The DW method is based on calculating two separate STFTs and then combining the results. The first STFT uses a broad spectral Gaussian window to obtain high temporal/depth resolution while the second STFT uses a narrow spectral window to generate high spectroscopic resolution. The two resulting TFDs are then multiplied together to obtain a single TFD with simultaneously high spectral and temporal resolutions.

To understand what the DW method is revealing, let us consider the FDOCT signal

where

*I(k)* is the total detected intensity,

*I*_{R} and

*I*_{S} are the intensities of the reference and sample fields, respectively, and

*d* is a constant optical path difference between the sample and reference arms. The STFT of the cross correlation term,

can be expressed as

Note that

*u*, the width or standard deviation of the Gaussian window, must be chosen carefully in order to obtain acceptable spectral or temporal resolution. If, for example,

*u* is chosen to be the same order of magnitude as the bandwidth of the source, then the STFT produces a TFD that has good temporal/depth resolution, but poor spectral resolution. On the other hand, if

*u* is chosen to be much smaller than the bandwidth of the source, then the STFT generates a TFD with good spectral resolution, but poor temporal resolution. The DW method, however, can avoid this resolution tradeoff.

Consider the TFDs resulting from two STFTs, S

_{1} and S

_{2}, generated by a narrow spectral window and a wide spectral window, respectively. Assuming that the reference field in

Eq. (1) is slowly varying over the frequencies of interest, the processed signal is given by

where

*a* and

*b* are independent parameters that set the widths of the windows, and

*b >> a*. In order to obtain a more insightful form of the processed signal, consider a coordinate change such that

where the Jacobian of the transform is unity. Thus, the processed signal DW can be written as

The term

from

Eq. (5) can be expressed in terms of a Wigner TFD by utilizing the ambiguity function [

12,

13]:

where

*W*_{S} (Ω, ζ) is the Wigner TFD of the sample field in the new coordinate system. After substituting

Eq. (6) into

Eq. (5) and simplifying, the processed signal yields

By integrating

Eq. (7) with respect to

*q* and assuming

*a* is small compared to

*b*, such that

*a*^{2}/

*b*^{2} << 1, the DW signal simplifies to

Equation (8) shows that the DW method is equivalent to probing the Wigner TFD of the sample field with two orthogonal Gaussian windows, one with a standard deviation of

*b/2* in the spectral dimension and another with a standard deviation of

*1/(2a)* in the spatial/temporal dimension. Furthermore,

*a* and

*b* independently tune the spectral and spatial/temporal resolutions, respectively, thus avoiding the tradeoff that hinders the STFT.

Equation (8) also shows that the processed signal is modulated by an oscillation that depends on the constant path difference,

*d*, between the sample and reference arms. This phenomenon is also observed in the cross terms of the Wigner TFD, which have been identified to contain valuable information about phase differences [

12]. We explore the utility of this oscillatory term below in section 4.

Another interesting result is obtained if

*a* approaches zero and

*b* is taken to be much larger than the bandwidth of the source, Δ

*k*. In these limits, the window with standard deviation

*a* →0 approaches the delta function, while the second window whose standard deviation

*b*>>Δ

*k*, becomes a constant across the spectrum. If our signal

*F(k)* = 2

*E*_{R}E_{S} · cos(

*k* ·

*d*), and

*f*(

*z*)

F(

*k*) is a Fourier transform pair,

Eq. (3) yields

Equation (9) is equivalent to the Kirkwood & Rihaczek TFD, and if the real part is taken, it is equal to the Margenau & Hill (MH) TFD [

13]. Either of these two distributions can be simply transformed to produce any of the Cohen’s class functions, such as the Wigner TFD [

13].