3.1 Channel Estimation
3.1.2 Proposed Scheme #2
Different from the previous algorithm, this one is based on well-known FFT technique, fast linear convolution. The linear convolution of two finite sequences is commonly performed by extending the input sequences with zeroes to length 2N, and then evaluating the circular convolution of these two sequences using FFT.
Because the architecture of FFT adopted here is 2-radix and is easily adjustable for power-of-2 FFTs, such as 16, 32 or 64 points in our simulation platform, N is the smallest power of two greater that or equal to the longer one of two input sequence.
Consider the linear convolution
{
c n( )
, 0≤ <n Na+Nb−1}
of two finiteTherefore, c1 can be written as the linear convolution in form of matrix operation,
(3.4)
Because of zeroes padding in later half of and , the above equation can also be written as circular convolution:
a1 b1
(3.5)
Consequently, the FFT of , , is directly multiplied by the 2N-point FFTs of and , and
In other words, linear convolution or deconvolution process can be accomplished fast by frequency domain multiplication or division, which does not require circular property by zeroes padding technique. Undoubtedly, the technique is very useful for a system without special design against ISI, like DSSS system, and we attempt to replace and with multipath channel impulse response and spreading sequence respectively.
a b
The key issue is that such technique is only for linear convolution of two “finite”
sequences; yet, signal transmission is continuous symbol by symbol. Concerning every received symbol, it is interfered by previous symbol and resulted interferences that exceed one symbol length will last to next symbol. A complete linear convolution of a spreading sequence and multipath cannot be obtained. If the received symbol can be reconstructed to pure convolution result of referenced sequences, channel will be estimated easily.
Now, let sk
( )
n be the kth spreading symbol with length L , and H, s, be channel matrix composed of h n
( )
, n=0 Lh− , where 1 (the channel spans about 2 spreading symbols). Note that andLh ≤Ls
h1 sk are 2N-long vectors of and with zeroes padding, where N is the smallest power of two greater that or equal to
( )
h n sk
( )
nL . Moreover, the channel matrix can be broken up into two parts: s
(3.8)
, where can be regarded as the shifted version of , and this representation is used to clarify effect caused by previous symbol and inter chip interference. At the same time, linear convolution can be thought as super-position of convolution of each fragment. We divided each linear convolution result of transmitted symbol and multipath into “body”, signals within one symbol length, and “ISI component”, effects on next symbol. Therefore, a receiving symbol
Hp Hn
( )
yk n , n=0 Lh− , that would be 1 extended to vector form yk by zero-padding can be illustrated as Figure 3.9, and its
( ) ( )
bodymatrix equation can be correspondingly formulated as follows:
1
k i k p k
y =H s +H s − (3.9)
In equation (3.9), the first term is the body from receiving symbol sk and multipath,
and the second is the ISI component caused by previous symbol sk−1. For the purpose of reconstruction, we want to remove the ISI component attributed to previous symbol, and concatenating the ISI component that present symbol leads to and influence next symbol as shown in Figure 3.10.
ISI
In vector expression, we can formulate above concept as
1 1 are all zeros — this means they are signals or effects within one symbol length; on the contrary, the first L entries of s H sn k are zeros, and the others are not — this represents ISI component. That is, after removing the ISI part attribute to last symbol (−H sp k−1) and concatenating the ISI part that the present received symbol leads to (+H sn k ), the result will have the relation in equation (3.6), and thus, the channel frequency response can be worked out by the reconstructed result and detected sk .
So far, the only thing remained is how to Figure out H sn k and H sp k−1. These two terms in essential stand for the same effect, ISI resulted from spreading sequence no matter sk or sk−1 . If we know the resulted ISI, we can do reconstruction by detected receiving spreading sequence. As far as DSSS mode in our platform is concerned, the preamble is modulated by DBPSK and spreading PN code is 11-chips Barker code. We denote the positive codeword — phase change 0
change 180 degrees—as B− . Receiving symbol sk can be either B or B− as
The ISI component and body of B convolved with channel eventually can be obtained respectively by inter-cancellation between different cases of signals, and it is able to be exploited not only for DBPSK but also for DQPSK or for other more complicated modulations (the larger code set is, the more complexity will be).
Consequently, let us summarize and introduce this algorithm to four steps in our platform:
a) Collect two sets of received spreading symbol:
One gathers or sums the symbols same with the previous one, and the other gathers or sums the symbols different from the previous one. By means of summation, noise can be suppressed.
b) Figure out the overall/average ISI component and body of B according to equation (3.13) and (3.14).
c) Reconstruction:
Use the result of b) to remove and concatenate the relative ISI components component based on different sets.
d) The multipath channel frequency response can be derived from
( ) ( )
2 1
2 N
N
F R
H
F B
= (3.15)
, where R is the reconstructed result from c), is 2N-point DFT matrix, and is the Fourier transform of that is extended from by zeroes padding.
F2 N
H1 h1 h n
( )
Above steps are carried out in data flow Figure 3.9.
Collected Pattern1 FFTM U X
peak triggerTiming & Operation Mode Control Frequency Response Buffer Multiplier Point-by-point
Overall Symbol with ICI Frequency Response of Pre-stored Long Preamble
CFR
sequential samples at 11 or 22 MHz/sec Frequency Response of Pre-stored Pattern
Collected Pattern2 Overall ISI OFDM signal
Pattern Recognition
DSSS signal
S E L ERP-OFDM ERP-DSSS/CCK 32 for 11 M Hz/sec 64 for 22 M Hz/sec
Figure 3.9 proposed scheme #2 channel estimation combined with OFDM channel estimation
The scheme is perform on platforms working at 11 MHz and 22 MHz for channel spacing 90.9 ns and 45.5ns respectively, as Figure 3.10 and Figure 3.11. They are simulated under a randomly generated IEEE multipath model with RMS delay 100 ns and SNR 7 dB by means of 32-point (N=16) FFT for the former, 64-point (N=32) FFT for the latter. We can obviously see that the accuracy of estimation result in Figure 3.10 is also very precise and the performance is almost the same with scheme #1 according simulation result. Moreover, scheme #2 is not restricted by sampling rate, that is, even if sampling is raised to 22 MHz, this algorithm is available, too. Although the accuracy under ISI resulted from transmit filter at sampling 22 MHz is still less than that at sampling 11 MHz, such estimation result can still make proposed receiver satisfy required system specs, remained to chapter 4, and prove the validation.
For advance consideration with longer multipath delay spread exceeding one spreading symbol length, scheme #2 can extend originally-considered one symbol spreading sequence to two or more ones against longer and more severe multipath, and then, the caused ISI to next two or more symbols can also be estimated by similar principle with more cases (more complicated algebraic equations).
Figure 3.10 estimation result of scheme #2 and ideal CIR over 11M Hz
Figure 3.11 estimation result of scheme #2 and ideal CIR over 22M Hz