  We study the limits on how accurately LISA will be able to estimate the
parameters of low-mass compact binaries, comprising white dwarfs (WDs), neutron
stars (NSs) or black holes (BHs), while battling the amplitude, frequency, and
phase modulations of their signals. We show that Doppler-phase modulation aids
sky-position resolution in every direction, improving it especially for sources
near the poles of the ecliptic coordinate system. However, it increases the
frequency estimation error by a factor of over 1.5 at any sky position, and at
f=3 mHz. Since accounting for Doppler-phase modulation is absolutely essential
at all LISA frequencies and for all chirp masses in order to avoid a fractional
loss of signal-to-noise ratio (SNR) of more than 30%, LISA science will be
simultaneously aided and limited by it. For a source with f > 2.5mHz, searching
for its frequency evolution for 1 year worsens the error in the frequency
estimation by a factor of over 3.5 relative to that of sources with f < 1mHz.
Increasing the integration time to 2 years reduces this relative error factor
to about 2, which still adversely affects the resolvability of the galactic
binary confusion noise. Thus, unless the mission lifetime is increased several
folds, the only other recourse available for reducing the errors is to exclude
the chirp parameter from ones search templates. Doing so improves the
SNR-normalized parameter estimates. This works for the lightest binaries since
their SNR itself does not suffer from that exclusion. However, for binaries
involving a neutron star, a black hole, or both, the SNR and, therefore, the
parameter estimation, can take a significant hit, thus, severely affecting the
ability to resolve such members in LISA's confusion noise.
