Optimized PR-QMF Based Codes For Multiuser Communications Kenneth Hetling, Gary Saulnier, and Pankaj Das Electrical, Computer, and Systems Engineering Department Rensselaer Polytechnic Institute Troy, New ork 80-3590 hetling@crl.ecse.rpi.edu PRESENTED AT SPIE AEROSENCE '95 ABSTRACT In communications systems, the message signal is sometimes spread over a large bandwidth in order to realize performance gains in the presence of narrowband interference, multipath propagation, and multiuser interference. The extent to which performance is improved is highly dependent upon the spreading code implemented. Traditionally, the spreading codes have consisted of pseudo-noise (PN) sequences whose chip values are limited to bipolar values. Recently, however, alternatives to the PN sequences have been studied including wavelet based and PR-QMF based spreading codes. The spreading codes implemented are the basis functions of a particular wavelet transform or PR-QMF bank. Since the choice of available basis functions is much larger than that of PN sequences, it is hoped that better performance can be achieved by choosing a basis tailored to the system requirements mentioned above. In this paper, a design method is presented to construct a PR-QMF bank which will generate spreading codes optimized for operating in a multiuser interference environment. Objective functions are developed for the design criteria and a multivariable constrained optimization problem is employed to generate the coecients used in the lter bank. Once the lter bank is complete, the spreading codes are extracted and implemented in the spread spectrum system. System Bit Error Rate (BER) curves are generated from computer simulation for analysis. Curves are generated for both the single user and the CDMA environment and performance is compared to that attained using Gold Codes.. Multiuser Systems and CDMA INTRODUCTION The goal of a multiple access channel is to maximize the number of users while maintaining a certain despread signal to noise ratio such that acceptable performance is retained. This is accomplished by ensuring that the energy transmitted by the users causes minimal interference to others in the communications channel. In other
words, each of the transmissions should, ideally, be completely orthogonal to each other. One simple method of accomplishing this, known as Time Division Multiple Access (TDMA), requires that each user transmit during a separate time interval. Since these time intervals do not overlap, orthogonality between user transmissions is ensured. Another method applies the same principle in the frequency domain. The available spectrum is divided into non-overlapping frequency bands and each user is assigned a band. This is known as Frequency Division Multiple Access (FDMA). In each of these cases, strict orthogonality cannot be attained since nite length transmissions require innite bandwidth and, conversely, a band limited signal requires an innitely long transmission time. Therefore, in order to increase the orthogonality between users, dead zones are inserted. In the TDMA system, guard time is placed between the transmissions of dierent users where no transmitting is permitted. Likewise, in FDMA, guard bands are implemented between the user bands. A third and more complex method of allowing multiple users to share a given time-frequency space is called Code Division Multiple Access (CDMA). 3 In this method of multiuser communications, the data transmissions are allowed to occur at the same time and over the same frequency band. This is accomplished by modulating the signal with a psuedorandom code which spreads the bandwidth of the signal beyond that which would normally be required for transmission. This is known as Direct Sequence Spread Spectrum (DSSS) communications. 4 The receiver of a DSSS signal demodulates and despreads the signal by correlating it with its own reference of the spreading code. By assigning each user in the system a unique spreading code, all transmissions can occur simultaneously over the same available bandwidth provided that the cross correlation between the spreading codes is suciently low. Under such conditions, the energy generated by the other users will be uncorrelated with the implemented spreading code and, therefore, the interference will not signicantly eect the signal to noise ratio and the bit error rate (BER). The most common spreading sequences used for CDMA applications are Gold Codes. 5 Gold codes are binary valued wide-band pseudo-noise sequences generated using maximal-length shift registers. They are attractive for use in a CDMA system since, in addition to their ease of generation in hardware, the cross correlation between any two codes is low enough such that interference between users is small. Specically, for any positive integer not divisible by 4, there exists a set of n + Gold codes whose length is n?. The normalized cross correlation between any two pairs of these maximal length sequences (m-sequences) consists of one of three values given by 4? N t(n)? N where N is the length of the code and t(n) = ( [t(n)? ] N + (n+) for n odd + (n+) for n even In addition to the cross correlation properties, a DSSS system exhibits other attractive properties. The wideband nature of the CDMA signal provides protection against multipath interference, common to a mobile wireless communications environment. Also, DSSS, as well as other spread spectrum methods, provides some protection against jamming and interference. A measure of the protection provided by a particular spreading code is called the processing gain, G p, and can be roughly estimated by 6 G p = W R
where W is the bandwidth of the spread signal and R is the bandwidth of the data. The larger the processing gain, the more resilience the code displays towards jamming, however, more bandwidth is required. These features, along with the cross correlation properties mentioned above, make a CDMA system attractive for a multiuser channel.. A PR-QMF Based Spread Spectrum System The PR-QMF based spread spectrum system was proposed as an alternative to the more traditional spread spectrum technique mentioned above. 7 Whereas the m-sequences are limited to binary valued chips, in the proposed system, no restriction is placed upon the individual chip values. Instead constraints and objectives are formed for the properties which the spreading codes should obey. A PR-QMF tree is used to provide a framework for designing the codes. A block diagram of the spread spectrum system under consideration is shown in Figure. The blocks labeled analysis bank and synthesis bank correspond to the two halves of a perfect reconstruction quadrature mirror lter bank (PR-QMF) commonly used in image and audio sub-band coding. 8 A typical lter bank structure is shown in Figure. The lter bank depicted is a full binary tree which will decompose the input signal x(n) into four Source Component Select Synthesis Bank Mod PN Code Channel Sink Decision Component Extract Analyis Bank Demod Figure : Proposed Spread Spectrum System sub-bands. The choice of the lter bank structure, however, is not limited to the full binary tree and other tree structures, such as the dyadic or irregular tree, may be implemented. At the transmitter, spreading waveforms are generated using the synthesis lter bank to perform a transform on an input vector which contains the message. This vector consists of all zeros except for one component. The position of the non-zero component is determined by a PN code and its value is set equal to the message signal which, in this case, will be either a plus or a minus one. In general, more than one lter can be referenced and thus the spreading code for a particular bit could be the linear combination of the codes generated by those lters. The length of the spreading code (which is related to the chip rate) is determined by the structure of the lter bank and the number of tap weights in each lter. Subsequent message bits are hopped to dierent positions in the input vector, each producing a dierent spreading waveform with the polarity being determined by the data bit. The receiver is structured as a mirror image of the transmitter and performs the dual operation to extract the message bit. Once the signal is removed from the carrier, the corrupted waveform is processed by the analysis
H 0 (z) G 0 (z) H 0 (z) + G 0 (z) x(n) H (z) G (z) x(n) + H 0 (z) G 0 (z) H (z) + G (z) Analysis H (z) G (z) Synthesis Figure : Four-band tree structure lter bank. If the analysis and synthesis stages exhibit the perfect reconstruction (PR) property, then all of the energy associated with the message will be isolated to a single lter output. A synchronized receiver PN code can then reference this lter output and extract the message signal. The message signal is then passed to a threshold device for a decision on its value. The above system is easily adapted for use in a multiuser channel. Similar to a CDMA system using Gold codes, whereby each user is assigned a dierent code, multiuser operation using the PR-QMF spreading system is accomplished by assigning each user a unique lter by which they can modulate their data. There are two cases of the multiuser channel which must be considered. The rst is known as the synchronous channel. Under synchronous conditions, the user transmit times all coincide with the same clock. This is also known as Code Division Multiplexing. In this application, there is no user interference using the PR-QMF codes since, as will be shown, each of the generated codes will be orthogonal. In the asynchronous channel, no restriction is placed on the transmit start times of any user. Therefore, orthogonality and the absence of interference from other users can not be guaranteed. This is the more general case that is usually referred to when discussing CDMA applications. The objective, therefore, is to design the lter bank such that the orthogonality between the spreading codes is maximized with respect to all delays in transmission times. CODE DESIGN FOR MULTIUSER COMMUNICATIONS Since the resulting waveforms are the impulse responses of each branch of the synthesis tree, the tap weights used in the FIR lters will have a direct result on the spreading codes and, therefore, on the performance of the system. A related study focused on the criteria required for the resulting spreading code to provide protection against jamming and interference. 9 Additionally, a design methodology which uses the criteria was presented for determining the lter coecients. In this paper, the same methodology will be applied, however, the design criteria considered will focus on producing spreading codes suited for a multiuser channel. The rst of these criteria is orthogonality between the spreading codes. This is required so that, in the presence of additive white Gaussian noise (AWGN), the noise power is equally distributed among all receiver lter outputs and the system can achieve the performance of binary antipodal signaling. The second criteria is the perfect reconstruction (PR)
property and is necessary for all of the message energy to be isolated to a single lter output at the receiver. The nal condition is that the designed spreading codes be orthogonal to each other to the maximum extent possible both in the synchronous and asynchronous scenarios.. Orthogonality and Perfect Reconstruction The orthogonality of the waveforms and the PR requirement are closely related and are more easily analyzed by considering the simplest sub-band lter bank shown in Figure 3. The perfect reconstruction property states H 0 (z) G 0 (z) x(n) + x(n) H (z) G (z) Analysis Synthesis Figure 3: Two-channel sub-band lter bank that if no processing is done between the analysis and synthesis stages, then ^x(n) will simply be a delayed version of x(n) or ^x(n) = x(n? n 0 ) where n 0 is a positive integer. If h(n) is the impulse response of the FIR lter H 0 (z), the conditions for perfect reconstruction are 0 N? X k=0 H (z) = z?(n?) H 0 (?z? ) G 0 (z) =?H (?z) G (z) = H 0 (?z) h(k)h(k + n) = n 8 n 0 where N is the lter order of H 0 (z) and n is the unit impulse function. The rst condition ensures H 0 (z) and H (z) are quadrature mirror lters. The next two conditions are required to eliminate aliasing eects during the reconstruction. The nal restriction places N= conditions on the H 0 (z) lter coecients and ensures perfect reconstruction. The remaining N= conditions which determine the lter coecients are application specic and, in this paper, will be chosen to minimize cross correlations. It can be shown that the PR-QMF two band structures can be used as the building blocks for the larger trees while still maintaining the PR property. The above conditions also guarantee that the resulting waveforms are orthogonal. The two waveforms, g 0 (n) and g (n), will be generated by supplying unit vectors to the lters in the synthesis bank. The cross correlation between these two waves is R g0g = N? X n=0 g 0 (n)g (n)
= = 0 N? X n=0 (?) n h(n)h(n?? n) In addition to dividing the input noise power among the basis functions, the cross correlation property also makes the waveforms attractive for use in a code division multiplexing system. 7 The orthogonality property can also be extended to multiple stage lter banks.. Matrix Representation of PR-QMF Banks Since the orthogonality and PR constraints place only N= conditions on H 0 (z), the remaining conditions can be used to specify lters which minimize the cross correlations. To accomplish this, consider the simple two channel tree of Figure 3. Each of the decimation operations can be represented by a matrix operator which in the ideal case of an innitely long input has the form A = 6 4. h 0 h h h 3 h N? h 0 h h h 3 h N? h 0 h h h 3 h N? with zeros inserted into any blank entries. If the column vector x is the input to the decimation, the output can now be calculated as. y = Ax Since, however, the spreading codes are of nite length and the analysis lter bank will be operating on received signals of nite length, the matrix is modied to have the form of A = 6 6 4 h 0 h h h 3 h N? h 0 h h N?3 h N? h N? 3 7 5. h 0 h h h 3 h N? h N? h N? h 0 h h N?3 h h N? h N? h 0 h This form provides an approximation for the edge eects and still produces perfect reconstruction if the PR conditions are implemented. The actual size of the matrix is dependent upon the length of the input vector to be decimated. Let A be an operator matrix for the decimation process associated with an N tap FIR lter A(z). The matrix for the corresponding interpolation associated with the reconstruction lter A r (z) is simply the transpose of A or A T. Let the column vector x be an unit vector whose rst element is a one with all others being zero. If A(z) is used in the simple two band structure of Figure 3, then the corresponding spreading code generated by the synthesis bank is. c = A T x 3 7 7 5
In general, for a tree of depth L, the spreading code along a branch with lters A (z); A (z); : : :; A L (z), where A (z) is the rst stage lter is c = [ L l= (A l ) T ]x where the matrices A to A L are the decimation matrices, each of appropriate dimension, and the product is carried out in ascending order as follows L l= (A l ) T = (A ) T (A ) T : : :(A L ) T.3 Cross Correlations Between Spreading Codes Using the notation specied above, the cross correlations between the spreading codes can now be dened. If only two codes are being considered at one time, then there are two dierent polarity cases which must be considered as shown in Figure 4. The two codes under consideration are labeled c and c with the polarity of the User Data bit +c +c +c Interferer Data bits Case User Data bit +c +c -c Interferer Data bits Case Figure 4: Data Bit Polarities transmitted code, and therefore the message bit, indicated by the sign. The cross correlations of all other polarity combinations dier only in sign from one of these two cases and, since it is the cross correlation magnitude which is of interest, they can be ignored. For case one, all of the data bits are of positive polarity. As in the previous section let the codes be represented by column vectors c and c. For a given delay time of chips, the cross correlation between the two codes is r = (c ) T c where (c ) is a circular shift down of the vector c by positions. In a similar fashion, the cross correlations for case two can be represented by r? = (c )?T c where (c )? is also a circular shift down by positions, however, the rst positions after the shift have an opposite sign.
.4 Progressive Optimality and Filter Bank Design It has been shown that recursively using the same set of QMF lters in the design of hierarchical subband structures does not necessarily produce an optimal tree. The task of determining the cross correlations between all generated codes, however, quickly becomes computationally complex. Therefore, a technique known as Progressive Optimality will be used. 3 The method of progressive optimality states that at each node in the tree, a new set of lters should be designed which specically incorporates the criteria at that particular stage. By applying this methodology to the system under consideration, only the cross correlations between the codes generated at a particular node will be optimized. Cross correlations between those codes and other codes generated by the tree will not be considered. For example, if at a particular node in the tree, the QMF pair of A(z) and A 0 (z) are being designed, then the optimization and constraint criteria will be based solely on the two spreading codes generated by the tree using A(z) and A 0 (z) as the input lters. Although this clearly does not minimize the cross correlation between all possible spreading codes, it greatly reduces the computational eort required for optimization. The optimization problem at each node can now be dened. Consider a node in a subband tree which occurs at stage L. Let the lters along the branch leading to that node be designated by A (z); A (z); : : : ; A L? (z) and the QMF pair to be designed as A L (z) and A 0 (z). The codes generated by these two branches are L and L? c = [ (A l ) T ]A T x L l= L? c 0 = [ (A l ) T ]A 0 T L To obtain the cross correlations, the circular shifts (c) and (c)? are and l= L? (c) = (A ) [ (A l ) T ]A 0 Lx l= L? (c)? = (A )?[ (A l ) T ]A 0 Lx l= must be generated. In matrix notation these where (A ) is a circular shift down by rows and (A )? is a circular shift down by rows and the rst rows in the matrix after the shift have opposite signs. For a given delay of chips, the cross correlation between the two codes is r = (c ) T c l= L? L? = ((A ) T[ (A l ) T ]A T L x)t (A T[ (A l ) T ]A 0 T x) L = x T A L [ l=l? = a T L PT Q Pa 0 L l= L? (A l )][(A ) A T ][ (A l ) T ]A 0 T x L l=
where a L and a 0 L are vectors containing the lter coecients of A L (z) and A 0 (z) respectively and L L? P = [ (A l ) T ] l= Both the coecient vectors and the the matrix P are not dependent on the shift. The shift is represented by Q = (A ) A T A similar formulation can be made for r?, however, in this case let so that Q? = (A )? AT r? = at L PT Q? Pa0 L Now let r be a column vector which contains as its elements each of the cross correlations between the two codes relative to a delay of chips. Dene r? in a similar way. The optimization problem can now be stated as subject to N? X k=0 minff g = r T r + r?t r? a k a k+n = n 8 n 0 In the above optimization problem, it is the sum of the squares of the cross correlations which is to be minimized. It is desirable, however, to have all of the cross correlations to be approximately equal. To accomplish this, additional constraints can be dened using r and r? by requiring the magnitude of each element in the two vectors to be less than a given cross correlation. If jrj represents the magnitude of each element of r then the constraints can be formulated by jrj jr? j The vector is a vector whose elements are all equal to the maximum tolerable cross correlation magnitude between the two codes. The above constraints can also be used if a dierent objective function such as energy compaction is desired. 9 3 IMPLEMENTATION AND RESULTS The above optimization problem was applied to generate eight codes each of 3 chips. This required a lter bank of three stages with each lter having 8-taps. The non-linear optimization problem was solved using a variation of Sequential Quadratic Programming. In each case, the maximum tolerable cross correlation was decreased until a problem for which no solution existed was encountered. Once the lter tap weights were determined, the spreading codes were extracted as discussed above and implemented in a system simulation. The frequency responses of the resulting codes were generated and are shown in Figure 5. As can be seen in
.5 0.5 0 0 0.5.5.5 3 radians Figure 5: Frequency Response of the Generated Codes the gure, each of the codes tends to display narrowband characteristics. This is due to the specication of the system and optimization criteria. The goal of the optimization was to generate completely orthogonal codes. As mentioned in Section, this can be accomplished by separating the users in time or in frequency. The structure of the system precludes any orthogonality in time since no restriction is placed on the transmit time of any user. Therefore, when attempting to achieve the optimization criteria, the results tended towards codes which were orthogonal in frequency. The guard bands present in an FDMA system, however, are not implemented. As described above, the methodology used to design the codes employed a variation of the progressive optimality concept. Therefore, the cross correlations between each combination of codes was not optimized or constrained. To see the eect of this, each of these cross correlations were calculated for all time shifts and all combinations of possible code pairs. The results were normalized for a code with an autocorrelation of one. A histogram of the results is shown in Figure 6A. As expected, there is a large occurrence of zero cross correlations due to the orthogonality of the codes at certain time shifts. Additionally, the non-zero values are well grouped. Since constraints were not placed on all possible combinations, there are some outlying values, however, their frequency of occurrence is small. If desired, these combinations can be traced back through the tree and constrained. For comparison, the same calculation and plot was computed for Gold Codes. Eight codes were chosen from the set of Gold codes consisting of 3 chip codes and again the cross correlations were normalized. The results are shown in Figure 6B. Each of the cross correlations results in one of fourteen values. Therefore, they are plotted as spikes as opposed to a histogram. Earlier it was mentioned that the cross correlations between Gold codes would result in one of three values, however, this did not account for both polarity cases of Figure 4 and, therefore, the other values result. Although the range for the Gold codes is smaller than that for the PR-QMF codes, the distribution is not nearly as concentrated around zero. In fact, a calculation of the sample variance of the cross correlation results for the two dierent codes shows that the variance of the Gold codes is between two and three times that of the PR-QMF based codes.
Number of Occurences Number of Occurences 000 800 600 400 00 Plot A: Cross Correlation Distribution for PR-QMF Codes 0 - -0.5 0 0.5 Normalized Cross Correlation Plot B: Cross Correlation Distribution for Gold Codes 500 400 300 00 00 0 - -0.5 0 0.5 Normalized Cross Correlation Figure 6: Histograms of Cross Correlation Possibilities Computer simulation results with the codes were generated for both the four user and the eight user cases. All simulations were performed at baseband and it is assumed that the modulation and demodulation blocks, depicted in Figure, is transparent to system performance. The results are shown in Figure 7. In the four user channel, the results are very dependent upon which lter inputs the users are assigned, therefore, simulation results were generated for a number of dierent cases and averaged to determine an overall expectation of the performance. The same is true for the eight user channel. The multiuser results for 3 chip Gold Codes are also plotted for comparison. The lter bank based codes show improvement in performance, however, the codes generated are still not as desirable as hoped due to the narrowband frequency characteristics they exhibited. As mentioned above, this is due to the structure of the system precluding any orthogonality in the time domain. Therefore, in order to achieve the objective, the system tended towards an FDMA conguration. There are several ways to achieve the desired broad band communication waveforms. The simplest is to not assign only one lter or spreading code to each user. Instead, each user could be assigned a set of lters in the synthesis bank which they are allowed to use to generate their codes. Using the narrowband codes generated above, this type of system would be similar to a Frequency Hopped Spread Spectrum (FHSS) system. 4 For such a system, however, the design criteria stated above is incomplete. The circular shifts used when calculating the cross correlations are not always applicable since the code for the subsequent bit in Figure 4 will most likely be dierent. A second method of generating broadband spreading codes is to specify that property as either an objective or constraint in the optimization problem. The objective is to have a spreading code which exhibits a spectrally at
E- Prob. of Err. E- E-3 E-4 8 Users 4 Users - Gold Codes 4 Users User E-5-0 3 4 5 6 7 8 9 Signal to Noise Ratio (db) Figure 7: Multiuser Simulation Results frequency response. This requires that the autocorrelation of the code over all time shifts be an impulse function. If a is a vector of the lter coecients, one of the PR conditions already requires that The remainder of the objective can be given as a T a = minff g = r T a r a where r a is a vector of the autocorrelation sequence of the resulting spreading code. As with the cross correlations, it may be desirable to additionally constrain the autocorrelation sequence values 4 CONCLUSION Objective functions and constraints were developed which allowed the design of spreading codes within the framework of a PR-QMF spread spectrum system. These objectives and constraints were developed for the case of a multiuser channel. A design methodology, adapted from subband image coding was employed to generate the tap weights for the lters in the PR-QMF tree. The codes were then extracted and implemented in a computer simulation of the system. Results were generated and compared to a CDMA system using Gold codes. Comparisons were made using criteria of code bandwidth, cross correlation distribution, and BER. Although improved performance was attained by the newly generated codes, the narrowband nature of the codes was undesirable. Several possible solutions, including a modication of the objectives, were given and are the source of further research.
5 ACKNOWLEDGMENTS This work is partially funded by Rome Laboratory, Griss AFB, Rome, N under contract #F3060-94-C- 04. 6 REFERENCES [] S. Haykin. Digital Communications. John Wiley and Sons, Inc., 988. [] T. Ha. Digital Satellite Communications. McGraw-Hill, Inc., 990. [3] R. Dixon. Spread Spectrum Systems With Commercial Applications. John Wiley and Sons, Inc., 994. [4] R. Ziemer and R. Peterson. Digital Communications and Spread Spectrum Systems. Macmillan, Inc., 985. [5] R. Gold. Optimal binary sequences for spread spectrum multiplexing. IEEE Transactions on Information Theory, pages 69{6, October 967. [6] A. Viterbi. Spread spectrum communications: Myths and realities. IEEE Communications Magazine, 7(5), May 979. [7] K. Hetling, M. Medley, G. Saulnier, and P. Das. A PR-QMF (wavelet) based spread spectrum communications system. In The Proceedings of the IEEE Conference on Military Communications, pages 760{764, 994. [8] R. Crochiere and L. Rabiner. Multirate Digital Signal Processing. Prentice Hall, Inc., 983. [9] K. Hetling, G. Saulnier, and P. Das. Optimized lter design for PR-QMF based spread spectrum communications. In The Proceedings of the IEEE International Conference on Communications, pages 350{354, 995. [0] M. Smith and T. Barnwell. A procedure for designing exact reconstruction lter banks for tree-structured subband coders. In The Proceedings of the IEEE ICASSP, 984. [] A. Akansu and R. Haddad. Multiresolution Signal Decomposition. Academic Press, Inc., 99. [] M. Tazebay and A. Akansu. Progressive optimization of time-frequency localization in subband trees. In The Proceedings of the IEEE International Symposium on Time-Frequency and Time-Scale Analysis, pages 8{3, 994. [3] M. Tazebay and A. Akansu. Progressive optimality in hierarchical lter banks. In The Proceedings of IEEE International Conference on Image Processing, pages 85{89, 994.