Random Processes and Analog Communications

AM power and modulation index

An AM wave s(t)=A_c[1+m\cos\omega_m t]\cos\omega_c t carries its power in three parts. The carrier power is P_c=A_c^2/2 across a 1 ohm load, and each of the two sidebands carries m^2P_c/4. The total power is therefore P_t=P_c(1+m^2/2). The carrier carries no message but takes most of the power when m is small. The sideband-to-total power ratio is \dfrac{m^2}{2+m^2}, which rises with m and equals 1/3 at m=1.

When the carrier and one sideband are suppressed, only one sideband remains, carrying m^2P_c/4. The percentage of power saved is (P_t-m^2P_c/4)/P_t\times 100. For m=1, P_t=1.5P_c and the remaining power is 0.25P_c, so the saving is (1.25/1.5)\times 100\approx 83.3\%.

Envelope detection depends on the sign of the message. For DSB-SC the envelope is A_c|m(t)|, which equals the message only when m(t)\ge 0 for all t. For AM the envelope is A_c|1+\mu m(t)|, and correct recovery needs 1+\mu m(t)\ge 0 everywhere, that is, no overmodulation. Requiring m(t)>0 is therefore sufficient but not necessary for AM, since negative message values are allowed as long as \mu|m(t)|\le 1.

Carson's rule FM bandwidth

Carson's rule estimates the bandwidth of an FM wave as B_T\approx 2(\Delta f+f_m)=2(\beta+1)f_m, where \Delta f is the peak frequency deviation, f_m is the highest message frequency and \beta=\Delta f/f_m is the modulation index. For narrowband FM with small \beta the bandwidth is close to 2f_m, while for wideband FM it approaches 2\Delta f. For a multitone angle, the peak deviation equals the sum of the individual deviations only when their peaks coincide. For \phi(t)=5\sin(2\pi f_1t)+2\sin(4\pi f_1t) the deviation is 5f_1\cos(2\pi f_1t)+4f_1\cos(4\pi f_1t), and its peak magnitude is 9f_1 because both cosines reach 1 at t=0.

Carson's bandwidth also decides whether spectra overlap after a nonlinear operation. Squaring an FM wave gives a term centred at 2\omega_c with doubled deviation, so its bandwidth is 2B_T and its lower edge is at 2\omega_c-B_T. The band of the original wave, 2x(t), lies around \omega_c with half-bandwidth B_T/2. Non-overlap requires 2\omega_c-B_T\ge\omega_c+B_T/2, that is, \omega_c\ge\tfrac{3}{2}B_T.

Ensemble mean and autocorrelation of random process

The ensemble mean of a random process is \mu_X(t)=E[X(t)], and its autocorrelation is R_X(t_1,t_2)=E[X(t_1)X(t_2)]. A process is wide-sense stationary (WSS) when \mu_X is constant and R_X depends only on the lag \tau=t_2-t_1. For X(t)=A\cos(\omega_0t+\theta), with A and \theta independent and \theta uniform over one period, the mean is zero and R_X(\tau)=\tfrac12E[A^2]\cos\omega_0\tau. This depends only on \tau, so the process is WSS.

A constant mean does not guarantee stationarity. Consider f(t)=\sum_n a_n p(t-nT) with zero-mean, pairwise independent, unit-variance coefficients. The mean is zero for every t, since E[a_n]=0. The autocorrelation, however, is \sum_n p(t-nT)p(t+\tau-nT), and at \tau=0 it switches between 0 and 1 depending on where t falls within the pulse grid. Hence the process is not WSS even though its mean is constant.

Average power from autocorrelation

For a WSS process the average power is the mean square value P_X=E[X^2(t)]=R_X(0), the autocorrelation at zero lag. To read power from a graph of R_X(\tau), take the value at \tau=0. The peak after some nonzero lag is not the power, and neither is the area under the curve. Equivalently, by the Wiener-Khinchin relation, P_X=\int S_X(f)\,df.

Phase modulation deviation and instantaneous frequency

In phase modulation the carrier angle is \theta(t)=\omega_c t+k_p m(t), where k_p is the phase deviation constant in rad/V. The instantaneous frequency is f_i(t)=f_c+\dfrac{k_p}{2\pi}\dfrac{dm}{dt}, so the frequency deviation is proportional to the derivative of the message. For a sinusoid m(t)=A_m\cos(2\pi f_mt), the peak frequency deviation is k_pA_mf_m. It therefore grows with message frequency, which is the opposite of FM, where the deviation depends only on the message amplitude.

When the message is given as an rms value V_{rms}, convert it to peak with A_m=\sqrt2\,V_{rms} before computing the deviation. For example, with k_p=3 rad/V, A_m=1 V and f_m=2 kHz, the peak deviation is 3\times1\times2=6 kHz, so the maximum instantaneous frequency is f_c+6 kHz.

Image frequency in superheterodyne receiver

A superheterodyne receiver mixes the incoming RF signal with a local oscillator (LO), producing the intermediate frequency f_{IF}=|f_{LO}-f_{RF}|. An image is an unwanted RF signal that the mixer also converts to the IF. When f_{LO}>f_{RF}, the image lies at f_{im}=f_{LO}+f_{IF}=f_{RF}+2f_{IF}. It sits on the far side of the oscillator and is separated from the wanted signal by twice the IF. For example, with f_{RF}=1200 kHz and an IF of 455 kHz, f_{LO}=1655 kHz and f_{im}=1655+455=2110 kHz.