  'The spectral leakage from windowing and the picket fence effect from
discretization' have been among the standard contents in textbooks for many
decades. The spectral leakage and picket fence effect would cause the
distortions in amplitude, frequency, and phase of signals, which have always
been of concern, and attempts have been made to solve them. This paper proposes
two novel decomposition theorems that can totally eliminate the spectral
leakage and picket fence effect, and could broaden the knowledge of signal
processing. First, two generalized eigenvalue equations are constructed for
multifrequency discrete real signals and complex signals. The two decomposition
theorems are then proved. On these bases, exact decomposition methods for real
and complex signals are proposed. For a noise-free multifrequency real signal
with m sinusoidal components, the frequency, amplitude, and phase of each
component can be exactly calculated by using just 4m-1 discrete values and its
second-order derivatives. For a multifrequency complex signal, only 2m-1
discrete values and its first-order derivatives are needed. The numerical
experiments show that the proposed methods have very high resolution, and the
sampling rate does not necessarily obey the Nyquist sampling theorem. With
noisy signals, the proposed methods have extraordinary accuracy.
