Chapter 2 MIMO Systems
2.5 Chase Detector
We already know the large gap in both performance and complexity between the maximum-likelihood (ML) and the other existed detectors, which are linear detectors or BLAST-ordered decision-feedback (BODF) [15] detectors, hence we have the motivated search for find out a favorable performance-complexity trade-off and a unified framework which is the chase family of detection. In the chase family of detection, there is an important class of reduced-complexity detectors called list-based detectors that adopt a two-step approach of first creating a list of candidate decision vectors, and second choosing the best candidate as its final decision. For the example, the parallel detector [8] generates its list by implementing a separate low-complexity detector for each possible value of the first symbol.
Numerical results suggest that if the first symbol detected is chosen so as to approximately minimize the probability of error for the remaining symbols, then the parallel detector achieves full receive diversity. This section proposes a family of Chase detectors, which includes as special cases the BODF [15], ML [5], parallel [8], PDF [16], B-CHASE [13]. Thus, the Chase family provides a unified framework for comparing a variety of existing detectors.
Furthermore, we propose the B-Chase detector as a new special case that performs well on fading channels. We will demonstrate that the B-Chase detector can approach ML performance with less complexity than previously reported detectors. The B-Chase detector
distinguishes itself from previous list-based detectors in the unique way it builds its list. We will see that the B-Chase detector achieves better performance with significantly smaller candidate lists, leading to a favorable performance-complexity trade-off. We introduce the Chase detector, a general detection strategy for MIMO channels that reduces to a variety of previously reported detectors as special cases. The Chase detector defines a simple framework for not only comparing existing MIMO detection algorithms but also proposing new ones. The Chase detector is described use five steps and that is shown.
Figure 2-7 Block diagram of the Chase detector [13]
Step 1) Selecting i∈ "{1, Nt}that the index of the first symbol to be detected.
Step 2) Generate a sorted list L of candidate values for the ith symbol, defined as the l elements of the alphabet nearest toy , i
2 1
( H α )− H
= + =
y H H I H r Fr (2.5.1)
where y is the output of either the zero-forcing (ZF) (α = or MMSE 0) (α2 =N0) linear filter.
Step 3) Generate a set of l residual vectors {r1,…rl }by cancelling the contribution to r from the ith symbol, assuming each candidate from the list is, in turn, correct:
j = − i js r r h
Step 4) Apply each of {r1,…rl } to its own independent subdetector, which makes decisions about the remaining Nt-1symbols (all but the ith symbol). Together with sj, the jth subdetector defines a candidate hard decision aˆj regarding the input a.
Step 5) Choose as the final hard decision ˆa the candidate hard decision { ,aˆ1" l that best aˆ } represents the observation r in a minimum mean-squared-error sense:
From these steps know that have four parameters be specified:
Parameter 1:select i algorithm that affact the system performance and complexity.
Parameter 2:set the list length l that affact the system performance and complexity.
Parameter 3:find the weighted filter ZF or MMSE.
Parameter 4:employ the subdetector algorithm to detect the received signal.
Table 2-3 Special cases of the Chase detector [13]
Detector First-Symbol index i
List Length l Filter type,α Subdetector
ML[14] any |A| ZF ML
BODF[15] ♦BLAST1 1 ZF or MMSE BODF
PDF[16] ♦BLAST1 1 ZF or MMSE Linear
Parallel[8] using Selection algorithm 1
|A| ZF any
B-Chase[13] using Selection 1 ≤ l ≤ |A| ZF or MMSE BODF
algorithm 1 or Selection algorithm 2
♦The index BLAST1 signifies the first index of the BLAST ordering [15]
Above that, the list length is maximal such that subdetector is likely ML detectors and the choice of which symbol to detect first is not critical to performance. The list length is one such that subdetector is likely BLAST-ordered decision-feedback (BODF) detectors and the choice of which symbol to detect first is critical to performance. The parallel detector is another Chase detector whose performance is highly sensitive to the choice of which symbol to detect first.
Chapter 3 B-Chase Detector
3.1 Introduce B-Chase Detector
We introduce the example for the B-Chase detector which is defined as a Chase detector that uses BODF as a subdetector and an SNR gain of a list detector that demonstrate the probability of error. We will see that the B-Chase detector achieves better performance with significantly smaller candidate lists, leading to a favorable performance-complexity trade-off.
We can demonstrate that the B-Chase detector can approach ML performance with less complexity than previously reported detectors. We show block diagram of the B-Chase detector.
Figure 3-1 Overall block diagram for the B-Chase detector
3.1.1 The SNR Gain of a List Detector for the B-Chase Detector
We say that a list detector makes an error when the actual transmitted symbol does not appear somewhere on the list. With this definition, when we increase the length of the list that leads to a decrease in the probability of error. Therefore, we can employthe 4-QAM alphabet
to describe the list detector. For the 4-QAM alphabet
3
the transmitted symbol isa ej4
π
= . For the ith symbol yi = +a n consider it as the input of the list detector and 12
[ ] SNRi
E n
= . Show the correct decision regions for lists lengths l∈{1, 2,3}
in the fig.3-2. Define the Pl as the list-error probability and the list length is l. Find that
Figure 3-2 Decision regions for a ej 4
= π and different list lengths: (a) l = 1; (b) l = 2; and (c) l = 3. The
decision list contains a whenever the input to the list detector falls within the shaded region. Also indicated is the minimum distance dl to the boundary [13]
In the high SNR case, we can approximate the list detector SNR gain and define the d ( A )l as the minimum distance from any element in A to the corresponding decision region boundary of the list detector with list length l, so define the SNR gain γl2 with a list length l in that
2 2 12
( ) ( )
d A
d A
γl = l (3.1.1.4)
Show the extreme case that is the maximal list length l = A and that have an infinite SNR gain 2
γ A = ∞ because the actual transmitted symbol is on the list with d ( A ) = ∞l .
3.1.2 The SNR of the B-Chase Detector
We will define the SNR for each symbol of the B-Chase detector and employ that to select which symbol is detected first .For describe that by doing the QR decomposition. Do the QR decomposition of the extended channel matrix and show that
α Nt
Figure 3-3 QR decomposition algorithm
Total complex of the QR decomposition algorithm is 3(Nt)3-3Nr(Nt)2-3(Nt)2 -3NrNt+2Nr+2Nt in the MMSE case. Where the matrix H are (Nr+Nt)×Nt, and where the columns of the matrix Q are orthonormal, and where L is a lower triangular Nt Nt× matrix with positive and real diagonal elements. Define the bottom rows of Q are the
matrix αL such that −1 αL L−1 =αI .Due to (3.1.2.1) write (2.5.1) as where u is the ith column of U. Define the SNR for the first symbol detected i
( ) 2
and then define the next symbol detected. That is defined by the QR decomposition of the extended channel matrix H whose columns are permuted, when employ the Π in the H , ( )i according to the detection order. Find the ordering and that shown.
( )i = ( ) ( )i i
HΠ Q L (3.1.2.5)
Where the columns of the (Nr+Nt)×Nt matrix Q( )i are orthonormal, and where L is a ( )i lower triangular Nt Nt× matrix with positive and real diagonal elements. For the case
( )i =
Q Q and L( )i =L whenΠ( )i =I .We can know that the Π is an ( )i Nt Nt× permutation
matrix that arranges the columns of H such that the ith column comes first, and the remaining columns are arranged according to the BLAST ordering. Use the QR decomposition ideal to construct SNR for B-Chase detector. First, show the SNR for the first symbol detected is
2 ( ) 2
For the first symbol detected can provide list-detection gain in the B-Chase detector.
Where lk k( )i, is the kth diagonal of L and the SNR of the final symbols can be shown. ( )i
3.1.3 The B-Chase Selection
In the B-Chase detector provide the selection algorithm that get two opposing goals. Now we argue that the choice of i must balance two opposing goals: (1) the SNR of the first symbol
1( )i
SNR is high that the list detector is likely to be correct, the actual transmitted symbol be on the list, that reduce the risk of error propagation, and (2) that the subsequent subdetectors can perform well. If our only concern is to ensure that the actual transmitted symbol can be on the list, we will choose i such that the SNR of the first symbol SNR1( )i is high. For that choose i so that hi is the column of H that is most orthogonal to the remaining columns which do not include hi in the remaining columns of H . On the other hand, if our only concern is to ensure that the subdetectors perform well when we make decisions about the remaining Nt–1 symbols, we will choose i so that the effective MIMO channel, we remove the hi in the column of H , seen by the subdetectors is as orthogonal as possible that we will get the
distance is likely the dfree [8]. So, we will choose i so that hi is the column of H that is least orthogonal to the remaining columns in the submatrix channel, that reduce the most co-planar vectors in the submatrix channel, which is precisely the i that corresponds to the SNR of the first symbol SNR1( )i is low. Therefore, to balance the two opposing goals, we should choose i
so that the SNR of the first symbol SNR1( )i is small, but not so small that the list does not contain the actual transmitted symbol. In other words, we should choose i so that the effective SNR of the list detector is neither too small nor too large.
That selection algorithm are shown
Selection Algorithm 1:
That maximizes the minimum SNR of the symbols. To implement the selection algorithm 1 can spend the complexity is O Nt( 4)computations when l >1. From the QR decomposition their complexity is O Nt( 3)computations, therefore the selection algorithm 1 implement Nt times. Due to the selection algorithm 1 complexity is high, so find the low-complexity to implement the selection algorithm. That will be shown the selection algorithm 2 which can reduce the complexity but can has the bad performance. Since the smallest SNR inside the subdetector is SNR2( )i when 1< <l A , select the symbol which maximizes the minimum of
1( )i
SNR and SNR2( )i . SNR2( )i is shown.
2( ) 2 2
Selection algorithm 2 is shown:
2 selection algorithm 2. The each squared-magnitude is need to compute the complexity is 5Nt.
Table 3-1 Complexity of the selection algorithm 1 and the selection algorithm 2
total complex Nr=Nt=N=4
The selection algorithm 1 3.5*Nr*(Nt)3+3.5(Nt)4+0.5Nr*(Nt)2+ 0.5(Nt)3
1856
The selection algorithm 2 3.5*Nr*(Nt)2+3.5(Nt)3+0.5Nr*Nt+0.5(Nt)2 +5(Nt)3-5(Nt)2
704
3.1.4 Implementing the B-Chase Detector
We will implement the B-Chase detector and show the block diagram in the fig.3-1, and the pseudocode in the fig.3-4, and fig.3-5. For the B-Chase detector use the selection algorithm 1 or the selection algorithm 2. Now it use the selection algorithm 1 to implement in the B-Chase detector. For the selection algorithm 1 we must compute the QR decomposition to get L such that use the selection algorithm 1 to decide which symbol to detect but we do ( )i not compute directly that. We use another method to compute the QR decomposition to get
( )i
L .
From the Π definition we know permute the columns of H by ( )i Π that is similar to ( )i permute the the rows of C = U QH HbyΠ( )i H. So we define the sorted-QR decomposition of
C and that is shown. H
( ) ( ) ( )
H i = i i
C Π Q U (3.1.4.1)
We can use the relation U( )i =(L( )i H)−1to get L .From the( )i Π( )i definition is the ith column of C comes first, so modify the sorted-QR decomposition. We can use the H algorithm of the sorted-QR decomposition to compute the sorted-QR decomposition after modify this such that the ith column of C firstly comes. Form the (3.1.4.1) equation we H can modify that
( ) ( ) ( ) ( ) ( )
H i = i = i i H i
C Π QUΠ QΘ Θ UΠ (3.1.4.2)
where Θ is a unitary matrix such that the ( )i U( )i =Θ( )i HUΠ is an upper triangular matrix ( )i with real and positive diagonals and form (3.1.4.2) and (3.1.4.1) equations we can define the relation Q( )i =QΘ ( )i .We can define the U sorted-QR decomposition and show
n = Fw - D U b . From the B-Chase preprocessing function we can get some parameters
( ) 2 2
1,1 ,
, { , }
i and d dNt Nt
F,M, Π " . Use these parameters in the B-Chase detector to implement that. We employ the list detector to generate an ordered list [ ,s1"sl ] of the l elements of A
that are nearest toy which is 1th element of y. For the ordered list1 [ ,s1 "sl], s is in the i ordered list and it is the ith closest element of A that is nearest toy . From the list detector 1 generate an l elements ordered list and then use y and the ordered list as inputs of the l DF detectors whose first symbol decisions are hard-wired to decide first outputs of DF detectors and then compute the first cost .The next steps use a decision-feedback process to decide other symbols and update the cost. For show that the lth subdetector cancels the intersymbol interference from the kth element of as follows:
1 decision vector of the l th subdetector, and where dec{x} quantizes x to the nearest element of A. From the outputs of subdetectors, B-Chase detector choose the minimum cost of the outputs of subdetectors as the decision vector. To express the cost of the l th decision vector as
( )i ˆ 2
{ }
Figure 3-4 Computationally efficient implementation of the B-Chase detector [13]
( ) 2 2
Figure 3-5 Preprocessing pseudocode for the proposed implementation of the B-Chase detector that uses selection algorithm 1 [13]
We can have two crucial thing that reduce the complexity.
z From compute the sorted-QR decomposition algorithm of U and the QR decomposition algorithm of H that we know the mk k, = element of the M matrix. 1 And then we can combine the equation (3.1.4.6) and the equation (3.1.4.7) that let we can rewrite the cost expression as
2 2
, , ,
1 Nt ˆ
l k k k l k l
k
c d x b
=
=
∑
− (3.1.4.8)From that we reduce computations in the cost equation (3.1.4.8) in the subdetector.
We can use the O Nt computations. ( )
z We can use a pruning and threshold-tightening strategy that can reduce the computations. A cost threshold can be established with the cost c1 of the first subdetector’s decision. In subsequent subdetectors, we can abort both the cost calculation (3.1.4.8) as well as the decision feedback process (3.1.4.7) whenever this threshold is exceeded the cost threshold. Furthermore, the threshold can be reduced each time a lower cost is found.
We will get the performance and complexity well .From the B-Chase detector know the channel parameters that Rayleigh-fading gain, and knowN . We can use B-Chase*(0 l) to denote the B-Chase detector with list length l , α2 =N0, and use selection algorithm (3.1.3.1).
We can use B-Chase(l ) to denote the B-Chase detector with list length l , α2 =N0,and use selection algorithm (3.1.3.3).We use input is 4 with 16-QAM and output is 4. And show figure the performance versus the number of antennas, where the SNR per bit is
2
For define the unit that is real multiplies (RMs) per bit to describe the complexity. We define the squared absolute value of a complex number is counted as two RM, and the complex multiplications are counted as three RMs. Now we define the preprocessing complexity that need to compute the computations that are required only once per channel estimation. And define the core-processing complexity need to compute the computations that must be implemented during every symbol period. In the B-Chase detector show the core-processing complexity when l =1 show their core-processing complexity is 3NrNt RM and whenl ≠1 show their core-processing complexity is 3(Nr+l)Nt RM. The overall complexity includes both core-processing complexity and preprocessing complexity. We assume that the channel estimate is updated in T symbol periods. That unit is real multiples per bit. We can show that as:
From preprocessing complexity we can know the state of the channel to compute complexity in the B-Chase detector. If the state of the channel changes quickly, then we can estimate the state of the channel is quick in the small symbol periods. That can affect the preprocessing complexity. If we have the small preprocessing complexity, then that reduce the complexity in the state of the channel changes quickly.
Table 3-2 System parameters
Transmit antenna 4
Receive antenna 4
Channel is updated in T symbol periods 8
Rayleigh-fading Mean=0,Varance=1
Channel order 0
Selection algorithm 1
List length l 1 ,2 ,and 16
5 10 15 20 25
10-5 10-4 10-3 10-2 10-1 100
SNR dB
BER
B-CHASE*(1) B-CHASE*(2) B-CHASE*(16)
Figure 3-6 The bit error rate versus SNR for the B-Chase detector* ( l ) with l =1, 2, 16 , T=8,and 16 QAM
From figure we can know when increase the length of the list that leads to a decrease in the probability of error. In other word shrink this gap and provide new solutions for managing the inherent performance-complexity trade-off in MIMO detection. We can find that shrink this gap quickly in the low the length of the list.
Table 3-3 System parameters
Transmit antenna 4
Receive antenna 4
Channel is updated in T symbol periods 8
Rayleigh-fading Mean=0,Varance=1
Channel order 0
Selection algorithm 1 and 2
List length l 1 ,and 2
5 10 15 20 25 10-3
10-2 10-1 100
SNR dB
BER
B-CHASE*(1) B-CHASE*(2) B-CHASE(1) B-CHASE(2)
Figure 3-7 Bit error rate versus SNR for the B-Chase detector* ( l ) and the B-Chase detector ( l ) with l =1,2 , T=8,and 16 QAM
From figure we can know selection algorithm 1 and selection algorithm 2 that have almost the same performance.
Table 3-4 System parameters
Transmit antenna 4
Receive antenna 4
Channel is updated in T symbol periods 8
Rayleigh-fading Mean=0,Varance=1
Channel order 0
Selection algorithm 1
List length l 1 ,and 2
0 2 4 6 8 10 12
10-4 10-3 10-2 10-1 100
SNR dB
BER
B-CHASE*(1) B-CHASE*(2) ML
Figure 3-8 Bit error rate versus SNR for the B-Chase detector* ( l ) with l =1,2 ,and the ML detector T=8,and BPSK
From figure we can know the B-Chase detector is nearly ML detector.
Table 3-5 Complexity for B-Chase Detector and ML Detector
Function B-Chase
3.1.5 The B-Chase Detector for Channel Estimation Errors
In previous sections, we always assumed that we have perfect the channel state information (CSI) at the receiver, which allows us to compare the performance. However, the channel information is typically not perfect. A channel estimator extracts from the received signal approximate channel coefficients during the transmission symbol. One method to accomplish this is to transmit the training signal prior to the transmission symbol. That are used as preamble at the start of each frame. Another way to estimate the channel fading coefficients is to embed the pilot bits, that is called pilot signal, inside the signal.
The impact from the channel estimation errors will degrade the performance of the system. To study the impact of the channel estimation errors on the B-CHASE detector algorithm, we introduce the error model at the receiver.
H = H + ΔH ′ (3.1.5.1)
where H represent the true channel matrix and ΔH denotes the channel estimation error. The elements ofΔH are assumed to be zero mean, variance is 0.01 and complex Gaussian. The B-CHASE*(16) is a measurement based on that we can accurately obtain the channel estimation. The B-CHASEer*(16) is a measurement based on that we can not accurately obtain the channel estimation. As shown in Figure, the channel estimation errors with The B-CHASEer*(16) given the B-CHASE decoding algorithm. It is clear from the figure, the B-CHASEer*(16) decoding algorithm starts to perform poorly. This poor performance is caused by inter-symbol interference (ISI).When we obtain the error channel matrix, find out the error outputs, ′F M etc., in the B-Chase preprocessing. From that obtain the error ′′ y = F r ′ produce the ISI. This cause a ISI problem since channel estimation error is the biggest contributor of the errors in the simulation at the high SNR region.
Table 3-6 System parameters
Modulation 16-QAM
Transmit antenna 4
Receive antenna 4
Channel is updated in T symbol periods 8
Rayleigh-fading Mean=0,Varance=1
Error of the Rayleigh-fading Mean=0,Varance=0.01
Channel order 0
Selection algorithm 1
List length l 16
5 10 15 20 25 30 35 10-5
10-4 10-3 10-2 10-1 100
SNR dB
BER
B-CHASE*(16) B-CHASEer*(16)
Figure 3-9 Bit error rate with channel estimation error and without channel estimation error
From figure we can know the channel estimation error demonstrate the error in the high SNR.