Since it is often difficult to identify the noise covariances for a Kalman filter, they are commonly considered design variables. If so, we can as well try to choose them so that the corresponding Kalman filter has some nice form. In this paper, we introduce a one-parameter subfamily of Kalman filters with the property that the covariance parameters cancel in the expression for the Kalman gain. We provide a simple criterion which guarantees that the implicitly defined process covariance matrix is positive definite.