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Thus higher data rates require more power and the limiting factor here is that the mobile
devices can only supply of the order of 200“300 mW. Therefore to achieve higher data
rates, the mobile device must be situated physically closer to the base station.

2.7.2 Orthogonal codes and signal separation
The signals that are all being transmitted at the same time and frequency must be separated
out into those from individual users. This is the second role of the code. Returning to
the party analogy, if this was a GSM party, then the problem is solved easily. All guests
must be quiet and each is then allowed to speak for a certain time period; no two guests
speak at the same time. At a CDMA party, all users are allowed to speak simultaneously,
and they are separated by speaking in different languages, which are the CDMA codes.

All of the codes that are used must be unique and have ideally no relationship to each
other. Mathematically speaking, this property is referred to as orthogonality. The system
can support as many simultaneous users as it has unique or orthogonal codes.
Orthogonal codes are used in CDMA systems to provide signal separation. As long
as the codes are perfectly synchronized, two users can be perfectly separated from each
other. To generate a tree of orthogonal codes, a Walsh“Hadamard matrix is used. The
matrix works on a simple principle, where the next level of the tree is generated from
the previous as shown in Figure 2.14(a). The tree is then built up following this rule,
with each new layer doubling the number of available codes, and the SF, as shown in
Figures 2.14(b) and 2.15.
For perfect orthogonality between two codes, for example, it is said that they have
a cross-correlation of zero when „ = 0. Consider a simple example using the following
two codes:
Code 1 = 1 ’ 1 1 ’ 1
Code 2 = 1 ’ 1 ’ 1 1

1 1 1 1 1 1 1

HM/2 HM/2 1 -1 1 -1 1 -1
HM =
HM/2 -HM/2 1 1 -1 -1

1 -1 -1 1

(a) (b)

Figure 2.14 Orthogonal code matrix

SF=1 SF=2 SF=4 SF=8

1 1 1 1 -1 -1 -1 -1
1 1 -1 -1 1 1 -1 -1
1 1 -1 -1
1 1 -1 -1 -1 -1 1 1
1 -1 1 -1 1 -1 1 -1
1 -1 1 -1
1 -1 1 -1 -1 1 -1 1
1 -1
1 -1 -1 1 1 -1 -1 1
1 -1 -1 1
1 -1 -1 1 -1 1 1 -1

Figure 2.15 Channelization code tree


code 1 1 -1 1 -1


code 2 1 -1 -1 1


integration 1 +1 + -1 + -1 = 0

Figure 2.16 CDMA cross-correlation

To verify if these two have a zero cross-correlation, they are tested in the above equation,
¬rst multiplied together and then integrated, as shown in Figure 2.16. The result is zero,
indicating that indeed they are orthogonal.
The number of chips which represent a symbol is known as the SF or the processing
gain. To support different data rates within the system, codes are taken from an appropriate
point in the tree. These types of orthogonal codes are known as orthogonal variable
spreading factors (OVSF).
In the 3G WCDMA system the chip rate is constant at 3.84 Mchips/s. However, the
number of chips that represent a symbol can vary. Within this system as laid down by the
speci¬cations, the minimum number of chips per symbol is 4 which would give a data
rate of 3 840 000/4 = 960 000 symbols per second. The maximum SF or number of chips
per symbol is 256,1 which would give a data rate of 3 840 000/256 = 15 000 symbols per
second. Thus it can be seen that the fewer chips used to represent a symbol, the higher
the user data rate. The actual user data rate must be rate matched to align with one of
these SF symbol rates. This process is described in more detail in Chapter 6.
Although orthogonal codes demonstrate perfect signal separation, they must be perfectly
synchronized to achieve this. Another drawback of orthogonal codes is that they do not
evenly spread signals across the wide frequency band, but rather concentrate the signal
at certain discrete frequencies. As an example, consider that the code ˜1 1 1 1™ will have
no spreading effect on a symbol.

2.7.3 PN sequences
Another code type used in CDMA systems is the pseudo-random noise (PN) sequence.
This is a binary sequence of ±1 that exhibits characteristics of a purely random sequence,
but is deterministic. Like a random sequence, a PN sequence has an equal number of +1s
and ’1s, with only ever a difference of 1. PN sequences are extremely useful as they
ful¬l two key roles in data transmission:

The speci¬cations actually allow for 512; however, a number of restrictions apply when this is

1. Even spreading of data: when multiplied by a PN sequence, the resultant signal is
spread evenly across the wideband. To other users who do not know the code, this
appears as white noise.
2. Signal separation: while PN sequences do not display perfect orthogonality
properties, nevertheless they can be used to separate signals. At the receiver, the
desired signal will show strong correlation, with the other user signals exhibiting
weak correlation.

Another property of PN sequences is that they exhibit what is known as autocorrelation.
This is de¬ned as the level of correlation between a signal and a time-shifted version of
the same signal, measured for a given time shift, i.e. …1 and …2 in the previous correlation
equation. For a PN sequence, the autocorrelation is at a maximum value, N , when perfectly
time aligned, i.e. „ = 0. N is the length in numbers of bits of the PN sequence. This
single peak drops off quickly at ±Tc , where Tc is the width of a chip of the code (see
Figure 2.17).
This allows a receiver to focus in on where the signal is, without a requirement for the
transmitter and receiver to be synchronized. In comparison, the autocorrelation of time-
shifted orthogonal codes results in several peaks, which means that this signal locking is
much more problematic.
PN sequences are generated using shift registers with a prede¬ned set of feedback taps.
The position of the taps is de¬ned by what is known as a generator polynomial. A simple
three-stage shift register arrangement is shown in Figure 2.18.


-Tc Tc

Figure 2.17 Autocorrelation of PN sequences

1 2 3 output


Figure 2.18 Three-stage shift register

clock output
1 2 3
1 0 1 0 0

2 1 0 1 1

3 1 1 0 0

4 1 1 1 1

5 0 1 1 1

6 0 0 1 1

7 1 0 0 0

8 0 1 0 0

9 ¦ ¦ ¦ ¦

Figure 2.19 Shift register states




Figure 2.20 Gold code generator

From a certain starting con¬guration in the registers, the outputs of stages 2 and 3 are
fed back to the input of the ¬rst stage via a modulo-2 adder. Any initial con¬guration
is allowed except 000, since this results in a constant output of zero. Consider that the
starting state is 010, then the stages for each clock cycle will be as shown in the state
diagram (Figure 2.19).
At clock cycle 8, the sequence repeats, so the generated output sequence is 0101110.
So for an M-stage shift register, a sequence of length 2M ’ 1 can be generated. These
are referred to as M-sequences.
An improved form of PN sequences known as Gold codes are generated by using two
such generators which are then combined (Figure 2.20). These Gold codes display better
autocorrelation properties and allow much larger numbers of codes to be generated.

A transmission from a mobile device is more or less omnidirectional, and this is also the
case for base stations which have only one cell. Base stations which are sectorized will
have directional antennas, which will transmit only over a certain range. For example, a
three-sectored site will have three antennas which each transmit over the range of 120

degrees. From the point of view of the mobile device, it would be ideal if a transmission
were unidirectional; however, this is impractical since it would mean that the antenna of
the mobile device would need to point towards the base station at all times. In this ideal
situation the device could transmit with reduced power, thus causing less interference to
other users and increasing the device™s battery life. In the cellular environment, much of
the power transmitted is actually in the wrong direction. In urban areas there is consid-
erable re¬‚ection of the signal from surrounding buildings. This is actually a reason why
cellular systems work, since the mobile device can thus be out of direct line of sight of
the BTS and its signal will still be received. The re¬‚ected signals travel further distances
than the direct line of sight transmission and therefore arrive slightly later, with greater
attenuation and possible phase difference (see Figure 2.21).
It would be advantageous if these time-shifted versions in the multipath signal could
be combined at the receiver with the effect that a much stronger signal is received. The
autocorrelation property of the PN sequence is again used. Since the received signal
resolves into a single peak around the chip width, then as long as the multipath pro¬le
is of a duration longer than the chip width, a number of peaks will be observed, each
one representing a particular multipath signal. Figure 2.22 shows an example where three
time-shifted paths have been resolved.
The number of paths that can be successfully resolved is related to the ratio of chip
width to multipath pro¬le. For WCDMA, the chip rate is constant at 3.84 Mchips/s, giving
a constant chip time of approximately 0.26 µs. Typically for an urban area, a multipath
pro¬le is of the order of 1“2 µs over which there are signals arriving with suf¬cient
power to be successfully resolved. Over this period, this means there is adequate time
to resolve about three or four signals. In terms of distance, a time difference of 0.26 µs
equates to 78 m, which means that to be resolved, a multipath must have a path length
of at least 78 m greater than the direct signal.
CDMA systems harness this property through the use of a rake receiver. The rake
receiver is so called since it has a number of ¬ngers which resemble a garden rake.
Figure 2.23 shows a simpli¬ed diagram of a rake receiver with three ¬ngers. A rake
receiver is a form of correlation receiver, so each ¬nger is fed the same received signal,


Figure 2.21 Multipath propagation

autocorrelation peaks


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