## 2 SNR Definition and Channel Models  
Referring to Figure 1, the transmitter-side SNR for a particular channel realization is defined as  
\[\text{SNR}=10\text{log}\frac{P_{\text{signal}}}{P_{N}} \tag{1}\]  
where \(P_{\text{signal}}\) is the power of \(x[n]\) and \(P_{N}\) is the power of \(w[n]\). On the other hand, the receiver-side SNR is defined as  
\[\text{SNR}=10\text{log}\frac{P_{r}}{P_{N}} \tag{2}\]  
where \(P_{r}\) is the signal power of \(r[n]\). Further details on the calculation of \(P_{\text{signal}}\), \(P_{r}\), and \(P_{N}\) will be provided later in this section. As previously stated, (1) equals (2) if the channel power gain in Figure 1 is one. To gain more insights into the channel gain, we need to describe the characteristics of a simulation channel. The multipath fading phenomenon in a mobile wireless channel is typically modeled as a tapped delay line [13; 14] with a constant delay for each tap. Specifically, the channel impulse response is modeled as:  
\[h(t)=\sum_{l=0}^{L-1}h_{l}\delta(t-\tau_{l}) \tag{3}\]  
where \(L\) is the number of paths in the channel, and \(h_{l}\) and \(\tau_{l}\) are the complex gain and the delay of path \(l\), respectively. Each \(\tau_{l}\) is a constant, and each \(h_{l}\) is an outcome of a complex-valued random variable \(\mathbf{h}_{l}\). Since uncorrelated scattering among paths is usually assumed in channel simulation, all \(\mathbf{h}_{l}\) are independent random variables. This model is also implemented as the wireless channel in simulations. As mentioned previously, ETSI defined three channel models--EPA, EVA, and ETU--for LTE DL simulations. The power-delay profiles of these models are listed in Table 1[12]. It is observed that the EPA model has a much shorter delay spread than the EVA and ETU models. Additionally, the sum of average powers over all taps is not 0 dB. During simulations, however, a normalization procedure is carried out to ensure that the expected value of the channel power gain is one. Specifically, let \(P_{l}\) denote the power of the \(l\)th tap listed in the table. The normalized power for each tap is given by  
\[\widetilde{p}_{l}=\frac{P_{l}}{\sum_{l=0}^{L-1}P_{l}} \tag{4}\]  
To generate a channel, the complex gains \(h_{l}\), \(l=0,1,\ldots,L-1\), are obtained as outcomes from independent complex Gaussian random variables \(\mathbf{h}_{l}\). Each of these complex random variables has independent and identically distributed (i.i.d.) Gaussian random variables in real and imaginary parts, with a mean of zero and a variance of \(\widetilde{p}_{l}/2\). For example, \(\widetilde{p}_{0}\approx 1/(1+0.794+0.631+0.501+0.158+0.019+0.08)=0.314\) for tap 1 in the EPA channel. Note that the values in the denominator are converted from dB (power). When applying (3) to the EPA model, \(\tau_{0}=0,\ \tau_{1}=30\) ns, etc., and \(E[\mathbf{h}_{0}\mathbf{h}_{0}^{*}]=\widetilde{\tau}_{0}=0.314\) for tap 1, \(E[\mathbf{h}_{1}\mathbf{h}_{1}^{*}]=\widetilde{p}_{1}=0.249\) for tap 2, etc., where \(E[\cdot]\) denotes the expectation operation and * denotes a complex conjugation. To conduct a simulated experiment, the trial is repeatedly carried out to produce different sets of \(h_{l}\). In the following, we refer to a set of \(h_{l}\) produced in one probability trial as one realization of the channel impulse response. In the simulations, unless otherwise specified, we set the speed to 0.01 km/h so that the channel impulse response remains almost unchanged within the duration of the transmitted signal. Referring to Figure 1, if the cyclic prefix (CP) is longer than the maximum delay of the multipath channel, the received signal within an OFDM symbol can be expressed as:  
\[y[m]=x[m]\bigotimes h[m]+w[m] \tag{5}\]  
where \(\bigotimes\) denotes the circular convolution, \(x[m]\) and \(h[m]\) are the sampled versions of the transmitted OFDM signal and channel impulse response, respectively, and \(w[m]\) is the complex-valued additive white Gaussian noise (AWGN) with mean zero and variance \(N_{0}/2\).