Side bands
Mathematically, the modulated wave described above would be expressed as \begin{equation} \label{Eq:I:48:9} S = (1 + b\cos\omega_mt)\cos\omega_ct, \end{equation} where $\omega_c$ represents the frequency of the carrier and $\omega_m$ is the frequency of the audio tone. Again we use all those theorems about the cosines, or we can use $e^{i\theta}$; it makes no difference—it is easier with $e^{i\theta}$, but it is the same thing. We then get  \begin{equation} \label{Eq:I:48:10} S = \cos\omega_ct + \tfrac{1}{2}b\cos\,(\omega_c + \omega_m)t + \tfrac{1}{2}b\cos\,(\omega_c - \omega_m)t. \end{equation} 
 \begin{align} \label{Eq:I:48:10} S = \cos\omega_ct &+ \tfrac{1}{2}b\cos\,(\omega_c + \omega_m)t\notag\\[.5ex] &+ \tfrac{1}{2}b\cos\,(\omega_c - \omega_m)t. \end{align}  So, from another point of view, we can say that the output wave of the system consists of three waves added in superposition: first, the regular wave at the frequency $\omega_c$, that is, at the carrier frequency, and then two new waves at two new frequencies. One is the carrier frequency plus the modulation frequency, and the other is the carrier frequency minus the modulation frequency. If, therefore, we make some kind of plot of the intensity being generated by the generator as a function of frequency, we would find a lot of intensity at the frequency of the carrier, naturally, but when a singer started to sing, we would suddenly also find intensity proportional to the strength of the singer, $b^2$, at frequency $\omega_c + \omega_m$ and $\omega_c - \omega_m$, as shown in Fig. 48–5. These are called side bands; when there is a modulated signal from the transmitter, there are side bands. If there is more than one note at the same time, say $\omega_m$ and $\omega_{m'}$, there are two instruments playing; or if there is any other complicated cosine wave, then, of course, we can see from the mathematics that we get some more waves that correspond to the frequencies $\omega_c \pm \omega_{m'}$. Therefore, when there is a complicated modulation that can be represented as the sum of many cosines,1 we find that the actual transmitter is transmitting over a range of frequencies, namely the carrier frequency plus or minus the maximum frequency that the modulation signal contains. Although at first we might believe that a radio transmitter transmits only at the nominal frequency of the carrier, since there are big, superstable crystal oscillators in there, and everything is adjusted to be at precisely $800$ kilocycles, the moment someone announces that they are at $800$ kilocycles, he modulates the $800$ kilocycles, and so they are no longer precisely at $800$ kilocycles!  Suppose that the amplifiers are so built that they are able to transmit over a good range of the ear’s sensitivity (the ear can hear up to $20{,}000$ cycles per second, but usually radio transmitters and receivers do not work beyond $10{,}000$, so we do not hear the highest parts), then, when the man speaks, his voice may contain frequencies ranging up, say, to $10{,}000$ cycles, so the transmitter is transmitting frequencies which may range from $790$ to $810$ kilocycles per second. Now if there were another station at $795$ kc/sec, there would be a lot of confusion. Also, if we made our receiver so sensitive that it picked up only $800$, and did not pick up the $10$ kilocycles on either side, we would not hear what the man was saying, because the information would be on these other frequencies!  Therefore it is absolutely essential to keep the stations a certain distance apart, so that their side bands do not overlap and, also, the receiver must not be so selective that it does not permit reception of the side bands as well as of the main nominal frequency. In the case of sound, this problem does not really cause much trouble. We can hear over a $\pm20$ kc/sec range, and we have usually from $500$ to $1500$ kc/sec in the broadcast band, so there is plenty of room for lots of stations. The television problem is more difficult. As the electron beam goes across the face of the picture tube, there are various little spots of light and dark. That “light” and “dark” is the “signal.” Now ordinarily the beam scans over the whole picture, $500$ lines, approximately, in a thirtieth of a second. Let us consider that the resolution of the picture vertically and horizontally is more or less the same, so that there are the same number of spots per inch along a scan line. We want to be able to distinguish dark from light, dark from light, dark from light, over, say, $500$ lines. In order to be able to do this with cosine waves, the shortest wavelength needed thus corresponds to a wavelength, from maximum to maximum, of one $250$th of the screen size. So we have $250\times500\times30$ pieces of information per second. The highest frequency that we are going to carry, therefore, is close to $4$ megacycles per second. Actually, to keep the television stations apart, we have to use a little bit more than this, about $6$ mc/sec; part of it is used to carry the sound signal, and other information. So, television channels are $6$ megacycles per second wide. It certainly would not be possible to transmit tv on an $800$ kc/sec carrier, since we cannot modulate at a higher frequency than the carrier. At any rate, the television band starts at $54$ megacycles. The first transmission channel, which is channel $2$ (!), has a frequency range from $54$ to $60$ mc/sec, which is $6$ mc/sec wide. “But,” one might say, “we have just proved that there were side bands on both sides, and therefore it should be twice that wide.” It turns out that the radio engineers are rather clever. If we analyze the modulation signal using not just cosine terms, but cosine and sine terms, to allow for phase differences, we then see that there is a definite, invariant relationship between the side band on the high-frequency side and the side band on the low-frequency side. What we mean is that there is no new information on that other side band. So what is done is to suppress one side band, and the receiver is wired inside such that the information which is missing is reconstituted by looking at the single side band and the carrier. Single side-band transmission is a clever scheme for decreasing the band widths needed to transmit information.