Pseudo-noise sequences (PN sequences) used in DSSS are generated in a deterministic manner although they appear to be random noise. From all the types of pseudo-noise sequences, the ones chosen for this study were m-sequences. The main purposes of spread-spectrum techniques are to make the signal less susceptible to jamming, to reduce energy density and to allow multiple access . In addition, a system using this technique has three main properties:

–  The signal will occupy a bandwidth greater than its baseband version.

–  The spreading sequence is independent of the data signal.

– The receiver correlates the received signal with a replica of the spreading sequence, a process known as despreading.

A chip is defined as the duration of each element inside the sequence and it is usually named Tc. This time is much smaller than the bit duration in the sequence that is going to be coded. In addition, PN sequences must satisfy the following properties:

1. Balance: the number of 1s in a sequence’s period differs by one from the number of 0s.

2. Run-length: there are runs of 1s and 0s inside one period. The number of runs of 1s must be the same as the number of 0s. In each period, half of the runs with the same sign have length 1, a quarter has length 2, an eighth have length 3 and so forth.

3. Autocorrelation: the autocorrelation function for these sequences is periodic and it has two values. For a given sequence ak of 1s and -1s, the autocorrelation C(K) is computed as:

Its values are 1 for k=0 and -1/N for the rest of the values of k. This is associated with the fact that comparing the periodic sequence with any shifted version of itself will result in a difference between matches and mismatches equal to 1.

Maximum-length sequences (m-sequences) are one of the most important PN sequences in use. They are called maximum-length because they can be generated by using a shift register of length equal to m; therefore, there are 2m-1 chips of length m. Typical length values are 7, 15, 31, 63, 127 and 255.

As the diagram shows, the reflected waves that arrive with different time delays are detected individually by the rake receiver. The diversity technique employed is maximal-ratio combining, where there is an amplification or attenuation factor proportional to the signal amplitude for each branch.

Let x(t) be the spread transmitted signal, then, the received signal will be:

Therefore, the signal is convolved by the channel impulse response, and AWGN is added. If the channel introduces k paths, we can model each of them by using the following expression:

where α_k is a complex number. Every correlator or finger is set with a time delay equal to or a multiple of the channel time delays. Therefore, assuming a delay of td, and two paths introduced by the channel, the signal at the output of the rake receiver is: