Passive control of trailing edge noise using complex periodic trailing edge serrations is investigated. The airfoil is modelled as a semi-infinite flat plate with a periodic trailing edge, set at zero angle of attack to a low Mach number flow. Analytical expressions have been derived for the far-field acoustic frequency spectrum for different serrations, namely, sawtooth, sinusoidal, slitted, slitted-sawtooth and sawtooth-sinusoidal. Numerical results have been presented for these serrations over a wide range of frequencies. It has been shown that the noise reduction from serrated trailing edges is a sensitive function of the complexity of the serration geometry and that the noise generation efficiency can be significantly reduced by applying complex periodic serrations to the trailing edge of the airfoil. Our numerical investigations have also shown that the slitted-sawtooth serration is the most effective design for reducing the trailing edge noise, particularly at low and mid frequencies. The theoretical results presented in this paper complement the experimental study presented by M. Gruber et al. .