  An explicit expression for the Kohn-Nirenberg symbol of a Weyl- Heisenberg
frame operator on $L^2(\mathbb{R})$ is obtained directly from the Gabor atom
coming from new classes of window functions. This new approach, using only
elementary Fourier analysis, is independent of the theory of distributions and
works strictly inside $L^2(\mathbb{R})$. Kohn-Nirenberg operators are
introduced and are shown to be Weyl-Heisenberg frame operators in suitable
cases.
