Chapter 3 B-Chase Detector
3.1 Introduce B-Chase Detector
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