Channel impulse response estimation with a linearly modulated sequence of symbols

The objective of this demonstration is to illustrate an algorithm for channel impulse response estimation with linearly modulated sequence of symbols with 2 USRP interfaced to matlab and 2 computers.

1. Description of the demonstration

To estimate the channel impulse response, a linearly modulated sequence of symbols has been generated. The following frame format has been used:

The following process is applied to the received signal:

1. As the received signal is composed of bursts, each being a replica of the signal of interest, the processing starts with the selection of one burst.
2. The adaptative filter is then applied to the signal to maximise the signal to noise ratio (SNR), and to suppress the intersymbol interference:

   

   The signal is then undersampled at the symbol rate.
3. The frequency offset is roughly estimated and compensated. This is done thanks to the first K=500 BPSK symbols according to the autocorrelation based method. A frequency offset of 0.0419 (normalised frequency) has been estimated.
4. The remaining frequency offset is then estimated and compensated according to the training sequence based method. A frequency offset of 0.000015 (normalised frequency) has been estimated
5. The channel impulse response is estimated (see below).
6. The constellation and its time variation of the obtained symbol are plotted (see below).

2. Channel impulse response algorithm

We assume that, after frequency offset compensation, the received signal writes where $h(k)$ are the coefficients of the channel impulse response to estimate and $w(n)$ stands for the additive Gaussian noise.

We also assume that the first $P$ coefficients of $x(n)$ are known from the receiver. The channel impulse response is then the following one:

$[y(0),...,y(P-1)]^T$ and $[x(0),...,x(P-1)]^T$ known, what are the $L$ most likely coefficients $[h(0),...,h(L-1)]^T$ given the above model ?

As $w(n)$ is assume to be Gaussian, it is well known that the solution is the vector $h$ that minimizes: where X is a $P \timesL$ Toeplitz matrix which first column equals $[x(0),...,x(P-1)]$ and first column $[x(0),0,...,0]$.

It is well known that the solution of this problem is The channel impulse response is estimated according to this algorithm.

3. Practical realisation and issues

To illustrate this method, two USRP interfaced with matlab have been used:

One the first computer, a linearly modulated sequence of symbols has been generated according to the frame described above. This signal has then been sent trough ISM bands to the second computer also connected to an USRP. The received signal has been processed according to the above steps.

The channel impulse response has been estimated according to the above method. The following result has been observed:

As expected the channel is flat.

The time synchronisation has then been performed, and the constellation of the obtained signal plotted. As illustrated on the figure below, the constellation is not time constant, meaning that the channel impulse response is a time dependant function:

To transmit sequence of symbols, it is hence necessary to regularly estimate the channel impulse response. More information about the channel variation can be found on the tutorial about the RF impairements of the USRP.