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By Simon Haykin

Bargains the main whole, up to date insurance to be had at the ideas of electronic communications. specializes in simple concerns, bearing on idea to perform anyplace attainable. various examples, labored out intimately, were incorporated to aid the reader improve an intuitive seize of the idea.

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The summation on the left-hand side of this Fourier-transform pair is a periodic signal with period T0. The summation on the right-hand side of the pair is a uniformly sampled version of the spectrum G(f). We may therefore make the following statement: Uniform sampling of the spectrum G(f) in the frequency domain introduces periodicity of the function g(t) in the time domain. 42), we may also write ∞ T0  m = –∞ g ( mT 0 ) δ ( t – mT 0 ) ⇌ ∞  G ( f – nf 0 ) n = –∞ in light of which we may make the following dual statement: Uniform sampling of the Fourier transformable function g(t) in the time domain introduces periodicity of the spectrum G(f) in the frequency domain.

To emphasize this point, we write g(t) ⇌ G(f) where the top arrow indicates the forward transformation from g(t) to G(f) and the bottom arrow indicates the inverse transformation. One other notation: the asterisk is used to denote complex conjugation. Tables of Fourier Tranformations To assist the user of this book, two tables of Fourier transformations are included: 1. 1 on page 23 summarizes the properties of Fourier transforms; proofs of them are presented as end-of-chapter problems. 2. 2 on page 24 presents a list of Fourier-transform pairs, where the items listed on the left-hand side of the table are time functions and those in the center column are their Fourier transforms.

The summation on the right-hand side of the pair is a uniformly sampled version of the spectrum G(f). We may therefore make the following statement: Uniform sampling of the spectrum G(f) in the frequency domain introduces periodicity of the function g(t) in the time domain. 42), we may also write ∞ T0  m = –∞ g ( mT 0 ) δ ( t – mT 0 ) ⇌ ∞  G ( f – nf 0 ) n = –∞ in light of which we may make the following dual statement: Uniform sampling of the Fourier transformable function g(t) in the time domain introduces periodicity of the spectrum G(f) in the frequency domain.

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