Given a digital modulation waveform $x(t)=\sum_i X[i]g_i(t)$ having bandwidth $B_x$, its waveform-level simulation is performed using discrete-time samples $x[k]\triangleq x\left(\frac{k}{\mathbb W}\right), k\in \mathbb Z$, where the sampling rate, namely the simulation bandwidth, is set to be $\mathbb W \gg B_x$. Taking into account the less stringent definition of $B_x$ and the frequency dispersion of the channel, we often choose $\mathbb W$ to be $8$ to $16$ times $B_x$ to ensure that the received waveform is oversampled as well. Then, other signals in the simulated communication link, including noise and interferences, are also needed to be represented by their own discrete-time samples spaced by $\frac{1}{\mathbb W}$. This simulation setup is well justified by a band-pass filter (BPF) or a baseband anti-aliasing LPF in the communication link, which reject signal components outside of $[-\frac{\mathbb W}{2}, \frac{\mathbb W}{2}]$ to satisfy the sampling theorem.     As shown in Fig. 1, at the receiver, we have $y[k]=y_{nl}[k]+z[k]$ to emulate $y(t)=y_{nl}(t)+z(t)$, where $y[k], y(t)$ and $y_{nl}[k],y_{nl}(t)$ are the received sequence/waveform and their noiseless version, respectively. Also, $z(t)$ is usually modelled as an ideal additive white Gaussian noise (AWGN) with infinite bandwidth and power, while $z[k]$ represents a band-limited AWGN with bandwidth $\mathbb W$ and power $N_0\mathbb W$, where $N_0$ is the one-sided power spectral density (PSD) of the AWGN [2] [3].

The sequence $x[k]$ with time interval $\frac{1}{\mathbb W}$ actually represents a signal with bandwidth $\mathbb W$. Although $x(t)$ only occupies a bandwidth of $B_x$ inside $\mathbb W$, the power of $x[k]$ is identical to that of $x(t)$ and is independent of $\mathbb W$, because theoretically no signal component of $x(t)$ exists outside of $B_x$. Any signal componets outside $B_x$ we observed from the PSD of $x[k]$ can be considered as the so-called out-of-band emission (OOBE).

    Note that the signal-to-noise ratio (SNR) is usually defined using $x(t)$ (or $y_{nl}(t)$) and in-band noise with power $N_0B_x$. Bearing in mind that different waveforms have different $B_x$ and $y_{nl}(t)$ may have a bandwidth wider than $B_x$ (due to channel freqeuncy dispersion), to make things easier, we often investigate various performance versus the energy per bit $E_b$ to noise PSD ratio given by $E_b/N_0$, because $E_b/N_0$ is independent of $B_x$ and $\mathbb W$.

    Let $E_X$ and $M_{sa}$ be the power (mean square value) and size of the signaling alphabet, respectively. Because the transmit pulses $g_i(t),i\in \mathbb Z,$ usually have the same energy $E_g$, we have $E_b=E_XE_g/\log_2 M_{sa}$. For a specified $\eta =E_b/N_0$, we can calculate $N_0=E_b/\eta$ and set the power of the noise sequence $z[k]$ to be $N_0\mathbb W$. Then, together with their own $y[k]$, a fair comparison of different waveforms may be achieved.

It should be noted that $y[k]$ represents a wideband signal containing not only high power noise $z[k]$ but also possible adjacent channel interferences (ACIs). Depending on receiver design, pre-filtering using a digital LPF (anti-aliasing filter) with cut-off frequency $\frac{B_x}{2}$ may be necessary to reject out-of-band noise and interferences. Otherwise, a downsampling of $y[k]$ may introduce significant aliasing of noise and interferences and then severely degrade receiver performance.