By Simon J. Godsill

The advent of top quality electronic audio media resembling the compact disc and electronic audio tape has significantly raised expectancies approximately sound caliber in all kinds of recordings. mixed with a up to date upsurge in curiosity in old and archive recordings of track, speech and different audio fabric, the recovery of degraded resources has develop into an more and more vital box. Containing study formerly unpublished in any shape, functions of recent sign processing innovations to audio recovery are provided right here in a complete and authoritative textual content. in addition to offering a large assurance of current methodologies the authors additionally examine either present components of analysis and destiny traits. additionally the textual content is supported by way of introductory fabric on electronic sign processing. geared toward researchers in sign processing, at communications industries and at track businesses and media libraries wishing to remarket or use archive sound recordings, it is a specified and crucial learn of audio recovery ideas.

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2) p=−∞ where we have defined gp = g(pT ). 3) p=−∞ Discrete time Fourier transform (DTFT) 20 2. 3) has the form of a Fourier series and the orthogonality of the complex exponential can be used to invert the transform. 3 Discrete time convolution Consider an analogue signal x(t) and its sampled representation ∞ ∞ xs (t) = x(t) n=−∞ δ(t − nT ) = n=−∞ x(nT )δ(t − nT ). 5) in which the impulse response needs only to be evaluated at integer multiples of T . 3, where x(nT ) represents values from an analogue signal x(t) sampled periodically at intervals of T and the outputs y(nT ) represent sample values which may be converted into an analogue signal by means of a digital to analogue converter.

6 The discrete Fourier transform (DFT) The Discrete Time Fourier Transform (DTFT) expresses the spectrum of a sampled signal in terms of the signal samples but is not computable on a digital machine for two reasons: 1. The frequency variable ω is continuous. 2. The summation involves an infinite number of samples The first problem may be overcome by simply choosing to evaluate the spectrum at a set of discrete frequencies. 2) so that it is only necessary to consider frequencies in the range: ωT = 0 → 2π Although the frequencies may be arbitrarily chosen in this range, important computational advantages are to be gained from choosing N uniformly spaced frequencies, ie.

7) where ∞ H(ejωT ) = hq . 8) q=−∞ is the system frequency response. 6) as follows. Let .. xp = yn = ejωpT ∞ hp . ejω(n−p)T p=−∞ .. yn ∞ = ejωnT hp . e−jpωT p=−∞ .. 4 The z-transform The Laplace transform is an important tool in continuous system theory as it enables one to deal with differential equations as algebraic equations, since it transforms convolutive functions into multiplicative functions. However, the natural mathematical description of sampled data systems is in terms of difference equations and it would be of considerable help to develop a method whereby difference equations can be treated as algebraic equations.