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21 results in Introduction to Orthogonal Transforms

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  • 2 - Vector spaces and signal representation

  • Ruye Wang, Harvey Mudd College, California
  • Book:
    Introduction to Orthogonal Transforms
    Published online:
    05 October 2012
    Print publication:
    08 March 2012, pp 34-104

  • Summary

    In this chapter we discuss some basic concepts of Hilbert space and the related operations and properties as the mathematical foundation for the topics of the subsequent chapters. Specifically, based on the concept of unitary transformation in a Hilbert space, all of the unitary transform methods to be specifically considered in the following chapters can be treated from a unified point of view: they are just a set of different rotations of the standard basis of the Hilbert space in which a given signal, as a vector, resides. By such a rotation the signal can be better represented in the sense that the various signal processing needs, such as noise filtering, information extraction and data compression, can all be carried out more effectively and efficiently.

    Inner product space

    Vector space

    In our future discussion, any signal, either a continuous one represented as a time function x(t), or a discrete one represented as a vector x = […, x[n], …]T, will be considered as a vector in a vector space, which is just a generalization of the familiar concept of N-dimensional (N-D) space, formally defined as below.

    Definition 2.1.A vector space is a set v with two operations of addition and scalar multiplication defined for its members, referred to as vectors.

  • 10 - Continuous- and discrete-time wavelet transforms

  • Ruye Wang, Harvey Mudd College, California
  • Book:
    Introduction to Orthogonal Transforms
    Published online:
    05 October 2012
    Print publication:
    08 March 2012, pp 461-491

  • Summary

    Why wavelet?

    Short-time Fourier transform and Gabor transform

    In Chapter 3, we learned that a signal can be represented as either a time function x(t) as the amplitude of the signal at any given moment t, or, alternatively and equivalently, as a spectrum X(f) = F[x(t)] representing the magnitude and phase of the frequency component at any given frequency f. However, no information in terms of the frequency contents is explicitly available in the time domain, and no information in terms of the temporal characteristics of the signal is explicitly available in the frequency domain. In this sense, neither x(t) in the time domain nor X(f) in the frequency domain provides complete description of the signal. In other words, we can have either temporal or spectral locality regarding the information contained in the signal, but never both at the same time.

    To address this dilemma, the short-time Fourier transform (STFT), also called windowed Fourier transform, can be used. The signal x(t) to be analyzed is first truncated by a window function w(t) before it is Fourier transformed to the frequency domain. As all frequency components in the spectrum are known to be contained in the signal segment inside this particular time window, certain temporal locality in the frequency domain is achieved.

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