Code Division Multiple Access (CDMA) phần 8 potx

C H A P T E R 5Multiuser Detection As discussed in previous chapters, the conventional matched filter receiver treats multiple accessinterference MAI, which is inherent in CDMA, as if it

0 50 100 150 200 250 300 350 400 0

0.05 0.1 0.15 0.2 0.25

Average loading (users)

Synchronous, fixed rate Synchronous, variable rate Asynchronous, fixed rate Asynchronous, variable rate

FIGURE 4.7: System throughput for FHMA system with no side information (N= 128, perfect codes).

wish to exploit the benefits of spread spectrum Such networks can be based on either directsequence or frequency hopping In such networks, a primary consideration is the assignment ofspreading waveforms, and we discussed the primary methods of code assignment Due to thepotential for severe near-far situations in ad hoc networks, frequency-hopped approaches aretypically more appropriate Thus, we focused on the network throughput for FHMA systems

126

C H A P T E R 5

Multiuser Detection

As discussed in previous chapters, the conventional matched filter receiver treats multiple accessinterference (MAI), which is inherent in CDMA, as if it were additive noise because, afterdespreading, the MAI tends toward a Gaussian distribution (see Chapter 2 for details) However,

we have also seen that this MAI is the limiting factor in the capacity of CDMA systems As

a result, capacities for single-cell CDMA systems can be substantially lower than those fororthogonal multiple access techniques such as TDMA or FDMA In addition, if one of thereceived signals is significantly stronger than the others, the stronger signal will substantiallydegrade the performance of the weaker signal in a conventional receiver due to the near-far problem Thus, CDMA performance can be greatly enhanced by receivers designed to

compensate for MAI Multiuser receivers (sometimes referred to as multiuser detection) is one

class of receivers that use the structure of MAI to improve link performance [55]

This chapter presents an overview of multiuser receivers and their usefulness for CDMA,particularly at the base station As detailed in Chapter 3, cellular or personal communicationsystem (PCS) design consists of two distinct problems: the design of the forward link (fromthe base station to the mobile) and the design of the reverse link (from the mobile to the basestation) The forward link can be designed so that all signals transmitted to the mobiles areorthogonal and all signals arrive at the mobile receiver with similar power levels Further, themobile receiver must be inexpensive and have low power requirements The reverse channel ispotentially more harsh but can support a more sophisticated receiver User signals arrive at thebase station receiver asynchronously and can have significantly different energies, resulting inthe near-far problem In contrast to the mobile receiver, the base station receiver can be largerand more complex, have higher power consumption, and use information available about theinterfering signals We focus on this latter situation because it is more feasible that the receivercan simultaneously detect signals from all users (i.e., implement multiuser detection)

To facilitate the discussion of the multiuser receiver structures presented in this chapter, werestate the model of DS-CDMA from Chapter 2 The received signal on the uplink can be

represented as

r (t)=K

k=1

where K users are independently transmitting bi-phase modulated signals in an AWGN

chan-nel,τ k is the delay of the kth user, n(t) is a bandpass Gaussian noise process with double-sided power spectral density N0/2, and

where Pk andθ k are the received power and phase of the kth user’s signal, respectively, bk(t) is the data waveform, and ak (t) is the spreading waveform with spreading gain N = T b /T c Asdiscussed in Chapter 2, the uplink of a CDMA system is generally asynchronous However,for ease of discussion, we will assume that signals are received synchronously (τ1= τ2= · · · =

τ K = 0; θ1= θ2= · · · = θk = 0) with random spreading codes We arbitrarily examine the

output of a filter matched to the kth user’s spreading waveform during the first bit interval

(assuming perfect carrier and PN code phase tracking) Assuming square symbol pulses, thecorrelator version of the matched filter is simply the integral of the received signal multiplied by

the spreading code of interest, ak (t) and a phase-synchronous carrier cos( ω c t) over the symbol

If this is repeated for each of K users, we can represent the set of matched filter outputs in

vector notation as,

where y= [y1, y2, , y K]T and R is a K × K matrix that represents the correlation between

spreading waveforms during the first bit interval Thus, ifρ j,k are the elements of R,

b= [b1, b2, , b K ] are the data bits from each of the K signals, and n = [n1, n2, , n K]T

is a vector of Gaussian noise samples with zero mean and covariance matrix n = σ2R.

σ2= No /4T bis the noise power after dispreading Decisions are then made as

b = sgny

(5.6)

where the function sgn(x) is applied element-by-element as

As discussed in Chapter 2, the SINR at the output of the matched filter depends on

the number and relative power of the interferers Specifically, the SINR for the kth signal is

 k = P k /2

N o 4T b

Now this result assumed asynchronous reception with random phases between users In general,

it can be shown that the SINR of the kth signal is [32]

1 synchronous, zero phase

1/2 synchronous, random phase

2/3 asynchronous, zero phase

1/3 asynchronous, random phase

(5.10)

Now, assuming synchronous reception of signals with equal (zero) phase ( = 1) and equal

received powers for each signal (Pi = Pk ∀i) the SINR for each signal is equal and equal to

It was widely believed for several years that because the MAI in a CDMA system tends toward

a Gaussian distribution (i.e., because y is accurately modeled as a Gaussian random vector),

the optimal receiver was the matched filter described above However, since the MAI is infact part of the desired signal, the optimal receiver is actually a joint detector that was first

addressed by Schneider [56] for both the synchronous and asynchronous AWGN channels.Verd´u expanded this work by more fully developing the mathematical model for the importantcase of the asynchronous channel and by determining the minimum receiver complexity [57].Furthermore, Verd´u developed probability of error bounds for the receiver.

For maximum likelihood sequence detection, we desire to maximize the joint a posteriori

probability

where r (t) is the observed signal defined in (5.1) If all input vectors b are equally likely, this is

equivalent to maximizing the a priori probability

1, ∀i), pseudo-random spreading codes, a spreading gain of N = 15, and E b /N o = 7dB What

is the performance advantage of the optimal receiver compared to the conventional matchedfilter in such a scenario?

that maximizes (5.17) for every bit interval A closed form expression for the BER performance

is not available and thus we must resort to simulation to determine the performance On theother hand, the BER performance of the matched filter is well approximated by the Gaussian

FIGURE 5.1: Comparison of the probability of bit error for the matched filter and optimal multiuser

detector for a CDMA system (synchronous, random codes, N = 15, E b /N o = 7dB)

approximation as discussed in Chapter 2:

this performance improvement has come at a considerable computational complexity of O(2 K)

due to the search over all possible vectors b For asynchronous cases, the performance gain is

equally dramatic although the complexity is even higher

The previous section mentioned that while significant performance gains could be achievedover the conventional matched filter receiver, the cost of this performance gain is exponentialcomplexity in the number of users In this section, we investigate receivers that can approachthe performance of the optimal receiver with significantly reduced computational complex-ity These sub-optimal receivers can be broken down into two general categories, linear andnon-linear, as shown in Figure 5.2 Linear sub-optimal receivers create data estimates based

CDMA receivers

Single user Multiuser

Matched filter

Adaptive filters

Optimal (MLSE) Sub-optimal

Linear Non-linear

Multistage Decorrelator MMSE feedbackDecision

Decorrelating decison feedback

Successive cancellation

Parallel interference cancellation

Decorrelating first stage

Succesive cancellation first stage

FIGURE 5.2: Receivers for CDMA systems

upon linear transformations of the sufficient statistics (i.e., the vector of matched filter

out-puts sampled at the symbol rate y), and the non-linear implementations make decisions using

non-linear transformations of the sufficient statistics

A linear detector is one that makes decisions based on a linear transformation of the matched

filter output vector y:

ˆb = sgn(Ty)

where T is a linear operator on y and sgn(x) is defined in equation (5.7) This detector is

illustrated in Figure 5.3 where the linear transformation is performed on the despread symbolsand the matched filter is defined in equation (5.3) However, since dispreading is a linearoperation, the linear transformation could obviously also be performed prior to dispreading or

in conjunction with dispreading as we will show shortly

Linear transformation

T

Matched Filter Bank

Matched filter user 1

Matched filter user K

^

^

^

FIGURE 5.3: Block diagram of generic linear sub-optimal receiver structures

If the multiple access channel (excluding noise) is viewed as a deterministic multi-input

multi-output linear filter with transfer function R, then we can remove the interference in each

of the matched filter outputs by applying the inverse transfer function In other words, we can

use the linear transformation T = R−1, which leads to

ˆb = sgn(R−1y)

which is known as the decorrelating detector [58–60] Since R is simply the normalized

cross-correlation between the users’ spreading codes, the decorrelating detector does not requireknowledge of the received signal energies In fact, the decorrelating detector is the optimallinear receiver when the signal energies are unknown [60] This obviates the need for estimates

of the received signal energies, which is a significant advantage since energy estimates tend to

be extremely noisy

Additionally, (5.21) shows that the data estimate of the kth user ˆbkis independent of the

interfering powers This can be seen from the fact that A is a diagonal matrix and eliminates

the near-far problem discussed in Chapter 2 We will more fully discuss near-far resistance inSection 5.5

There are, however, two main disadvantages of this receiver The first is the need to

calculate the inverse of the cross-correlation matrix R−1to obtain the decorrelation coefficients

If the correlation matrix changes infrequently (i.e., the spreading codes change infrequently),this may not be a serious issue However, if the matrix changes frequently or perhaps everysymbol (as with long pseudo-random spreading codes), the complexity will be very high.The second disadvantage is that in high noise situations (i.e., low Eb/No), the receiverperformance can be severely degraded due to the enhancement of the noise power In fact, theperformance can actually be worse than that of the matched filter More specifically, similar tothe intersymbol interference (ISI) analog known as the zero-forcing equalizer, the application ofthe channel inverse results in increased noise power that is dependent on the cross-correlation be-tween users To show this, we examine the covariance matrixzof the decision metrics z = R−1y:

hood function or, equivalently, the minimization of (y − Rb)TR−1(y − Rb) [61] The probability

of symbol error (equivalent to bit error in BPSK) of the kth user can be written as [60]

where zk is the decision metric of the kth signal, E [zk|bk]= Akbk, var[zk ] is the (k , k)th

diagonal element ofz, and Q(·) is the standard Q-function Using these values and (5.22) in

where we have substituted for2and N0is the one-sided noise power spectral density and wehave assumed all data symbols are transmitted with equal probability Thus, the performance ofthe decorrelator is identical to the single-user case with the exception of the noise enhancement

factor (R−1)kk Since all the elements of R are less than or equal to one, we find that (R−1)kk

> 1 Unfortunately, general statistics of R−1 are not easily found, and thus predicting errorprobabilities is best done using the actual correlation matrix of a known set of user codes Onecan obtain an estimate of the performance with random spreading codes by calculating the

average of the elements along the diagonal of R−1from simulations

Example 5.2 Determine the BER performance of a two-user synchronous CDMA system

with spreading codes that are the same every symbol interval when using a decorrelating detectorreceiver

Solution: The correlation matrix for two users is written as

The decorrelator as described requires no estimates of the users’ received signal energies.However, it does require estimates of the timing and phase of each user along with knowledge

of the spreading waveforms Varanasi’s work [62, 63] generalized the decorrelator to the case ofnoncoherent demodulation (specifically, differentially coherent phase shift keying) where neitherthe received amplitudes nor the received phases need to be estimated This receiver, termedthe bi-linear detector, simply performs the decorrelating operation followed by differentialdetection The performance is again invariant to the interfering powers, thus obtaining near-far resistance Additionally the receiver requires no phase tracking, assuming that the phase isconstant over at least two consecutive symbol intervals

The decorrelator can also be extended to non-AWGN channels For example, the ceiver can be used in flat and frequency selective Rician fading channels [64, 65] as well as flatand frequency selective Rayleigh fading channels (both coherent and differentially coherent)[66–71] The most notable difference between these detectors and the AWGN case describedearlier is that the multipath versions treat resolvable multipath as individual signals until a fi-

re-nal decision is made That is, the transform matrix R is now KL × KL, incorporating each

of the L paths from each user The outputs for each path are then combined according to

the desired diversity combination scheme (e.g., maximal ratio combining) after tion Further investigations have been published regarding the decorrelator in fading chan-nels [64–71], temporal dispersion [72, 73], asynchronism [74–76], adaptive [77, 78] and re-duced complexity [79–82] Other variants of the decorrelating detector have also been reported[83–90]

decorrela-Example 5.3 Show that the decorrelating detector can be formulated as a single-user detector

and that the resulting receiver is simply a despreading operation with a modified spreading code

Solution: The decision metric of the kth user is simply the kth element of the vector z

z k = (R−1y)k (5.28)

Defininga, b as the inner product of the vectors a and b, the matched filter output yk can be

written as the dot product of N samples of the filtered received signal and the spreading code

y k = r, ak where rkis kth chip-matched filter output