  This paper studies transmit beamforming design in an integrated sensing and
communication (ISAC) system, where a base station sends symbols to perform
downlink multi-user communication and sense an extended target simultaneously.
We first model the extended target contour with truncated Fourier series. By
considering echo signals as reflections from the valid elements on the target
contour, a novel Cram\'er-Rao bound (CRB) on the direction estimation of
extended target is derived. We then formulate the transmit beamforming design
as an optimization problem by minimizing the CRB of radar sensing, and
satisfying a minimum signal-to-interference-plus-noise ratio requirement for
each communication user, as well as a 3-dB beam coverage requirement tailored
for the extended sensing target under a total transmit power constraint. In
view of the non-convexity of the above problem, we employ semidefinite
relaxation (SDR) technique for convex relaxation, followed by a rank-one
extraction scheme for non-tight relaxation circumstances. Numerical results
show that the proposed SDR beamforming scheme outperforms benchmark beampattern
design methods with lower CRBs for the circumstances considered.
