This chapter treats the detection of digitally modulated waveforms on ideal channels that are corrupted with additive white Gaussian noise (AWGN). The received bandpass noise corrupted waveform is processed with a quadrature demodulator to extract its complex envelope. For each message sent, the complex envelope is projected onto the same complete set of orthonomal basis used to obtain the message vectors. This generates an observation vector that is a sufficient statistic for the receiver to decide on the message that was sent. Afterwards coherent detection is considered where the complex channel gain \(\tilde {h}\) of the ideal channel is assumed known to the receiver, resulting in the correlator and matched filter detectors. This is followed by the analytical error rate performance of various modulation schemes with coherent detection. Afterwards, a noncoherent detector is considered where no attempt is made to estimate the received carrier phase and applied to the detection of orthogonal MFSK signals. Finally, various bounds on error probability are derived, along with bit versus message error probabilities, and the effect of rotations and translations of a signal constellation on the error probability performance.