Handling peak-to-average power ratio is a major challenge in the design of communications systems, as current signal designs constrain the power of the generated signal and therefore its peak amplitude is considered as an uncontrolled outcome of the power-constrained signal generation scheme. An alternative signal design approach would be to restrict the peak of the signal’s amplitude. The capacity of continuous-time bandlimited linear channels with additive Gaussian noise and peak input amplitude constraint is unknown to date; however, if the channel impulse response has finite energy, then any rate achieved by peak-amplitude constrained waveforms can be achieved by binary waveforms (unit processes). This fact is the basis for the two major previous works that have derived lower bounds on the achievable rate of this channel for the ideal bandlimited case. In this work we propose a different approach for obtaining lower bounds on the capacity of this channel, particularly relevant for linear, time-invariant channels with non-ideal frequency responses. Our approach is based on modulating a subset of the Walsh basis functions and using a fundamental relationship between the peak amplitude and the power of such signals. This approach yields achievable rates for general linear channels.