Caner Ozdemir - Inverse Synthetic Aperture Radar Imaging With MATLAB Algorithms

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(− j 2 πft ) and operators (multiplication and integration) on the right of side of Eq. 1.1should 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.”

This transformation is the inverse operation of the FT. IFT, therefore, synthesizes a frequency‐domain signal from its spectrum of frequency components to its time domain form. The IFT of a continuous signal G ( f ) where −∞ < f < ∞ is described as

(1.2)

where the IFT operation from frequency domain to time domain is represented by .

There are many useful Fourier rules and pairs that can be very helpful when applying the FT or IFT to different real‐world applications. We will briefly revisit them to remind the properties of the FT to the reader. Provided that FT and IFT are defined as in Eqs. 1.1and 1.2, respectively, FT pair is denoted as

(1.3)

and the corresponding alternative pair is given by

(1.4)

Based on these notations, the properties of FT are listed briefly below.

If G ( f ) and H ( f ) are the FTs of the time signals g ( t ) and h ( t ), respectively, the following equation is valid for the scalars a and b .

(1.5)

Therefore, the FT is a linear operator.

If the signal is shifted in time with a value of t o, then the corresponding frequency signal will have the form of

(1.6)

If the time signal is multiplied by a phase term of , then the FT of this time signal is shifted in frequency by f oas given below

(1.7)

If the time signal is scaled by a constant a , then the spectrum is also scaled with the following rule

(1.8)

If the spectrum signal G ( f ) is taken as a time signal G ( t ), then, the corresponding frequency domain signal will be the time reversal equivalent of the original time domain signal, g ( t ) as

(1.9)

If the time is reversed for the time‐domain signal, then the frequency is also reversed in the frequency domain signal.

(1.10)

If the conjugate of the time‐domain signal is taken, then the frequency‐domain signal conjugated and frequency‐reversed.

(1.11)

If the time‐domain signals, g ( t ) and h ( t ) are multiplied in time, then their spectrum signals G ( f ) and H ( f ) are convolved in frequency.

(1.12)

If the time‐domain signals, g ( t ) and h ( t ) are convolved in time, then their spectrum signals G ( f ) and H ( f ) are multiplied in the frequency domain.

(1.13)