  By analyzing the Lyapunov exponent (LE), we develop a rigorous, fundamental
scheme for the study of general non-Hermitian quasicrystals with both complex
phase factor and non-reciprocal hopping. Specially, the
localization-delocalization transition point, $\mathcal{PT}$-symmetry-breaking
point and the winding number transition points are determined by LEs of its
dual Hermitian model. The analysis was based on Avila's global theory, and we
found that winding number is directly related to the acceleration, the slope of
the LE, while quantization of acceleration is the crucial ingredient of Avila's
global theory. This result applies as well to the models with higher winding,
not only the simplest Aubry-Andr\'{e} model. As typical examples, we obtain the
analytical phase boundaries of localization transition for non-Hermitian
Aubry-Andr\'{e} model in the whole parameter space, and the complete phase
diagram is straightforwardly determined. For the non-Hermitian
Soukoulis-Economou model, a high winding model, we show how the phase
boundaries of localization transition and winding number transitions relate to
the LEs of its dual Hermitian model. Moreover, we discover an intriguing
feature of robust spectrum, i.e., the spectrum keeps invariant when one changes
the complex phase parameter $h$ or non-reciprocal parameter $g$ in the region
of $h<|h_c|$ or $g<|g_c|$ if the system is in the extended or localized state,
respectively.
