Code-division multiple access

The technology of code-division multiple access channels has long been known.

A CDMA2000 mobile phone

• Synchronous CDM (code-division 'multiplexing', an early generation of CDMA) was implemented in the Global Positioning System (GPS). This predates and is distinct from its use in mobile phones.
• The Qualcomm standard IS-95, marketed as cdmaOne.
• The Qualcomm standard IS-2000, known as CDMA2000, is used by several mobile phone companies, including the Globalstar network.[nb 1]
• The UMTS 3G mobile phone standard, which uses W-CDMA.[nb 2]
• CDMA has been used in the OmniTRACS satellite system for transportation logistics.

CDMA is a spread-spectrum multiple-access technique. A spread-spectrum technique spreads the bandwidth of the data uniformly for the same transmitted power. A spreading code is a pseudo-random code that has a narrow ambiguity function, unlike other narrow pulse codes. In CDMA a locally generated code runs at a much higher rate than the data to be transmitted. Data for transmission is combined by bitwise XOR (exclusive OR) with the faster code. The figure shows how a spread-spectrum signal is generated. The data signal with pulse duration of ${\displaystyle T_{b}}$ (symbol period) is XORed with the code signal with pulse duration of ${\displaystyle T_{c}}$ (chip period). (Note: bandwidth is proportional to ${\displaystyle 1/T}$, where ${\displaystyle T}$ = bit time.) Therefore, the bandwidth of the data signal is ${\displaystyle 1/T_{b}}$ and the bandwidth of the spread spectrum signal is ${\displaystyle 1/T_{c}}$. Since ${\displaystyle T_{c}}$ is much smaller than ${\displaystyle T_{b}}$, the bandwidth of the spread-spectrum signal is much larger than the bandwidth of the original signal. The ratio ${\displaystyle T_{b}/T_{c}}$ is called the spreading factor or processing gain and determines to a certain extent the upper limit of the total number of users supported simultaneously by a base station.[1][2]

Generation of a CDMA signal

Each user in a CDMA system uses a different code to modulate their signal. Choosing the codes used to modulate the signal is very important in the performance of CDMA systems. The best performance occurs when there is good separation between the signal of a desired user and the signals of other users. The separation of the signals is made by correlating the received signal with the locally generated code of the desired user. If the signal matches the desired user's code, then the correlation function will be high and the system can extract that signal. If the desired user's code has nothing in common with the signal, the correlation should be as close to zero as possible (thus eliminating the signal); this is referred to as cross-correlation. If the code is correlated with the signal at any time offset other than zero, the correlation should be as close to zero as possible. This is referred to as auto-correlation and is used to reject multi-path interference.[18][19]

An analogy to the problem of multiple access is a room (channel) in which people wish to talk to each other simultaneously. To avoid confusion, people could take turns speaking (time division), speak at different pitches (frequency division), or speak in different languages (code division). CDMA is analogous to the last example where people speaking the same language can understand each other, but other languages are perceived as noise and rejected. Similarly, in radio CDMA, each group of users is given a shared code. Many codes occupy the same channel, but only users associated with a particular code can communicate.

In general, CDMA belongs to two basic categories: synchronous (orthogonal codes) and asynchronous (pseudorandom codes).

The digital modulation method is analogous to those used in simple radio transceivers. In the analog case, a low-frequency data signal is time-multiplied with a high-frequency pure sine-wave carrier and transmitted. This is effectively a frequency convolution (Wiener–Khinchin theorem) of the two signals, resulting in a carrier with narrow sidebands. In the digital case, the sinusoidal carrier is replaced by Walsh functions. These are binary square waves that form a complete orthonormal set. The data signal is also binary and the time multiplication is achieved with a simple XOR function. This is usually a Gilbert cell mixer in the circuitry.

Synchronous CDMA exploits mathematical properties of orthogonality between vectors representing the data strings. For example, the binary string 1011 is represented by the vector (1, 0, 1, 1). Vectors can be multiplied by taking their dot product, by summing the products of their respective components (for example, if u = (a, b) and v = (c, d), then their dot product u·v = ac + bd). If the dot product is zero, the two vectors are said to be orthogonal to each other. Some properties of the dot product aid understanding of how W-CDMA works. If vectors a and b are orthogonal, then ${\displaystyle \mathbf {a} \cdot \mathbf {b} =0}$ and:

${\displaystyle \mathbf {a} \cdot (\mathbf {a} +\mathbf {b} )=\|\mathbf {a} \|^{2},\ {\text{since}}\ \mathbf {a} \cdot \mathbf {a} +\mathbf {a} \cdot \mathbf {b} =\|\mathbf {a} \|^{2}+0,}$

${\displaystyle \mathbf {a} \cdot (-\mathbf {a} +\mathbf {b} )=-\|\mathbf {a} \|^{2},\ {\text{since}}\ {-\mathbf {a} }\cdot \mathbf {a} +\mathbf {a} \cdot \mathbf {b} =-\|\mathbf {a} \|^{2}+0,}$

${\displaystyle \mathbf {b} \cdot (\mathbf {a} +\mathbf {b} )=\|\mathbf {b} \|^{2},\ {\text{since}}\ \mathbf {b} \cdot \mathbf {a} +\mathbf {b} \cdot \mathbf {b} =0+\|\mathbf {b} \|^{2},}$

${\displaystyle \mathbf {b} \cdot (\mathbf {a} -\mathbf {b} )=-\|\mathbf {b} \|^{2},\ {\text{since}}\ \mathbf {b} \cdot \mathbf {a} -\mathbf {b} \cdot \mathbf {b} =0-\|\mathbf {b} \|^{2}.}$

Each user in synchronous CDMA uses a code orthogonal to the others' codes to modulate their signal. An example of 4 mutually orthogonal digital signals is shown in the figure below. Orthogonal codes have a cross-correlation equal to zero; in other words, they do not interfere with each other. In the case of IS-95, 64-bit Walsh codes are used to encode the signal to separate different users. Since each of the 64 Walsh codes is orthogonal to all other, the signals are channelized into 64 orthogonal signals. The following example demonstrates how each user's signal can be encoded and decoded.

Example

An example of 4 mutually orthogonal digital signals

Start with a set of vectors that are mutually orthogonal. (Although mutual orthogonality is the only condition, these vectors are usually constructed for ease of decoding, for example columns or rows from Walsh matrices.) An example of orthogonal functions is shown in the adjacent picture. These vectors will be assigned to individual users and are called the code, chip code, or chipping code. In the interest of brevity, the rest of this example uses codes v with only two bits.

Each user is associated with a different code, say v. A 1 bit is represented by transmitting a positive code v, and a 0 bit is represented by a negative code −v. For example, if v = (v₀, v₁) = (1, −1) and the data that the user wishes to transmit is (1, 0, 1, 1), then the transmitted symbols would be

(v, −v, v, v) = (v₀, v₁, −v₀, −v₁, v₀, v₁, v₀, v₁) = (1, −1, −1, 1, 1, −1, 1, −1).

For the purposes of this article, we call this constructed vector the transmitted vector.

Each sender has a different, unique vector v chosen from that set, but the construction method of the transmitted vector is identical.

Now, due to physical properties of interference, if two signals at a point are in phase, they add to give twice the amplitude of each signal, but if they are out of phase, they subtract and give a signal that is the difference of the amplitudes. Digitally, this behaviour can be modelled by the addition of the transmission vectors, component by component.

If sender0 has code (1, −1) and data (1, 0, 1, 1), and sender1 has code (1, 1) and data (0, 0, 1, 1), and both senders transmit simultaneously, then this table describes the coding steps:

Step	Encode sender0	Encode sender1
0	code0 = (1, −1), data0 = (1, 0, 1, 1)	code1 = (1, 1), data1 = (0, 0, 1, 1)
1	encode0 = 2(1, 0, 1, 1) − (1, 1, 1, 1) = (1, −1, 1, 1)	encode1 = 2(0, 0, 1, 1) − (1, 1, 1, 1) = (−1, −1, 1, 1)
2	signal0 = encode0 ⊗ code0 = (1, −1, 1, 1) ⊗ (1, −1) = (1, −1, −1, 1, 1, −1, 1, −1)	signal1 = encode1 ⊗ code1 = (−1, −1, 1, 1) ⊗ (1, 1) = (−1, −1, −1, −1, 1, 1, 1, 1)

Because signal0 and signal1 are transmitted at the same time into the air, they add to produce the raw signal

(1, −1, −1, 1, 1, −1, 1, −1) + (−1, −1, −1, −1, 1, 1, 1, 1) = (0, −2, −2, 0, 2, 0, 2, 0).

This raw signal is called an interference pattern. The receiver then extracts an intelligible signal for any known sender by combining the sender's code with the interference pattern. The following table explains how this works and shows that the signals do not interfere with one another:

Step	Decode sender0	Decode sender1
0	code0 = (1, −1), signal = (0, −2, −2, 0, 2, 0, 2, 0)	code1 = (1, 1), signal = (0, −2, −2, 0, 2, 0, 2, 0)
1	decode0 = pattern.vector0	decode1 = pattern.vector1
2	decode0 = ((0, −2), (−2, 0), (2, 0), (2, 0)) · (1, −1)	decode1 = ((0, −2), (−2, 0), (2, 0), (2, 0)) · (1, 1)
3	decode0 = ((0 + 2), (−2 + 0), (2 + 0), (2 + 0))	decode1 = ((0 − 2), (−2 + 0), (2 + 0), (2 + 0))
4	data0=(2, −2, 2, 2), meaning (1, 0, 1, 1)	data1=(−2, −2, 2, 2), meaning (0, 0, 1, 1)

Further, after decoding, all values greater than 0 are interpreted as 1, while all values less than zero are interpreted as 0. For example, after decoding, data0 is (2, −2, 2, 2), but the receiver interprets this as (1, 0, 1, 1). Values of exactly 0 mean that the sender did not transmit any data, as in the following example:

Assume signal0 = (1, −1, −1, 1, 1, −1, 1, −1) is transmitted alone. The following table shows the decode at the receiver:

Step	Decode sender0	Decode sender1
0	code0 = (1, −1), signal = (1, −1, −1, 1, 1, −1, 1, −1)	code1 = (1, 1), signal = (1, −1, −1, 1, 1, −1, 1, −1)
1	decode0 = pattern.vector0	decode1 = pattern.vector1
2	decode0 = ((1, −1), (−1, 1), (1, −1), (1, −1)) · (1, −1)	decode1 = ((1, −1), (−1, 1), (1, −1), (1, −1)) · (1, 1)
3	decode0 = ((1 + 1), (−1 − 1), (1 + 1), (1 + 1))	decode1 = ((1 − 1), (−1 + 1), (1 − 1), (1 − 1))
4	data0 = (2, −2, 2, 2), meaning (1, 0, 1, 1)	data1 = (0, 0, 0, 0), meaning no data

When the receiver attempts to decode the signal using sender1's code, the data is all zeros; therefore the cross-correlation is equal to zero and it is clear that sender1 did not transmit any data.

When mobile-to-base links cannot be precisely coordinated, particularly due to the mobility of the handsets, a different approach is required. Since it is not mathematically possible to create signature sequences that are both orthogonal for arbitrarily random starting points and which make full use of the code space, unique "pseudo-random" or "pseudo-noise" sequences called spreading sequences are used in asynchronous CDMA systems. A spreading sequence is a binary sequence that appears random but can be reproduced in a deterministic manner by intended receivers. These spreading sequences are used to encode and decode a user's signal in asynchronous CDMA in the same manner as the orthogonal codes in synchronous CDMA (shown in the example above). These spreading sequences are statistically uncorrelated, and the sum of a large number of spreading sequences results in multiple access interference (MAI) that is approximated by a Gaussian noise process (following the central limit theorem in statistics). Gold codes are an example of a spreading sequence suitable for this purpose, as there is low correlation between the codes. If all of the users are received with the same power level, then the variance (e.g., the noise power) of the MAI increases in direct proportion to the number of users. In other words, unlike synchronous CDMA, the signals of other users will appear as noise to the signal of interest and interfere slightly with the desired signal in proportion to number of users.

All forms of CDMA use the spread-spectrum spreading factor to allow receivers to partially discriminate against unwanted signals. Signals encoded with the specified spreading sequences are received, while signals with different sequences (or the same sequences but different timing offsets) appear as wideband noise reduced by the spreading factor.

Since each user generates MAI, controlling the signal strength is an important issue with CDMA transmitters. A CDM (synchronous CDMA), TDMA, or FDMA receiver can in theory completely reject arbitrarily strong signals using different codes, time slots or frequency channels due to the orthogonality of these systems. This is not true for asynchronous CDMA; rejection of unwanted signals is only partial. If any or all of the unwanted signals are much stronger than the desired signal, they will overwhelm it. This leads to a general requirement in any asynchronous CDMA system to approximately match the various signal power levels as seen at the receiver. In CDMA cellular, the base station uses a fast closed-loop power-control scheme to tightly control each mobile's transmit power.

In 2019, schemes to precisely estimate the required length of the codes in dependence of Doppler and delay characteristics have been developed.[20] Soon after, machine learning based techniques that generate sequences of a desired length and spreading properties have been published as well. These are highly competitive with the classic Gold and Welch sequences. These are not generated by linear-feedback-shift-registers, but have to be stored in lookup tables.

A novel collaborative multi-user transmission and detection scheme called collaborative CDMA[21] has been investigated for the uplink that exploits the differences between users' fading channel signatures to increase the user capacity well beyond the spreading length in the MAI-limited environment. The authors show that it is possible to achieve this increase at a low complexity and high bit error rate performance in flat fading channels, which is a major research challenge for overloaded CDMA systems. In this approach, instead of using one sequence per user as in conventional CDMA, the authors group a small number of users to share the same spreading sequence and enable group spreading and despreading operations. The new collaborative multi-user receiver consists of two stages: group multi-user detection (MUD) stage to suppress the MAI between the groups and a low-complexity maximum-likelihood detection stage to recover jointly the co-spread users' data using minimal Euclidean-distance measure and users' channel-gain coefficients. An enhanced CDMA version known as interleave-division multiple access (IDMA) uses the orthogonal interleaving as the only means of user separation in place of signature sequence used in CDMA system.

• Papathanassiou, A., Salkintzis, A. K., & Mathiopoulos, P. T. (2001). "A comparison study of the uplink performance of W-CDMA and OFDM for mobile multimedia communications via LEO satellites". IEEE Personal Communications, 8(3), 35–43.

Wikimedia Commons has media related to CDMA.

• Talk at Princeton Institute for Advanced Study on Solomon Golomb's work on pseudorandom sequences