List Sphere decoding (LSD) is modified from sphere decoding [17]. LSD gives a list L which contains Ncand candidates of X with smaller values in eq. (2.15) for generating the soft information. In order to generate L , the sphere decoder needs to be modified in two ways. Every time it finds a point inside the initial radius r：
1) it does not decrease r to correspond to the radius of this new point;
2) adding this point to L if the list is not full; or if L is full, it compares this point with the point in L with the largest radius and replaces this point if the new point has smaller radius.
By the changes above, List Sphere Decoding searches all the points which are inside of the sphere with given initial radius, and gives the Ncand best point. Sphere Decoding changes to search the point in new sphere (the center of the sphere is still the same) when its find a new point whose distance to the center of the sphere is smaller than initial radius (mentioned in section 2.3.1). So the points which are inside the old sphere and outside the new sphere will not be considered in Sphere Decoding and the searching tree of SD goes smaller and smaller (actually, another smaller tree) which is not the case in LSD. The searching tree of LSD is still the same and all the branches must be gone through.
Fig. 2.5 Sphere of SD or LSD in received signal domain.
Here, we use Fig. 2.5 to explain the different between SD and LSD. In Fig.2.5, the circle mark represents the center of the sphere and cross mark means the feasible solution in received signal domain where the number beside of them is used to indicate. For SD with
initial radius r , if the cross mark 4 is first found and radius 1 r is used to be the new radius 2 of the searching tree then cross mark 1、2 and 3 will not be presented in new searching tree with radius r . Again, if the cross mark 6 is found in the new searching tree with radius 2 r 2
then cross mark 5 will not be considered for next searching tree. However, for LSD with initial radius r , if the cross mark 4 is first found, cross mark 1、2、3、5、6 are still in the 1 searching tress with radius r because LSD will not change the radius. Back to fig. 2.3, when 1 LSD finishes the searching operation at layer 1 successfully, it puts this point into the list, backs to layer 2, and keeps the algorithm going. Then all the feasible points in the sphere will be found out successively (the finding order is independent of the distance to the center of the sphere).
Chapter 3
Grouped Based ICI Cancellation Method
3.1 System Model
In section 2.2, we have known the ICI effect on OFDM systems and its mathematic representation. Now, we change the mathematic representation in section 2.2 into matrix form which is convenient to be used in proposed method. Eq. (2.12) can be rewrote as follows：
H
= N
x F X (3.1) where x=[ ,...,x0 xN−1]T, X=[X0,...,XN−1]T as showed in Fig. 2.1 and FN is the N point FFT matrix which can be represented as follows：
2 0 0 2 0 1 2 0 ( 2) 2 0 ( 1)
Assume that the maximum delay spread of the channel always less than or equal to the length of cyclic prefix be used, so the received signal y=[y0,...,yN−1]T and its FFT becomes：
= +
3.2 Group based ICI Cancellation Method with Sphere Decoding
From [8], we know that most of the ICI effect on a subcarrier comes from neighboring
subcarriers, so we assume the window length of the ICI effect is Q=2q+1 that is Xi causes
Fig. 3.1 Group Method for ICI with window length Q=2q+1.
The basic ideal of our method is to cancel the component of Xj, i≠ in j Yi. Again, form
[8], most of the ICI effect on a subcarrier comes from neighboring subcarriers. We use
2 1
nx > q− so that Yi is affected by Xi−1, Xi and Xi+1 only. The relationship of Yi,
i−1
X , Xi and Xi+1 can be represented as follows：
1 1 1 1 have been cleaned, we expect that the more correct Xi can be obtained. Using the procedure
as described above iteratively, we expect that the recover performance will be better and better.
Fig 3.2 is the block diagram of the algorithm for one group Yi. At initial state, switch links to the path which input a zero vector to do ICI cancellation (i.e. no ICI cancellation). At the (j+1)th iteration, switch links to the path which input [Aii−1](Xˆi−1)j and [Aii+1](Xˆi+1)j where
ˆ 1
(Xi−)j represents the output made from the (i-1)th group at the jth iteration for doing ICI cancellation. After ICI cancellation, Sphere Decoding generates (Xˆi−1)j+1 can be used in ICI
cancellation for next iteration.
Fig. 3.2 Block diagram of group based ICI cancellation method with Sphere Decoding.
3.3 Group based ICI Cancellation Method with List Sphere Decoding
Different from section 3.2, we apply List Sphere Decoding to generate some candidates of Xi and use these candidates to compute the soft symbols for doing ICI cancellation.
3.3.1 Soft Symbol
For QPSK modulation, soft symbol can be calculated as follows：
, , , is selected form the list generated by List Sphere Decoding [22]. Assume that the prior probability of every bit is equal probability and is independent to each other, so we can go further from eq. (3.15) as follows：
, , 1
3.3.2 Conditional Probability
From above derivations, if we want to compute the soft symbol then we need to find ( i| i)
assume n and I i n are independent Gaussian random variable [4], so we need compute the Qi
mean and variance of n in order to get i P Y X ( |i i)
( ) ( )
Above is the case of initial state which ICI cancellation has not been done yet. There are some changes to be done after ICI cancellation. Fortunately, we only need to change the mean and variance of n . Again, let’s see the subcarrier j Y after ICI cancellation, i
[ ] (
ˆ)
ˆFig. 3.3 Block diagram of group based ICI cancellation with List Sphere Decoding
The block diagram of the group based ICI cancellation with List Sphere Decoding is showed in fig. 3.3. The procedure is as the same as the one with Sphere Decoding except that Sphere Decoding in fig. 3.2 is replaced by List Sphere Decoding with a soft decision device
followed to compute the soft symbol for ICI cancellation here.
3.4 Radius of the Sphere
In chapter 2, there are two key questions of Sphere Decoding are mentioned. The second one can be solved by the algorithm of Sphere Decoding, but the first one how to choose the radius does not have exactly solution. Here, we use a simple method to decide the initial radius for each application of Sphere Decoding in our group based ICI cancellation method. From eq. (2.21), we use least square solution to be the center of the sphere. We know that least square solution is the best solution to satisfy the ML criterion, but it may not be the feasible solution that is, it may not be a point on the signal constellation. Because of the channel effect, the hard decision of the least square solution may not be the best solution within the all feasible solutions. We can use Fig 3.4 to interpret the problem above. In Fig 3.4, left hand side is the feasible point in transmitted signal space and right hand side is the transformation of left hand side by the channel matrix. On the left hand side, square is the least square solution and the circle is the hard decision of the least square solution. On the right hand side, square Yˆi is the transformation of the least square solution and circle is the transformation of the hard decision value. Hard decision of the least square is the nearest point of the least square solution within the all feasible points (on the left hand side), but after
transform hard decision is not the nearest point of least square solution’s transformation.
Fig. 3.4 signal space in transmitter and receiver.
If hard decision of least square solution after transformation is not the nearest feasible to the center of the sphere Yˆi in the received signal space then the solution which satisfies eq.
(2.21) must be inside of the sphere with radius which is equal to the distance between Yˆi and the hard decision of least square solution after transformation. By the notion above, it is reasonable to set the initial radius of the sphere to be the distance between Yˆi and the hard decision of least square solution after transformation.
Chapter 4
Simulation Result
4.1 Simulation Environment
Simulation results are given for the proposed group based ICI cancellation with Sphere Decoding and List Sphere Decoding under two kinds of simulation environments.
For the first one, a 6 taps channel is considered, and each tap of the channel are modeled as independently complex Gaussian random process which is generated by the Jakes’
Doppler spectrum with 120、240 or 300 km/hr relative velocity. The relative delay of the first tap is zero and others are uniformly distributed in the interval [1,NG] where N is the length G of cyclic prefix we used. The power of the path 2, 3, 4, 5 and 6 is 1dB, 9dB, 10dB, 15dB and 20dB smaller than the first path. An OFDM system with N=256 subcarriers and quarter phase shift keying (QPSK) are simulated. The carrier frequency is 2.5GHz and the bandwidth of the system is 5MHz. According to the relative velocity and parameter above, f T equal to d s 0.0142、0.0284 and 0.0356. The detail of the parameters can be checked in Table 4.1.
For the second one, a 6 taps channel which is as the same as the first one is considered, but the relatively velocity of the Jakes’ Doppler spectrum changes to be 85、170 or340 km/hr.
An OFDM system with N=64 subcarriers and quarter phase shift keying (QPSK) are simulated. The carrier frequency is 2GHz and the bandwidth of the system is 200 kHz.
According to the relative velocity and parameter above, f T equal to 0.05、0.1、0.2. The d s detail of the parameters can be checked in Table 4.2.
Table 4.1 Simulation parameters of the first kind of simulation environment Modulation QPSK
Path 6
Relative power (dB) 0,-1,-9,-10,-15,-20
Cyclic prefix length 16
Carrier frequency 2.5 GHz
Subcarriers 256
Bandwidth 5MHz
Vehicle speed (km/hr) 120 240 300
d s
f T 0.0142 0.0284 0.0356
Table 4.2 Simulation parameters of the second kind of simulation environment Modulation QPSK
Path 6
Relative power (dB) 0,-1,-9,-10,-15,-20
Cyclic prefix length 16
Carrier frequency 2GHz
Subcarriers 64
Bandwidth 200kHz
Vehicle speed (km/hr) 85 170 340
d s
f T 0.05 0.1 0.2
4.2 Simulation Result Discussions
4.2.1 Simulation in Environment I
Fig. 4.1 shows the bit error rate of the OFDM system which applies group based ICI cancellation method with Sphere Decoding, and group size is equal to 8 and Speed is equal to 120 km/hr. The fist curve we use group based method without ICI cancellation. We can see that error floor appear when SNR is high if ICI cancellation is not used, and the curve of iteration number equal to 1 and 4 is almost the same as the perfect one which use the correct data of one neighboring groups on each side to do ICI cancellation.
Fig. 4.2 shows the bit error rate of group based ICI cancellation method with Sphere Decoding in different speed with iteration number equals to 4. For high speed case, we can get the diversity gain from ICI but error floor occurs when SNR is high. The performance difference between perfect ICI cancellation and not perfect ICI cancellation is getting larger and larger when speed is getting faster and faster.
Fig. 4.3 shows the bit error rate of group based ICI cancellation method with Sphere Decoding in different group size and speed with iteration number equals to 4. We can see that a larger size of group implies a better performance, but the error floor problem is still not improved by using a larger group size.
Fig. 4.4 shows the difference of bit error rate between group based ICI cancellation method with Sphere Decoding and List Sphere Decoding. LSD can get better performance than SD, but the improvement is not good enough. No matter LSD or SD is applied, the performance still has a gap between perfect one. At last, it seems that group based ICI cancellation method with LSD need less iteration number than SD to get the same performance.
Fig. 4.5 shows the bit error rate of the group based ICI cancellation method with LSD in different group size and speed. Just like the SD case, the performance improvement is not very well for using larger group size. Back to the Fig. 4.3 which use SD, the curve with speed equal to 300km/hr has worse BER than the one with speed equal to 120km/hr at high SNR, but it is different in LSD case the curve with speed equal to 300km/hr still has better BER than the one with speed equal to 120km/hr at high SNR.
Fig. 4.6 and Fig. 4.7 shows the comparison of BER in different group sizes with SD and LSD when the speed is 240 km/hr. Both of SD and LSD, the performances become worse and worse when group size gets smaller and smaller.
Fig. 4.8 and Fig. 4.9 are the comparison between group size equal to 2 and 4 in different speed with SD and LSD. From these two figures, we can find that the performance of group size equals to 2 and 4 are almost the same when speed is equal to 120 km/hr, but the performance of group size equals to 4 is better than equals to 2 when speed equals to 240
km/hr and 300 km/hr. The phenomenon above shows that if ICI effect becomes severer, then we need a larger group.
Fig. 4.1 Comparison of BER in different iteration number (I).
Fig. 4.2 Comparison of BER in different speed (I).
Fig. 4.3 Comparison of BER in different group size (I).
Fig. 4.4 Comparison of BER in different method (I).
Fig. 4.5 Comparison of BER in different speed and group size (I).
Fig. 4.6 Comparison of BER in different group sizes with SD (I).
Fig 4.7 Comparison of BER in different group sizes with LSD.
Fig. 4.8 Comparison of BER in different speed and group sizes with SD (I).
Fig. 4.9 Comparison of BER in different speed and group sizes with LSD (I).
4.2.2 Simulation in Environment II
Fig 4.10 is bits error rate of the group based ICI cancellation method with Sphere Decoding. The sizes of every groups is 8 and we set the window length of ICI is 9. The first line only use group based method without ICI cancellation. We can see that performance gets a lot of improvement by using ICI cancellation. Performance goes better and better with numbers of iteration increasing. Unfortunately, the performance saturates at numbers of iteration equal to 3 and the this performance still much worse than the one (the last curve in Fig. 4.1) which ICI effect comes from two groups nearby ( i.e. Xi−1 and Xi+1 for Xi ) are perfect canceled. By the mention above, the decision made by each groups is not good enough to make the performance closes to the last curve in Fig. 4.1.
In Fig. 4.11, we try different sizes of group. We use 4, 8, 16 three kinds of size and the corresponding ICI windows length are 5, 9 and 17. With numbers of iterations equal to 4, the larger size of group has better performance and higher diversity gain. However, as mention in Fig. 4.1 the error floor still exist because of the accuracy of others group decision.
In Fig. 4.12, we try different kinds of f T . As we know the higher d s f T causes d s
severer ICI effect and Fig. 4.3 show the same result. The performance can be improved by using larger size of group, but the improvement still not good enough when f T is equal to d s 2 or higher. We also can find that it seems no error floor when f T equal to 0.05 and size of d s
group equal to 16. One more thing to be mentioned is that the method we proposed here can get diversity gain form higher f T in low d s Eb/N . Compare with the method in [6] which 0 use all the subcarriers to do Sphere Decoding. In the case of f T equals to 0.1, although we d s have error floor effect, our method has better performance before Eb/N reaches to 32dB. 0 However, in the case of f T equals to 0.2, our method has better performance before d s
/ 0
Eb N reaches to 22dB.
In Fig. 4.13, we show the performance comparison of group based ICI cancellation method with SD and LSD. As the same as the case of Fig. 4.1, the size of the group is equal to 8 and f T is equal to 0.1. The solid line represents the method with LSD and dash line d s represents the method with SD. Different marks represent different numbers of iteration. We can find that the performance of the method with LSD is better the one with SD in every kind of numbers of iteration. We can see that the performance of the method with LSD whose number of iteration is equal to 2 is better than the method with SD whose number of iteration is equal to 3, so the method with LSD has faster speed of being saturation than the method with SD.
In Fig. 4.14, we show that the error floor effect comes from the accuracy of the decision made by groups nearby (we use them to do ICI cancellation) not comes from others group which we do not consider the ICI effect comes from them. The group’s size of the solid curve in this figure is equal to 8 and 16 for dash curve. The curve with star mark cancels the ICI
effect perfectly comes form the nearby groups and the curve with diamond mark cancels all the ICI perfectly from other groups expect itself. We can see that the dash curve with star and diamond are very close, it means that it does not matter how many groups is taken to do ICI cancellation if the size of group is large enough (also see the curves with star mark). Now look back to the dash curve with circle mark which use the decision of the nearby groups to do ICI cancellation. The different between the dash curve with circle mark and star mark is that the decision of nearby groups is correct or not.
Fig. 4.10 Comparison of BER in different numbers of iteration (II).
Fig. 4.11 Comparison of BER in different sizes of group (II).
Fig. 4.12 Comparison of BER in different f T (II). d s
Fig. 4.13 Comparison of BER in LSD and SD (II).
Fig. 4.14 BER performance of group based method with perfect ICI cancellation (II).
4.3 Computational Complexity
The expected complexity of the sphere decoding algorithm is O N( 3) when the signal-to-noise ratio (SNR) is high [23]. Where the sphere decoding algorithm is applied in a subspace of S , N S is a set corresponding to the modulation scheme and N corresponding to number of subcarrers or group size in group based method. By the notion above, we can get a brief computational complexity comparison (Table 4.3) between the method which using all subcarriers to do Sphere Decoding and the method we propose. Let N denotes number of subcarrers of OFDM system, N denotes group size of group based ICI cancellation method, x
and I denotes the iteration number in group based ICI cancellation method. As we know group based method has N N groups and every group will do Sphere Decoding with / x N x subcarriers. The operation of group based method iterative I+ times (include initial state). 1 In table 4.3 we use an example to make above notion more clearly.
Table 4.4 show the numbers of adder and multiplier are used in group based ICI cancellation method with different sizes of group and general Sphere Decoding without group based method [6] under the second kind of environment which the number of subcarriers N of the OFDM systems is equal to 64. This simulation considers the case which Eb/N is equal 0 to 32dB and for every group based ICI cancellation method the numbers of iteration is equal to 4. The same as we expect, as the group size goes larger and larger the complexity goes
larger and larger. Although, Sphere Decoding without group based method (group size is
larger and larger. Although, Sphere Decoding without group based method (group size is