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Fourier Transform and Convolution

Fourier Transform and Convolution

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In this chapter, both the Fourier transform and Fourier series will be discussed. The properties of the Fourier transform will be presented and the concept of impulse function will be introduced. The definition of convolution and its relation with Fourier transform will be presented. Examples of some commonly used Fourier transforms are given and the results are presented in a table for quick reference. There are many good books on this subject. A few of the books are listed at the end of this chapter. Readers with a background in this area can skip this chapter. However, Examples 3.10 and 3.11 may be of interest, because the former one is related to the radar pulse train and the latter will be used in the Hilbert transform.

Chapter Contents:

  • 3.1 Introduction
  • 3.2 Fourier Transform
  • 3.3 Impulse Function
  • 3.4 Properties of Fourier Transform
  • 3.4.1 Linearity
  • 3.4.2 Even and Odd Functions
  • 3.4.3 Duality
  • 3.4.4 Scaling
  • 3.4.5 Time Shift
  • 3.4.6 Frequency Shift
  • 3.4.7 Derivative
  • 3.4.8 Integral
  • 3.5 Fourier Series
  • 3.6 Comb Function
  • 3.7 Convolution
  • 3.8 Parseval's Theorem
  • 3.9 Examples
  • 3.10 Summary
  • References

Inspec keywords: convolution; Fourier series; Fourier transforms

Other keywords: Hilbert transform; convolution; radar pulse train; Fourier series; Fourier transform; impulse function

Subjects: Signal processing and detection; Signal processing theory; Integral transforms; Integral transforms

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