This paper is concerned with alpha- and D-negative imaginary systems. The definition of alpha-negative imaginary transfer functions is firstly introduced. Then, the relationship between negative imaginary and alpha-negative imaginary transfer functions is studied. By means of the generalised inverse, an alpha-negative imaginary lemma is proposed to test the alpha-negative imaginary property of transfer functions. Also, a necessary and sufficient condition is provided for the alpha-stability of interconnection of negative imaginary systems. A state-feedback controller design condition is established such that the resulting system is alpha-negative imaginary. Moreover, the concept of alpha-negative imaginary transfer functions is extended to that of D-negative imaginary transfer functions. Finally, the developed results are illustrated by numerical examples.