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Sparse Fourier Transform Algorithm for Real-Time Applications
Sparse Fourier Transform (SFT) is an algorithm used to solve frequency analysis problems. This algorithm is designed to estimate the frequency components of sparse signals in the time-frequency domain. This is often useful in applications such as sensor data or speech processing.
To understand this, we first need to understand the Fourier Transform. The Fourier Transform is a way of transforming a signal in the time domain to the frequency domain. This transformation decomposes a continuous time signal into its frequency components. Thus, it can be used to decompose a complex signal in the time domain into simpler frequency components.
However, the Fourier Transform requires a large amount of computation and is not suitable for processing large amounts of data. This is why the SFT was developed. The SFT is specialized in estimating sparse frequency components and can obtain the results of the Fourier Transform with less computation.
The SFT only computes some of the frequency components and ignores the rest. To do this, the SFT starts by selecting some frequency components. This selection is used to estimate the frequency components from the data. Then, the SFT is performed on the selected frequency components.
The mathematical principles of the SFT are similar to those of the Fourier Transform, but there are some differences. The SFT only computes the selected frequency components. Therefore, the SFT can reduce the computation by using the results of the previous frequency estimation. This allows the SFT to operate faster than the Fourier Transform.
There are many applications of SFT. For example, SFT can be used to process large amounts of sensor data. In addition, SFT can be used to estimate sparse frequency components. In such cases, it provides better results than the Fourier Transform.
Abstract
The Fourier transform is one of the most widely used computational tools. It is used in signal processing, communications, audio and video compression, medical imaging, genomics, astronomy, etc. The fastest algorithm for computing the Fourier transform is FFT, which has O(n log n) time complexity. The near-linear time of the FFT made it an indispensable tool in many applications. However, with the emergence of big data problems and the need for real-time decision making, FFT’s runtime is no longer sufficient and faster algorithms that do not sample every data point are required. The Sparse Fourier Transform is a family of algorithms that compute the frequency spectrum faster than FFT. The Sparse Fourier Transform is based on the insight that many real-world signals are sparse –i.e., most of the frequencies have a negligible contribution to the overall signal. Exploiting this sparsity, we can compute the Fourier Transform of sparse signals very quickly (in sub-linear time). In this talk, I will give an overview of the Sparse Fourier Transform algorithms with a focus on real-time applications like GPS receivers, dynamic spectrum access, 5G wireless network, and radio astronomy.