## 10 Soft Decision Cognitive Radio and Hybrid Overlay/Underlay Cognitive Radio Waveform Design
### Hybrid Overlay/Underlay Waveform Design for Soft Decision Cognitive Radio  
Previous work provides a general analytic framework for SMSE signals that accommodates multi-carrier, CR-based waveforms [159]. Specifically, an arbitrary CR waveform can be expressed in terms of its amplitude (**A**), phase (\(\mathbf{\Theta}\)) and frequency (**F**) characteristics. These three factors aid in SMSE waveform design through six design variables, namely data modulation (**d**), Code (**c**), window (**w**), orthogonality (**o**) and two frequency allocation variables. An in-depth treatment of the SMSE analytic development and the family of SMSE waveforms is provided in [159, 160]. Considering \(N_{f}\) total frequency components, the coding \(\mathbf{c}=[c_{1},c_{2},\ldots,c_{N_{f}}],c_{i}\in\mathbb{C}\), data modulation, \(\mathbf{d}=[d_{1},d_{2},\ldots,d_{N_{f}}],d_{i}\in\mathbb{C}\), and windowing, \(\mathbf{w}=[w_{1},w_{2},\ldots,w_{N_{f}}],w_{i}\in\mathbb{C}\) vectors account for component-by-component amplitude and/or phase variations. A phase only variable \(\mathbf{\phi}=[o_{1},o_{2},\ldots,o_{N_{f}}],o_{i}\in\mathbb{C}\) is used for orthogonality between symbol streams and facilitate multiple access. The analytic SMSE framework development begins by considering data, code and window variables. The \(m^{th}\) frequency component of the \(k^{th}\) symbol is given by  
\[S_{k}[m]=\ c_{m}d_{m,k}w_{m}e^{j(\theta_{d_{m,k}}+\theta_{c_{m}}+\theta_{w_{m}})} \tag{87}\]  
where \(m=0,1,...,N_{F}-1\) is the frequency index and \(c_{m},d_{m}\) are magnitude and phase design variables. The expression in (87) is next modified to incorporate frequency and orthogonality variables. Frequency component selection is a function of two factors, including an _available_ variable \(\mathbf{a}=[a_{1},a_{2},\ldots,a_{N_{f}}]\), \(a_{i}\in\{0,1\}\) and a _use_ variable \(\mathbf{u}=[u_{1},u_{2},\ldots,u_{N_{f}}]\), \(u_{i}\in\{0,1\}\). Given an \(N_{f}\)-point fast Fourier transform (FFT) process, \(N_{f}\) frequency components or spectral bands are available for waveform design. It is important to note that the frequency assignment variable takes on binary values \(0\) or \(1\) to indicate the spectrum availability for secondary users. As a direct result, this pool of frequencies is reduced by component selection to create a number of CR available frequencies _and_ usable frequencies. The \(m^{th}\) component of the \(k^{th}\) CR symbol corresponds to  
\[S_{k}[m]=\ a_{m}u_{m}c_{m}d_{m,k}w_{m}e^{j(\theta_{d_{m,k}}+\theta_{c_{m}}+ \theta_{w_{m}}+\theta_{o_{m,k}})} \tag{88}\]  
where the product \(a_{i}u_{i}\in\{0,1\}\). The discrete time domain SMSE waveform is obtained by taking the Inverse Discrete Fourier Transform (IDFT) of (88) according to  
\[s_{k}[n]=\frac{1}{N_{f}}Re\left\{\sum_{m=0}^{N_{f}-1}a_{m}u_{m}c_{m}d_{m,k}w_ {m}\ e^{j(2\pi f_{m}t_{n}+\theta_{d_{m,k}}+\theta_{c_{m}}+\theta_{w_{m}}+ \theta_{o_{m,k}})}\right\} \tag{89}\]  
Figure 48: Cognitive radio. where \(t_{k}\leq t_{n}\leq t_{k}+T\), \(f_{m}=f_{c}+m\Delta\!f\), \(T\) is the symbol duration and \(\Delta\!f=1/T\) is the frequency resolution [159]. The SMSE framework provides a unified expression for generating and implementing a host of multi-carrier type waveforms (e.g., OFDM [161], MC-CDMA [162], CI/MC-CDMA [163, 164], TDCS [165, 166], etc.) and satisfies current CR goals of exploiting unused spectral bands. However, it does not exploit _underused_ spectrum. This section re-visits the original SMSE framework development and the frequency assignment variables to exploit both _unused_ and _underused_ spectrum to generate both overlay-CR and underlay-CR type waveforms. Figure 49 illustrates a conceptual view of the unused and _underused_ spectrum utilization using an arbitrary interference threshold (IT). IT is assumed to be a limit set forth by the primary users based on the measured power spectrum density in a given bandwidth.