Fuzzy Methods in Nonlinear Signal Processing: Part II—Applications - Nonlinear Signal and Image Processing: Theory, Methods and Applications - page 109

algorithms based on ranks can be designed that are robust to varying sam-
ple statistics. This is very important if little or nothing is known about the
noise (or signal) statistics, or if the noise environment is changing rapidly
as is often the case in wireless communications. The majority of the detec-
tion algorithms developed to date have been based on crisp ranks that con-
tain only very coarse information about the observation samples, and thus
lead to detection schemes that exhibit low efficiency when compared to opti-
mal techniques. The use of fuzzy ranks can help to overcome this drawback.
We illustrate this through the use of fuzzy rank order (FRO) detectors 49-52 for
fast-frequency-hopping multiple access networks.
In a frequency-hopping spread spectrum network, K users transmit se-
quences of L tones called frames. Each tone is chosen from a set of Q possible
tones. At the receiver, the signals of all users are superimposed. The frames of
the received signal are noncoherently detected at Q possible frequencies for
each of the L time slots. The results are arranged in a Q L received matrix,

where the rows represent the frequencies and the columns represent the time
slots. To detect user K , the rows are de-spread with the K th user's spreading
sequence, yielding a decoded matrix with entries x i, j . The decoded matrix
will contain one row at the level of the desired user's symbol. This row is
referred to as the correct row. The samples in the correct row correspond to
the desired signal plus noise and the entries in the remaining rows are due to
noise only, where the noise is due to a combination of background noise and
contributions from interfering users (multiple access interference). Denoting
the cumulative distribution of the samples in the correct row and the spurious
rows by F S + N and F N , and assuming that all elements in the received matrix
are mutually independent, 53 the decision of which Q rowisthe correct row
can be put in the framework of a Q -ary hypothesis test:

×

H k : x k, ∼ F S + N , = 1 , 2 , ... ,L and

x i, j ∼ F N , j = 1 , 2 , ... ,L,

i = 1 , ... ,k − 1 ,k + 1 , ... ,Q . (3.24)

Thus, the samples in the row under test have to be compared to the remaining
pooled data in some fashion.
The conventional detector forms a binary detection matrix by detecting
any energy (above the noise floor) in a given time-frequency slot in x as a
hit, regardless of the actual amount of energy present. To obtain a decision
regarding which symbol was transmitted, the number of hits is determined
and the row with the maximum number of hits is chosen to be the correct row.
Figure 3.30a depicts the energy samples seen in a typical received matrix, for
Q 25 dB, by gray tones where white is assigned to the

maximum energy sample and black is assigned to the element with the small-
est value. In this example, the first row is the correct row. This is reflected in the
received matrix by two clear hits on the third and fifth hop. Aside from those

= 8 ,L = 5 , and SNR =