3.2 Equalization
3.2.2 Frequency Domain Equalization Combined with Data Demapping
Traditional equalization approaches separate equalization and CCK decoding, and are limited in performance especially when a DFE is used, due to error propagation in the feedback filter. Integrating problems mentioned before, we try an opposite train of thought; consider convolution instead of deconvolution. Now, we attempt to develop a joint equalization algorithm that demodulation is conditional on estimated channel impulse response before. The concept of joint equalization has
been realized by different algorithm in [13] and [14], and been presented demonstrating the greatly improved performance attained using this algorithm to the performance obtained using separate equalization and decoding block. So, the code set used to demodulation is, in the beginning, convolved with channel, and that is the place we claim that it is convolution-directed instead of deconvolution, and it is also called “joint equalization and decoding” or “combined equalization and decoding” in [13] and [14]. with definition in equation (3.7) and (3.8). And
k
c and j are corresponding column vectors of and with extension of zeroes like section 3.1.2. So, the received vector
,where is zero-mean white complex Gaussian vector process with components of variance
Nk 2
σn. For simplicity, assume that the last symbol is decoded correctly, and thus, the term
1 k
j
H cp − can be removed in advance. So, the processed received vector yk′ we will face becomes
k
j
k i
y ′ =H c +Nk (3.17)
According to optimum maximum a priori (MAP) principle for transmission of CCK code words over ISI channel, the probability density function (pdf)
k
j
H c conditioned i
on a hypothesis yk′ is
2
Without loss of generality, we assume 64 CCK code words are uniformly distributed, and equation (3.17) can be reduced to optimum maximum likelihood principle
2
It is clear that the input signal, yk′, will be sent to the 64 correlators with coefficients
k
j
H c and subtract power of each coefficients i
k
j
H c individually. Then, select the i
CCK code word with maximum correlation that means maximum possibility. However, not only the computation of coefficients
k almost kinds of time domain convolution (not really convolution). In order to reuse the equalization module of OFDM and avoid additional module that computes
k
H cp − , we pass the received signal into DFT symbol by symbol. Equation (3.15) and (3.16) can be written as
, where F means 16-point (2 times length of CCK symbol) DFT matrix, and
j
FH cp can be removed as mentioned before. Especially, since Fourier transform is
a kind of linear transformation, is still a zero-mean white complex Gaussian vector process with a variance different from
FNk
2
σn. Similar to equation (3.18), we can obtain a frequency domain maximum likelihood principle,
2
And we can design a frequency domain ML demapper, which is composed of 64 correlators each with coefficient
k
Figure 3.12 joint equalization decoder
Before going on, we clarify the remaining parts that are not solved. One is how to generate coefficients
k
j
FH c , which is used as coefficients for proposed demapper. i
The other is ISI term resulted from last CCK code word,
1 k
j
FH cp − , which is used to make received signal F yk eliminate ISI effects and become F yk′. Review
equations (3.4)~(3.6) and (3.10), and multiply DFT matrix at two sides of equation (3.10),
, where is direct multiplication entry by entry, is estimated channel frequency response, and
Fh1
k
F c is Fourier transforms of CCK code words. In equation j
(3.23), the first term
k
FH c is the Fourier transform of linear convolution of channel j
impulse response and one CCK code word, and the right most two terms are frequency domain body, need for ML demapper and ISI elimination is hided in
k
FH c , and it can be obtained j
by multiplication of Fh1 and
k
F c . However, it is really a big problem to separate j
these two terms. Although they can be separated easily in time domain, our processing domain is frequency domain, and these two terms will be mixed due to Fourier transform.
Now we have to find a way to filter out these two terms in frequency domain.
body ISI
others are zeroes; so,
k
We attempt to design an operator G′, which can filter out
k
Combined with equation (3.24), we have a set of simultaneous equations, and the operator G′ is introduced by
The other part, ISI component affecting next CCK symbol, can be derived from
Consequently, all information is generated for proposed frequency domain maximum likelihood decoder conditioned on CIR. Notice that although
1 k
j
FH cp − is truly ISI component we want to remove when receiving a symbol instead of
1
is just a shifted version of
1 equal to multiply each entry of
1 k
j
FH cn − by
( )
−1 m−1, m is entry index, due to DFT, and it can also be achieved by changing sign bit on odd indices of vector1 k
j
FH cn − . After channel estimation is complete, processes are summarized as:
a) 64 sets of coefficients
k
j
FH c and corresponding ISI effects can be computed i
sequentially according to equation (3.25) and (3.27), and at the same time, coefficients
k
j
FH c are updated to correlators in Figure 3.12. This process will i
be finished before PSDU (CCK symbols) come.
b) As soon as the PSDU comes, the receiving CCK symbols will be sent into DFT first, and the result is F yk . Then, F yk′ is obtained by removing the ISI effect this demapper eventually outputs the most likely data bits and its ISI effect to feedback when next symbol comes.
Figure 3.13 is the corresponding data flow after timing synchronization, frequency synchronization, and channel estimation.
FFT M U X Timing & Operation Mode Control
Frequency Response Buffer Multiplier Point-by-point CFR
OFDM signal
DSSS signal ERP-OFDM
ERP-DSSS/CCK Channel Estimation preparation Joint Equalization Demapper Separation Operator G’
Update
Data OFDM Payload Modules
ISI feedback
Figure 3.13 joint equalization data flow
So far, 16-point FFT was used because of the length of CCK symbol and applied FFT technique, and each resulted coefficient was sixteen long; i.e. the spacing is 90.9ns (11 M Hz/sc). However, it is insufficient to suffer from smaller spacing or severe multipath fading as mentioned in the beginning even if we can estimate correctly. For a given estimated channel at 22 MHz, the relative 22 MHz computations of coefficients, against channel fading, but the results of
k presented in 22 MHz, either. As general DFE structure, the feedforward equalizer is operated at 22 MHz (45.5ns-spaced), but the feedback equalizer is 90.9ns-spaced.
The resulted difference from 11 MHz and 22 MHz is different correlator length. The correlator length at 22 MHz is two times as large as at 11 MHz, and two times hardware cost has to be taken into account. Thus, we try to implement
k
− at both operating rate 11 and 22 MHz— correlator lengths will be 16 and 32 individually—for maximum likelihood demapper under receiver with 22 MHz sampling rate and do some analysis in chapter 4.
It is simple for consideration of 22 MHz
k lengths of operations are merely doubled, and the separation operator G′ becomes
16 16 16 16
− derived from 22 M Hz computation, the lengths of prepared frequency domain CCK code words and DFT are, respectively, 16 (two times upsampled and it will be extended to 32) and 32 during coefficient computing, while the resulted coefficient length is maintain at 16 instead of 32 and incoming CCK symbol is received at 11 M Hz as well. This “down conversion” technique here lies in the operator G that produces separation operator G′ again and is according to interpolation concept. Recalling an application of DFT signal processing, when we interpolate zero behind each element of x
( ) {
i = x x0, ,1 ,xN−1}
, the Fourier transform of x i( )
is equal to the first half of Fourier transform of x′( ) {
i = x0, 0, , 0,x1 ,xN−1, 0}
that is interpolated from x; that is,
For this reason, the separation operator G′ is therefore designed as
16 16
As a result, the coefficients used for 11M Hz maximum likelihood demapper under 22 MHz receiver is obtained by retaining the first half after the output of separation operator . Worthy to notice that since we only need first half of the results of separation operator, the computation of separation are substantially the same with that under 11 MHz receiver. Although sampling rate and the process of separation are raised two times to 22 MHz, the hardware cost does not increase. Only half of information to be considered leads to performance loss definitely, and we will show how the loss is by simulation with kinds of channel models in chapter 4.
G′
Ch4
Simulation Results
The packet error rate (PER) of two estimation methods with proposed demapper is regarded as our judgment and simulated under indoor channel models proposed by IEEE, JTC, and SPW with different RMS delays, different spacings of channel, and different sampling rate of receiver. The results presented in this section are carried out with ERP-DSSS/CCK 11 Mbps mode. We also assume a fading channel varying from packet to packet; i.e., it is constant within each packet, and the required performance is measured at a PER of 8%.