3.9 Phase Delays and Walsh Orthogonality

In section 3.2 we introduced the technique of convolving our bit stream with the orthogonal Walsh functions, enabling our bit stream to inherit the benefits of their orthogonality, and thereby remain isolated and distinct from other signals in code space. We mumbled something quickly under our breath about one slight caveat in connection to Doppler shifts and multipath, but quickly shuffled off to discuss other topics. Now that we've gotten a few more fundamentals under our belt, including the effects of multipath on a CDMA spread signal, we are probably as ready as we will ever be to deal with the effects of phase delays on the Walsh orthogonality.

The problem these phase delays introduce is related to the fact that two orthogonal functions remain orthogonal only as long as they retain their original timing relation. For example, our sine and cosine waves cease to be orthogonal if we delay one by, say, 30 or 50 or 90 degrees. In the latter case, if you delay a sine wave by 90 degrees, you have a cosine wave, and suddenly your delayed sine wave is no longer orthogonal to the original cosine wave, as shown below (using the trig relation Cos ²(x) = ¹/ ₂ (1 + Cos(2x) ):

₀ ² pCos(x) Cos(x) dx = ₀ ² p ¹/ ₂ (1 + Cos(2x)) dx = p

A similar effect occurs with the Walsh codes. Walsh-coded channels on...