Digital Signal Processing: Mathematical and Computational by Jonathan M. Blackledge

By Jonathan M. Blackledge

This ebook kinds the 1st a part of a whole MSc path in a space that's primary to the continued revolution in info know-how and communique platforms. vastly exhaustive, authoritative and complete and bolstered with software program, this is often an advent to fashionable equipment within the constructing box of electronic sign Processing (DSP). the focal point is at the layout of algorithms and the processing of electronic signs in components of communications and keep watch over, supplying the reader with a accomplished creation to the underlying rules and mathematical versions.

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Extra info for Digital Signal Processing: Mathematical and Computational Methods, Software Development and Applications (Woodhead Publishing Series in Optical and Electronic Materials)

Example text

1) n Zl = . sm 1 . Hence, (cos () + i sin ())n = cos( n()) + i sin( n()). 6 The Complex Exponential Unlike other functions, whose differentiation and/or integration yields different functions (with varying levels of complexity), the exponential function retains is functional form under differentiation and/or integration. Thus, we can define the exponential function as that function f (x) say, such that d f'(x) == dxf(x) = f(x). What form should such a function have? Suppose we consider the power series x2 x3 xn f(x) = 1 + x + I"" + I"" + ...

Such a solution is determined by considering how the process of diffusion responds to a single point source (a space-time dependent impulse) which yields the Green's function (in this case, a Gaussian function). The connection between the basic convolution model for describing signals and systems and the Green's function solution to PDEs that describe these systems is fundamental. Thus, the convolution model that is the basis for so much of the material discussed in this work is not phenomenological but based on intrinsic methods of analysis in mathematical physics via the application of Green's function solutions.

1 ~ B ~ 27f. Green's Theorem in the Plane Theorem If 5 is a closed region in the x - y plane bounded by a simple closed curve C and if P and Q are continuous function of x and y having continuous derivatives in 5, then f(PdX c + Qdy) = JJ s (~~ - ~:) dxdy CHAPTER 1. 10). _ B A C t--t-----~... 10: Path of integration for the proof of Green's theorem in the plane. Then, b = b J J a a P(x, Y2)dx - P(x, Y1)dx b =- J P(x, Y1)dx - a J P(x, Y2)dx =- f a b e Pdx. 4. 2 d Q(X 1 , y)dy = c J Q(X 2 , y)dy + J Q(X 1 , y)dy d c = f Qdy.

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