Let B p v(R d ), 1 ≤ p< ∞, be the space of all bounded bandlimited functions from Lp(R d ). The uniform bounds for truncated multi-dimensional Whittaker-Shannon se- ries based on local sampling are derived for signal functions f ∈ B p v(R d ) without decay assumption. Then the optimal bounds of aliasing and truncation errors for non-bandlimited signal functions from Sobolev classes U(W r p(R d )) with r ≥ d are obtained up to a logarithmic factor.