ABSTRACT
Since the introduction of Cooley-Tukey fast Fourier transform (FFT) [1], the Fourier transform has been widely used in different areas of signal and image processing, communication systems, data compression, pattern recognition and image reconstruction, interpolation, linear filtering, and spectral analysis [2]-[6]. The Fourier transform determines all frequencies in the function (signal), and transfers the data defined on the real space into the complex, while simplifying the realization of the operation of linear convolution. We start with the definition and properties of the discrete Fourier transformation in the one-dimensional case, and then we will try to reveal the mathematical structure of this transformation for better understanding the transformation and using it in practical applications. The splitting of the transform is based on the paired representation of the signals, in a form of sets of short signals which can be analyzed and processed separately. The paired representation of signals is referred to as a time-frequency representation; however, the paired transform is not the wavelet transform, different types of which were developed after the Haar transform. It is interesting to note that the matrices of the paired and Haar transformations are equal up to a permutation of rows and columns. To show that, we describe the complete set of the one-dimensional paired functions and, then, analyze the relation between the paired and Haar transformations.
