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G a b give twice the average value of the given

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the real part of inner and outer conjugate products in phasor notation (e.g. A × B ) give twice the average value of the given product. This will be important when we discuss average power flow and the time-average Poynting vector. 1.4 Mathematical preliminaries The remainder of this chapter concerns powerful mathematical concepts that will be helpful for understanding material in successive chapters of the text. A brief treatment of Fourier analysis is presented. After that, basic ideas in prob- ability theory are reviewed. Fourier analysis and probability theory culminate in the definition of the power spectral density of a signal, something that plays a central role in radar signal processing. 1.4.1 Fourier analysis Phasor notation can be viewed as a kind of poor-man’s Fourier analysis, a fundamental mathematical tool used in a great many fields of science and engineering, including signal processing. The basis of Fourier analysis is the fact that a periodic function in time, for example, can be expressed as an infinite sum of sinusoidal basis functions: f ( t ) = 1 2 a + a 1 cos ω t + a 2 cos 2 ω t + · · · + b 1 sin ω t + b 2 sin 2 ω t + · · · where τ is the period of f ( t ) and ω 2 π/τ . The series converges to f ( t ) where the function is continuous and converges to the midpoint where it has discontinuous jumps. The Fourier coefficients can be found by utilizing the orthogonality of sin and cosine functions over the period τ : a n = ω π integraldisplay τ/ 2 τ/ 2 f ( t ) cos tdt b n = ω π integraldisplay τ/ 2 τ/ 2 f ( t ) sin tdt where the integrals can be evaluated over any complete period of f ( t ) . Fourier decomposition applies equally well to real and complex functions. With the aid of the Euler theorem, the decomposition above can be shown to be equivalent to: f ( t ) = summationdisplay n = −∞ c n e jnω t (1.1) c n = ω 2 π integraldisplay τ/ 2 τ/ 2 f ( t ) e jnω t dt (1.2) Written this way, it is clear that the Fourier coefficients c n can be interpreted as the amplitudes of the various frequency components or harmonics that compose the periodic function f ( t ) . The periodicity of the function ultimately permits its representation in terms of discrete frequencies. 14
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It can be verified easily that the sum of the squares of the harmonic amplitudes gives the mean squared average of the function, (| f ( t ) | 2 ) = summationdisplay n = −∞ | c n | 2 (1.3) which is a simplified form of Parseval’s theorem. This shows that the power in the function f ( t ) is the sum of the powers in all the individual harmonics, which are the normal modes of the function. Parseval’s theorem is sometimes called the completeness relation because it shows how all the harmonic terms in the Fourier expansion are required in general to completely represent the function. Other forms of Parseval’s theorem are discussed below.
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