DSSS Multiple Access Channel (Cont.)

DSSS Multiple Access Channel (Cont.)
Objectives Understand the Concept of Multiuser Detector (MUD) Understand the Impact of Suboptimal Channel on Spectral efficiency Outline MUD Concept and Examples of MUD detectors Spectral efficiency with Random-Spreading and MUD detectors Lecture 10: Sepectral efficiency Adv. Wireless Comm. Systems DSSS -

Adv. Wireless Comm. Systems - DSSS -
Spectral Bit Rate in More Realistic Conditions Shannon’s channel coding theorem in (9.10) provides maximum spectral bit rate as a function of SNR (per bit) in ideal conditions: Gaussian channel Single user (i.e., no spreading) Unlimited bandwidth ( ) Optimal coding We want to get bounds in more realistic conditions as function of: The spreading sequence, The chip rate, The number of users, K Modulation, detection & coding Finite bandwidth, SNR Such bounds were obtained in the following paper and references there S. Verdu and S. Shamai, Spectral Efficiency of CDMA with Random Spreading, IEEE Trans. On Information Theory, Vol. 45, No. 2, pp , 1999. Lecture 10: Sepectral efficiency Adv. Wireless Comm. Systems DSSS -

Spectral Bit Rate in More Realistic Conditions (cont.) The paper considers several multiuser detectors The Optimal multiuser detector A bank of single-user matched filters (with a large number of users) A multiuser Decorrelator detector The Minimum Mean Square Error (MMSE) multiuser detector followed by autonomous single-user decoders Lecture 10: Sepectral efficiency Adv. Wireless Comm. Systems DSSS -

Receivers for AWGN Channels
A receiver is usually subdivided the into a demodulator and a detector The demodulator converts the received waveform r(t) into a K-dimensional vector , where K is the dimension of the transmitted signal waveform (users in our spreading case) The detector decides which of all possible signal waveforms was transmitted based upon observation of vector r Single or Multi User Detector User decoder User decoder Demodulator Channel gain Lecture 10: Sepectral efficiency Adv. Wireless Comm. Systems DSSS -

Multiuser Detection (MUD)
Multiuser detection considers all users as signals for each other joint detection Reduced interference leads to capacity increase Alleviates the “near-far” problem MUD can be implemented in the BS or mobile, or both In a cellular system, base station (BS) has knowledge of all the chip sequences Size and weight requirement for BS is not stringent Therefore MUD is currently being envisioned for the uplink (mobile to BS) Lecture 10: Sepectral efficiency Adv. Wireless Comm. Systems DSSS -

Concept of MUD Consider the following system model Baseband signal for the kth user with PAM and no multi-path is: T is the symbol interval xk(i) is the ith input symbol of the kth user gk(i) is the real, positive channel gain ak(t) is the signature waveform containing the PN sequence k is the transmission delay; for synchronous CDMA, k=0 for all users Received signal at baseband K number of users z(t) is the AWGN - Spreading Attenuation ISI (10.1) Lecture 10: Sepectral efficiency Adv. Wireless Comm. Systems DSSS -

Concept of MUD (cont.) For simplicity consider the synchronous case The output signal r(t) in (10.1) during the period does not depend on the input the user and other users sent during the past or future symbol periods Consequently we consider a snap-shot with an input vector Lecture 10: Sepectral efficiency Adv. Wireless Comm. Systems DSSS -

Concept of MUD – Single User Matched Filter
Sampled symbol output of the kth user with a matched filter is: A receiver that treats each individually and regards MAI as noise, although interferences are cross-correlated, is sub-optimal. That is, is not a sufficient statistic for recovering the kth user data. It is customary (as we did earlier) to approximate the MAI as independent Gaussian sequences In fact it is not Gaussian as the cross-correlation is known at receiver (they are PN sequences) and the other user signals need not be Gaussian Despreading + Sampling via single user Matched Filter user k data MAI Independent Gaussian noise Lecture 10: Sepectral efficiency Adv. Wireless Comm. Systems DSSS -

Concept of MUD – Single User Matched Filter (cont.)
Thus achieving the maximum capacity just with a bank of single user matched filters requires the knowledge of the cross-correlations and input distributions of all other interferers – knowledge of the latter not practical! However, it turns out that this information becomes useless as the number of users K grows unbounded and received power is equal. This is the matched filter MUD case considered in the paper. Lecture 10: Sepectral efficiency Adv. Wireless Comm. Systems DSSS -

Concept of MUD – Utilizing Cross Correlation
Recall - sampled output of the kth user matched filter is: How can we do better than a single user matched filter? Consider the two-user case (K=2) with BPSK modulation and let Despreading + Sampling via single user Matched Filter user k data MAI colored Gaussian noise Cross Correlation Lecture 10: Sepectral efficiency Adv. Wireless Comm. Systems DSSS -

Concept of MUD – Utilizing Cross Correlation (cont.)
Outputs of the matched filters are: Detected symbol for user k: If user 1 is much stronger than user 2 (the near-far problem), the MAI term r g1x1 present in the signal of user 2 is very large We therefore can use Successive Interference Cancellation Decision is first made for the stronger user 1: Then, subtract the estimate of MAI from the signal of user 2: All MAI is subtracted from user 2 signal provided estimate is correct MAI is reduced and near-far problem is alleviated (10.2) need to estimate channel gain (10.3) Lecture 10: Sepectral efficiency Adv. Wireless Comm. Systems DSSS -

MUD Algorithms Maximum-likelihood sequence estimation (MLSE) is the optimal detector (Verdú, 1984) that achieve maximum capacity of the channel given in (10.1) Lecture 10: Sepectral efficiency Adv. Wireless Comm. Systems DSSS -

The Optimal MUD Algorithm
x1+I1 MF 1 Viterbi Algorithm r1(t)+r2(t)+r3(t) x2+I2 MF 2 Searches for ML bit sequence x3+I3 MF 3 MLSE is done by using the Viterbi algorithm - a dynamic program with states Complexity is too high and for the async case it extends over more than one symbol duration Next we introduce the Decorrelator MUD Lecture 10: Sepectral efficiency Adv. Wireless Comm. Systems DSSS -

The Cross-Correlation Function
Matrix representation of the received signal : where r = [r1,r2,…,rK]T, R and W are KxK matrices Components of are given by cross-correlations between signature waveforms ak(t) within a symbol period W is diagonal with component Wk,k given by the channel gain gk of the kth user z is a Colored Gaussian noise vector For random spreading where with equal prob., and i.i.d for all user k and chip n Where are i.i.d. r.v. taking values 1 and –1 equally likely Lecture 10: Sepectral efficiency Adv. Wireless Comm. Systems DSSS -

The Cross-Correlation Function (cont.)
For every is binomially distributed, namely, For every From DeMoivre-Laplace central limit theorem The distribution is used in the paper (see latter) to compute the average maximum capacity With the moments Lecture 10: Sepectral efficiency Adv. Wireless Comm. Systems DSSS -

The Decorrelator (Linear) MUD
Estimates x by inverting R When the signatures are linearly independent, R is invertible Eliminates MAI interference Pros: Does not require knowledge of users’ powers Cons: Noise enhancement Element-wise we get Recovered data Lecture 10: Sepectral efficiency Adv. Wireless Comm. Systems DSSS -

The Linear MMSE MUD Its objective is to minimize the mean square error between original and decoded data – therefore must accounts for the noise too. That is, For equal received SNR (defined later) of all users, MMSE estimates x by inverting the matrix , namely It doesn’t eliminate MAI but it maximizes each user’s SNR, given by The mean square error for user k is given by Lecture 10: Sepectral efficiency Adv. Wireless Comm. Systems DSSS -

Multistage Detector (Non-Linear)
A Two User Example Decisions produced by 1st stage are (see (10.2)-(10.3)): By 2nd stage: and so on… need to estimate channel gain r Lecture 10: Sepectral efficiency Adv. Wireless Comm. Systems DSSS -

Spectral Bit Rate with Random Sequence Spreading for Various MUD Receivers Lecture 10: Sepectral efficiency Adv. Wireless Comm. Systems DSSS -

Spectral Bit Rate – Normalized SNR The following relates between SNR in terms of the “user transmitted energy per symbol ” (during G transmitted chips), , and SNR in terms of “energy per bit ”, as a function of K, G & total # bits/chip, (10.4) processing gain total # of bits/chip that can be transmitted arbitrarily reliably by K users one-sided Gaussian noise spectral level ( ) symbol period chip period Lecture 10: Sepectral efficiency Adv. Wireless Comm. Systems DSSS -

Spectral Bit Rates with K Synchronized Users K users with no-spreading, optimal coding and unlimited W Absolutely best MAC (Shannon) Sync DSSS with identical signatures to all K users and unlimited W Orthogonal DSSS with users and unlimited W Lecture 10: Sepectral efficiency Adv. Wireless Comm. Systems DSSS -

Spectral Bit Rates with K Synchronized Users (cont.) For we obtain and Exercise 10.1: Prove the equality above. For K=G, orthogonal signatures incurs no loss relative to best MAC Can be shown that for K>G, there are spreading codes with best MAC In spite of their overlap in time and frequencies, the K users can completely be separated at the RCV with matched filters or a Decorrelator K>G, asynchronous users and distorted channels destroys orthogonality Optimal spectral bit rate for non-orthogonal signatures requires joint processing and decoding of all users. Lecture 10: Sepectral efficiency Adv. Wireless Comm. Systems DSSS -

Spectral Bit Rates with K Synchronized Users (cont.) A complexity-performance tradeoff is a front-end multiuser detector such: Bank of single-user matched filters Decorrelator Minimum Mean Square Error (MMSE) demodulator followed by autonomous single-user decoders Lecture 10: Sepectral efficiency Adv. Wireless Comm. Systems DSSS -

Spectral Bit Rates with K Users and Random Spreading The DSSS with PN sequences (as IS-95) is nicely modeled by signatures are chosen equally likely and independent for all Since the signatures are r.v.’s, the spectral bit rate is also a r.v. Averaging it with respect to the random sequence choices provides a lower bound to the optimal deterministic choice (from convexity) A ratio that plays a key role is that we saw in All spectral bit rates are almost surely limits where and is held fixed (10.4) Lecture 10: Sepectral efficiency Adv. Wireless Comm. Systems DSSS -

Random Spreading with Optimal Decoder Spectral bit rate with optimal MUD where When or , the loss due to random choice of signatures (as opposed to ) vanishes for large K. Maximum loss of 50% occurs for K=N and Lecture 10: Sepectral efficiency Adv. Wireless Comm. Systems DSSS -

Random Spreading with Sub-optimal Decoders Spectral bit rate with a bank of single-user matched filters (MF) The maximum is achieved when where with Unless is relatively low and is high, random signatures with “MF” incurs substantial losses with respect to E.g., For K=G, is at most 1/3 of Lecture 10: Sepectral efficiency Adv. Wireless Comm. Systems DSSS -

Random Spreading with Sub-optimal Decoders (cont.) Spectral bit rate with a Decorrelator MUD for This yields (10.5) That is, is a fraction of with a fraction SNR/bit Exercise 10.2: Prove equation (10.5). (Hint: express both spectral bit rates as a fixed-point solution for some function, i.e., x=f(x). ) Lecture 10: Sepectral efficiency Adv. Wireless Comm. Systems DSSS -

Random Spreading with Sub-optimal Decoders (cont.) Spectral bit rate with an MMSE MUD The right hand-side is as for but without the last 2 terms It can be shown that for , the loss of with respect to grows without bound with Namely, unlimited loss due to linear processing at the RCV (MMSE Multiuser detector followed by a single-use decoding) compared with joint detection and decoding of all users, i.e., optimal Lecture 10: Sepectral efficiency Adv. Wireless Comm. Systems DSSS -

Spectral Bit Rates - Explained A multiuser channel with K users and spreading sequence of G chips per symbol (codeword) transmission can be viewed in two ways: G narrow baseband channels (one per chip) used by K users A single channel with a composite transmission of K waveforms depending on the detector Recall the basic Shannon capacity bits/symbol - Received signal bandwidth (Hz) - Average received symbol power (watts/symbol) - Noise power density (across channel bandwidth units) For the total bandwidth used by the channel, W, satisfies (#chips/symbol - processing gain) (10.6) Using the notations of (total bits/chip) and (bit energy) we have (10.7) Lecture 10: Sepectral efficiency Adv. Wireless Comm. Systems DSSS -

Spectral Bit Rates - Explained (cont.) Orthogonal DSSS with users Equivalent to K independent narrow baseband channels, where the entire bandwidth W (~G) is split into G baseband channels each channel uses a 1/W (~1/G) fraction of the entire bandwidth Using Shannon’s capacity with W = 1 for each user i, we get bits/symbol for user i For K independent users we get bits/symbol for K users Dividing by G and using , we get Substituting (10.7) we obtain the spectral bit rate fixed point equation Lecture 10: Sepectral efficiency Adv. Wireless Comm. Systems DSSS -

Spectral Bit Rates - Explained (cont.) DSSS with a Matched Filter per User Equivalent to K narrow baseband channels, where the entire bandwidth W (~G) is split into G baseband channels each channel uses a 1/W (~1/G) fraction of the entire bandwidth for , MAI appears as Gaussian noise with power density of on the single user baseband channel As for the orthogonal case, the total Shannon capacity of K users is bits/symbol for K users Dividing by G and using , we get Substituting (10.7) we obtain the spectral bit rate fixed point equation Lecture 10: Sepectral efficiency Adv. Wireless Comm. Systems DSSS -

Spectral Bit Rates - Explained (cont.) DSSS with a Decorrelator and Users Equivalent to K narrow baseband channels, where the entire bandwidth W (~G) is split into G baseband channels each channel uses a 1/W (~1/G) fraction of the entire bandwidth MAI is entirely eliminated Gaussian noise on each user channel i is enhanced by As for the orthogonal case, the total Shannon capacity of K users is bits/symbol for K users Dividing by G and using and the fact that Substituting (10.7) we obtain the spectral bit rate fixed point equation Lecture 10: Sepectral efficiency Adv. Wireless Comm. Systems DSSS -

Optimum Coding-Spreading Tradeoff Large-K spectral bit rate with The optimal spreading gain G, heavily depends on the RCV type For Optimal and “MF” it is better to let Thus, coding is preferred over spreading (more users compared with gain) Lecture 10: Sepectral efficiency Adv. Wireless Comm. Systems DSSS -

Optimum K/G for Large K Optimum K/G for large-K as function of 0.75 4 The optimum K/G for optimal and “MF” is obtained when The optimum for the Decorrelator varies from 0 to 1 The optimum for the MMSE is very large for low and reaches the minimum of 0.75 at Lecture 10: Sepectral efficiency Adv. Wireless Comm. Systems DSSS -

Spectral Bit Rate with Optimum K/G Large-K spectral bit rate with optimum K/G The Optimal and “MF” behaves very differently as The Optimal spectral bit rate grows unlimited with , whereas the “MF” approaches 0.72 bits/chip Lecture 10: Sepectral efficiency Adv. Wireless Comm. Systems DSSS -

Spectral Bit Rate with Optimum K/G Large-K spectral bit rate with optimum K/G 5.2 Decorrelator gets better than “MF” for and grows to The MMSE and the Decorrelator get similar for Lecture 10: Sepectral efficiency Adv. Wireless Comm. Systems DSSS -

Spectral Bit Rate - Optimal Random vs. Orthogonal Optimum with orthogonal and Random with large K=G The slopes of both with respect to the log of are asymptotically equal. However, Lecture 10: Sepectral efficiency Adv. Wireless Comm. Systems DSSS -

Spectral Bit Rate – Optimal Random vs. Orthogonal (cont.) G=3K or K=3G G=2K or K=2G G=K Ratio between random and orthogonal spreading for various K and G 0.5 -1.6 Random sequences are asymptotically optimal (optimal decoding) for Large but fixed and Fixed and Lecture 10: Sepectral efficiency Adv. Wireless Comm. Systems DSSS -

Spectral Bit Rate - “MF” Random vs. Orthogonal Ratio between MF random and orthogonal spreading for various K and G G=2K K=2G G=K 0.333 -1.6 This is monotonically decreasing with At the best the ratio is 0.333 Lecture 10: Sepectral efficiency Adv. Wireless Comm. Systems DSSS -

Spectral Bit Rate - Decorrelator Random vs. Orthogonal Ratio between Decorrelator random and orthogonal spreading for various K and G G=3K G=2K 0.75G=K 0.9G=K This is monotonically increasing with For every it becomes zero for too low Lecture 10: Sepectral efficiency Adv. Wireless Comm. Systems DSSS - -1.6

Spectral Bit Rate - MMSE Random vs. Orthogonal Ratio between MMSE random and orthogonal spreading for various K and G G=2K G=2K G=K 3G=K For K=G, MMSE is about 40% of the orthogonal For K>G, it suffers substantial losses for high At low , MMSE and orth behave similarly (between ) Lecture 10: Sepectral efficiency Adv. Wireless Comm. Systems DSSS - -1.6

DSSS Multiple Access Channel (Cont.)

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