HOS (Hierarchy of orthogonal sequences)

The Hierarchy of Orthogonal Sequences (HOS) is a mathematical framework used in digital communication systems to generate a set of signals that are mutually orthogonal to each other. The HOS is particularly useful in communication systems that use spread spectrum techniques, such as Code Division Multiple Access (CDMA), which require multiple users to transmit simultaneously over the same frequency band.

In this essay, we will explain the principles of HOS, its applications, and its limitations. We will start by introducing the concept of orthogonality and how it is used in digital communication systems. Then, we will explain how the HOS is used to generate orthogonal sequences, and we will discuss some of the advantages and limitations of this approach.

Orthogonality in Digital Communication

In digital communication systems, signals are transmitted over a channel that can introduce noise and interference. To minimize the effects of noise and interference, it is desirable to use signals that are as distinct from each other as possible. One way to achieve this is by using orthogonal signals.

Two signals are said to be orthogonal if their inner product is zero. Mathematically, the inner product of two signals x(t) and y(t) over a time interval T is given by:scssCopy code<x(t), y(t)> = ∫_0^T x(t) y*(t) dt

where y*(t) is the complex conjugate of y(t). If <x(t), y(t)> = 0, then x(t) and y(t) are orthogonal.

Orthogonal signals have a number of advantages in digital communication systems. First, they can be easily distinguished from each other, even in the presence of noise and interference. Second, they can be transmitted simultaneously without interfering with each other. Third, they can be combined at the receiver to improve the signal-to-noise ratio (SNR) of the received signal.

The Hierarchy of Orthogonal Sequences

The Hierarchy of Orthogonal Sequences (HOS) is a method for generating a set of orthogonal sequences that can be used in digital communication systems. The HOS is based on the concept of Hadamard matrices, which are matrices with elements +1 and -1 that are mutually orthogonal.

A Hadamard matrix of order N is an N × N matrix H such that:

• The elements of H are +1 or -1.
• The rows of H are mutually orthogonal.
• The columns of H are mutually orthogonal.

For example, the following is a Hadamard matrix of order 4:cssCopy codeH = [ 1  1  1  1      1 -1  1 -1      1  1 -1 -1      1 -1 -1  1 ]

The first row of the matrix is [1 1 1 1], the second row is [1 -1 1 -1], the third row is [1 1 -1 -1], and the fourth row is [1 -1 -1 1]. It can be shown that the rows and columns of H are mutually orthogonal.

The HOS is a hierarchy of Hadamard matrices of increasing order. The first matrix in the hierarchy is a Hadamard matrix of order 1, which is simply the number +1. The second matrix is a Hadamard matrix of order 2, which is given by:cssCopy codeH_2 = [ 1  1       1 -1 ]

The third matrix is a Hadamard matrix of order 4, which we have already seen. The fourth matrix is a Hadamard matrix of order 8, which can be generated recursively from the third matrix as follows:

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To generate the fourth matrix, we first write the third matrix as four blocks of size 2x2:cssCopy codeH_4 = [ H_2  H_2        H_2 -H_2 ]

We then replace each block in the matrix with a copy of the third matrix:cssCopy codeH_4 = [ H_2  H_2  H_2  H_2        H_2 -H_2  H_2 -H_2        H_2  H_2 -H_2 -H_2        H_2 -H_2 -H_2  H_2 ]

It can be shown that the rows and columns of H_4 are mutually orthogonal, and that H_4 contains all possible binary sequences of length 3 as its rows.

The HOS continues in this way, with each matrix of order 2^n being generated recursively from the matrix of order 2^(n-1). The resulting matrices form a hierarchy that can be used to generate a set of orthogonal sequences that can be used in digital communication systems.

Applications of HOS

The HOS has a number of applications in digital communication systems. One of the most important is in Code Division Multiple Access (CDMA) systems, which are used in cellular networks to allow multiple users to transmit simultaneously over the same frequency band.

In a CDMA system, each user is assigned a unique code sequence that is used to spread the user's signal over a wider bandwidth. The code sequence is chosen to be orthogonal to the code sequences of other users, so that the signals can be transmitted simultaneously without interfering with each other.

The HOS can be used to generate a set of code sequences that are mutually orthogonal to each other. For example, the first four matrices in the HOS can be used to generate a set of four orthogonal code sequences that can be assigned to four different users in a CDMA system. The code sequences can be combined with the user's data to generate a spread spectrum signal that can be transmitted over the cellular network.

Another application of the HOS is in the design of pulse compression filters, which are used in radar systems to improve the range resolution of the radar. A pulse compression filter is a matched filter that is designed to compress a transmitted pulse into a shorter pulse with higher peak power. The filter is designed to be orthogonal to other filters used in the radar system, so that the compressed pulse can be easily distinguished from other signals.

The HOS can be used to generate a set of orthogonal pulse compression filters that can be used in a radar system. The filters can be combined with the transmitted pulse to generate a compressed pulse that can be transmitted over the radar. The compressed pulse can then be detected by a matched filter at the receiver to improve the range resolution of the radar.

Limitations of HOS

While the HOS has many advantages in digital communication systems, it also has some limitations. One of the main limitations is that the number of orthogonal sequences generated by the HOS is limited by the order of the highest Hadamard matrix in the hierarchy. For example, the fourth matrix in the HOS generates 8 orthogonal sequences, but the fifth matrix only generates 16 orthogonal sequences.

This limitation can be overcome by using other methods to generate orthogonal sequences, such as Gold sequences and Kasami sequences. These sequences are generated using feedback shift registers, and can generate a large number of orthogonal sequences with good correlation properties.

Another limitation of the HOS is that it requires a large amount of computational resources to generate the Hadamard matrices. The computation of a Hadamard matrix of order N requires O(N^2) operations, which can be prohibitively expensive for large values of N.

Conclusion

The Hierarchy of Orthogonal Sequences (HOS) is a mathematical concept used in digital communication systems to generate a set of orthogonal sequences that can be used for various applications such as Code Division Multiple Access (CDMA) systems and pulse compression filters in radar systems. The HOS is generated recursively from the Hadamard matrices, which are square matrices with entries +1 and -1 that have orthogonal rows and columns.

The HOS has limitations such as a limited number of generated orthogonal sequences and a large computational cost to generate the Hadamard matrices. However, the HOS remains an important tool in digital communication systems and can be used in combination with other techniques to generate a large number of orthogonal sequences with good correlation properties.