The high order statistics used in this algorithm are the cumulants. Given a set of n real-valued random variables, their cumulants of r-th order are defined as [2]

where is the joint characteristic function of.

We can express the cumulants in terms of the moments [9] . Now, if, then

It is pointed out in [10] that the third and higher order cumulants of a Gaussian process are zero. That is one of the reasons why cumulants are chosen to be the “criterion of goodness” in an algorithm mitigating non- Gaussian interference in the presence of Gaussian noise.

Because the n-th order cumulant of any Gaussian process is identically zero for. This general scheme applies to all high order statistics for.

Let denote the n-th order cross-cumulant of the primary input and the reference input, we get

in which the cumulant operator is defined as [9]

i.e. a special case of Equation (6) that.

Let denote the n-th order cross-cumulant of the adaptive filter output and the reference input, then, if the j-th coefficient of the adaptive filter denoted by,

Comparing (11) and (12) leads to the conclusion that if, then and will be identical and the adaptive filter will produce the optimum estimate of the interference. This is the reason why we use the difference between and to measure the error. The new “criterion of goodness” can be written as

while can be defined to include the whole -dimensional space, it is not practical because of the computational complexity. Instead, we choose a proper domain P that is a subset of the -di- mensional space. Thus, the simplified error measurement is

Finally, it can be shown that

Thus, the “criterion of goodness” is finally defined as

where P is a subset of the -dimensional space.

In matrix form,

To obtain the optimum filter coefficients, we minimize with respect to using the steepest descent algorithm, namely

If we denote the number of points in set P as M and N as the number of taps of the FIR filter, then is an column vector and is an matrix. It is noted in [3] that we need to guarantee the reliability of the filter coefficients. is an vector of filter coefficients, namely

The filter coefficients are updated recursively by

The value of the step size m depends on the order of the algorithm and the behavior of the noise signal. The strategy to find an appropriate value of m will be discussed later under the condition that a FOS algorithm is applied on K-distributed interference. Figure 3 shows the structure of the proposed HOS interference canceller.

Since the fourth order statistic is used, we only need a domain such that.

Shin and Nikias [7] provided the following domain selection when cancelling a sinusoidal interference with uniformly distributed random phase:, where the domain size L is a positive integer that is chosen to determine the number of elements in the domain. This domain is also used in our work when the interference is K-distributed.

Note that the number of points M satisfies

Assuming the input process is ergodic, we can compute an estimate of the fourth-order cumulant as follows:

The step-size parameter applied to the FOS cancelleris [7]

in which is the adaptation parameter constant that can be altered to set a proper step-size.

Now, a small step-size may result in long convergence time when the SIR is large, while a large step-size may not permit the algorithm to converge when the SIR is low and a variable step-size scheme may be needed. The details of the adaptive scheme will be given shortly.

Although the FOS interference canceller works fairly well when the interference is K-distributed, it still needs some modification to work properly in an OFDM communication environment. First, the received OFDM- modulated signal is complex but the FOS canceller only works on real-valued signals. Second, similar to the NMLS algorithm, we need to normalize the input signal to make sure the algorithm will converge. The time domain cancellation scheme is depicted in Figure 4.

If the received signal is denoted as, the prime and reference input signal are normalized by the scale factor

where is the average symbol energy.

We first evaluate performance of the proposed algorithm design in a platform that uses a binary phase shift keyed (BPSK) digital communication system. The signal of interest used in the BPSK-based platform is obviously different than that used in a computer platform that uses OFDM communications in their transceivers.

The simulation parameters are listed in Table 1.

The variable step-size parameter is set to

From Figures 5-8, we can clearly see on a qualitative basis that the canceller works well for SIRs of 5 dB and 12.5 dB. Quantitatively, after cancellation, the power of the K-distributed interference is reduced by 36% and 32% at 5 and 12.5 dB, respectively. The cancellation algorithm converges before the first 2000 samples, which takes microseconds in most modern wireless communication systems implemented in computer platforms. The BER improvement after interference cancellation can be observed in Figure 9. This figure shows 5 simulation results, namely, one before cancellation and four after cancellation, in the SIR range from 0 dB to 15 dB. The improvement can be as large as 6 dB at SIR of 5 dB. Hence, we can conclude that the FOS-canceller with variable step- size works well in a computer platform that uses BPSK communications in the presence of K-distributed interference.

Table 2 lists the parameters used in the OFDM modulation simulation environment. The composition of each frame during the ODFM modulation is now described. After the serial to parallel conversion, the pilot symbols are inserted evenly throughout the frame. If the n-th pilot symbol is denoted as P_n and the n-th QAM modulated symbol is denoted as X_n, the frame is described by.

Note that in an actual OFDM modulator, the location of pilot symbol should not be the same for each frame. However, since the locations of pilot symbols have nothing to do with the cancellation algorithm in this problem, the pilot symbols are set at fixed locations in each frame.

The IFFT is performed right after the pilot symbol insertion. The IFFT size applied here is 1024 and it is greater than the actual number of symbols in a single frame, i.e. 840. Zeros are added at the beginning and the end of each frame to make each frame contains exactly 1024 symbols, as follows:

.

After the IFFT of length 1024, the frame is described by, where denotes the nth symbol in modulated domain.

Finally, the prefix symbols are added and the OFDM modulated frame is described by

.

a) Performance evaluation for fixed K-distribution parameter

The first scenario we consider to evaluate OFDM system performance uses a K-distribution with a fixed set of parameters, namely,. The simulation uses samples and performance is evaluated for SIRs of 2.5, 5, 7.5 and 10 dB. For SIR = 12.5 dB, the simulation uses samples, in order to observe enough errors to improve low bit error rate(BER) statistics reliability. The step-size parameter is set to

.

System performance is shown in Figure 10. We can see that the FOS canceller results in consistent BER improvement for all SIR values. Also, at low SIR (2.5 dB) the improvement is better than traditional cancellation schemes.

b) Performance evaluation for different K-distribution parameters

Alban [6] pointed out that a change in the K-distribution parameters may result in different BER performances for the same cancellation algorithm. In case a) the cancellation algorithm described in [7] can improve the BER by 6 dB while in case b), the improvement is less than 4 dB. Figure 11 shows OFDM system performance of the FOS canceller for both cases.

It is clear from this figure that the shape parameters b and v do affect system performance. The larger the b, the wider the probability density function of the interference. Regardless of their values, the interference cancellation strategy herein proposed consistently results in performance improvement. We can also see from the figure that case a) outperforms case b). This difference is not as high as that achieved in [6] , however, the BER performance is much better than the result in [6] at low SIR values. When SIR = 2.5, FOS canceller and the discrete Kalman filter both can produce a BER between 0.1 and 0.01. But when SIR = 10, the BER produced by

our proposed FOS canceller is around 10⁻⁴ while the result of discrete Kalman filter is in the neighborhood of 0.01. This improvement is at the cost of computational complexity and the additional requirement of knowing the reference signal.