We study the observation of a stochastic gravitational-wave background (SGWB) made by a pulsar-timing array in the spherical harmonic space of the observable. The observable is a timing residual which is the time-averaged redshift fluctuation of a pulsar over the duration of observation. Using the Sachs-Wolfe line-of-sight integral for the redshift fluctuation, we derive the power spectrum of the timing residual, from which we develop a fast algorithm to compute the overlap reduction functions for the SGWB intensity and polarization anisotropies. We find that the algorithm is less complicated and more efficient than our previous work which is based on the expansion of the polarization basis tensors in terms of spherical harmonics. Also, we use the power spectrum to construct the bipolar spherical harmonic coefficients that characterize the statistical isotropy of the SGWB. In particular, the coefficients for the linear-polarization anisotropy are worked out for the first time. Our harmonic-space method is useful for the data analysis in future pulsar-timing-array observation on a large number of pulsars as well as for the measurement of the statistical isotropy of the SGWB.
