WHITE NOISE

White noise is defined to be a stationary random process having a constant spectral density function:
$$S_{wn}(j\omega)=\int_{-\infty}^{\infty}R_X(\tau)e^{-j\omega\tau}d(\tau)=A=const$$
where $$R_X(\tau)=\int_{-\infty}^{\infty}x(t)x(t+\tau)$$
$$R_X(\tau)=A\frac1{2\pi}\int_{-\infty}^{\infty}e^{j\omega t}d(\omega)=A\delta(\tau)$$
$$N(t)=n(t)e^{2\pi f_ct}$$
$$S_N(\omega)=\int_{-\infty}^{\infty}R_N(\tau)e^{-j2\pi\omega t}d\tau$$
$$=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}n(t)n(t+\tau)e^{2\pi f_ct}e^{2\pi f_c(t+\tau)}e^{-j2\pi f\tau}dtd\tau$$
$$=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}n(t)n(t+\tau)e^{2\pi 2f_ct}dtd\tau$$
$$=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}n(t)n(t+\tau)d\tau e^{2\pi 2f_ct}dt$$
$$=\int_{-\infty}^{\infty}R_N(t)e^{2\pi 2f_ct}dt$$
$$=\int_{-\infty}^{\infty}A\delta(t) e^{2\pi 2f_ct}dt=A$$

Matlab:

 N=100000;  % Number of samples  
 fc=4e6;   % sinusoidal signal frequency  
 Fs=10e6;  % Sampling frequency  
 n=randn(1,N);% noise with standard deviation -->1  
 s=exp(j*2*pi*fc/Fs*[0:N-1]); % noise * exp(j2pi*fc*t)  
 y=n.*s;           %y=ns  
 var(y)-var(n) %power of y (RY(t)) = power of x (RN(t))?