## Pseudo-random sequences

#### One of the most common ways of spreading a CDMA signal is with a pseudo-random noise sequence (often called a PN sequence). This is a sequence generated by a shift register with feedback (see digaram) which repeats itself after every N = 2n – 1 clock cycles. The 3-bit shift register shown would thus generate a sequence that repeats every seven bits, (often called 'chips' in spread spectrum applications).

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### Simple 3 stage PN Sequence Generator

#### Of particular interest in CDMA is the auto-correlation of a PN sequence with itself, and the cross-correlation with other PN sequences generated using different length shift registers, or registers with different 'taps' from which the feedback signals are obtained.

#### The result of auto-correlation of a PN sequence is shown in the figure. Auto-correlation is this application involves comparing the similarity of one sequence with a time displaced version of the same sequence for all possible time offsets until the sequence repeats itself. A maximum output is achieved only when the two sequences are exactly time aligned, and falls to a minimum value of 1 / N for all sequence offsets greater than 1 chip period. Auto-correlation is achieved practically by mixing the incoming spread spectrum signal with a locally generated replica of the spreading PN sequence and sliding the timing of the local PN sequence until a correlation peak is reached. At this point, the modulation, which has been superimposed on the spreading code in the transmitter, can be extracted.

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### Auto-correlation of a PN Sequence

#### If all other spread spectrum signals are operating with unique and carefully chosen spreading codes, then there will be no cross-correlation between them and the contribution from all these other users will also be 1 / N times that of the wanted user correlation peak. Clearly, the larger the sequence length N, the larger the wanted correlation peak will be with respect to all other signals. This is very important if a large number of users are to be accommodated on the same frequency and still allow the de-spreading of any individual signal with sufficient signal to noise ratio for acceptable communications quality. If the sequence length N is made too big however, the time taken to find the correlation peak by sliding one sequence against another can be prohibitively long.