTION OF EMBODIMENTS In the following description, reference is made to the accompanying drawings, which form part of the disclosure, and in which are shown, by way of illustration, specific aspects in which the present invention may be placed. It is understood that other aspects may be utilized and structural or logical changes may be made without departing from the scope of the present invention. The following detailed description, therefore, is not to be taken in a limiting sense, as the scope of the present invention is defined by the appended claims. For instance, it is understood that a disclosure in connection with a described method may also hold true for a corresponding device or system configured to perform the method and vice versa. For example, if a specific method step is described, a corresponding device may include a unit to perform the described method step, even if such unit is not explicitly described or illustrated in the figures. Further, it is understood that the features of the various exemplary aspects described herein may be combined with each other, unless specifically noted otherwise.

Figure 1 shows a schematic diagram illustrating a communication system 100 comprising a communication apparatus 1 10 for encoding a message to be transmitted over a

communication channel, and a communication apparatus 120 for decoding the message according to an embodiment.

The communication apparatus 1 10 comprises a precoder 102 configured to generate on the basis of the message and a polar transform a pre-coded message, and a channel encoder 104 configured to encode the pre-coded message based on the polar transform into a code word, wherein the code word comprises a plurality of bits which have a probability distribution, wherein the precoder 102 is configured to generate the pre-coded message such that the channel encoder 104 generates the plurality of bits of the code word with the probability distribution matching a target probability distribution, as will be described in more detail further below. rresponding to the Matching the probability distribution to a target probability distribution can comprise that the precoder 102 generates the bits and/or symbols based on the target probability distribution. Further, this can comprise that the bits and/or symbols generated by the precoder 102 (or the channel encoder 104 that processes the output of the precoder) have a probability distribution that is substantially equal to the target distribution, in particular that the probability distribution has substantially the same mean and/or variance as the target distribution. For example, the distance between the probability distribution of the plurality of bits of the code word and the target probability distribution can be given by the Kullback-Leibler distance.

The communication apparatus 1 10 can easily be configured to adapt the generation of the code word with other modifications of polar codes, such as rate adaptation (puncturing, shortening and incremental redundancy methods) and other modifications for polar codes (like parity check (PC)-polar codes, cyclic redundancy check (CRC) assisted/CA-polar codes, sub-polar codes). Furthermore, the frozen bits of the polar transform can be chosen randomly or pseudo-randomly, wherein the randomness is shared between the encoder and the decoder.

The communication apparatus 120 for decoding a message according to an embodiment comprises a channel decoder 120a configured to generate an estimate of a vector d

containing a vector of information bits e and a vector of shaping bits s by decoding the message sent by the communication apparatus 1 10 on the basis of the polar transform on the basis of a vector of frozen bits /, as will be described in more detail further below.

Figure 2 shows a schematic diagram illustrating the channel encoder 104 comprising a unit 104b for implementing the polar transform according to an embodiment.

In this embodiment, the channel encoder 104 is a polar encoder comprising a demultiplexer 104a and the polar transform unit 104b according to an embodiment.

The demultiplexer 104a can be configured to combine the vector d representing the pre- coded message with a vector of frozen bits / in order to form a vector u of length N. In an embodiment, the obtained vector u has the following structure:

Ui, i E I contains the elements of d

Uj,j 6 F contains the elements of f

/ U F = {0,1, ... , ]V - 1}

I n F = 0, namely, the vector u comprises, at the indices /, the bits of d (that contains a vector of information bits e representing the message and a vector of shaping bits s), and at the indices F, the frozen bits, that are also known at the receiver side (i.e., the communication apparatus 120). The vector u can then be converted into the code word c by using the polar transform implemented in the unit 104b, which can be seen as a multiplication of the vector u with a N x N polarization matrix.

Figure 3 shows a schematic diagram of a communication apparatus 120 for decoding a message comprising a polar decoder 120a, in particular successive cancellation (SC) or a successive cancellation list (SCL) decoder 120a according to an embodiment.

In the embodiment shown in figure 3, the SC or SCL polar decoder 120a has four inputs: the channel output vector or received symbols y (also called 'channel decoder input vector'), e.g., the log-likelihood values (Inp1 ), the vector of frozen bits / (Inp2), the set of indices of the information bits / (Inp3), and the set of indices of the frozen bits F (Inp4). Moreover, the SC or SCL polar decoder 120a can have one output, namely the decoded information bits (Out1 ), i.e., the vector d representing the pre-coded message. In embodiments of the invention, the channel encoder 104 shown in figure 2 and/or the SC or SCL decoder 120b shown in figure 3 can have additional parameters depending on the implementation, and some of the input parameters can be related, such that one can obtain some input parameters depending on the other input parameters. For example, if the set of the information bits / is known, then the set of frozen bits indices F can be obtained as well.

Figure 4 shows a schematic diagram illustrating a polar transform of length N implemented in the unit 104b according to an embodiment.

The polar transform of length N implemented in the unit 104b can be used to generate the code word or polar code word c of length N by means of \og ₂ (N) polarization steps, which can be performed successively. Therefore, a polar code word c of length N can also be seen as a code word obtained by using two polar codes of length N/2, which can further be polarized in one more step, as shown in figure 4. The communication apparatus 1 10 can further comprise an interleaver 500 configured to interleave the plurality of bits of the code word and/or a modulator, in particular a symbol mapper 502 configured to map the code word to one or more symbols for transmission over the communication channel. Figure 5 shows a schematic diagram illustrating the communication apparatus 1 10 for encoding a message comprising the precoder 102, the channel encoder 104, the interleaver 500, and the symbol mapper 502 according to an embodiment. In the embodiment shown in figure 5, the vector of information bits e representing the message is first processed by the precoder 102, which generates a vector d representing the pre-coded message. In this embodiment, the vector d = (d ₀ , ... , d _n ) representing the pre- coded message can be encoded by the channel encoder 104 in order to generate the code word, in particular a binary code word c. In an embodiment, P _c (l)≠ 0.5, i.e., the binary code word c can contain ones and zeros with an unequal probability distribution (note that P _z (a) denotes the probability of occurrence of the element a in the vector z.).

As already described above, in an embodiment, the precoder 102 is configured to generate the vector d in such a way that, after channel encoding by the channel encoder 104, the resulting code word c has certain characteristics in terms of probability distributions.

In embodiments of the invention, the characteristics in terms of probability distributions can be described as follows: the code word c comprises m subvectors (for example subvectors of the same length) c , ... , c _m , and each of these m subvectors has a respective target probability, i.e.,

In embodiments of the invention, the vector d comprises the vector of information bits e, i.e, the vector of information bits e is a subvector of the vector d, i.e., the precoder 102 is a systematic precoder. This provides the advantage that, at the receiver or communication apparatus 120, the inversion of the precoding operation is not required, allowing for a less complex receiver. Moreover, the binary code word c can be shuffled or interleaved via the interleaver 500 in order to obtain a vector b. Then, the vector b can be fed to a symbol mapper 502.

The symbol mapper 502 can be configured to map the binary code word c to one or more symbols for transmission over the communication channel. In particular, the symbol mapper 502 can output these symbols from the alphabet X. Therefore, in this embodiment, the output vector x of the symbol mapper 502 has a nonuniform probability distribution.

Figure 6 shows a schematic diagram illustrating an embodiment where the precoder 102 comprises a first and a second sub-precoder.

In embodiments of the invention, m different parts of the vector u of length N are generated in such a way that each of these m parts have some dependencies within their bits or different probability distributions. In particular, m can correspond to the number of bit-levels for higher-order modulation.

In particular, in figure 6, an example for m = 2 is shown, wherein the vector u of length N consists of two subvectors u _x and u ₂ of length N/2. The subvectors u _x and u ₂ can be generated using the first and the second sub-precoder. This can imply that the bits comprised in Uj have some dependencies. If [ι½ u ₂ ] is encoded via a single polar transform implemented in the unit 104b of length N, then the resulting code word c can have the desired properties mentioned above.

The subvectors u _x and u ₂ can comprise information bits and frozen bits. Moreover, in embodiments of the invention, the subvectors u _x and u ₂ can comprise shaping bits as well, which are based on the information bits and frozen bits, and which help the output distribution to converge to the target probability distribution.

The exemplary procedure explained in the context of figure 6 can be extended to larger m values as well. For example, by diving the vector u into m = 4 and using two extra polarization steps, one can obtain a single polar code word c with four different bit distributions.

Figure 7 shows a schematic diagram illustrating the precoder 102 of the communication apparatus 1 10 according to an embodiment.

In the embodiment shown in figure 7, the precoder 102 comprises two multiplexing units 102a-1 , 102a-2, m sub-precoders 102b-1 ,...,102b-m, and a demultiplexing unit 104a. The precoder 102 can be configured to decompose a vector v comprising the vector of information bits e and the vector of frozen bits / into m subvectors v wherein each of the subvectors v _t comprises a vector e _i which is a subvector of the vector of information bits e and a vector /;, which is a subvector of the vector of frozen bits /.

In an embodiment, the multiplexing unit 102a-1 is configured to decompose the vector of information bits e into m vectors e _t , and the multiplexing unit 102a-1 is configured to decompose the vector of frozen bits / into m vectors

In an embodiment, each of the m sub-precoders 102b-1 ,...,102b-m is configured to generate each of the respective subvectors d _t of the vector d representing the pre-coded message on the basis of the respective vector e _t , the vector /; and a respective target probability p _t .

In an embodiment, the demultiplexing unit 104a is configured to generate the vector d representing the pre-coded message by combining the m subvectors d _t . In an embodiment the precoder 102 contains a SC or SCL polar decoder 120a as already described in the context of figure 3.

Figure 8 shows a schematic diagram illustrating a more detailed view of one of the m sub- precoders 102b-1 ,...,102b-m shown in figure 7.

In the embodiment shown in figure 8, the sub-precoder 102b-1 comprises a unit 800 configured to generate a respective auxiliary channel decoder input vector y'i on the basis of the respective target probability p _t . In an embodiment, the channel decoder input vector is a vector of length N (same length as the code word length), with the elements having the same value log(pi/(l - i)).

Moreover, in this embodiment, the respective subvector of information bits e _t and the respective subvector of frozen bits f _t are combined in the demultiplexer 802 in order to generate a vector / The bits composing the vector /'; can be used as auxiliary frozen bits for a SC/SCL decoder 804 in the sub-precoder 102b (Inp2). Moreover, the SC/SCL decoder 804 in the sub-precoder 102b-1 can use the respective auxiliary channel decoder input vector y'i as the channel output vector (Inp1 ). The output of the SC/SCL polar decoder 804 can contain a respective shaping bit vector s; (Out1 ). Furthermore, a demultiplexing unit 806 can be configured to concatenate the respective shaping bit vector s _t to the respective information bits subvector e _t in order to generate the respective vector d;. The sub-precoder 102b-1 provides the advantage that the vector d contains e as a subvector, which provides the advantage of an easy decoding at the side of the receiving communication apparatus 120.

Moreover, the sub-precoder 102b-1 provides the advantage of using a SC or SCL decoder 804 for generating the code word c, which is already a part of the polar transmission chain, namely the decoder 120a of the receiving communication apparatus 120 shown in figure 3. In other words, the precoder 102 can be realized by using existing hardware on a chip comprising the communication apparatus 1 10 and the communication apparatus 120. Figure 9 shows a schematic diagram of a method 900 for encoding a message to be transmitted over a communication channel according to an embodiment.

The method 900 comprises the steps of: generating 902 on the basis of the message and a polar transform a pre-coded message, and encoding 904 the pre-coded message based on the polar transform into a code word, wherein the code word comprises a plurality of bits which have a probability distribution, wherein the pre-coded message is generated in such a way that the plurality of bits of the code word are generated with the probability distribution matching a target probability distribution. While a particular feature or aspect of the disclosure may have been disclosed with respect to only one of several implementations or embodiments, such feature or aspect may be combined with one or more other features or aspects of the other implementations or embodiments as may be desired and advantageous for any given or particular application. Furthermore, to the extent that the terms "include", "have", "with", or other variants thereof are used in either the detailed description or the claims, such terms are intended to be inclusive in a manner similar to the term "comprise". Also, the terms "exemplary", "for example" and "e.g." are merely meant as an example, rather than the best or optimal. The terms "coupled" and "connected", along with derivatives may have been used. It should be understood that these terms may have been used to indicate that two elements cooperate or interact with each other regardless whether they are in direct physical or electrical contact, or they are not in direct contact with each other. Although specific aspects have been illustrated and described herein, it will be appreciated by those of ordinary skill in the art that a variety of alternate and/or equivalent

implementations may be substituted for the specific aspects shown and described without departing from the scope of the present disclosure. This application is intended to cover any adaptations or variations of the specific aspects discussed herein.

Although the elements in the following claims are recited in a particular sequence with corresponding labeling, unless the claim recitations otherwise imply a particular sequence for implementing some or all of those elements, those elements are not necessarily intended to be limited to being implemented in that particular sequence.

Many alternatives, modifications, and variations will be apparent to those skilled in the art in light of the above teachings. Of course, those skilled in the art readily recognize that there are numerous applications of the invention beyond those described herein. While the present invention has been described with reference to one or more particular embodiments, those skilled in the art recognize that many changes may be made thereto without departing from the scope of the present invention. It is therefore to be understood that within the scope of the appended claims and their equivalents, the invention may be practiced otherwise than as specifically described herein.