Theory and applications of ofdm and cdma wideband wireless communications phần 9 pot

The discrete transmission model including the decorrelation receiver is given by the relevant part r♦ of the receive vector r with respect to the signature base vectors gk.. We see that,

We note that the same formula can easily be obtained in the discrete matched filter model

v = RCs + m with v = G†r In the noise-free channel, x = Cs holds and we simply obtain x by applying

where the index k indicates the kth element of the vector Thus, we need only to know

the signature waveforms of all the other users It is not necessary to know their fadingamplitudes

The discrete transmission model including the decorrelation receiver is given by

the relevant part r♦ of the receive vector r with respect to the signature base vectors gk.

Without losing generality, we consider user number 1 An error occurs if the first coordinate

of the transmit vector corresponding to the base vector g1 has a positive sign, but the firstcoordinate of the receive vector has a negative sign (or vice versa) We can visualize the

Transmit vector

threshold Decision

g1





g2

Figure 5.35 Distance to the decision threshold for the decorrelator forc1 = c2= 1

geometrical situation as follows TheK− 1-dimensional hyperplane spanned by g2, , g K

divides theK-dimensional hyperplane given by the vector space V = span(g1, , g K ) into

two half planes An error occurs if the (relevant) receive vector r♦ lies in the other halfplane ofV than the transmit vector For K = 2 and c1= c2= 1, the situation is depicted

in Figure 5.35 We may thus regard theK − 1-dimensional hyperplane span(g2, , g K ) as

a decision threshold Let be the distance between the transmit vector and this threshold.

The bit error probability for user 1 is then given by

P b= Q



 σ



.

We note that the coordinates of all users are independent and uniquely defined, and thedecision about the transmit symbols1 of user 1 does not depend on the values of the othertransmit symbols s2, , s K For the performance analysis, we may thus set s2= · · · =

s K= 0 This means that the distance of the transmit vector to the threshold is the same asthe distance ofc1g1 to the threshold (see Figure 5.35) It is evident from the figure and thePythagorean theorem that

In that case, the projector P is given by the matrix

P = g2g†2.

For the general case of more dimensions, we calculate by using the orthogonality

prin-ciple The formalism is similar to the derivation of the decorrelating receiver We define

is the orthogonal projector onto the hyperplane span (g2, , g K ) An orthogonal projector

P is a matrix with the following properties: it is Hermitian, that is,

As a secondary result, we have thus seen that

is just the column of theK− 1 detector outputs of g1 for all other detectors g2, , g K.

From Equation (5.45), we see that the performance compared to ideal single-user BPSK

is degraded by the channel attenuation|c1|2 and the geometrical factor

1−&&Pg1&&2

ForRayleigh fading, we simply average over the fading amplitude c1 as shown in Subsec-tion 2.4.3 to get

P b= 12





1 −

012

for the two-user performance

in the AWGN channel and compare it with the SUMF receiver We set the fading amplitude

c1 = 1 Figure 5.36 shows the bit error curves for both receivers for equal fading amplitudes

c1 = c2 = 1 and different values of the correlation coefficient ρ12 We see that for theserelatively high correlation coefficients, the decorrelator performs several decibels betterthan the SUMF receiver The difference becomes smaller for small correlations, but thedecorrelator is always better In fact, one can show that for real-valued coefficientsc1,c2and

ρ12, the decorrelator is better than the SUMF receiver (see Problem 3) For complex values

of the fading amplitudes, it may occasionally happen that c∗1c2ρ12))in Equation (5.30)becomes smaller than |c1ρ12|2 in Equation (5.45) because of favorable channel phases.Comparing the performance of the decorrelator for ρ12= 0.6 with MLSE curve for the

sameρ12in Figure 5.30, we note that the loss is always less than 1 dB Figure 5.37 showsthe bit error curves for both receivers for a fixed correlation coefficient ρ12= 0.6 and

the different values of the fading amplitudes For small interferer amplitudes|c2|  |c1|,the performance curves of the SUMF receiver approach the limit of the ideal single-user(SU) BPSK curve However, for this relatively high correlation coefficient, the degradationbecomes severe if the interferer power is of the same order (or even higher) as that of theuser under consideration The curve for the decorrelator does not depend on the amplitude

of the interferer It has a fixed degradation (caused by the geometry) of approximately 2 dBcompared to the ideal BPSK Therefore, for very low values of the interferer amplitude,the SUMF receiver performs better

Figure 5.36 Bit error curves for the SUMF receiver (−o−) and the decorrelator (− + −)for equal fading amplitudesc1= c2= 1 and ρ12= 1/2, 3/5, 4/5.

The MMSE receiver

If an estimate for the SNR is available, a linear receiver that is better than the decorrelator

can be obtained from the minimum mean square error (MMSE) condition We consider the

orthonormal base discrete channel model

with(B) ik = bik The method to find the matrix B is quite similar to the derivation of the

Wiener estimator in Subsection 4.3.3 We apply the orthogonality theorem of probabilitytheory that says that the MMSE condition is equivalent to the orthogonality condition

holds for any real numbert (if the inverses mentioned above exist) (see Problem 4) We

can thus write

5.3.4 Suboptimal nonlinear receiver structures

In this subsection, we present the method of successive interference cancellation (SIC) (see e.g (Frenger et al 1999)) There are two variations of that theme The serial SIC should

be preferred if the different users have very different power levels as is the case in theabsence of power control We see that, in this case, the SUMF receiver may totally fail

for the users with low signal level The parallel SIC is the better choice if the different

users have approximately the same power levels as it is the case in the presence of powercontrol For this method, it is necessary that the SUMF receiver gives correct results atleast for a significant part of the users

Serial successive interference cancellation

The idea is simple: if the reception is corrupted by strong interferers with known signaturepulses and known fading amplitudes, one can combat the near–far problem by adjustingthe decision threshold to the situation

We illustrate the idea for the case of two users with the same parameters as in Figure 5.27and Figure 5.29 In those figures, the fading amplitudes are c1= 1 and c2= 2, and thetransmit symbol energy has been normalized toE S= 1 The signature vectors are given by

g1 =

10



, g2= 15

34

where n is the two-dimensional AWGN Because the signal of user 2 is stronger than that

of user 1, the SUMF receiver will work quite well for that user by utilizing Threshold 2 inFigure 5.38 to provide us with a reliable estimate

Threshold 2

00 10

01 11

c1g1

c2g2

Region 1 Region 2

Figure 5.38 Two-user BPSK SIC receiver for unequal fading amplitudes

corresponding to Region 3 and Region 4, respectively These regions are separated byThreshold 1b Formally, the decision ons1 is given by

ˆ

s1= sign (g1· (r − 2ˆs2g2))

Ifs2 has been correctly decided, this corresponds to the maximum likelihood decision for

s1 However, the estimate ˆs2 may be wrong Comparing Figure 5.29 and Figure 5.38, wenote that there are two stripe-like shaped regions where the decisions ons2differ However,there are only two small triangular regions where this affects the decision ˆs1 Therefore,s1

may be correctly decided even if the decision on s2 is erroneous This correct decision ˆs1

now provides us with a better estimate fors2, which is given by

ˆ

s2 = sign (g2· (r − ˆs1g1))

Thus, this successive interference cancellation also improves the decisions for the strongersignal

We now formalize the procedure and generalize toK users For simplicity, we restrict

ourselves to BPSK The generalization on QPSK is straightforward The receive signal is

After this first iteration, estimates ˆs1, , ˆs K for allK symbols are available We may now

start a second iteration We replace the old estimate ˆs K with the new one

Note that the estimates ˆs i , i < k of the last iteration are used together with the already

available new estimates ˆs i (new) , i > k When all estimates are calculated in descending

order, we may start a third iteration, and so forth

The method described above is called serial SIC because the estimates are obtained

serially in an order given by the signal strengths It is especially useful if there are significantdifferences in the signal strengths This is the case in the absence of power control In case

of an efficient power control, all signal powers are of the same order, and there is no reasonfor one of these signals to be privileged In that case, we should apply the parallel SICreceiver

Parallel successive interference cancellation

In a parallel SIC receiver, all iterations are performed in parallel Again, we consider BPSKtransmission and the discrete channel model

In the first iteration, the SUMF receiver outputs for all users are calculated in parallel to

obtain the estimates

ˆ

s k = signc∗kg†kr

for k = 1, , K After this first iteration, estimates ˆs1, , ˆs K for all K symbols are

available We may now start a second iteration We replace the old estimate ˆs k by thenew one

the last iteration step

It is obvious that this method needs a reliable first SUMF estimate for most users Itwill fail if most users have a power level far below the level of one or more high-powerlevel users

Asynchronous Wideband CDMA Transmission

As already pointed out at the beginning of this chapter, spreading by itself does not improvethe power efficiency (i.e the BER performance as a function of E b /N0) in a Gaussianchannel It only uses a spectrally inefficient pulse shape that spreads the same powerover a wider transmission band However, in a fading environment, the receiver may takeadvantage from this higher transmission bandwidth As we have seen in Subsection 4.4.3,

a wideband channel has an inherent diversity degree that is significantly greater than 1 ifthe transmission bandwidthB significantly exceeds the correlation bandwidth fcorr Since

B ≈ 1/Tc, this means that the delay spread τ must significantly exceed the chip duration

T c The interpretation is that the signal arrives at the receiver via several uncorrelated or

only weekly correlated transmission paths corresponding to different delays We may thus

speak of path diversity of the channel In the heuristic discussion in Subsection 5.1.1, we

introduced the RAKE receiver as a device that can exploit this diversity In the followingdiscussion, we shall derive this receiver structure as a special case of the maximum ratiocombiner if the following model assumptions are made:

• The signal arrives at the receiver via L discrete transmission paths with delays τl , l=

1, , L.

• We may consider the signal of one time slot separately Intersymbol interferencemay be neglected due to orthogonality or may be regarded as an additional noise-likedisturbance MAI is treated similarly

• We neglect the time variance of the channel during the time slot under consideration

For reasons of receiver implementation, we will assume at a later stage that eachτ l is aninteger multiple of the chip duration T c We should keep in mind that the assumption of

L discrete transmission paths is a very rough reflection of a real physical channel impulse

response and that there is some arbitrariness in the model as to how these delays are chosen

We now assume that user number k with the signature pulse g k (t) transmits a BPSK

symbols k∈±√E b

 The transmit pulse is normalized to

g2=

 ∞

−∞|g(t)|2 dt = 1.

Because we only look at this one user numberk, we may drop the index k and write sg(t)

instead of s k g k (t) The signal will be transmitted over a multipath channel with complex

fading amplitudesc l corresponding to the delaysτ l The receive signal is then given by

We may interpret h(t) as a one-dimensional transmit base Sufficient statistics is trivially

given by the detector output

is the detector for the delayed signature pulseg(t − τl ) Thus, the receiver multiplies the

detector outputs for all the delayed pulses by the complex conjugate channel amplitudesc∗l

and sums them up (combines them) (see Figure 5.39).

The combiner output is given by

by Equation (5.51) in the same way as, for instance, the detector outputs obtained fromdifferent antennas.

We note that

D l[ r] = g∗ −t) ∗ r(t)))

t =τ l

is just the output of one filter (i.e a correlator) matched to the signature pulse and sampled

at the timest = τ1, , τ L Thus, we do not have to implement a bank of L correlators as

depicted in Figure 5.3 We only need one correlator (matched filter) to store the matchedfilter outputs and combine them with the appropriate multiplicative factors If the delaysτ l

are integer multiples of the chip durationT c, we may implement the RAKE receiver by a

filter matched to the chip pulse followed by a digital FIR (finite impulse response) filterstructure

Figure 5.39 Block diagram for the BPSK RAKE receiver

3 Up to a factor that can be ignored for that decision.

Performance of the RAKE receiver

To evaluate the performance of the RAKE receiver, we divide Equation (5.52) byh to

obtain the normalized combiner outputu = v h−1 as

u = h s + n,

where the noise termn is a complex mean zero Gaussian random variable with variance

σ2= N0 We assume independent Rayleigh fading for the different transmission paths.Then the coefficients c l are independent complex Gaussian random variables with meanzero The respective transfer powers of the paths are given by the variances

where Ec1, ,c L{·} means averaging over the fading amplitudes The same derivation aspresented in that subsection now results in the integral expression

which can easily be evaluated by numerical methods

We finally note that the well-known formulas for L-fold diversity cited at the end of

subsection 2.4.6 cannot be applied here because the transmit powers of the different pathstypically are (in general) different A formula for diversity with different powers can befound in (Proakis 2001) However, that formula can only be applied if no pair of diversitybranches has the same power The formula presented above is more flexible because it can

be applied in any case

The RAKE receiver and MAI

In the above derivation, we treated the RAKE receiver as a SUMF receiver with a matchedfilter h∗ −t) that includes the impulse response of the channel The RAKE receiver is

the optimal receiver under the assumptions made at the beginning of the subsection Evenfor one user only, there is the approximation that the signal for only one time slot and

its delayed versions have to be taken into account Intersymbol interference (ISI) from

neighboring signals has been ignored This may be justified by good correlation properties

of the sequence

For the RAKE receiver as a SUMF receiver, we may regard the MAI due to otherusers as a noise-like interference with Gaussian statistics As already discussed for thesynchronous case, this can be justified for the case of Rayleigh fading The near–far problem

is still present If there is a fast power control mechanism that is able to follow the time

variance of the channel, we do not have Rayleigh fading any more and the situation comescloser to a Gaussian channel

Even though the RAKE receiver is essentially a detector matched to one user (includingits signature waveform and the corresponding channel impulse response), we may think ofmultiuser detection for K users by using K RAKE receivers matched to all these users

(including their signature waveforms and their respective channels impulse responses) If

we restrict this treatment again to the signal for only one time slot and its delayed versions,

we may write the receive signal as

where s k is the transmit symbol of user number k and h(t) is the signature waveform

convolved with the channel impulse response for user number k The RAKE combiner

output for user numberi is then given by

withρ ik = hi , h k and detector outputs micorresponding to the noise or, in vector notation,

v = Rs + m

with(R) ik = ρik This is just like the synchronous matched filter model, and we may thusapply any of the MUD receivers discussed in the preceding, e.g successive interferencecancellation Optimality will never be achieved as other time slots are ignored, but someimprovements are possible This may help to reduce the effect of the near–far problem,but the problem is not resolved because the other time slots of the unwanted strong signalare not taken into account This may severely influence the detector output for the wantedweek signal An optimal receiver must take all the time slots into account We discuss such

a receiver in the following subsection

In this subsection, we study optimal receiver structures The maximum likelihood sequence

estimation (MLSE) for jointly optimum decisions and the bitwise maximum a posteriori

(MAP) receiver for estimating the most probable individual bit transmitted by one userhave already been discussed in some detail The formalism can directly be applied to achannel that is formally described by

r= Hs + n.

This channel model applies to synchronous wideband MC-CDMA and asynchronous

wide-band DS-CDMA as well However, the interpretation of matrix H is different and both

cases have to be discussed separately

The optimal receiver for asynchronous DS-CDMA

We recall that the above compact notation for the discrete transmission model includestwo degrees of freedom (or dimensions), and all quantities are labeled by two indices.One is the user index k that already occurs in the synchronous model The other one is

the time index l Thus, the complexity of an exhaustive search for a maximum likelihood

receiver would grow exponentially in the number of usersK and with the frame duration

L Fortunately, one can usually assume that the time dispersion of the channel is finite.

For single user transmission over a channel with finite time dispersion, an MLSE receivercan be implemented by using the Viterbi algorithm (see e.g (Benedetto and Biglieri 1999;

Kammeyer 2004; Proakis 2001)) This receiver type is sometimes called Viterbi equalizer.

We may generalize that receiver structure for multiuser detection Here, we note that thegeneral case of different time-dispersive channels for different users already includes anycase of asynchronous transmission

The maximum likelihood receiver has to find the most probable transmit sequence ˆs

Let us first consider the single-user case in a time-dispersive channel, that is, we rederive

the well-known MLSE equalizer structure for that channel The autocorrelation matrix R

has elements given by

We assume that the time-dispersive channel has finite memory M, hence ρ[m]= 0 for

m > M The metric can now be written as

M

m =−M ρ[m]s l −m

is a convolution and can be interpreted as a digital finite impulse response (FIR) filter

described by a shift register5 Assuming BPSK transmission, the shift register has 22Mstates characterized by

(s l −M+1 , s l −M+2 , , s l +M ),

and the MLSE receiver can easily be implemented by the Viterbi algorithm in the sameway as for convolutional codes as described in Subsection 3.2.2, but with the metric definedabove

4 We writeρ instead of ρ11for a single user.

5 We ignore that the filter is noncausal This can easily be removed by a time shift.

The complexity of the trellis can be further reduced by noting that the matrix R is

Hermitian Consider the simplest caseM= 1 The autocorrelation matrix is then given by

Thus, we have a trellis diagram with four states labeled by the signs of(s l−1, s l−2) The

actual input bit iss l

For general values of M, we readily find the expression

which is described by a trellis diagram with 2M states

6 Here and in the following text, we start with the indexl= 1.

is already the combiner output of the RAKE receiver for time slotl, this single-user MLSE

receiver is a generalization of the former because it takes intersymbol interference intoaccount The RAKE receiver by itself detects only one symbol However, for a single userthis intersymbol interference will not be severe if sequences with good correlation propertiesare chosen Problems will occur because of intersymbol interference by a stronger user.ForK asynchronous users, the autocorrelation matrix is given by the general expression

(5.25) As for the single-user case, for a channel with finite memory, the matrix elementswill be zero outside the inner subdiagonals We may then proceed in a similar fashion as

for the single-user case and calculate s†Rs to obtain metric expressions corresponding to

that case In contrast to the single-user case the subdiagonals do not consist of identicalelements, and thus the metric increment calculation will be different from time-step totime-step, but it will be repeated afterK time-steps For K= 2, there will be two differentexpressions for the metric calculation, one for even and one for odd time indices

We shall now evaluate the metric expression for K= 2 and finite memory such that

R[l] = 0 for l > 1 Thus we have to consider only the submatrices R[0] and R[1] with

that is, we have assumed for simplicity thatρ21[1]= 0 This is the case if the time dispersion

is small We note that for the asynchronous case without time dispersion, the diagonal

s†Rs= Eb L (ρ11[0]+ ρ22[0])

12[0]} (s1s2+ s3s4 11[1]} (s1s3+ s3s5+ · · · )

[1]} (s s + s s [1]} (s s + s s + · · · )

Ignoring the first term that is not relevant for the decision, the metric increment expressionsare

µ l l−2(v l−2− sl−1ρ12[0]− sl ρ11[1])}for odd values ofl and

µ l l−2(v l−2− sl−1ρ12[1]− sl ρ22[1])}for even values ofl.

The expressions for the metric increments for more users and more channel memorycan be derived in a similar manner Given these metric increment expressions, the MLSEreceiver can be implemented by means of the Viterbi algorithm (see Subsection 3.2.2).Alternatively, the bitwise MAP receiver can be implemented by the BCJR algorithm (seeSubsection 3.2.4) The first receiver makes joint optimal decisions, and the second onemakes individually optimal bit decisions The decisions may be different, but typically this

is a very rare event The MAP algorithm is computationally more complex, but it providesreliability information that may be used by an outer decoder Alternatively, the suboptimalSOVA (see Subsection 3.2.3) may be used to obtain such reliability information

We note that, since the complexity of both optimal receivers grows exponentially withthe number of users, it can only be applied ifK is not too large However, one may think

of a receiver that utilizes only the most significant entries in the matrix R, which will

then be an intermediate solution between the SUMF receiver (implemented by a RAKEreceiver) and the optimal receiver The most significant matrix entries (outside the maindiagonal) correspond to the strongest interferers Since the main task of multiuser detection

is to mitigate the near–far problem, such a strategy will certainly improve the performancesignificantly

The optimal receiver for synchronous wideband MC-CDMA

For synchronous MC-CDMA, the discrete transmission model is formally the same as aboveand thus the maximum likelihood estimate is formally the same, that is, it is given by

However, the interpretation is different Since H = C ◦ G, the column vector hk of H is

the signature vector of userk with random signs in frequency direction multiplied with the

fading amplitudes for that user at the respective subcarrier frequencies The component

v k= h†

r

of v is the detector output corresponding to that composed pulse shape It acts as a back

rotation of the channel phase and the PN sequence and weights with the fading amplitude

In contrast to the DS-CDMA time-dispersive channel, there is no trellis structure volved here Hence, the MLSE must be implemented by exhaustive search Thus, thecomplexity grows exponentially with the number of users and with the length of the spread-ing sequence Therefore, the MLSE receiver seems to be too complex for implementation

in-in most cases The same holds for the bitwise MAP receiver

As mentioned in Section 5.1, the origin of spread spectrum is in the field of military cations and radar systems Today, there are also some specialized markets for CDMA aswireless microphones However, in this section we focus on the main commercial applica-tions as wireless local area networks, satellite navigation and mobile radio networks

In 1997, the Institute of Electrical and Electronics Engineers (IEEE) released the standardIEEE 802.11 for wireless local area network (WLAN) applications Systems according tothis standard offer a wireless access to existing local area computer networks via the so-

called access points as well as direct wireless interconnections within a small group of

computers

Within the standard, three transmission modes are specified:

• an infrared (IR) mode,

• a frequency hopping (FH) mode,

• a direct sequence spread spectrum (DSSS) mode

However, only products using the DSSS mode have been established on the market TheDSSS mode is operating in the ISM frequency band 2400–2485 MHz ISM means thatthis band is free for unlicensed wireless systems for industrial, medical and scientific ap-plications Hence, it is not exclusively reserved for IEEE 802.11 Another prominent andimportant system operating in the same band is Bluetooth Therefore, the spread spectrumtechnique is used within IEEE 802.11 to lower the power spectral density to reduce the

interference to other systems and as a kind of antijamming method with respect to

inter-ference experienced from other systems with a smaller bandwidth Bluetooth, for example,

is using frequency hopping instead It should be noted that IEEE 802.11 is not a CDMAsystem; separation of different connections and radio resource allocation is managed by acarrier sense multiple access (CSMA) scheme, which is known for wired Ethernet (IEEE802.3) networks and which is adapted to the special needs of wireless networks

Within the DSSS mode, two transmission schemes are defined working at data rates

of 1 Mbit/s and 2 Mbit/s Spreading within IEEE 802.11 is performed using an 11-chipBarker code word at a chip rate of 11 Mchip/s Spreading and modulation is illustrated inFigure 5.40: for the data rate of 1 Mbit/s the bits are modulated by a DBPSK changing the

wide-band DS -CDMA as well However, the interpretation of matrix H is different and both

cases have to... applica-tions as wireless local area networks, satellite navigation and mobile radio networks

In 199 7, the Institute of Electrical and Electronics Engineers (IEEE) released the standardIEEE... level far below the level of one or more high-powerlevel users

Asynchronous Wideband CDMA Transmission

As already pointed out at the beginning of this chapter, spreading