Rectangular Power Spectral Density Orthogonal Signals
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Solution Overview
Problem
Existing digital communication systems face challenges in transmitting multiple orthogonal signals over a single allocated bandwidth while maintaining orthogonality, especially after sampling and truncation, which disrupts the rectangular power spectral density required for high-fidelity transmission.
Innovation Solution
The use of Haar orthogonal functions in the frequency domain, combined with the Singular Value Decomposition (SVD) method, to generate and transmit signals with rectangular power spectral densities, ensuring orthogonality is restored after sampling and truncation, and employing these signals in ultra-wideband and orthogonal frequency division multiplexing (OFDM) systems.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If sampling and truncation are applied to digital communication signals, then the signals can be transmitted over discrete channels, but orthogonality is destroyed and rectangular power spectral density is lost
Solution Approach 1:
The patent applies Singular Value Decomposition (SVD) as a preliminary orthogonalization process before sampling and truncation. By pre-processing the continuous orthogonal signals with SVD to establish proper orthogonality relationships, the discrete sampled signals maintain orthogonality even after truncation, thus preserving both transmission capability and orthogonality
Solution Approach 2:
The patent transforms the orthogonal signals from continuous time domain to discrete time domain by changing sampling parameters and truncation length. By carefully selecting discrete time parameters (sampling rate, truncation point) based on the continuous signal characteristics, the rectangular power spectral density and orthogonality are preserved in the discrete domain
2Productivity
If multiple orthogonal signals are transmitted over a single allocated bandwidth, then spectral efficiency is improved, but maintaining rectangular power spectral density becomes difficult
Solution Approach 1:
The patent segments the allocated bandwidth into multiple orthogonal frequency channels, each carrying independent modulated signals. By using orthogonal frequency division multiplexing (OFDM), the total bandwidth is divided into subcarriers that are orthogonal to each other, allowing multiple signals to coexist while maintaining rectangular power spectral density through proper subcarrier spacing and windowing
Solution Approach 2:
The patent transitions from time-domain signal multiplication to frequency-domain signal superposition. By transforming the problem into the frequency domain using Fourier transforms, multiple orthogonal signals can be transmitted simultaneously over the same bandwidth without interfering with each other, preserving the rectangular power spectral density through proper frequency domain filtering
3Reliability
If continuous orthogonal signals are used, then orthogonality and rectangular power spectral density are maintained, but discrete transmission and digital processing become difficult
Solution Approach 1:
The patent replaces continuous mechanical signal processing with discrete digital signal processing. By using digital sampling, quantization, and computer-based SVD orthogonalization, the continuous orthogonal signals are converted to discrete digital signals that can be processed by digital communication systems while maintaining orthogonality through algorithmic computation rather than physical analog processes
Data Source
AI summary
In this application, a set of orthogonal functions is introduced whose power spectral densities are all rectangular shape. To find the orthogonal function set, it was considered that their spectrums (Fourier transforms of the functions) are either real-valued or imaginary-valued, which are corresponding to even and odd real-valued time domain signals, respectively. The time domain functions are all considered real-valued because they are actually physical signals. The shape of the power spectral densities of the signals are rectangular thus, the Haar orthogonal function set can be employed in the frequency domain to decompose them to several orthogonal functions. Based on the inverse Fourier transform of the Haar orthogonal functions, the time domain functions with rectangular power spectral densities can be determined. This is equivalent to finding the time-domain functions by taking the inverse Fourier transform of the frequency domain Walsh functions. The obtained functions are sampled and truncated to generate finite-length discrete signals. Truncation destroys the orthogonality of the signals. The Singular Value Decomposition method is used to restore the orthogonality of the truncated discrete signals.