  We study the observation of stochastic gravitational-wave background (SGWB)
made by pulsar-timing arrays in the spherical harmonic space. Instead of using
the Shapiro time delay, we keep the Sachs-Wolfe line-of-sight integral for the
timing residual of an observed pulsar. We derive the power spectrum of the
timing residual, from which the overlap reduction functions and the bipolar
spherical harmonics coefficients are constructed for the SGWB intensity and
polarization anisotropies. We have reproduced the previous results, noting that
we have developed a fast algorithm for computing accurate overlap reduction
functions and the bipolar spherical harmonics coefficients for the
linear-polarization anisotropy are worked out for the first time. Our
harmonic-space method is useful for future pulsar-timing-array observation on a
few thousand pulsars and provides optimal estimators for testing the
statistical isotropy of the SGWB.
