How do signals at double the frequency indicate a video signal?
I don't claim to be an expert, but I can try to explain this in my limited fashion.
Some old radio receivers used to have two modes of operation: AM and FM. AM stands for "amplitude modulation" and FM stands for "frequency modulation". We can emit radio waves of different amplitudes and wavelengths, and by changing them frequently we can encode signals. E.g. a region of higher amplitude or greater frequency would indicate
1, and the reverse would indicate
0. Using this we can encode anything, provided enough bandwidth.
There's also another type of modulation - polarisation modulation. The novel explains it well, in my opinion:
"When a wave of light comes at you--visible light, radio light, any kind of light--it's vibrating at right angles to your line of sight. If that vibration rotates, the wave is said to be elliptically polarized. If it rotates clockwise, the polarization is called right-handed; counterclockwise, it's left-handed. I know it's a dumb designation. Anyway, by varying between the two kinds of polarization, you could transmit information. A little right polarization and that's a zero; a little left and it's a one. Follow? It's perfectly possible. We have amplitude modulation and frequency modulation, but our civilization, by convention, ordinarily just doesn't do polarization modulation."
Contact by Carl Sagan
As you see, with three different types of polarisation one can have three signals in one. As far as I understand, the signal received from Vega was an AM signal containing the prime numbers. It's commonly believed that a natural phenomenon could not generate the prime sequence just like that, so it must be a sign of intelligent life (from the script, with a hat tip to Valorum for finding it):
FISHER Linearly polarized; a set of moving pulses restricted to two different amplitudes --
So the two amplitudes corresponded to binary numbers, and when converted to the decimal system they yielded the prime numbers. Later, David Drumlin suggests they also check the polarisation modulation:
ELLIE ... could it be a nested code of some sort? DRUMLIN You must have checked the signal for polarization modulation already... Ellie hesitates. Drumlin looks at her. Whoops. ELLIE Dr. Lunacharsky...? LUNACHARSKY (takes a drag) Analyzing signal polarization shifts. He type in a series of commands. Something begins to happen on the main display -- LUNACHARSKY Bingo. POV - THROUGH WINDOW Kitz, on the phone, sees the activity around the console. BACK TO SCENE Lunacharsky types in commands as he talks. LUNACHARSKY A second layer, nested within the main signal; possibly... a picture? Product of three primes... KITZ (joining them) What. LUNACHARSKY ... definitely three dimensions, either a hologram or a two- dimensional picture that moves in time; a movie. DRUMLIN (dryly) Hope there's a cartoon. KITZ How is that possible? How could all that information be encoded in -- KENT Well, sir, some bits of the signal are bits that tell us how to interpret the other bits. Technically speaking. JURYRIGGED 35-INCH MONITOR Strings of zeros and ones fill the screen as the group assembles around it. ELLIE Enlarge. Willie types. Ellie moves closer, studying it. Intuitively: ELLIE Try plotting values in a three dimensional coordinate system. A pattern begins to emerge. DRUMLIN Throw a gray scale on it; standard interpolation. ELLIE Rotate 90 degrees counterclock wise. Willie enters commands. All are mesmerized by the shadows taking form on the screen. ELLIE It has to be an image. Stack it up, string-breaks every 60th character. On the screen a distinct black and white moving image forms; grays define it even further. The group is transfixed. Kitz whispers to an aide who makes a call in a hand radio. KENT Um... I've got an auxiliary sideband channel here. I think it's audio.
The "auxiliary sideband" Kent refers to is also a nifty trick. The modulation process isn't 100% efficient, so the power that was supposed to go to one specific frequency or amplitude may "leak" to a slightly higher or lower frequency or amplitude. By changing the amount of leakage, one could encode another message using the sideband.
Later Arroway says there's another, weaker signal "underneath" the previous two, though she doesn't disclose the method:
"Ms. President, sorry to be late, but I think we've just hit the cosmic jackpot. We've.. It's... Let me try and explain it this way: In classical times, thousands of years ago, when parchment was in short supply, people would write over an old parchment, making what's called a palimpsest. There was writing under writing under writing. This signal from Vega is, of course, very strong. As you know, there's the prime numbers, and `underneath' them, in what's called polarization modulation, this eerie Hitler business. But underneath the sequence of prime numbers and underneath the retransmitted Olympic broadcast, we've just uncovered an incredibly rich message--at least we're pretty sure it's a message. As far as we can tell, it's been there all along. We've just detected it. It's weaker than the announcement signal, but I'm embarrassed we didn't find it sooner."
From the novel