Code-Banding of Spectrally Phase Encoded OCDMA based on Walsh

R. Menendez, et al., “ Code-Grouping of Spectrally Phase Encoded OCDMA Based on Walsh-Decomposition of Hadamard Codes,” IEEE ICC 2006

Code-Grouping of Spectrally Phase Encoded OCDMA based on Walsh-Decomposition of Hadamard Codes R. Menendez, S. Galli, P. Toliver, J. Jackel, S. Etemad Telcordia Technologies, Applied Research, One Telcordia Dr., Piscataway, New Jersey 08854, USA Abstract( 1 ) –We propose a novel optical networking concept enabling the routing of groups of spectrally phase encoded (SPE) optical code division multiple access (OCDMA) signals. This code grouping is based on particular properties of the Hadamard sequences, i.e. the existence of a multiplicative basis based on Walsh sequences. We describe here how this code grouping concept permits groups of SPE codes to be passively "labeled" and routed as groups on the basis of those labels and provide simulation results demonstrating the process. The ability to deal with groups of codes has important implications for routing and for the overall signal obscurity provided by OCDMA.

I. INTRODUCTION Recently, there has been a renewed interest in OCDMA due to its potential for offering increased levels of security at ultra-high data rates as well as simplifying key networking functions such as passive all-optical code translation (CT), routing based on code assignment, and physical layer code scrambling . Although OCDMA operates at the physical layer in many ways [1]-[6], the most common form of OCDMA network operates in a broadcast-and-select configuration in which communication is established between matching encoders and decoders (Figure 1, top). Signals from all encoders are broadcast to every decoder and the desired signal is discriminated from the other users (interferers) on the basis of their differing code signatures. To establish arbitrary connectivity amongst all users, the encoders and/or decoders at the edges of the network must be tunable. However, since these networks operate in a broadcast-and-select mode, tunable decoders make eavesdropping on a given transmission relatively easy. At the same time, tunable encoders with fixed decoders obviate simple multicast operation and make it possible for two transmitters to attempt to send to the same decoder simultaneously, thus resulting in code collision and data loss. As shown in the lower part of Figure 1, the addition of an appropriate CT stage at the midpoint of a conventional OCDMA network can passively route communication between mismatched encoders and decoders. In a sense, CT can be viewed as analogous to wavelength translation in a WDM-based network with the key difference that CT can be accomplished with a passive encoding device while wavelength translations are active in nature. Strictly speaking, the CT stage converts the desired code to the code format required at the receiver but it does not block the second code from arriving at that receiver; instead the (1)

This material is based upon work supported by DARPA under Contract No.MDA972-03-C-0078. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of DARPA.

second code is converted into another code mismatched to that receiver. Unencoded message 1

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Figure 1. A conventional OCDMA configuration is shown at the top in which messages are exchanged between matched encoder/decoder pairs. The addition of an appropriate intermediate CT stage, as shown at bottom, can interchange the recipients of the two input signals.

In prior work, CT and routing of OCDMA signals focused on dealing with one code at a time. Here we describe how sets of codes can be assigned to groups and labeled such that these groups can be passive routed as a composite entity on the basis of these labels. In a sense, code-grouping can be viewed as analogous to wavebanding in a WDM-based network with the key difference that here the labeling and CT can be accomplished with passive encoding devices and can enhance the overall obscurity of the signals. In the following, we first briefly introduce the SPE OCDMA system concept and, then, we describe how the Walsh basis underlying Hadamard codes can be used for code grouping. We next describe optical networks based on “group routing,” provide simulation results as a proof of concept and explore the implications of these code groups in the context of signal obscurity. II. DESCRIPTION OF SPE OCDMA SYSTEM In a SPE-OCDMA system, the optical source is a MLL which produces a comb of phase-locked lines. In our experiments, we have used a set of 16 lines spaced at 5 GHz occupying a total optical bandwidth of 80 GHz. The lines are the longitudinal modes of the MLL and are separated by a frequency interval equal to the temporal pulse repetition rate.


Their phase-locked addition generates the mode-locked pulse train at a repetition rate with a pulse width inversely proportional to the total frequency window. The electric field E(t) output of our MLL is well represented as a set of N equi-amplitude phase-locked laser lines:

E (t ) =

1 N j (2πf n t ) ∑e N n =1

(1)

where fn = f0 + (n-1)∆f are equally spaced frequencies and f0 ~193THz. Signal E(t) is a periodic signal constituted of a train of pulses spaced at a period T = 1/∆f seconds and each pulse has a width approximately equal to 1/(N∆f) seconds. In (1), E(t) is normalized to 1 at its peaks. The pulse train is “spectrally phase-encoded” by applying a spectral phase mask that multiplies each of its N frequencies by a phase function that effectively multiplies the field amplitudes of the individual phase-locked frequencies by [+1,1], i.e. changes their relative phases by [0,π]. Here we use the well-known orthogonal Hadamard matrices HN to determine the sequence of 1’s and –1s across the frequency spectrum and we refer to the columns (or rows) of these matrices as Hadamard codes. In general, all of the elements of Hadamard matrices are complex roots of unity and HN satisfies the following condition H N H ′N = NI N (2) where ′ indicates the Hilbert operator, and HN and IN represent the Hadamard and identity matrices of order N, respectively. An important subset of the Hadamard matrices (the Sylvester type) can be generated recursively as follows:

⎡H H2N = ⎢ N ⎣H N

HN ⎤ − H N ⎥⎦

(3)

where the recursion starts posing H1=1. In the absence of data modulation, the effects of encoding with code j and decoding with code i can be expressed as (here we represent element (i,j) of HN as Hi,j) E (t ) =

1 i 2πf 0 t N i 2πn∆ft ∑ H n,i H n, j e e N n =1

(4)

As a consequence of (2), we can write:

1 N ∑ H n,i H n, j = δ i , j N n =1 where δi,j =1 iff i=j, 0 otherwise If i=j (matched encoder and decoder), (4) reduces to (1) and again |E(t)|=1 at integer multiples of the period T=1/∆f. When i≠j (mismatched encoder and decoder), |E(t)|=0 at integer multiples of the period T. The effect of the spectral phase coding is that matched codes recover the original MLL pulses while mismatched coders result in a null at the time of the peak of the desired signal. With appropriate synchronization between SPE transmitters, a receiver can discriminate its matching coded signal from the (N-1) other signals by sampling the decoded signal at integer multiples of T where the desired signal is maximized and the interfering signals are nulled. In practice, an optical time gate

(OTG) is used to temporally select a narrow time window at integer multiples of T. Essentially the same process operates when all transmitters are sending signals that are data modulated. Cascaded application of two such coders is equivalent to element-by-element multiplication of their respective phase codes to yield the output code [5]. The interesting observation is that the set of Hadamard codes is closed under such element-by-element multiplication: cascading two Hadamard codes results in another code in the set. Thus, a signal encoded with code i, and then passed through a code-j coder emerges in code k, where code k is another element of the Hadamard set. One consequence of the closure of the Hadamard codes is that any of the Hadamard codes can be passively converted to any other code in the set by means of passing it through an appropriately chosen phase coder. As shown in the lower part of Figure 1, the addition of an appropriate code translation stage in a conventional OCDMA network can passively route communication between mismatched encoders and decoder. Experimental demonstration of cascaded passive optical CT of SPE signals is found in [2], [6]. In the latter, an SPE OCDMA signal propagated through a cascade of four different coders while incurring a power penalty of about 1-2 dB per stage. III. WALSH DECOMPOSITION OF HADAMARD CODES The Walsh codes Wm of length N (m = 1, 2,…, M=log2(N)) constitute a sub-set of the Hadamard sequences of length N. In particular, Walsh codes are the Hadamard sequences extracted from the (power-of-two -1) columns of HN. Thus for N=32, there are 5 Walsh codes Wk (k=2l, l=1,2,…, log2(32)) and they correspond to the (k+1)-th column of HN. In addition, the W0 code is the all-ones code, i.e. the first column of HN. As described in [7], the Sylvester-type Hadamard codes of length 2N can be represented as the Schur-Hadamard product of up to N Walsh codes (i.e., element-by-element multiplication); specifically the ith element of code n of Hadamard N: N

H n,i = W0 ∏ ⎛⎜W j −1 ⎞⎟ 2 ,i ⎠ j =1⎝

bj

(5)

where bj is the (j-1)th digit of the binary representation of (n-1) counting from the least significant bit of b. Thus, for n=6, (n-1)2 = b = 0101 and H6 = W4 ⊗ W1, where ⊗ denotes the Schur-Hadamard product and, for simplicity, we neglect W0 in the notation. Note that the Walsh codes form a proper subset of the Hadamard codes. It should be noted that many (but not all) other sub-groups of size log2(N) selected from the Hadamard codes also form a multiplicative basis that can span the complete set of Hadamard codes in sense of equation (5). All these spanning groups can be used as the basis for code grouping and we focus here on the Walsh subset for specificity. In the context of Walsh codes, the cascading of two Hadamard codes Ha and Hb can be determined from the


digit-by-digit XOR of the two binary numbers (a-1)2 and (b-1)2. The XOR operation yields a third binary number c2 which corresponds to the effective code Hc+1 represented by the cascade. Thus, for N=32, H6⊗H16 can be represented by (00101)2 XOR (01111)2 = (01010)11 or equivalently H3. III.a Applications of Walsh Decomposition As mentioned above, cascading SPE coders is equivalent to multiplication of their respective phase codes. Thus we can interpret the product function in (5) as a recipe for cascading Walsh coders to generate the larger set of Hadamard coders. We have identified two possible network applications of Walsh decomposition. First, a variable cascade of fixed Walsh coders, Figure 2, can yield a dynamically adjustable Hadamard coder. Depending on the settings of the five 2x2 optical crossbar switches, an incoming optical signal encounters or bypasses each of the four Walsh coders and can thereby any of 16 Hadamard codes can be realized. The reconfiguration timescale for this configuration is determined by the switching times for the optical crossbar switches (µsec to nsec). 2x2 crossbar switches

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Figure 2. A variable cascade of Walsh coders.

The focus of this paper is a second application; exploiting Walsh decomposition to create and route groups of codes. IV. CODE GROUPING To approach the notion of Hadamard code grouping, it is helpful to introduce a simpler notation for the Walsh product representation. Specifically, for N=32, the code H12 can be written: H12 = W16 W8 W4 W2 W1 = W8 W2 W1 where the underscored Walsh terms are understood to be absent from the product and the Schur-Hadamard product is implicit. In general, we represent the set of N Hadamard codes in terms of the presence or absence of the log2(N) different Walsh terms. From this viewpoint, the XOR operations described in Section III take the form of adding or dropping terms in the overall product according to simple rules for the addition or deletion of a Walsh term • Wa Wa = Wa Wa =1 (6a) • Wa Wa = Wa Wa= Wa (6b) We are now in a position to identify groups of Hadamard codes. In general, we are free to select any combination of Walsh terms and use them as a code group label. The remaining combinations of Walsh terms can then be thought of as identifiers for the members of each group. As a specific example, consider

Figure 2. In this example, we have chosen the first two Walsh terms, W1 and W2, as the labels for the code groups. We have therefore four different code labels and four different code groups. (W1W2, W1W2, W1W2, W1W2). Thus, all those codes that include both W1 and W2 terms are in the W1W2 group; codes that include W1 but not W2 terms are in the W1W2 group, and so on. 2N-2 :Number of available codes in each code group W1 W2 W4 W8 W16 …WN Code Group Label

Figure 3. One possible way of segmenting the space of Walsh codes into group labels and group member identifiers; here the code group label size is two resulting in four possible code groups. The remaining Walsh terms serve as group member identifiers, that is, these terms differentiate the members of the groups. So, for this example of N=32 and selecting two Walsh terms as the group label, we arrive at a maximum of four groups with up to 8 codes each. We emphasize that the choice of which Walsh terms to use as labels appears to be entirely arbitrary at this point, that is, any pair of Walsh terms would seem to be as good as any other pair as group labels. However, there may be physical implementation-specific reasons for preferring some terms over others (see the discussion of “bin-edge effects” in [4]). As mentioned earlier, previous code-routing schemes considered what amounts to groups of one (code-by-code routing and CT). In light of the formulation described above, it is now clear that effectively the code group label was the entire code. Thus code-by-code routing can be viewed as a special case of this more general view. It is worth pointing out that there are intrinsically no CT operations occurring here that are not included in the code translation table provided in [6]. But owing to the complexity of the code translation operations, it is difficult to see this opportunity for code grouping in that approach. Viewing the Hadamard codes in terms of their Walsh decomposition makes this opportunity much more immediate. V. CODE-GROUP ROUTED NETWORKS An optical broadcast-and-select network as is typical in OCDMA is depicted in Figure 3, but in this case the network is configured to route groups of codes. In each of the boxes on the left (right) of the figure a different subset of the N Hadamard encoders (decoders) is grouped according to their code label. For N=32, up to 8 codes could be in each such group identified by a label size of 2. Within each box on the left, these encoded signals would be passively combined and sent to the star coupler. The outbound legs of the star coupler connect to matching groupings of decoders on the right. Variable coders adjustable to any of the four label states are positioned on the outbound legs of the star coupler. The


variable coders can, in effect, re-label any of the four incoming code groups so that it is compatible with its associated decoder group. At the same time, this re-labeling changes any of the other incoming code groups to a form mismatched to that same decoder group. The re-labeling proceeds according to the rules outlined in equations (6a,b). Note that the variable coders affect only the code label and not the group member identifier portions of the codes. More than one decoder group can detect a given signals from one encoder group (multicasting). H code group defined by W1W2

H code group defined by W1W2







Variable W1W2 coders Figure 4. An optical network configured to route groups of Hadamard codes according to their code label.

Although Figure 3 depicts the variable coders positioned near the star coupler at the center of the network, these coders could equivalently be positioned near the edges of the network. A centralized location puts control of the routing function in the hands of a centralized management system and thereby might prevent unwanted eavesdropping on some signal groups. Edge placement de-centralizes the control of the routing function but opens the door to possibly unwanted eavesdropping. Likewise, the variable coders used for routing/re-labeling could be placed on the inbound fibers on the other side of the star coupler As was pointed out earlier, in the network configuration in Figure 2, the encoders and decoders in each of the boxes on the left and right, respectively, are associated with a unique subset of the N Hadamard codes. However, as shown in Figure 4, it possible to physically separate the code labeling functions from the assembly of the code group. In each of the four encoder (decoder) groups on the left (right) an identical set of coders can be used. This set consists of all those codes which include none of the Walsh terms used in the group label—in short, those codes using some combination of the group member identifier set of Walsh codes. Once this set of codes is assembled, all of the codes can be simultaneously and passively “labeled” by passing them through one of the four “labeling” coders (shown in the dotted ellipse on the left of the figure). Each of these four labeling encoders is in one of the four possible code label states. In Figure 4, the labeling coders on the left are depicted as fixed coders, but here again there is latitude: the variable coder function could be on the inbound fibers and the fixed coder on the outbound fibers. And as is the case for code-by-code

broadcast-and-select networks, variable coders could be used on both the transmit and receive ends of the network. In short, these configuration look a great deal like the basic OCDMA configuration (top of Figure 1). Given these similarities, it is clear that these networks operate vary similarly to conventional OCDMA networks with the exception that here we are dealing with groups of codes. The next section addresses the importance of code groups in maintaining physical-layer obscurity of OCDMA signals. H code group without W1 or W2

H code group without W1 or W2



W1W2


W 1W 2



W1W2

W 1W 2

Fixed coders used for labeling


Variable W1W2 coders Figure 5. An optical network in which identical groups of codes on the left are differently labeled by a passive encoder and then routed to identical groups of decoders on the basis of those labels.

VI. SIMULATION RESULTS As a demonstration of the code-grouping concept, we simulated the temporal response the network shown in Figure 6. For the sake of simplicity, we consider two groups of codes with two codes each. In this simulation, the MLL (not shown in Figure 6) generates 16 spectral lines with a spacing of 5 GHz resulting in a pulse stream with 200 ps pulse spacing. The MLL signal is split four ways and set to four PR1 duobinary modulators each of which is driven by an independent pseudo-random NRZ data stream operating at 5 Gbps. The modulated pulse streams pass through the indicated series of Hadamard encoders where the spectral filtering properties of the encoders are closely matched to those of the physically realizable encoders described in [4]-[6]. The variable encoders in the configuration in Figure 6 are trivially set to the “bar” state such that both of the codes in encoder-group 1 (or 2) are decoded by the pair of matching decoders on the right. Following the rightmost decoders, four OTGs (not shown) each extracts a narrow time window centered at the proper sampling time in the received temporal signal. Since the focus here is on code groupings and cascaded code translations, the MLL, data modulators and OTGs are idealized in their performance, but the coder themselves reflect achievable performance with real components.

R. Menendez, et al., “ Code-Grouping of Spectrally Phase Encoded OCDMA Based on Walsh-Decomposition of Hadamard Codes,” IEEE ICC 2006 Group 1

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Figure 6. Two groups to two codes each in a network with both of the variable coders (on the right) set to their corresponding “bar” states (H11 and H13).

The four received RZ temporal optical signals emerging from the OTGs are shown in Figure 7 where optical intensity is plotted versus time and the four original NRZ PRBS data streams (Data 1-4) are also indicated. Note that a “1” in the NRZ data is indicated by the presence of an RZ optical pulse and a “0” is matched to a low-intensity signal at the sampling times. Intersymbol interference (ISI) arising from the multiple stages of encoding gives rise to the observed variations in the amplitudes of the “1” pulses and the residual optical signal present at the “0”s is attributed to the finite width of the OTG. Since these signals are simulated representations of the optical intensity versus time, these traces correspond to detection with an idealized noiseless receiver with very high bandwidth.

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t (ps) Figure 8. Temporal eye diagram for one of the four post-gate received optical signals. Normalized optical intensity is plotted modulo the data period in ps.

The ability to shift a group of transmitted codes from one receiver group to another is central to the notion of code groups. If the two variable encoders in Figure 6 are now set to their respective “cross” states (H11 is changed to H5 and H13 is changed to H3), decoder group 1 will be able to recover the signals sent by encoder group 2 and vice versa. While we are simulating a cross-connection between two code groups, recall that since the variable encoders can be changed independently and this is a broadcast-and-select network, both decoder groups can also be set to simultaneously receive one encoder group. Group 1

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Figure 7. The four simulated received temporal RZ waveforms in comparison with the original NRZ data. Here the two variable encoders are set to their “bar” states (H11 and H13, respectively).

Figure 9. The four simulated received temporal RZ waveforms in comparison with the original NRZ data. Here the two variable encoders are set to their “cross” states (H5 and H3, respectively).

One of the four simulated eye diagrams of the signal following the OTG is shown in Figure 8. The simulated post-gate eye diagrams of all four data signals are quite similar and are well represented by this example. The eye is open and the ISI and time-gate impairments mentioned earlier are evident.

Comparing Figure 7 and Figure 9, it is clear that data signals that were originally recovered at decoder group 1 are now recovered at decoder group 2 and vice versa. In addition, the eye diagrams associated with the four recovered temporal signals are substantially the same as that shown in Figure 8. This demonstrates that groups of codes (here groups of two)


can be routed to decoder groups by changing a single variable coder in the path. The behaviors described here are in accordance with the code-grouping concepts described in Section IV.

complement to use of inverse multiplexing to form a self-obscuring signal and would be particularly advantageous in these scenarios. In addition, this code routing approach is fully compatible with the code-scrambling defined in [6].

VII. SIGNAL OBSCURITY AND CODE GROUPS One of the reasons for interest in OCDMA is that it may provide physical-layer obscurity for high-bandwidth signals. Recent analysis [1] of the eavesdropping protection provided by SPE and other types of OCDMA point out that such systems are most vulnerable to eavesdropping when only a single code is present on a link (as would typically be case on the inbound links to a central star coupler). However, as shown in Figure 10 and as discussed in [6], one can envision a network scenario consisting of multiple secure islands within which Hadamard coding would be used for signal routing/addressing purposes, But before these signal groups exit the secure islands they would be scrambled by passing through a shared randomly-chosen scrambling stage. Note that such a network configuration, with two stages of cascaded splitters, would operate with a total of four encoders in cascade, much as in the experimental demonstration reported in [6].We emphasize that this architecture can satisfy the basic obscurity conditions in [1] by precluding eavesdropper access to a single code and is a natural match to the code grouping issues discussed here. Specifically, we must assume that the groups of codes leaving the secure island are always sent simultaneously, as would be the case if large bandwidth signals were being sent on several codes in parallel by inverse multiplexing. Finally, note that the code-labeling coders in Figure 5 are positioned at the same point in the network as the scrambling encoders in Figure 10. This would permit the sets of coders within the secure islands to be identical and to route signals on the basis of passively added code labels. Given the commutative nature of the Schur-Hadamard product, it is possible to integrate both the scrambling and labeling functions into a single combined coder.

Star Coupler Regular encoder (fixed) Scrambling encoder Figure 10: Secure islands with conventional Hadamard coders interconnected with dynamic “scrambling” encoders for increased data obscurity between islands. Compare with Figure 5. For clarity, the links in this figure hide the fact that there would be two fibers (inbound and outbound) on each link.

REFERENCES [1] [2]

[3]

[4]

VIII.

CONCLUSION

We describe a novel optical networking concept enabling the routing of groups of SPE OCDMA Hadamard-coded signals. The foundation of code grouping is the set of Walsh functions which form a multiplicative basis underlying the Hadamard codes. We have described how this code-grouping concept permits groups of SPE Hadamard codes to be passively "labeled" and routed as groups on the basis of those labels. Furthermore, it possible to physically separate the code labeling functions from the assembly of the code group and thereby use identical sets of coders within each group. We have presented simulation results that demonstrate the principle of code-grouping and routing of code groups. The ability to deal with groups of codes has important implications for the overall signal obscurity provided by OCDMA. In large measure, OCDMA signal obscurity rests on codes obscuring one another [1]. If several lower-data-rate codes are used to transmit one larger data rate signal (via inverse multiplexing), signal obscurity is enhanced at the physical layer because the group of codes forms a mutually self-obscuring set. Thus, the ability to perform "group routing" is a natural

[5] [6]

[7] [8]

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