Chapter 4 Channel Estimation Algorithms
4.3 Channel Compensation and Demapping
Intuitively, the channel compensation is implemented by using a divider, where the denominator is the value of estimated channel responses and the numerator is the received sub-carriers, and the output is for the demapper. The hard demapping process uses the result from divider and finds the distance to each point in QAM constellation map. After this, the point with minimum distance represents the result of hard demapping.
In [27] and [28], a divider free idea is referred. The idea is to multiply the denominator to both sides as shown in Eqn. (4.6).
) ( ) ( ) , ) (
( ) , ) (
( SC n m X m CR m
m CR
m n m SC
X = ⇒ = × (4.6)
where X(m) is supposed to be the answer of SC(n,m)/CE(m) which is unknown yet.
Since the value of points in constellation map is known, replacing the X(m) by using all possible points in constellation map and finding the distances to all points are practicable. The point in constellation map with the minimum distance indicates that it is the most like-hood answer. Then the demapper outputs the demapping result which the point represents. This derivation can be thought as magnifying the constellation map by a scale which value equals to the denominator. However for the 64-QAM application in DVB-T, using exhaustive method means at least 63 comparators and 64 multipliers are necessary. Therefore, in [28] a three-stage demapping algorithm is jointed to reduce the hardware cost. By using dichotomy method, the X(m) can be replaced by axes and middle line of two neighboring points.
Therefore the system needs few reference values stored, three multipliers and three comparators. Eqn. (4.7) illustrates the idea of Eqn. (4.6) in detail.
[ ] [ ]
CRxx(m) and Xxx(m). There are two possible choices to cancel the denominator. One is to multiply the denominator to the right side directly. Therefore, the left side is a complex number and right side becomes to a complex number multiplies a complex number. The second choice is to multiply the conjugate of denominator to the numerator and denominator. After multiply operation, the numerator is still a complex number but the denominator turns to an integer. Then move the integer denominator to right side and the operation is done. This transfer operation makes a complex division turn to two integer divisions and then makes the right side multiply the denominator as a scale. Comparing to the first one, the second operation uses an integer multiplier, which can be simplified by several adders, to scale the constellation map instead of a complex multiplier. Therefore, the power consumption and hardware cost are able to be reduced in the hard demapping process which will be discussed later.After partitioning the equation into real and image parts, the demapping process is able to demap the real part and image part separately. In [1], all possible situations were described and the values of all positions were also defined. As a result, all the
possible middle line values are stored previously. Fig. 4.15 represents the values of F1
in Eqn. (4.7).
Fig. 4.15 Division simplification results in QAM
A divider free and fast three-stage real and image individual hard demapping process is proposed below:
1. Get the sign of Re: if Re<0 y0=1, else y0=0
2. If constellation mode is 16-QAM or 64-QAM go to 3, otherwise go to 7.
3. If 16-QAM B=α+1, else B=α+3 4. If |Re|>B×NF×INT y2=0, else y2=1.
5. If constellation mode is 64-QAM go to 6, otherwise go to 7.
6. If y2=0, B=B+1: if |Re|>B×NF×INT y4=0, else y4=1
else B=B-1: if |Re|>B×NF×INT y4=1 else y4=0 7. y0,y2,y4 is demapped.
Repeat the same procedure to demap the y1,y3,y5.
In this procedure, the Re is the real part of the numerator, INT is the denominator F2 of Eqn. (4.7), NF is the normalized factor referred in [1] and B is the decision boundary (the X-axis/Y-axis value between two neighboring points in X-axis/Y-axis aspect). As a result of the derivation above, the system can magnify the constellation map by a scale which value equals to the denominator (INT). Then detect what region in the magnified constellation map does the F1 locates on.
Fig. 4.16 (b) is the block diagram of proposed three-stage hard demapper, where the function of stage 1 is the as same with Eqn. (4.7), which will first examine the sign of F1 to decode the y0 and y1. Because of the first reference values are X axe and Y axe, the multiplier is able to be saved. After stage 1, stage 2 uses the information from stage 1 to choose a decision boundary value from Fig. 4.17 (a). The system will tell the F1 value is bigger or less then the value of decision boundary multiplies the denominator (INT×B×NF) to decide the region F1 locates. Stage 3 repeats the actions of stage 2 and uses the information from stage 2 and decision boundary shown in Fig.
4.17 (b).
As shown in Fig. 4.16 (a), before the constellation mode is detected by TPS decoder, the system use stage 1 to decode y0 and y1 as QPSK mode. If the constellation mode is 16-QAM, stage 2 starts to decode y2 and y3. Stage 3 only works at 64-QAM mode. By using this hierarchical decoding scheme, the system can disable the useless stages to reduce the unnecessary power consumption. For example, stage 2 and stage 3 are disabled while doing QPSK demodulation.
Stage 1 (y0y1)
16 or 64 QAM?
Stage 2 (y2y3)
64 QAM?
Stage 3 (y4y5) Start
End Y
N
Y
N
(a)
(b)
Fig. 4.16 (a) FSM and (b) block diagram of hard demapping
DB S2 -DB S2
DB S2
-DB S2
(a) (b) Fig. 4.17 Decision boundary of (a) stage 2 and (b) stage 3
Fig. 4.18 is an example of the three-stage demapping process. The three-stage demapper demapps the real and image parts individually and using their intersection to finding the demapping results.
Location region after stage 1 Location region after stage 2 Location region after stage 3
Fig. 4.18 Example of three-stage demapping process