eld Connections Cell 8: : 1000000011010101011011101110001101100000111 Cell 7:0100000010111111110110011001001011010000100 Cell 6: 0010000001011111111011001100100101101000010 Cell 5: (0001000011111010100110001000011111010100110 Cell 4: 0000100010101000001000101010000010001010100 Cell 3: : 0000010001010100000100010101000001000101010 Cell 2:0000001011111111011001100100101101000010010 Cell 1:0000000110101010110111011100011011000001110 Figure 5.9. A Galois LFSR producing sequences with differing periods. Single peri- ods are underlined. configuration and memory cell sequences for this example are shown in Figure 5.9. 5.5.4 The Trace Representation of M-Sequences A fundamental mathematical tool used in the further investigation of PN generators is a particular linear mapping from a finite field onto a subfield. This mapping, called the trace function, will now be reviewed, and an explicit expression for the elements of an m-sequence will be constructed. Let GF(q) be any finite field contained within a larger field GF(q). Then the trace polynomial Tra(z) from GF(q) to GF(q) is defined as (5.121) The trace (function) in GF(q) of an element a in GF(q) is defined as Tra(a), i.e., the trace polynomial evaluated at a. The values of q and d are often obvious in the context of a particular application, in which case the cumbersome superscript and subscript are dropped from the trace notation. The trace function has the following useful properties which are proved in Appendix 5A.9. Property T-1. When a is in GF(q), then Tra(a) is in GF(q). Property T-2. All roots of an irreducible polynomial ma(z) over GF(q), with root a in GF(q), have the same trace, i.e., Tra(a8)=TrA(a) for all i. (5.122) Pseudonoise Generators Property T-3. The trace function is linear. That is, for a and b in GF(q), and a and B in GF(q), (5.123) Property T-4. For each choice of b in GF(q) there are qd-1 elements a in GF(q) for which Tra(a) = b. (5.124) Property T-5. If GF(q) GF(q) (GF(q), then (5.125) for all a in GF(q. Those familiar with finite fields will