he right circumstances, we can attenuate same-sector signals by more than the processing gain. We do this by selecting bit sequences specifically designed to cancel each other out. These are called orthogonal sequences or orthogonal codes. Consider two signals A and B being sent on the same channel with two chips per bit. Let ak and bk be the A bits and B bits for time period k. If the two chips are Ck1 = ak + bk and Ck2 - then the receiver can compute the sum Ck1 + Ck2 for the A bits and the difference Ck1 - Ck2 for the B-bits.4 In each of these calculations, the cancellation of the other signal is not merely statistical; it is absolute. These two 2-bit codes agree ex- actly half the time and disagree exactly half the time. The theme matrix of Hadamard-Walsh orthogonal codes is H2 in Eq. (8.12): (8.12) Note that if we take any two rows of H2 and add their pairwise products, then we get zero. 5 The sum of pairwise products is called the inner product of vectors a and b. The inner product is also called the dot product. When two nonzero vectors a and b have an inner product of zero, they are said to be orthogonal. We can take another step and build H4 from four copies of H2 in Eq. (8.13): (8.13) The four copies of H2 in H4 are the exact same structure as the four copies of +1 in H2. Filling in the +) 1 and -1 elements of H2 yields four orthogonal rows of H4 in Eq. (8.14): +1 +1 +1 +1 +1 -1 +1 -1 (8.14) +1 +1 +1 -1 -1 +1 4This calculation gets 2ak and 2bk. Since our receiver is using anything positive for +1 and any- thing negative for - 1, the extra factor of 2 does not matter. 5Not a lot of choice here, but bigger matrices have the same property. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. The CDMA Principle The CDMA Principle Any two of the four rows of H4 are orthogonal, so the entire H4 matrix is called an orthogonal matrix. The next 