Comparison with the Fourier Transform (*)

The Fourier Transform pair

$$\tilde{\underline{E}}(\underline{r}, \nu) =\int_{-\infty}^{\infty} \underline{\mathcal{E}}(\underline{r},t) e^{-j \nu t} \, dt \space ,$$ (2.66)

$$\underline{\mathcal{ E}}(\underline{r},t) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} \tilde{\underline{E}} e^{j \nu t} \, d\nu$$ (2.67)

allows us to transform a physical quantity, such as the electric field, from the time domain, where the appropriate field vector is $\underline{\mathcal{ E}}(\underline{r},t)$, to the frequency domain, where the appropriate field vector is $\tilde{\underline{E}}(\underline{r},\nu)$, and viceversa. The equations in the frequency domain are formally identical to the equations in the phasor domain, and this may create some confusion between the two. The difference between frequency-domain quantities is illustrated by the following example.

Consider the electric field given by (A.51), whose phasor is given by (A.55). The corresponding field in the frequency domain is obtained by substituting (A.51) into (A.66) and using the integral representation

$$\delta(\nu)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} e^{-j \nu t} \, dt$$ (2.68)

for the one-dimensional delta function. It is found that

$$\tilde{\underline{E}}(\underline{r},\nu)=\pi \underline{E}_0(\und... ...u-\omega) + e^{-j \varphi(\underline{r})} \delta (\nu +\omega) \right] \space .$$ (2.69)

Now let us compare (A.55) and (A.69). Basically, both equations contain the same information, i.e. the amplitude $\underline{E}_0(\underline{r})$ and the initial phase $\varphi(\underline{r})$ of the electric field; however, the phasor formula is simpler then the frequency domain formula. Additionally, the phasor $\underline{E}$ is measured in $V/m$ just as $\underline{\mathcal{E}}$ is, whereas the frequency-domain quantity $\tilde{\underline{E}}$ is measured in $V \cdot s/m$.

In conclusion, although calculations in th phasor domain and frequency domains are the same because the transformed equations are identical, the transformed quantities have different meaning and physical dimensions, and the formulas to find the time-domain quantities from the corresponding transformed quantities are different in the two cases.