  We give several descriptions of positive quadrature formulas which are exact
for trigonometric -, respectively, Laurent polynomials of degree less or equal
$n-1-m$, $0\leq m\leq n-1$. A complete and simple description is obtained with
the help of orthogonal polynomials on the unit circle. In particular it is
shown that the nodes polynomial can be generated by a simple recurrence
relation. As a byproduct interlacing properties of zeros of para-orthogonal
polynomials are obtained. Finally, asymptotics for the quadrature weights are
presented.
