Chapter 4 Temporal and spatial frequency domain representation 4.1 Interpretation of the temporal transform When a field is represented by a continuous superposition of elemental components, the esulting decomposition can simplify computation and provide physical insight. Such rep- esentation is usually accomplished through the use of an integral transform. Although everal different transforms are used in electromagnetics. we shall concentrate on the powerful and efficient Fourier transform Let us consider the Fourier transform of the electromagnetic field. The field depends on x, y, z, t, and we can transform with respect to any or all of these variables. However, a consideration of units leads us to consider a transform over t separately. Let y(r, t) represent any rectangular component of the electric or magnetic field. Then the temporal transform will be designated by y(r, a) ψ(r,1)←v(r,o) Here o is the transform variable. The transform field y is calculated using(A. The inverse transform is, by(A. 2) ψ(r,o) ejon d e Since y is complex it may be written in amplitude-phase form y(r, o)=ly(r, o). Since y(r, t) must be real, (4. 1)shows that ψ(r,-)=v’(r,ω) Furthermore, the transform of the derivative of y may be found by differentiating(4.2) r(r, =2/ 2001 by CRC Press LLCChapter 4 Temporal and spatial frequency domain representation 4.1 Interpretation of the temporal transform When a field is represented by a continuous superposition of elemental components, the resulting decomposition can simplify computation and provide physical insight. Such rep￾resentation is usually accomplished through the use of an integral transform. Although several different transforms are used in electromagnetics, we shall concentrate on the powerful and efficient Fourier transform. Let us consider the Fourier transform of the electromagnetic field. The field depends on x, y,z, t, and we can transform with respect to any or all of these variables. However, a consideration of units leads us to consider a transform over t separately. Let ψ(r, t) represent any rectangular component of the electric or magnetic field. Then the temporal transform will be designated by ψ( ˜ r,ω): ψ(r, t) ↔ ψ( ˜ r, ω). Here ω is the transform variable. The transform field ψ˜ is calculated using (A.1): ψ( ˜ r,ω) = ∞ −∞ ψ(r, t) e− jωt dt. (4.1) The inverse transform is, by (A.2), ψ(r, t) = 1 2π ∞ −∞ ψ( ˜ r,ω) e jωt dω. (4.2) Since ψ˜ is complex it may be written in amplitude–phase form: ψ( ˜ r,ω) = |ψ( ˜ r,ω)|e jξ ψ (r,ω), where we take −π<ξ ψ (r,ω) ≤ π. Since ψ(r, t) must be real, (4.1) shows that ψ( ˜ r, −ω) = ψ˜ ∗(r, ω). (4.3) Furthermore, the transform of the derivative of ψ may be found by differentiating (4.2). We have ∂ ∂t ψ(r, t) = 1 2π ∞ −∞ jωψ( ˜ r,ω) e jωt dω,