Appendix    ( blind problem )

Linear blind problems are categorized in two types.

[A]  Blind equalization (deconvolution)

Find inverse of the channel only from the received signal.


Such problem has double freedom between channel distortion and transmitted sequence, and requires the following conditions.

    1.   is IID except Gaussian.

    2.  Linear system is possible to have an inverse around a time-origin.
    In wireless systems, spectrum null occurs due to multi-path effect. Diversity or MIMO with multiple antennas are installed to avoid this unequalizability. In such cases, signal and channel response are defined respectively by vector and matrix. The scalar system below can be straightly extended to multi-dimensional cases.

In 1975, Sato proposed the following strategy makes the total system transparent for uniformly distributed transmitted sequence.

In 1980, Benveniste showed that a wide class of cost function given by

is blindly equalizable for any distributions except Gaussian. In the same year, Godard proposed a strategy for complex signal, i.e, the constant modulus blind equalization for QAM.

Intuitive interpretation of blind strategy can be equivalently said in following two ways.

    1. To untangle time-dependency in the signal

    2. To sharpen the distribution diverged through the linear system.

For Godard's cost function, under assumptions;

real valued signal,
infinite length of equalizer,
not Gaussian,

we can derive an explicit form;

where . It can be derived that the above has the following stationary points.

A minimum is realized at , where j is arbitrary.
Saddle points are realized at .

The algorithm for QAM based on Sato's cost function is given by


The computer simulation succeeds as follows.


[B]  Blind identification

In mobile system, the channel varies rapidly and its inverse doesn't always exist. Then we must pursue other methods based on identification principle.

Find channel response and/or the transmitted sequence
only from the received signal.

The algebraic structure in noiseless case that estimates jointly channel response and the transmitted sequence should be described as follows.

step1: For each (2N-1)-dimensional candidate vector ,  check whether set of and satisfies the following equation or not.

step2: If the solution exists, let be the survivor.

step3: By continuing above steps, find a coherent chain of survivors.

In general, the coherent chain of survivors cannot be uniquely determined due to possibility of branch and merge of survived path. Such ill cases should be listed up and referred in practical extension to the noisy cases.

Ref.  Y.Sato, "A Blind Sequence Detection and Its Application to Digital Mobile Communication" IEEE Journal on Selected Areas in Communications 13(1), pp.49-58, Jan. 1995

RAKE receiver employed in CDMA is the matched filter which is obtained by estimates of channel response by long correlation. The diverged response can be completely concentrated at time origin, only if the whitening filter is put before RAKE receiver.