Canal surfaces are defined and divided into nine types in Minkowski 3-space ${\mathbb{E}}_1^3$ , which are obtained as the envelope of a family of pseudospheres ${\mathbb{S}}_1^2$ , pseudohyperbolic spheres ${\mathbb{H}}_0^2$ , or lightlike cones ${\mathbb{Q}}^2$ , whose centers lie on a space curve (resp. spacelike curve, timelike curve, or null curve). This paper focuses on canal surfaces foliated by pseudohyperbolic spheres ${\mathbb{H}}_0^2$ along three kinds of space curves in ${\mathbb{E}}_1^3$ . The geometric properties of such surfaces are presented by classifying the linear Weingarten canal surfaces, especially the relationship between the Gaussian curvature and the mean curvature of canal surfaces. Last but not least, two examples are shown to illustrate the construction of such surfaces.