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Frequency offset estimation with linearly modulated sequence of symbols
1. Goal of this tutorial
$y(n) = (\sum_kh(k) x(n-k) + w(n)) exp(2 i \pi\delta_fn)$
where
- $\delta_f$is the frequency offset
- $h(k)$ are the coefficients of the channel impulse response
- $x(n)$ are the transmitted symbols
- $w(n)$ stands for the additive noise
2. Rough frequency offset estimation with autocorrelation function
2.1. Theoretical part
The first method assumes that the transmitted symbols, $x(n)$ are BPSK (or real) symbols. Note that this method is the one used in the other tutorials to estimate and compensate the frequency offset.In this context, the autocorrelation function of the received samples equals
$E[y^2(n)]= \sum_kh(k)^2 exp(4 i \pin \delta_f)$
Hence, the Fourier transform of the function $E[y^2(n)]^$ shows a peak at the frequency $2 \delta_f$.This method gives good results has long as $\delta_f$is greater than the frequency step of the Fourier transform. This method can hence be used for rough estimation of the frequency offset, prior to fine estimation.
2.2. Some remarks
This method can directly be applied to the oversampled received signal. Assuming that the shaping filter is a square root cosine filter with a bandwith excess lower than 1, the Fourier transform of the autocorrelation function has 3 peaks, at $-T_e/T_c+ 2 \delta_f,2\delta_f,T_e/T_c+2 \delta_f$.3. Fine frequency offset estimation with training sequences
An alternative method for frequency offset estimation is to use training sequences. We therefore assume that:$\forallp \in[0,P-1], x(p) = x(N+p)$
and that the training symbols are constant modulus. In other words, the training sequence inserted at the beginning of the transmitted signal
is also inserted at the end of the this signal. With this assumption, and with the above received signal model, we get:
$\sum_py(p+P+N)y(p)^* = \sum_k|h(k)|^2 exp(2 i \pi(N+P) \delta_f)$
To work, this method requieres that $(N+P) \delta_f$belongs to [-1/2,1/2]. It can then only be used for fine frequency offset estimation.
4. Illustration with two USRP interfaced to matlab
These methods have been illustrated with two USRP interfaced with matlab.
4.1. Signal generation
A linearly modulated sequences of symbols has been generated with the following sequence of symbols:
- The first K symbols are BPSK symbols used for rough frequency offset estimation. K equals 500. Note that the receiver does not need to know these 500 BPSK symbols to estimate the frequency offset.
- The next P symbols are QPSK symbols used for fine frequency offset estimation. P equals 200. These symbols are also not known by the receiver, but are the same symbols than the one at the end of the frame.
- The frame is then composed of N=9000 symbols of information. These symbols are QPSK symbols.
The received signal is composed of bursts, each being a replica of the signal of interest. The signal processing at the receiver starts with a burst selection. The selection part of the signal has then been resampled at the symbol rate.
4.2. Rough frequency offset estimation
The frequency offset has first been roughly estimated with the autocorrelation based method. The Fourier transform of the 600 first samples of the square signal (not modulus !) has then be computed. As expected, there is a peak at a frequency which is not zero:
4.3. Fine frequency offset estimation
The frequency offset estimation has then been again estimated but with the fine frequency offset method. The following value has been estimated: $0.00004$. This value would have been very hard to be estimated with the autocorrelation based method since it requieres a very small frequency step. But the fine method would not have been able to estimate the rough frequency offset 0.045 (0.091/2) since $(N+P)*0.045 > 1$.