nel matrix with in which NTX and NRX denote the number of transmit and receive antennas, respectively, the received signal vector y can be calculated as y = Hx + V where complex-valued Gaussian-distributed noise vec- tor V ~ C(0,021) denotes the noise vector with respect to the size of the FFT NFFT and the number of used subcarriers before the detector Nused. We define the complex-valued Gaussian-distributed noise vector n ~ C(0,021) to be noise after the FFT operation. The receive SNR before the detector is given = the SNR after the FFT operation is ivenc as can be observed that the difference between pre-FFT and post-FFT SNRs (preF/(/ Nused NFFT is always positive, imply- ing that the FFT operation suppresses the noise and enhances the SNR. 3.2.1.3 Peak-to-Average Power Ratio The peak-to-average power ratio (PAPR) for a single-carrier modulation signal depends on its constellation and the pulse-shaping filter roll-off factor. For a Gaussian distributed p-norm of matrix H is defined as In the special case when p = 2 the norm is called the Frobenius norm and forp=`ois called the maximum norm. The Frobenius norm or Hilbert-Schmidt norm of matrix H is similar, although the latter term is often reserved for the operators on a Hilbert space. In general, this norm can be defined in the following forms: wher = denotes the conjugate trans- pose of matrix H and Si are the singular values of matrix H. The Frobenius norm is further similar to the Euclidean norm on RN and is obtained from an inner product on the space of all matrices. The Frobenius norm is submultiplicative and is very useful for numerical linear algebra. 322 Chapter 3 OFDM signal, the cumulative distribution function (CDF) 8 of PAPR for 99.0%, 99.9%, and 99.99% are approximately 8.3, 10.3, and 11.8 dB, respectively. Since the OFDM signal has a high PAPR, it could be clipped in the transmitter PA, because of its limited dynamic range or nonlinearity. Higher output back-off is required to prevent performance degradation and inter- modulation products spillin