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The introduction of complex notation may seem like an

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Unformatted text preview: The introduction of complex notation may seem like an unnecessary complication, but it simplifies things enormously. This is because differentiation and integration become algebraic operations. Henceforth, we regard the real part of the phasor quantity as being physically interesting. The last step in our calculations will usually be to evaluate the real part of some expression. It is usually convenient to combine φ and A into a complex constant representing both the amplitude and phase of the signal, i.e. C = A exp( jφ ) . C can be viewed as a vector in the complex plane. When dealing with waves, we 13 expect both the amplitude and phase (the magnitude and angle of C ) to vary in space. Our task is then usually to solve for C everywhere. There is no explicit time variation — that is contained entirely in the implicit exp( jωt ) term, which we often even neglect to write. Note that the components of vectors too can be expressed using phasor notation. This notation can be confusing at times but really represents no special complications. Be sure not to mistake the complex plane with the plane or volume containing the vectors. As usual, the real part of a phasor vector is what is physically interesting. The rules of arithmetic for complex numbers are similar to those for real numbers, although there are important differences. For example, the norm of a vector in phasor notation is defined as | E | 2 = E · E ∗ . Consider the norm of the electric field associated with a circularly polarized wave traveling in the ˆ z direction such that E = (ˆ x ± j ˆ y ) exp( j ( ωt − kz )) . The length of the electric field vector is clearly unity, and evaluating the Euclidean length of the real part of the electric field indeed produces a value of 1. The phasor norm of the electric field meanwhile gives E · E ∗ = 2. In fact, 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 cos2 ω ◦ t + ···...
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