This paper presents an analytical study of noise-induced triggering, which is the spontaneous transition of a thermoacoustic system from a linearly stable state to an unstable state. Our aim is to find out whether there are some frequency bands in the noise spectrum that are more effective at triggering than others. We model the complete combustion system as a combination of three elements: a combustion chamber, and within it, a flame and a noise source. The empty combustion chamber is described by its impulse response (also called tailored Green's function). The flame is described by its heat release rate, which has a nonlinear relationship with the acoustic velocity. The noise is narrow-band and described in terms of its frequency spectrum; it is filtered white noise and has three variable parameters: noise intensity, bandwidth and centre frequency. We use Green's function approach to combine the three elements; this leads to an integral equation for the time history of the acoustic field in the combustion chamber. The calculated time histories reveal the consequences of varying noise properties. The noise level is an important parameter: if it is sufficiently high, triggering occurs; this manifests itself by a rapid increase in amplitude, which is soon followed by a limit cycle. The bandwidth of the noise also plays a role: if the bandwidth is increased (without changing the centre frequency), triggering occurs more readily. The parameter with the most remarkable behaviour is the centre frequency of a frequency band: if this is in the very low-frequency range, triggering is achieved with a low noise level, but as the frequency increases, substantially higher noise levels are required. Preferential triggering at the resonance frequency of the combustion chamber has not been observed.