Help understand: diversity, interleaving, SFFT, additive Gaussian white noise

Variety

Diversity is a technique used in wireless communications to increase signal reliability and reduce the effects of fading. The core idea is to improve communication quality through multiple independent or nearly independent signal replication. Because each signal replica experiences a different propagation path or condition, they may not all be affected by fading at the same time. By merging these copies, the strength or quality of the signal can be improved, thereby increasing the reliability of communications.

The main types of diversity technologies include:

  1. Spatial Diversity:

    • Use multiple antennas on the transmitter and/or receiver.
    • When each antenna receives or transmits a signal, it may experience different multipath conditions due to its physical distance from other antennas.
    • Common technology: MIMO (Multiple Input Multiple Output).

  2. Frequency Diversity:

    • This is achieved by sending signal copies on different frequency bandwidths.
    • Different frequencies may be subject to varying degrees of frequency selective fading.
    • Common technology: OFDM (Orthogonal Frequency Division Multiplexing).

  3. Time Diversity:

    • This is achieved by sending signal copies at different time intervals.
    • When the communication environment changes in time (such as the Doppler effect caused by movement speed), there may be different channel conditions at different points in time.
    • Common technologies: Rake receiver, signal interleaving.

  4. Polarization Diversity:

    • Different polarizations of electromagnetic waves are used to send and receive signals.
    • Vertically and horizontally polarized signals may experience different propagation conditions.

  5. Angular Diversity:

    • Based on the difference in the arrival angle or emission angle of the signal.
    • Especially useful in systems with directional antennas.

When these independent or nearly independent signal copies arrive at the receiver, the receiver combines them in some way to obtain the best signal quality. This can be accomplished by maximum ratio binning, averaging, or other methods.

Hinchin Fast Fourier Transform (SFFT) is an efficient algorithm designed for processing the Fourier transform of sparse signals. Its goal is to find non-zero or significant frequency components in a signal rather than counting all possible frequency components.

“Interleaving” is an important technology in digital communications and data storage. Its purpose is to improve the error correction performance of data or cope with the impact of bit errors. Here are some basic introductions to interleaving:

Interweaving:

  1. Definition: Interleaving is a process that rearranges the order of data blocks or data symbols, usually to ensure that consecutive errors do not affect a consecutive block of data.

  2. Purpose: The main purpose of interleaving is to protect against so-called sudden errors. In digital communications, many errors tend to be bursty, meaning that errors can occur continuously over a short period of time. Through interleaving, these consecutive errors are “scattered” into different blocks of data, making it easier for error-correcting coding to correct these errors.

How it works:

Imagine you have a piece of data, for example:

ABCDEFGHIJ

A simple interleaving strategy might split this data into two rows:

ABCDE
FGHIJ

Then reading by column, we get:

AFBGCHDIEJ

In this way, data that was continuous in the original data is now separated in the interleaved data.

Deinterleave:

In order to recover the original data, deinterleaving is required. This is the inverse of the interleaving process. In the above example, deinterleaving would rearrange AFBGCHDIEJ back to ABCDEFGHIJ.

Application:

  1. Digital Communication: Interleaving is widely used in digital communication systems, especially wireless communication systems, such as 3G, 4G and 5G. Since wireless communication channels are often affected by factors such as multipath attenuation and interference, data may encounter sudden errors during transmission. Interleaving ensures that these errors are not concentrated on a single block of data, thereby improving the performance of error correction coding.

  2. Digital TV and Broadcasting: Interleaving is also used in digital TV and broadcasting systems, such as DVB-T, DVB-S, etc.

  3. Data Storage: In some data storage solutions, such as optical disks and some solid-state drives, interleaving is also used to ensure that sequential physical damage does not result in sequential data loss.

In summary, interleaving is a key technology in digital communications and data storage, which improves the system’s tolerance to errors by changing the order of data.

Hinchin Fast Fourier Transform (SFFT)

Traditional FFT:
For a signal of length N, the traditional FFT requires

O

(

N

log

?

N

)

O(N \log N)

Computational complexity of O(NlogN). If we were to Fourier transform this signal, we would get

N

N

N frequency components, even if the signal has only a few non-zero frequency components.

SFFT:
Assume that the signal has K significant non-zero components in the frequency domain, where

K

?

N

K \ll N

K?N. The goal of SFFT is to find only this

K

K

K significant components, rather than all

N

N

N components. Therefore, the computational complexity of SFFT is proportional to K rather than N. For most practical applications,

K

K

K is much smaller than

N

N

N, which means that the SFFT is computationally more efficient than the traditional FFT.

To achieve this goal, SFFT employs a variety of strategies, such as hashing, subsampling, and iterative techniques, to efficiently identify and calculate these significant non-zero components.

In short, the basic idea of SFFT is:

  1. Use subsampling to reduce the dimensionality of your data.
  2. Use hashing or other techniques to identify the location of possible non-zero frequency components.
  3. The exact location and amplitude of these non-zero frequency components are determined progressively more precisely through iterations.

Therefore, when we know that a signal is sparse or partially sparse in the frequency domain (i.e., there are only a few significant frequency components), SFFT can provide a more efficient Fourier transform method than the traditional FFT.

Additive Gaussian white noise

The purpose of assuming that the noise is zero-mean cyclic symmetric complex Gaussian noise (often referred to as AWGN, additive white Gaussian noise) is mainly to simplify analysis and design. This assumption brings several advantages:

  1. Mathematical Simplification: The Gaussian noise model is simple for analysis and mathematical processing. For example, the probability density function (PDF) associated with Gaussian noise is known and has a simple form.

  2. Central Limit Theorem: In many practical situations, noise superimposed by many small, independent random processes tends to tend to a Gaussian distribution. The central limit theorem proves this.

  3. Worst Case: Gaussian noise is generally considered the most destructive for a given signal power. Therefore, designing a system for AWGN can ensure the system’s robustness in more real-world situations.

  4. Linear system and AWGN: When a linear system is affected by AWGN, the output is still AWGN. This means that for a series of linear operations (such as filtering, amplification, etc.), we can expect the noise to still maintain its Gaussian properties.

  5. Common Model: Due to its simplicity and practicality, AWGN is a standard model in many communication books and research articles. Therefore, using this model allows comparison and reference with existing literature.

  6. Practical Applications: Although real-world noise may not be entirely Gaussian, the AWGN model is still a fairly accurate approximation in many practical applications.

In summary, the assumption of zero-mean cyclic symmetric complex Gaussian noise is mainly used to simplify the problem and provide a baseline model that is applicable in a variety of situations.