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Discrete Fourier Transform

Discrete Fourier Transform

The Discrete Fourier Transform (DFT) is a fundamental tool in signal processing, extending the discrete-time Fourier transform by evaluating discrete signals at uniformly spaced frequency intervals. This transformation converts a finite sequence of time-domain samples into frequency components, each representing complex sinusoids ordered by frequency. The DFT translates these sequences into the frequency domain, effectively indicating the magnitude and phase of each frequency component present...

Properties of DTFT I

Properties of DTFT I

In signal processing, Discrete-Time Fourier Transforms (DTFTs) play a critical role in analyzing discrete-time signals in the frequency domain. Various properties of the DTFTs such as linearity, time-shifting, frequency-shifting, time reversal, conjugation, and time scaling help understand and manipulate these signals for different applications.
The linearity property of DTFTs is fundamental. If two discrete-time signals are multiplied by constants a and b respectively, and then combined to...

Discrete-time Fourier transform

Discrete-time Fourier transform

The Discrete-Time Fourier Transform (DTFT) is an essential mathematical tool for analyzing discrete-time signals, converting them from the time domain to the frequency domain. This transformation allows for examining the frequency components of discrete signals, providing insights into their spectral characteristics. In the DTFT, the continuous integral used in the continuous-time Fourier transform is replaced by a summation to accommodate the discrete nature of the signal.
One of the notable...

Discrete-Time Fourier Series

Discrete-Time Fourier Series

The Discrete-Time Fourier Series (DTFS) is a fundamental concept in signal processing, serving as the discrete-time counterpart to the continuous-time Fourier series. It allows for the representation and analysis of discrete-time periodic signals in terms of their frequency components. Unlike its continuous counterpart, which utilizes integrals, the calculation of DTFS expansion coefficients involves summations due to the discrete nature of the signal.
For a discrete-time periodic signal x[n]...

Basic signals of Fourier Transform

Basic signals of Fourier Transform

The Fourier Transform is a pivotal mathematical tool in signal processing, enabling the transformation of time-domain signals into their frequency-domain representations. Among the numerous elements within this domain, certain functions like the sinc function, delta function, and exponential signals hold significant importance due to their unique properties and implications.
The sinc function, defined as sinc(x) = sin(πx)/(πx), is particularly notable for its symmetry and behavior at zero. It...

Properties of Fourier Transform II

Properties of Fourier Transform II

The Fourier Transform (FT) is an essential mathematical tool in signal processing, transforming a time-domain signal into its frequency-domain representation. This transformation elucidates the relationship between time and frequency domains through several properties, each revealing unique aspects of signal behavior.
The Frequency Shifting property of Fourier Transforms highlights that a shift in the frequency domain corresponds to a phase shift in the time domain. Mathematically, if x(t) has...

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Related Experiment Videos

Discrete wavelet transform and data expansion reduction in homomorphic encrypted domain.

Peijia Zheng¹, Jiwu Huang

¹School of Information Science and Technology, Sun Yat-Sen University, Guangzhou 510006, China. zhengpj@mail2.sysu.edu.cn

IEEE Transactions on Image Processing : a Publication of the IEEE Signal Processing Society

|March 27, 2013

Summary

This study introduces a novel method for secure signal processing using homomorphic encryption. It enables discrete wavelet transform (DWT) and multiresolution analysis (MRA) while minimizing data expansion for enhanced privacy.

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Area of Science:

Cryptography and Signal Processing
Secure data analysis and transformation

Background:

Signal processing in the encrypted domain is crucial for protecting sensitive data.
Existing methods face challenges with data expansion and quantization errors.

Purpose of the Study:

To implement discrete wavelet transform (DWT) and multiresolution analysis (MRA) in the homomorphic encrypted domain.
To address data expansion issues in encrypted domain signal processing.
To enhance secure image processing capabilities.

Main Methods:

A framework for encrypted domain DWT and inverse DWT (IDWT) is proposed.
Data expansion and quantization errors are analyzed.
A multiplicative inverse method is employed to reduce data expansion.

Main Results:

The proposed method enables multilevel DWT/IDWT with reduced data expansion in the encrypted domain.
Experimental results with 2-D Haar wavelet transform demonstrate advantages in secure image processing.
Computational complexity is analyzed and compared.

Conclusions:

This work presents the first implementation of DWT and MRA in the encrypted domain.
The method offers a practical solution for secure signal processing with manageable data expansion.
The findings advance the field of privacy-preserving signal analysis.