Orthogonal frequency division multiplexing (OFDM) has become a very popular method for high-data-rate communication. However, it is well known that OFDM is plagued by a large peak-to-average power ratio (PAPR) problem. This high PAPR results in overdesigned power amplifiers, which amongst other things leads to inefficient amplifier usage, which is undesirable. Various methods have been recommended to reduce the PAPR of an OFDM transmission; however, all these methods result in a number of drawbacks. In this thesis, a novel method called offset modulation (OM-OFDM) is proposed to control the PAPR of an OFDM signal. The proposed OM-OFDM method does not result in a number of the drawbacks being experienced by current methods in the field. The theoretical bandwidth occupancy and theoretical bit error rate (BER) expression for an OM-OFDM transmission is derived. A newly applied power performance decision metric is also introduced, which can be utilised throughout the PAPR field, in order to compare various methods. The proposed OM-OFDM method appears to be similar to a well-known constant envelope OFDM (CE-OFDM) transmission. The modulation, structural and performance differences between an OM-OFDM and a CE-OFDM method are discussed. By applying the power performance decision metric, the OM-OFDM method is shown to offer significant performance gains when compared to CE-OFDM and traditional OFDM transmissions. In addition, the OM-OFDM method is able to accurately control the PAPR of a transmission for a targeted BER. By applying the power performance decision metric and complementary cumulative distribution function (CCDF), the proposed OM-OFDM method is shown to offer further performance gains when compared to existing PAPR methods, under frequency selective fading conditions. In this thesis, the OM-OFDM method has been combined with an existing active constellation extended (ACE) PAPR reduction method. To introduce a novel method called offset modulation with active constellation extension (OM-ACE), to control the PAPR of an OFDM signal. The theoretical BER expression for an OM-ACE transmission is presented and validated. Thereafter, by applying the decision metric and CCDF, the OM-ACE method is shown to offer performance improvements when compared to various PAPR methods. The use of OM-OFDM for cognitive radio applications is also investigated. Cognitive radio applications require transmissions that are easily detectable. The detection characteristics of an OM-OFDM and OFDM transmission are studied by using receiver operating characteristic curves. A derivation of a simplified theoretical closed-form expression, which relates the probability of a missed detection to the probability of a false alarm, for an unknown deterministic signal, at various signal-to-noise ratio (SNR) values is derived and validated. Previous expressions have been derived, which relate the probability of a missed detection to the probability of a false alarm. However, they have not been presented in such a generic closed-form expression that can be used for any unknown deterministic signal (for instance OFDM and OM-OFDM). Thereafter, an examination of the spectrum characteristics of an OM-OFDM transmission indicates its attractive detection characteristics. The proposed OM-OFDM method is further shown to operate at a significantly lower SNR value than an OFDM transmission, while still offering better detection characteristics than that of an OFDM transmission under Rician, Rayleigh and frequency selective fading channel conditions. In addition to its attractive PAPR properties, OM-OFDM also offers good detection characteristics for cognitive radio applications. These aspects make OM-OFDM a promising candidate for future deployment.