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The FT can be simply defined as a certain linear operator that maps functions or signals defined in one domain to other functions or signals in another domain. The common use of FT in electrical engineering is to transform signals from time domain to frequency domain or vice‐versa. More precisely, forward FT decomposes a signal into a continuous spectrum of its frequency components such that the time signal is transformed to a frequency domain signal. In radar applications, these two opposing domains are usually represented as “spatial‐frequency (or wave‐number)” and “range (distance).” Such use of FT will be often examined and applied throughout this book.

The forward FT of a continuous signal g(t) where −∞ < t < ∞ is described as

(1.1)

where represents the forward FT operation that is defined from time domain to frequency domain.

To appreciate the meaning of FT, the multiplying function exp(−j2πft) and operators (multiplication and integration) on the right of side of Eq. 1.1 should be examined carefully: The term is a complex phasor representation for a sinusoidal function with the single frequency of “f_i.” This signal oscillates with the single frequency of “f_i” and does not contain any other frequency component. Multiplying the signal in interest, g(t) with provides the similarity between each signal, that is, how much of g(t) has the frequency content of “f_i.” Integrating this multiplication over all time instants from −∞ to ∞ will sum the “f_i” contents of g(t) over all time instants to give G(f_i) that is the amplitude of the signal at the particular frequency of “f_i.” Repeating this process for all the frequencies from −∞ to ∞ will provide the frequency spectrum of the signal represented as G(f). Therefore, the transformed signal represents the continuous spectrum of frequency components; i.e. representation of the signal in “frequency domain.”