5

Code Division for Multiple

Access (CDMA)

5.1 Introduction to CDMA

CDMA stands for Code Division for Multiple Access and is considered a path-breaking
wireless technology due to its several superior properties. It was first employed in the 2nd
generation IS-95 cellular standard, which was predominantly used in North America, under
the brand name cdmaOne. It also forms the basis for several advanced 3rd Generation i.e., 3G
cellular standards such as Wideband CDMA (WCDMA), High-Speed Downlink Packet Access
(HSDPA), High Speed Uplink Packet Access (HSUPA), CDMA 2000, and 1x Evolution
Data Optimized (1xEV-DO). In order to understand the concepts in CDMA, it is critical to
understand the concept of multiple access. In conventional wired communication systems,
there is a dedicated wireline communication channel which is allocated exclusively to the
particular device such as a telephone, etc. However, in a wireless network, mobile phones and
other wireless-communication devices are required to share the common radio channel over
the air. This is shown in Figure 5.1. This is because the radio channel is common for all the
users/ devices and the available wireless frequency bands are limited. Thus, it is necessary to
device a mechanism for multiple users to access this common radio channel, which is termed
as a Multiple Access (MA) technology. Thus, multiple access is at the heart of modern wireless
technologies, especially 3G and 4G cellular technologies.
Several multiple-access technologies have been developed and employed for cellular
applications. In fact, each generation of cellular standards is characterized by a particular
multiple-access technology. For instance, the first generation, i.e., 1G cellular standards were
based on Frequency Division Multiple Access (FDMA). In FDMA, different users are alloted
different frequency bands. Thus, the users are multiplexed in the frequency domain and
120 Principles of Modern Wireless Communication Systems

Figure 5.1 Multiple access for wireless cellular networks

User 0 User 1 User 3

f
B B B

Figure 5.2 Frequency division for multiple access

they access the radio channel in their respective frequency bands of bandwidth B . This is
schematically shown in Figure 5.2. On the other hand, the second generation or 2G cellular
standards are based on digital Time Division for Multiple Access (TDMA) in which different
users are allocated different time slots of duration T for accessing the wireless channel.
Thus, the different users are multiplexed in the time domain as shown in Figure 5.3. These
technologies were replaced by CDMA in successive 3G wireless technologies. The motivation
and basic mechanism of CDMA is described in the next section.

User 0 User 1 User 2

T T T

Figure 5.3 Time division for multiple access

 Code Division for Multiple Access (CDMA) 121

5.2 Basic CDMA Mechanism

CDMA, as the name suggests, is a multiple-access technology based on code division. In other
words, different users are multiplexed using different codes. Consider a two-user scenario, i.e.,
two users accessing the radio channel simultaneously. Let a0 denote the symbol of the user 0,
while a1 denotes the transmit symbol corresponding to the user 1. Let the code c0 of the user 0
be given as c0 = [1, 1, 1, 1]. The above code c0 is of length N = 4 chips. Each element of the
code is termed as a chip. The transmitted signal x0 of the user 0 is then given by multiplying
the code c0 with the symbol a0 as

x0 = a0 × [1, 1, 1, 1]
= [a0 , a0 , a0 , a0 ] (5.1)

The structure of the above transmit signal x0 can be interpreted as follows. The symbol a0,
of the user 0, is multiplied by the code c0 to yield 4 chips x0 (i), 0 ≤ i ≤ N − 1. Similarly,
let the code c1 , given as c1 = [1, −1, −1, 1], correspond to the code of the user 1. Hence, the
sequence of chips corresponding to the user 1 transmission is given as

x1 = a1 × [1, −1, −1, 1]

= [a1 , −a1 , −a1 , a1 ] (5.2)

The signals x0 , x1 corresponding to users 1, 2 respectively are now summed to yield the net
signal x as

x = x1 + x2 = [(a0 + a1 ) , (a0 − a1 ) , (a0 − a1 ) , (a0 + a1 )] (5.3)

This sum, or composite, signal is then transmitted on the downlink from which each of the users
0, 1 detect their own signal. This is done as follows. User 1 correlates the received signal x with
his code c0 , i.e., basically multiplies each chip of the received signal x with the corresponding
chip of the code c0 = [1, 1, 1, 1] and sums across the chips as follows.

a0 + a1 a0 − a1 a0 − a1 a0 + a1
× 1 1 1 1
(5.4)
(a0 + a1 ) + (a0 − a1 ) + (a0 − a1 ) + (a0 + a1 ) = 4a0

Thus, the result of the above correlation is 4a0 , which is proportional to the transmitted symbol
a0 . Similarly, at the user 2, the received signal x is correlated with the chip sequence c1 =
122 Principles of Modern Wireless Communication Systems

[1, −1, −1, 1] of the user 1 as

a0 + a1 a0 − a1 a0 − a1 a0 + a1
× 1 −1 −1 1
(5.5)
(a0 + a1 ) − (a0 − a1 ) − (a0 − a1 ) + (a0 + a1 ) = 4a1

to yield 4a1, which is proportional to the transmitted symbol a1 of the user 1. Thus, unlike in
GSM or FDMA, in which the signals of different users are transmitted in different time slots
or frequency bands, in CDMA, all the signals of the different users are contained in the single
signal x over all time and frequency. However, in CDMA, the symbols of the different users
are combined using different codes. For instance, in the above example, the symbols a0 , a1 of
users 0, 1 are multiplied with codes c0 , c1 prior to transmission. Thus, the users of the different
signals are multiplexed over the common wireless channel employing different codes. Hence,
this is termed Code Divison for Multiple Access, i.e., multiple access based on different codes.
The key operations in CDMA can be summarized as follows.
1. Multiplying or modulation the symbols of the different users with the corresponding
assigned unique code, similar to the procedure illustrated in equations (5.1), (5.2).
2. Combining or adding the code-modulated signals of all the users to form the composite
signal as shown in Eq. (5.3), followed by subsequent transmission of the signal.
3. Finally, correlation of the composite received signal x at each user with the corresponding
code of the user to recover the transmitted symbol. This is described in Eqs (5.4), (5.5).

5.3 Fundamentals of CDMA Codes

In fact, from the example illustrated in the previous section, the astute reader will realize that
it is no accident by which we are able to recover the signals of users 0, 1. Computing the
correlation r01 of the user codes c0 , c1 yields
3
r01 = c0 (k) c1 (k)
k=0
 Code Division for Multiple Access (CDMA) 123

= 1 × 1 + 1 × (−1) + 1 × (−1) + 1 × 1
= 1 + (−1) + (−1) + 1
=0

Thus, since the correlation between the codes c0, c1 is zero, the codes are, in fact, orthogonal.
This is what helps us recover the symbols of the different users from the composite signal.
This is a key property of the codes employed in CDMA wireless systems, and a fundamental
principle on which the theory of CDMA is based.
Further, consider a fundamental property of the CDMA system arising because of the
employment of these codes. Let the symbol rate for the symbols a0 of the user 0 be 1 kbps.
Hence, the time period T per symbol is

1
T = = 1 ms
1 kbps

Hence, the corresponding bandwidth required for transmission is

1
B= = 1 kHz
T

However, now consider the transmission of the symbol a0 multiplied with the corresponding
code c0, i.e., a0 × [1, 1, 1, 1] = [a0 , a0 , a0 , a0]. Thus, for each symbol a0, one has to transmit
4 chips. Thus, to keep the symbol rate constant at 1 kbps, the time of each chip Tc has to be set
as Tc = 14 T = 0.25 ms. Thus, the bandwidth required for this system is

1 1
BCDMA = = = 4 kHz
Tc 0.25 ms

Thus, modulating with the code c0 of length N = 4, results in an increase of the required
bandwidth by a factor of N , i.e., from 1 kHz to 4 kHz. This is shown schematically in
Figure 5.4. Thus, it basically results in a spreading of the original signal bandwidth and, hence,
is termed a spreading code. Also, since the resulting signal occupies a large bandwidth, CDMA
systems are also termed spread spectrum or wideband systems.
Also, another interesting question the reader might be interested in is the following: How
many such orthogonals exist for a given spreading code length N ? The answer is there are N
such orthogonal codes. For instance, consider the case N = 4.
124 Principles of Modern Wireless Communication Systems

Figure 5.4 Spread spectrum communication

The different orthogonal spreading codes are

c0 = 1 1 1 1
c1 = 1 − 1 − 1 1
c2 = 1 − 1 1 − 1
c3 = 1 1 − 1 − 1

The reader can verify that the codes c0, c1 , c2 , c3 are orthogonal to each other. For example,
consider c1 , c2 . The correlation r12 between codes c1, c2 is
3
r12 = c0 (k) c1 (k)
k=0

= 1 × 1 + (−1) × (−1) + (−1) × 1 + 1 × br −1

= 1 + 1 + (−1) + (−1)
=0

This implies that given a spreading sequence length N , there exist N orthogonal codes and
hence, N users can be multiplexed together. This is important, since the bandwidth increases
by a factor of N due to transmission employing the codes as described earlier. However, it
is important to note that no inefficiency is introduced in the system because of the increase
in bandwidth, because this increase in bandwidth by a factor of N is compensated by the
parallel transmission of the signals corresponding to the N users over the same bandwidth.
Thus, the spectral efficiency of the system is not compromised. This is schematically illustrated
in Figure 5.5.
 Code Division for Multiple Access (CDMA) 125

a0 Code 0
a1 Code 1
N Symbols
a2 Code 2
a3 Code 3

a0 a1 a2 a3
T = NTc
T Symbols

Figure 5.5 Parallel transmission of N symbols over N codes in CDMA over

time interval T = N Tc (above) and comparison with transmission
of N symbols in time T = N Tc in a conventional single carrier or
time division system

5.4 Spreading Codes based on Pseudo-Noise (PN) Sequences

Consider the code c2 = [1, −1, 1, −1]. Observe that the code looks like a random sequence of
+1, −1, or a pseudo-noise (PN) sequence. This is so termed since it only resembles a noise
sequence, but is not actually a noise sequence. One method to generate such long spreading
codes based on PN sequences for a significantly large N is through the employment of a Linear
Feedback Shift Register (LFSR). This is described next.
Consider the shift register architecture shown in Figure 5.6, where the element D
represents delays. Thus, the digital circuit therein contains D = 4 delay elements or shift
registers. The input on the left is denoted by Xi , and the outputs of the different delays are
Xi−1 , Xi−2 , Xi−3 , Xi−4 . Let Xi−4 also denote the final output of the system. Also observe
that the xor Xi−4 ⊕ Xi−3 is fed back as Xi which is the input to the first shift register. Thus,
the governing equation of the circuit is

Xi = Xi−3 ⊕ Xi−4

which is a linear equation. Thus, since it implements a linear relation, with feedback and uses
delay elements or shift registers, such a circuit is also termed a Linear Feedback Shift Register
(LFSR) architecture. Since the next inpur, i.e., Xi depends on Xi−1, Xi−2, Xi−3, Xi−4, this
can also be thought of as the current state of the system. Consider initializing the system in the
state Xi−1 = 1, Xi−2 = 1, Xi−3 = 1, Xi−4 = 1. Thus, we have the corresponding Xi given
126 Principles of Modern Wireless Communication Systems

as

Xi = Xi−3 ⊕ Xi−4 = 1 ⊕ 1 = 0

This Xi becomes Xi−1 at the next instant and similarly, Xi−2, Xi−3 are shifted to the right as
Xi−3 , Xi−4 respectively. Continuing in this fashion, the entire sequence of state of the above
LFSR is summarized. It can be seen that the LFSR goes through the sequence of 15 states
1111, 0111, 0011, 0001, 1000, 0100, 0010, 1001, 1100, 0110, 1011, 0101, 1010, 1101, 1110,
before reentering the state 1111. Subsequently, the entire sequence of states repeats again.
Observe that this goes through 2D − 1 = 24 − 1 = 15 states. Also note that the maximum
number of possible states for D = 4 is 2D = 16. However, the LFSR can be seen to go
through all the possible states except one, which is the 0000 or the all-zero state.

Xi Xi - 1 Xi - 2 Xi - 3 Xi - 4
D D D D

Feedback

Xi = Xi - 3 ≈ Xi - 4

Figure 5.6 Linear feedback shift register

Xi = Xi−3 ⊕ Xi−4 = 0 ⊕ 0 = 0

Further, observe that if the LFSR is initialized in the 0000 state, it continues in the 0000 state,
since the corresponding Xi is leading to the next state of 0000. Thus, the LFSR never gets out
of the all zero states! Therefore, it is desired that the LFSR never enter the all-zero state. Such
an LFSR circuit which goes through the maximum possible 2D − 1 states, without entering the
all-zero state is termed a maximum-length shift register circuit or maximum length LFSR. The
generated PN sequence is termed a maximum-length PN sequence. Thus, the maximum-length
PN sequence is of length 2D − 1. For instance, for the above LFSR, the maximum-length PN
sequence is the sequence of outputs Xi−4 given as

PN Sequence = 1 1 1 1 0 0 0 1 0 0 1 1 0 1 0
 Code Division for Multiple Access (CDMA) 127

We can map the bits 1, 0 to the BPSK symbols −1, +1 to get the modulated PN sequence,

PN sequence = −1 − 1 − 1 − 1 + 1 + 1 + 1 − 1 + 1 + 1 − 1 − 1 + 1 − 1 + 1 (5.6)

Next, we examine the properties of such PN sequences derived above.

5.4.1 Properties of PN Sequences

Property 1-Balance Property: Consider the BPSK-modulated PN sequence shown in
Eq. (5.6). As already described, the PN sequence is of maximal length 2D − 1 = 15
corresponding to D = 4. Counting the number of −1 and +1 chips in the sequence, it can
be seen that the number of −1s is one more than the number of +1s. This is termed the
balance property of the PN sequence. This fundamentally arisies from the noiselike properties
of PN sequences. If we are generating random noise of +1, −1 chips, with P (Xi = +1) =
P (Xi = −1) = 12 , we expect to find on an average that half the chips are +1 and the rest are
−1. In the above case, however, as the total number of chips is an odd number, i.e., 15, it is
not possible to have an exactly even number of +1, −1s. Hence, we observe that the number
of +1, −1s is close to half the total number, i.e., eight −1s and seven +1s. Thus, the balance
property basically supports the notion of a noiselike PN chip sequence.
Property 2-Run-Length Property: A run is defined as a string of continuous values. There
are a total of 8 runs in this PN sequences. For instance, the first run −1, −1, −1, −1 is a run
of length 4. Thus, there is one run of length 4. Similarly, there is one run +1, +1, +1 of length
3, and two runs of length 2, viz., −1, −1, +1, +1. Finally, it can also be seen that there are 4
runs of length 1, viz., two runs of +1 and two runs of −1. Thus, there are a total of 2(D−1) = 8
runs. Out of the 8 runs, it can be seen that 1, i.e., 18 of the runs are of length 3, 14 of the runs
are of length 2 and 12 of the runs are of length 1. This is termed the run-length property of
PN sequences and can be generalized as follows. Consider a maximal length PN sequence of
length 2D − 1. Out of the total number of runs in the sequence, 12 of the runs are of length 1, 14
of the runs are of length 2, 18 of the runs are of length 3, and so on. This is again in tune with
the noiselike properties of PN sequences. For instance, consider a random IID sequence of
+1, −1. In such a sequence, one would expect the average number of +1 or −1 to be half the
total chips. Further, the number of strings +1, +1 or −1, −1, i.e., runs of length two would
be expected to comprises 14 of the total runs. This arises since the probability of seeing two
128 Principles of Modern Wireless Communication Systems

consecutive +1, +1 symbols is

1 1 1
P (Xi = +1, Xi+1 = +1) = × =
2 2 4

Similarly, one can explain the fraction 18 corresponding to runs of length 3. Thus, this further
supports the noiselike properties of PN sequences.

Property 3-Correlation Property: The correlation property is one of the most important
properties of PN sequences. Consider again the BPSK chip sequence shown in Eq. (5.6) and
denote it by c0 (n). Let us now look at the correlation properties of this sequence. Consider the
correlation r00 (0), i.e., the correlation of the sequences c0 with itself (the meaning of the (0)
will become clear soon). This correlation is given as
N −1
1
r00 (0) = c0 (n) c0 (n)
N
i=0
N −1
1
= 1
N
i=0
1
= ×N = 1
N
Now, consider a circularly shifted version of the PN sequence, shifted by n◦ = 2. Let it be
denoted by c0 (n − 2). This circularly shifted sequence by 2 chips can be readily seen to be
given as

PN Sequence = −1 + 1, −1 − 1 − 1 − 1 + 1 + 1
+1 − 1 + 1 + 1 − 1 − 1 + 1 (5.7)

Let us denote the correlation between c0 (n) and c0 (n − 2) by r00 (2), where the (2) can now
be seen to represent a circular shift of 2. The correlation can be seen to be given as
N −1
1
r00 (2) = c0 (n) c0 (n − 2)
N
i=0
 Code Division for Multiple Access (CDMA) 129

1
= {(−1) × (−1) + (−1) × (1) + (−1) × (−1) + (−1) × (−1) + (1) × (−1) +
15

(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)
15
1 1
= × (−1) = −
15 15
1
=−
N
In fact, one can compute the correlation for other such nonzero delays, and can demonstrate
the the correlation is always − N1 . This autocorrelation property of the PN sequence, i.e., of
the sequence with a delayed version of itself, is shown pictorially in Figure 5.7. Thus, it can
be seen that while the correlation of the sequence with itself corresponding to a lag of 0 is
1, for any other nonzero shift, it assumes a very low value of − N1 , which tends to the limit
0 as the spreading length N → ∞. This autocorrelation property of the PN sequences can be
summarized as follows.

Shift n0

-1/N

Figure 5.7 Autocorrelation of PN sequence

⎧
N −1 ⎨ 1 if n = 0
1 ◦
r00 (n◦ ) = =
N ⎩ − 1 otherwise
i=0 N

With this background, let us investigate the properties of random spreading sequences in the
next section.
130 Principles of Modern Wireless Communication Systems

5.5 Correlation Properties of Random CDMA Spreading Sequences

In the previous section, we have seen that CDMA spreading sequences can be chosen as PN
sequences, which have noiselike properties. In other words, one can choose a chip sequence
ck (i) , 0 ≤ i ≤ N − 1 for the user k such that P (ck (i) = +1) = P (ck (i) = −1) = 12 . Thus,
we have,

1 1
E {ck (i)} = × (+1) + (−1) = 0.
2 2

Further, another important aspect is to choose such sequences as containing Independent

Identically Distributed (IID) chips, i.e., satisfying the property

E {ck (i) ck (j)} = E {ck (i)} E {ck (j)} = 0 × 0 = 0

The above property implies that each chip ck (i) is uncorrelated with chip ck (j). Further, one
can choose independent sequences for different users, that is, to say

E {ck (i) cl (j)} = E {ck (i)} E {cl (j)}

Let us examine the correlation properties of such random spreading sequences. As before, let
r0 0 (k) denote the autocorrelation of the chip sequence of the user k = 0, corresponding to a
lag k = 0. This can be expressed as

N −1
1
r00 (k) = c0 (i) c0 (i − k)
N
i=0

The average or expected valued of r00 (k) can be seen to be given as

N −1
1
E {r00 (k)} = E c0 (i) c0 (i − k)
N
i=0

N −1
1
= E {c0 (i) c0 (i − k)}
N
i=0
 Code Division for Multiple Access (CDMA) 131

N −1
1
= E {c0 (i)} E {c0 (i − k)}
N
i=0

N −1
1
= 0=0
N
i=0

Thus, the average value or the expected value of the correlation E {r00 (k)} is zero for lags
k = 0. This is expected from the random properties of the spreading sequence. To compute the
2 (k) given as
variance of the autocorrelation r00 (k), consider r00
⎛ ⎞
N −1 N −1
2 1
r00 (k) = 2 c0 (i) c0 (i − k) ⎝ c0 (j) c0 (j − k)⎠
N
i=0 j=0

N −1 N −1
1
= 2 c0 (i) c0 (i − k) c0 (j) c0 (j − k)
N
i=0 j=0

Now, let us consider the quantity c0 (i) c0 (i − k) c0 (j) c0 (j − k). It can be seen that if i = j ,
the expected value of this quantity can be simplified as

E {c0 (i) c0 (i − k) c0 (j) c0 (j − k)} = E {c0 (i) c0 (i − k)} E {c0 (j) c0 (j − k)}

= E {c0 (i)} E {c0 (i − k)} E {c0 (j)} E {c0 (j − k)}

=0

On the other hand, if i = j , the same quantity can be simplified as

E {c0 (i) c0 (i − k) c0 (j) c0 (j − k)} = E {c0 (i) c0 (i − k) c0 (i) c0 (i − k)}

= E (c0 (i))2 E (c0 (i − k))2

= 1×1 = 1
2 (k) can be simplified as
Thus, the variance of r00 (k), i.e., E r00
N −1 N −1
2 1
E r00 (k) = 2 E {c0 (i) c0 (i − k) c0 (j) c0 (j − k)}
N
i=0 j=0
132 Principles of Modern Wireless Communication Systems

N −1
1
= E c20 (i) c20 (i − k)
N2
i=0

N −1
1 1
= 2 1= ×N
N N2
i=1

1
=
N
Thus, the variance or basically the power of r00 (k), the autocorrelation of the random CDMA
spreading sequence is E r00 2
(k) = N1 . Also, once again, the autocorrelation corresponding
to a lag of k = 0 can be readily seen to be given as
N −1
1
E {r00 (0)} = E c0 (i) c0 (i)
N
i=0

N −1
1
= E c20 (i)
N
i=0

N −1
1
= 1
N
i=0

1
= ×N = 1
N
Therefore, one can succinctly summarize the autocorrelation properties of the random
spreading sequence as follows. For k = 0, r00 (k) = 1. For k = 0, r00 (k) is a random variable
with E {r00 (k)} = 0 and variance E r002
(k) = N1 . Let us now examine the cross-correlation
properties of the random CDMA spreading sequences, i.e., the correlation between the
spreading sequences c0 (i) , 0 ≤ i ≤ N − 1 and c1 (j) , 0 ≤ j ≤ N − 1. We denote by r01 (k)
the cross-correlation between spreading sequences c0, c1 corresponding to a lag k as

N −1
1
r01 (k) = c0 (i) c1 (i − k)
N
i=0

Once again, the expected value for any lag k can be computed as
N −1
1
E {r01 (k)} = E c0 (i) c1 (i − k)
N
i=0
 Code Division for Multiple Access (CDMA) 133

N −1
1
= E {c0 (i) c1 (i − k)}
N
i=0

N −1
1
= E {c0 (i)} E {c1 (i − k)}
N
i=0

N −1
1
= 0×0=0
N
i=0
2
Further, the variance E r01 (k) for any delay k is given as
⎧ ⎛ ⎞⎫
⎨ N −1 N −1 ⎬
2 1
E r01 (k) = 2 E c0 (i) c1 (i − k) ⎝ c0 (j) c1 (j − k) ⎠
N ⎩ ⎭
i=0 j=0

N −1 N −1
1
= E {c0 (i) c1 (i − k) c0 (j) c1 (j − k)}
N2
i=0 j=0

N −1
1
= E c20 (i) E c21 (i − k)
N2
i=0

N −1
1
= 2 1
N
i=0

1 1
= 2
×N =
N N
where we have again used the fact E {c0 (i) c1 (i − k) c0 (j) c1 (j − k)} is nonzero only if
i = j in the above derivation. Thus, once again, it can be seen that the cross-correlation
r01 (k) between two random CDMA spreading sequences c0 c1 is a random variable
with E {r01 (k)} = 0 and variance E r01 2
(k) = N1 . Thus, unlike the codes introduced in
Section 5.3, these random spreading codes do not satisfy the definition of exact orthogonality.
However, they are approximately orthogonal, in that the average value of the correlation is
zero and the power in the correlation is proportional to N1 which tends to 0 as N → ∞.

5.6 Multi-User CDMA

Now let us analyze the performance of a multi-user CDMA system using the properties
of the spreading sequences described above. Let a0 , a1 denote the symbols of users 0, 1