Chapter 3 Proposed Algorithm
3.3 The Proposed Algorithm
The timing uncertainty often arises when using the time domain algorithms, except the more complicated ML ones, are used. This implies the ISI will be introduced if the estimated results of the time domain algorithms are directly applied. Often an ad hoc correction term is applied to achieve better results after the location of the peak correlation coefficient being determined. But the compensating rules are ad hoc as well and the optimal timing estimation is usually not obtained. Frequency domain algorithms, on the other hand, may avoid this problem. The receiver structure is listed below, Fig 3.3. In the figure, the receiver generates the received packet from the antenna first. After the S/P, the signals in the packet pass to the FFT directly. Then from the outputs of the FFT, the symbol timing offset estimator collects the information to estimate the symbol timing offset. The received data is also passed to the demodulation
Figure 3.3 Block diagram of the receiver.
Before going to the details of the algorithm, the OFDM signal model and the concept of differential phase offset (DPO) are introduced first, especially the DPO is taken as a metric that contains information of any misalignment of symbol timing, i.e., The DPO is the data used for determining the symbol timing offset. DPO is introduced first, and the OFDM signal model is followed.
For the convenience and distinguishing with all other phase terms talked in the algorithm, DPO is defined to represent the phase difference of two adjacent received subcarriers. Note that the phase of each received subcarrier has made some modification before generating DPOs. More delicate details are given in the following.
First, suppose there is a time domain transmitted signal x(t), then the pair of time domain signal x(t) and its FFT Xk is represented as:
( ) k k exp( k)
x t
⇔X
=A j ϕ
. (3.2)Therefore, after the transmission, the received signal R(t) and its frequency domain signal can be expressed as:
( ) k kexp( k)
R t ⇔R =B jΨ . (3.3)
In the last two equations, Ak and
φ
kare the amplitude and the original phase of X
k respectively, likewise, Bk andΨ
k are the amplitude and the phase of Rkindividually. Because the phase After the phase
Ψ
k is modified by the phaseφ
k , finally, DPO (denoted byΔ
k) is the modified phase difference of the neighboring k-th and (k+1)-th subcarrier, i.e.,1 1
( ) (
k k+ ϕk+ k ϕk)
∆ = Ψ − − Ψ − . (3.4)
The definition of DPO is stated above and the reason why DPOs could be used for determining the symbol timing will be given in the later. For now, the focus is changed to the signal model. Through this model, the core idea of the proposed algorithm is also introduced.
The OFDM signal is generated at baseband by taking the inverse fast Fourier transform (IFFT) of quadrature phase shift keyed (QPSK) subsymbols.
The samples of the transmitted signal can be expressed as:
1
where cn is the modulated data with the phase
φ
k, N is the number of IFFT points, Ng is the number of guard samples, and n stands for the index of subcarrier. All the parameters are all defined in the Fig 3.4. Meanwhile, at the receiver side, there exist impairments caused by carrier-frequency offset, sampling clock errors, and symbol timing offset. Usually, the frequency offset and timing errors are more dominant than the sampling clock inaccuracy.Hence, in this paper, the symbol timing synchronization will be considered assuming a perfect sampling clock.
Then, suppose the channel is modeled with a timing delay (D) and a phase offset (θ), then it can be written as:
h( k )=exp( j ) ( kθ δ −D ). (3.6) Therefore, the received OFDM signal, r(k), through the equivalent channel, h(k), will be
( ) exp( ) ( )
r k = j s k Dθ − . (3.7)
To simplify the illustration of the algorithm, the noise term is ignored. As a result, the i-th received OFDM symbol will be
1
This received OFDM symbol will be passed to FFT directly. At the output, the symbol will be
Figure 3.4 Inputs and outputs of IFFT and the OFDM signal.
where k is a index of subcarrier, and i is a received symbol index. Carrying equation 3.8 r (ki
) into equation 3.9 R
i,k to replace r (ni), then, the received OFDM
It’s obvious that the index n and the index l are the same in equation 3.10. As a
result, the received i-th OFDM symbol at k-th subcarrier can be expressed as
Until now, the definition of DPO and the system model are both introduced.
The next step is the key player of proposed algorithm. How to generaet the DPOs from the received signals and why the proposed algorithm works are given in the following paragraph.
First, extract the phase of Ri,k out, denoted as
Ψ
k. Note the index i is omitted because the operations made in the following all in the same OFDM symbol, so only the index k of subcarrier need to taking into consideration. So, the phase of the i-th OFDM symbol at k-th subcarrier is2 computational result is
1 1
On the other hand, there is an effect need to consider if the symbol timing doesn’t fetch the ideal FFT window, as mentioned in 3.2. Because of symbol timing effect induced by the non-ideal FFT window, the received i-th OFDM symbol at k-th subcarrier after FFT will be
, ,
where Td , normalized to sample based, is the symbol timing offset.
Follow same steps of generating DPO, the DPO will be
1 1 Hence, the timing offset introduced by D and Td can be estimated through the DPOs as well.
Clearly, the information of symbol timing offset is encoded in the DPOs.
Assuming there are N (indexed by n = 1 to N) subcarriers in an OFDM symbol, then there will be (N-1) DPOs (indexed by k = 1 to N-1) available for one OFDM symbol. From the relation between time and frequency domain, a delay/ahead in time domain will introduce a negative/positive phase in frequency domain. So, DPO is ranged from -π to π. But for the simplicity of histogram, DPO maps to the new range of 0 to 2π temporally. In this way, the histogram will be done by simple comparisons. Once the phase introduced by a timing delay is estimated, the result still needs to re-map to the range between -π and π while computing the value of time delay D.
The estimator is shown in figure 3.5. In summary, the gap between the
phase angle and the correct phase of each tone is calculated, i.e., for each k, the difference (
Ψ
k- φ
k)
is calculated. Note that since the calculation is done with mod 2π arithmetic, the result will be wrapped within the range of 0 to 2π. Then the DPO is calculated for k = 1 to N-1. This calculation is done with same operation, mod 2π arithmetic, so the result is between 0 and 2π as well. There are (N-1) DPOs calculated for one OFDM symbol, but the receiver is not restricted to use all of them for the purpose of symbol timing synchronization. A subset of the calculated DPOs may be used. Nor is the receiver restricted to use one OFDM symbol’s worth of data to estimate the symbol timing offset. More than one OFDM symbol can be used if conditions permit. For example, if K OFDM symbols and (N-1) DPOs per symbol are used, there will be K(N-1) DPOs to be processed. In this paper, only the case of one symbol (K = 1) is described for the sake of simplicity. The case of multiple symbols is a straightforward generalization that can be done easily. Next, the histogram of DPOs is obtained. After the interval from 0 to 2π is normalized to a new range from 0 to 2 -1N , this new interval, from 0 to 2 -1N , is divided into 2L bins. The key idea of the method is to fetch the first L bits of each normalized DPO to be the selector of a MUX and a DEMUX (Fig. 3.6). Each address of register, which is used for saving the current value to a corresponding bin, is treated as the lower boundary of each bin.Figure 3.5 Block diagram of the symbol timing offset estimator.
The value stored in each register will be fetched and add 1 to the value, then the new value will be returned to the same register when a register is selected by a truncating DPOs. After all DPOs are checked, the histogram operation is done at the same time. The histogram of DPOs is further processed to determine the timing delay D. The peak of histogram, i.e., the bin which has the highest frequency, is chosen. Assume the peak occurs at the l-th bin. The l-th bin ranged
from ( 1) 2l
average of the possible values. Note the estimation of D will be (l 0.5 1 N L
− − ) if
the lower boundary of the l-th bin is greater than π. The reason why (l 0.5) L
−
minuses 1 comes form the phase need to be wrapped between -π to 0. Once the symbol timing offset is estimated, the symbol timing synchronization can be easily achieved by adjusting the boundaries of each time domain OFDM symbol.
A positive D means that the current boundaries are behind the correct boundaries and need to be moved ahead by D samples. Inversely, a negative D means the boundaries need to be delayed by D samples.
Figure 3.6 The diagram of the simple histogram operation.
As an example, consider a linear-phase channel, whose impulse response is
And the received signal after FFT is
exp( 2 0/ ).
k k k k
R =A C⋅ ⋅ −jφ − j πkn N+β . (3.17) By the proposed algorithm, the DPO is 2 n0
N
π if no noise exists. At last, the
symbol timing estimation is n0.
Besides, a simple simulation is given here. Consider an OFDM system with a FFT window size is 50 transmits the signal through an AWGN channel with SNR = 20dB and the optimal delay of the channel is 25. From the context above, this delay will introduce a phase shift on each subcarrier of an OFDM symbol.
From equation 3.13, the DPO should lie around -π( 2 (25) 50 ) π
= − . Then, taking
these DPOs through a histogram operation. Figure 3.7 is the histogram result of the DPOs. This histogram is operated under 64 bins, i.e. the range from –π to π is divided into 64 subintervals. From the figure, the peak is occurred around -3.09. There are not only one bin containing all the DPOs because the presence of noise. So, the delay will be estimated at 25 ( 3.09 50
2π
= × ).
The center of this container is -3.09
Frequency
Phase (radian)
Figure 3.7 The histogram of the DPOs through an AWGN channel with SNR=20dB. This histogram is divided into 64 bins.
The computation cost of the proposed algorithm, taking one complete OFDM signal with N subcarriers of the OFDM system into account, is 2N subtractions and N comparisons. It is better than the correlation method [1], which calls for 2N multiplications, N additions, and N subtractions. As for the computation amount of phase extraction is not counted here because the phase extraction operation is a task of demodulators in M-PSK OFDM systems.