Multi-user Detection

1. Multi-user Detection Gwo-Ruey Lee

2. Outlines • Multiple Access Communication • Synchronous CDMA Model/ Asynchronous CDMA Model • Single-user Matched Filter • Optimum Multi-user Detection • Decorrelating Detector • Non-Decorrelating Linear Multi-user Detection • Decision-Driven Multi-user Detection

3. Multiple Access Communication • Several transmitters share a common channel, e.g., • mobile telephones transmitting to a base station • ground stations communicating with a satellite, ...

4. Multiple Access Communication • The receiver obtains the superposition of the signals sent by the active transmitters

5. Multiple Access Communication • Frequency Division Multiple Access (FDMA) • FDMA assigns a different carrier frequency to each user so that the resulting spectra so not overlap

6. Multiple Access Communication • Time Division Multiple Access (TDMA) • In TDMA, time is partitioned into slots assigned to each incoming digital stream in round-robin fashion. Synchronization is required.

7. Multiple Access Communication • Code Division Multiple Access (CDMA) • Users are assigned different signature waveforms. Each transmitter send its data stream by modulating its own signature waveform as in a single-user digital communication system.

8. Multiple Access Communication • Code Division Multiple Access (CDMA) • Direct Sequence Spread Spectrum (DS-SS)

9. Multiple Access Communication • Code Division Multiple Access (CDMA) • Frequency Hopping Spread Spectrum (FH-SS)

10. Multiple Access Communication • Near-far problem: • Any interferer that is sufficiently powerful receiver causes arbitrarily high performance degradation. • The objective of multi-user detection is: • the design and analysis of digital demodulation in the presence of multi-access interference (MAI).

11. Synchronous CDMA Model • Basic Synchronous CDMA Model where • is the inverse of the data rate. • is the deterministic signature waveform assigned to the k-th user. It is normalized such that • is the received amplitude of the k-th user's signal. • is the bit transimitted by the k-th user. • is the white Gaussian noise, which is uncorrelated with the transmitted signals, and has unit power spectral density.

12. s1(t) A1 b1 s2(t) A2 n(t) b2 y(t) . . . sK(t) AK bK Synchronous CDMA Model

13. Synchronous CDMA Model • The crosscorrelation of two signature waveforms, and , is • By Cauchy-Schwarz inequality, the crosscorrelation satisfies • The cross correlation matrix, defined by has diagonal elements equal to 1 [see (29) and (30)], and is symmetric nonnegative definite, i.e.,

14. Asynchronous CDMA Model • Basic Asynchronous CDMA Model • where are the time offsets that correspond to users • One special case happens when then asynchronous model reduces tosynchronous model • Another special case happens when and (a single user undergoes multipaths), it becomes

15. CDMA Model • Direct-sequence spread spectrum • Direct-sequence waveforms where • is the chip waveform that satisfies and • N is the number of chips per bit N,

16. Single-user Matched Filter • Consider the synchronous CDMA model, where only a single user exist: • The signal listed above is passed through a linear filter, the output of which is then sampled at T

17. Single-user Matched Filter • One problem is: Find the linear filter h(t) that maximize the signal-to-noise ratio at the filter output Y , i.e., • By Cauchy-Schwarz inequality, we have

18. Single-user Matched Filter • The objective function satisfies , where the equality holds when • Notice that in this derivation, we did not invoke the fact that noise is Gaussian. • Note that is a Gaussian r.v. with zero-mean and unit variance.

19. Single-user Matched Filter • The probability of error, in determining from , is

20. Single-user Matched Filter • Single-user Matched Filter in Rayleigh Fading • single user model • Assuming that A and s(t) are given, we want to find the estimate of b, , that minimizes • The first and second terms on the RHS of above equation are irrelevant to b, and we can write the minimization problem as a maximization problem:

21. Single-user Matched Filter • The solution to

22. Matched Filter User 1 . . . Sync 1 y1 . . . Sync 2 Matched Filter User 2 y2 . . . Sync 3 y(t) Matched Filter User 3 y3 . . . . . . yK Matched Filter User K . . . Sync K Single-user Matched Filter • Discrete-time Synchronous Models • Multi-user detection commonly have a front-end, whose objective is to obtain a discrete-time process from the received continuous-time waveform y(t). • Matched filter outputs

23. Single-user Matched Filter • In the synchronous case, the outputs of the bank of matched filters are

24. Single-user Matched Filter • The vector form of above equation is where and n is a zero-mean Gaussian random vector with covariance matrix equal to , i.e.,

25. Maximum A Posteriori (MAP) and Maximum Likelihood (ML) Detectors • The MAP-detector chooses the hypothesis that maximizes the a posteriori probability, and achieves the minimum probability of error. • The ML-detector chooses the hypothesis that maximizes the likelihood function, it achieves the minimum probability of error, when the hypotheses are equally probable (P0 = P1).

26. Maximum Likelihood (ML) Detectors

27. Maximum Likelihood (ML) Detectors

28. Maximum Likelihood (ML) Detectors

29. Maximum Likelihood (ML) Detectors

30. Maximum Likelihood (ML) Detectors • Are they the same?

31. Individual Optimum ML-Detector

32. Individual Optimum ML-Detector

33. Individual Optimum ML-Detector

34. Joint and Individual Optimum ML-Detector for the K-User Scenario • Recall the discrete-time synchronous CDMA model that • The joint optimum ML-detector is the solution to

35. Joint and Individual Optimum ML-Detector for the K-User Scenario • The maximization problem is a combinatorial optimization one, which means that the set of possible arguments comprises a finite set. • Combinatorial optimization problems can always be solved by exhaustive search, i.e., we evaluate the objective function at all possible arguments, and select our detected value to be the argument that produces the maximum. • Joint optimum decisions would be preferable to minimum bit-error-rate decisions due to their complexity.

36. Decorrelation Detector • Recall that the output vector of the bank of K matched filters is • Assume that R is invertible. • Premultiplying by give • In the absence of noise n, the k-th component of is • The decorrelating detector detects through

37. R-1 Matched Filter User 1 . . . Sync 1 . . . Sync 2 Matched Filter User 2 . . . Sync 3 y(t) Matched Filter User 3 . . . . . . Matched Filter User K . . . Sync K Decorrelation Detector

38. Decorrelation Detector • Note that the decorrelating detector is influenced by additive noise, and not by other interferers ( ). • Two features of the decorrelating detectors are • 1. It does not need to know the received amplitudes ( ). • 2. Detection of each user can be implemented independently. • Note that

39. Decorrelation Detector • From the fact that • We know that is orthogonal to any linear combination of . • If is linearly independent, we can find from for all k, and can have the modified decorrelating detector.

40. Matched Filter . . . Sync 1 . . . Sync 2 Matched Filter . . . Sync 3 y(t) Matched Filter . . . . . . Matched Filter . . . Sync K Decorrelation Detector • Modified decorrelating detector

41. Decorrelation Detector • In the two user scenario,

42. Decorrelating Detector and ML-Criterion

43. Non-Decorrelating Detector - LMMSE

44. [R+2A-2]-1 Matched Filter User 1 . . . Sync 1 . . . Sync 2 Matched Filter User 2 . . . Sync 3 y(t) Matched Filter User 3 . . . . . . Matched Filter User K . . . Sync K Non-Decorrelating Detector - LMMSE

45. Properties of the LMMSE Detector

46. LMMSE Detector for the Bank of Orthonormalized Filters

47. LMMSE Detector Maximizes Signal to Interference Ratio

48. Minimum Output Energy Detector

49. Successive Cancellation

50. Successive Cancellation