## 1 Introduction
In a DSA network [7], the unused frequency bands may be non-contiguously available for the secondary users. Using MCM schemes, the subcarriers coinciding with the portions of the spectrum that are not used by the primary users are assigned to secondary users in order to fill the spectrum holes [8]. In spread spectrum modulation the transmitted signal bandwidth is much larger than the minimum bandwidth required for the signal. Spreading of the signal is accomplished by multiplying the data bits with pseudo-noise sequence or by varying the frequency of transmission according to the pseudo-noise sequence. DS-CDMA is a spread spectrum modulation technique that is commonly used in underlay cognitive radio systems. In DS-CDMA, data rate and power level are controlled based on the spectrum availability and interference constraint on the primary user. In DS-CDMA based underlay cognitive radio systems, the secondary CDMA users spread their data using a signature sequence and the primary users can be CDMA or non-CDMA users. The transmission power and signature sequence of the secondary users are varied adaptively in a dynamic spectrum access (DSA) network. There are various multicarrier spread spectrum modulation (MCSSM) schemes proposed in the literature, such as Multicarrier Code Division Multiple Access (MC-CDMA) and Multicarrier Direct Sequence Code Division Multiple Access (MC-DS-CDMA), which combine the advantages of both MCM and SS schemes. MCSSM schemes are used in both interweave and underlay CR systems. This article gives an exhaustive survey of various SS, MCM and MCSSM schemes. The two basic elements of cognitive radio systems are spectrum sensing mechanism and adaptive transceiver architecture. Through spectrum sensing mechanism, a secondary user finds the unused spectral holes which can be used for its transmission without causing interference to the primary users. When the nature of the primary user's signal is known, spectrum sensing can be performed based on feature detection and when such knowledge is not available, energy detection is used for spectral analysis. An energy detector can be implemented using Fast Fourier Transform (FFT) by averaging the frequency bins. By increasing the FFT size \(N\), the frequency resolution can be improved and it helps in detection of narrowband signals. FFT, as a part of multicarrier modem can be used for spectral analysis [6] in order to identify the spectral holes, without any extra hardware or software. IFFT and FFT are the most computationally intensive blocks in multicarrier modulation schemes. CR systems require efficient FFT/IFFT algorithms in order to save time and energy. In this paper, various efficient FFT/IFFT algorithms proposed in the literature for cognitive radio applications are investigated. The remainder of this contribution is organized as follows. In Section 2, variants of OFDM based MCM schemes are described. In Section 3, variants of multicarrier CDMA based modulation schemes are described. Section 4 elaborates the efficient FFT algorithms for multicarrier modulation schemes. In Section 5, a summary of efficient FFT algorithms is provided. Section 6 contains the conclusion.