  We illustrate, through a series of prototypical examples, that linear
parity-time (PT) symmetric lattices with extended gain/loss profiles are
generically unstable, for any non-zero value of the gain/loss coefficient. Our
examples include a parabolic real potential with a linear imaginary part and
the cases of no real and constant or linear imaginary potentials. On the other
hand, this instability can be avoided and the spectrum can be real for
localized or compact PT-symmetric potentials. The linear lattices are analyzed
through discrete Fourier transform techniques complemented by numerical
computations.
