Quasimonochromatic and Complex Signals

For a strictly monochromatic signal, electrical engineers have known for a long time that it is very convenient to use a complex voltage $V(t)=E_0 \exp(i(\omega t+\varphi))$  whose real part gives the actual signal $E_r(t)=E_0 \cos(\omega t +\varphi)$. One need not think of the imaginary part as a pure fiction since it can be obtained from the given signal by a phase shift of $\pi/2$, viz. as $E_i(t)=E_0 \cos(\omega t+\phi-\pi/2)$. In practice, since one invariably deals with signals at an intermediate frequency derived by beating with a local oscillator, both the real and imaginary parts are available by using two such oscillators $\pi/2$ out of phase. Squaring and adding the real and imaginary parts give $E_r^2(t)+ E_i^2(t)=V(t)^*V(t)=E_0^2$ which is the power averaged over a cycle. This is actually closer to what is practically measured than the instantaneous power, which fluctuates at a frequency $2\omega$.

These ideas go through even when we have a range of frequencies present, by simply imagining the complex voltages corresponding to each of the monochromatic components to be added. In mathematical terms, this operation of deriving $E_i(t)$ from $E_r(t)$ goes by the name of the ``Hilbert Transform'', and the time domain equivalent is described in Section 1.8 But the physical interpretation is easiest when the different components occupy a range $\Delta \omega$ - the so called ``bandwidth'' - which is small compared to the ``centre frequency'' $\omega_0$. Such a signal is called ``quasimonochromatic'', and can be represented as below

$$E_q(t)=Re~\exp(i\omega_0 t) \sum_{-\Delta \omega/2<\omega_1<\Delta \omega/2} E(\omega_1) ~\exp(i\omega_1 t+i\varphi(\omega_1))$$

In this expression, $\omega_1$ is a frequency offset from the chosen centre $\omega_0$, so that $E(\omega_1)$ actually represents the amplitude at a frequency $\omega_0+\omega_1$, and $\varphi (\omega_1)$ the phase. We can now think of our quasimonochromatic signal as a rapidly varying phasor at the centre frequency $\omega_0$, modulated by a complex voltage

$$V_m(t)= \sum_{-\Delta \omega/2<\omega_1<\Delta \omega/2} E(\omega_1) ~\exp(i\omega_1 t+i\varphi\omega_1)$$

This latter phasor varies much more slowly than $\exp(-i\omega_0 t)$. In fact, it takes a time $\Delta \omega^{-1}$ for $V_m(t)$ to vary significantly since the highest frequencies present are of order $\Delta \omega$. This time scale is much longer than the timescale $\omega^{-1}$ associated with the centre frequency. Writing $V_m(t)$ in the polar form as $R(t) \exp(i\alpha(t))$, our original real signal reads

$$E_q(t)=R(t)\cos(\omega_0 t +\alpha(t))$$

.

We can think of $R$ and $\alpha$ as time dependent, slowly varying, amplitude and phase modulation of an otherwise (hence ``quasi'') monochromatic signal.

While the mathematics did not assume smallness of $\Delta \omega$, the physical interpretation does. If $R(t)$ changes significantly during a cycle, some of its values may not be attained as maxima and hence its square cannot be regarded as measuring average power. This is as it should be. No amount of algebra can uniquely extract two real functions $R(t)$ and $\alpha(t)$ from a single real signal without further conditions (and the condition imposed is explained in section 1.8).

But returning to the quasimonochromatic case, we can now think of $V_m(t)^*V_m(t)$ as the (slowly) time varying power in the signal. Likewise we can think of $\langle V_m^*(t)V_m(t+\tau)\rangle$ as the autocorrelation. (A little algebra checks that this is the same as the autocorrelation of the original real signal). One advantage in working with the complex signal is that the centre frequency cancels in any such product containing one voltage and one complex conjugate voltage. We can therefore think of such products as referring to properties of the fluctuations of the signal amplitude and phase, and measure them even after heterodyning has changed the centre frequency.