Application of Orthogonal vectors in CDMA

Mathematics has been learned since kindergarten. For most people, except for exams and accounting, it is of little use. Everyone knows that technological progress is closely related to mathematics, but we don't feel it. This article describes how CDMA uses orthogonal vector features to perform channel multiplexing tasks.

What is CDMA?

CDMA (Code Division Multiple Access) is translated into "Code Division Multiple Access", which is a method of channel multiplexing. Initially, it was used for military communication because the system has strong anti-interference capabilities. With the advancement of technology, the price and size of CDMA equipment have been greatly reduced, so it is now widely used in civil mobile communications, especially in Wireless LAN.

What is an orthogonal vector? What features does it have?

N-dimensional vector: N ordered numbers A1, A2 ,... And an array is called an n-dimensional vector.
Inner Product of a vector: x = (x1, x2, X3 ,..., XN) T, y = (Y1, Y2, Y3 ,..., YN) T, so [x, y] = x1y1 + x2y2 +... + Xnyn is called [x, y] As the Inner Product of the vectors x and y.
Some features of vector accumulation: [x, y] = [y, X]; [ax, y] = A [x, y] (A is a real number ); [x + y, z] = [x, y] + [y, z]. If [x, y] = 0, x is orthogonal to y. If x = 0, x is orthogonal to any vector.

How does CDMA use orthogonal vectors?

CDMA divides each bit time into m short intervals, which is called a piece of code. M is usually 64 or 128. Each station using CDMA is assigned a unique M-bit code sequence. How does a station send bit1, then it sends its own code segment sequence. If you want to send bits 0, the binary anticode of the code sequence is sent. For example, if the serial number assigned to the S station is 10101101, when s wants to send bits 1, it will send the serial number 10101101, and when it wants to send bits 0, send 01010010. For convenience, write 0 in the Code slice as-1 and 1 as + 1. Therefore, the serial number of the S station is (+ 1-1 + 1-1 + 1 + 1-1 ).
+ 1 ).
In CDMA, the serial numbers assigned by each station are not only different, but also orthogonal to each other. We can know from the characteristics of the orthogonal vector above that the inner product of the code sequence between any two stations is 0. In addition, the inner product of the reverse code of a station's code sequence and other station's code sequence is also 0, and the inner product of the reverse code of its own code sequence is-1. This allows the sites to separate their own information from the information they receive.
How does the receiving station separate the information sent to itself? Now, if Station X wants to receive data sent by station S, Station X must know the sequence of chunks unique to station S. Station X uses the bitwise sequence it obtains to inner product with the received signal (note that Station X receives the sum of the sent sequences of each station ). The signals of other stations are filtered out, but only the signals sent by S stations are left. When the S station sends bits 1, the result of calculating the inner product on the X station is + 1. When the S station sends bits 0, this result is-1.
It is the working principle of CDMA. In this example, we assume that each source code is spread-out with eight chunks, and the S station sends 110. The serial number of the chunks is (-1-1-1 + 1 + 1-1 + 1 ), the spread spectrum signal sent is SX; the T station sends 110, and the code segment sequence is (-1-1 + 1-1 + 1 + 1 + 1-1 ), the spreading signal sent is Tx. This means that each station receives a superimposed signal Tx + SX.

From the above description, we can clearly understand how orthogonal vectors and their characteristics are used in CDMA. In fact, this principle is also useful when we write specific applications.
This article is based on Xie xiiren's "Computer Network" and "Engineering Mathematics: Linear Algebra" compiled by the Mathematics Teaching and Research Section of Tongji University.