Two words that sound alike but aren't
In the previous guide we watched a 0 or a 1 become a real signal on a wire. The obvious next question is: how fast can we shovel those bits across? Before the formulas, we have to untangle two words people use loosely as if they were the same: bandwidth and data rate. They are not. Bandwidth is a width of frequencies, measured in hertz (Hz) — the physical span the channel can carry, like the range of pitches a singer can hit. The data rate is how many bits per second you actually push through, measured in bps, Mbps, or Gbps. Bandwidth is the road's width; data rate is how many cars per second you drive down it.
There is a second pair to keep straight, this one inside the data rate itself: bit rate versus baud. A baud, or symbol rate, counts how many distinct signal states (symbols) you send per second; the bit rate counts how many bits those symbols carry. If each symbol is just high-or-low, one symbol equals one bit and the two numbers match. But if each symbol can take one of four levels, it carries two bits, and the bit rate is double the baud rate. The rule is bit rate = baud rate times log2(M), where M is the number of distinct symbols. People say "baud" when they mean "bit rate" all the time; they are equal only in the simplest, two-level case.
Nyquist: how fast a clean channel lets you flash
Imagine signalling to a friend across a valley by flashing a torch. Even if your hand were tireless and the night perfectly clear, there is a fastest rate at which your friend can still tell one flash from the next — flash too quickly and the flashes blur into a smear. The Nyquist theorem makes that intuition exact for a perfect, noiseless channel: a channel of bandwidth B hertz can carry at most 2B symbols per second cleanly. Cram symbols closer than that and they smear into each other so badly no receiver can separate them.
Turn symbols into bits and you get Nyquist's famous formula: maximum data rate = 2B times log2(M) bits per second, where B is the bandwidth in hertz and M is the number of distinct signal levels per symbol. The 2B part is the symbols-per-second ceiling that bandwidth alone fixes. The log2(M) part is how many bits you stuff into each symbol by using more levels. Notice what this says: in a perfect channel you could raise M forever — go from 2 levels to 4 to 16 — and the data rate would just keep doubling, for free. That should set off an alarm: a formula that promises unlimited speed is missing something. What it is missing is noise.
Nyquist (no noise): R_max = 2 * B * log2(M) B = 3000 Hz, M = 2 levels -> 2*3000*log2(2) = 6,000 bps B = 3000 Hz, M = 4 levels -> 2*3000*log2(4) = 12,000 bps B = 3000 Hz, M = 16 levels -> 2*3000*log2(16) = 24,000 bps (... and on, for free??)
Noise, attenuation, and the SNR margin
Recall from the last guide that a digital signal is really a wobbly analog voltage we agree to read as a few clean levels. The wobble comes from noise: thermal jitter in the electronics, interference from a neighbour's gear, a microwave oven leaking onto your Wi-Fi band. The question that decides everything is not how loud the noise is on its own, but how the wanted signal compares to it. That comparison is the signal-to-noise ratio (SNR): signal power divided by noise power. Because the range is huge, engineers quote it in decibels: SNR(dB) = 10 times log10(S/N). A ratio of 1,000 is 30 dB and is healthy; a ratio of 1 is 0 dB and is hopeless.
SNR matters because the more signal levels you try to pack in, the closer they must sit, and the easier it becomes for noise to nudge a received level into a neighbour's slot — an error. So noise is exactly what caps how big M can be. Worse, the signal does not stay strong: attenuation fades it as it travels, while the noise floor stays put. A wire turns some signal energy into heat; a radio wave thins as it spreads. As the signal weakens but the noise does not, SNR falls, and once it drops too low the receiver can no longer tell the levels apart. That is why a copper Ethernet run is limited to about 100 metres — not because the bits run out, but because beyond that the signal is too faint relative to the noise.
Shannon: the true ceiling, drawn by noise
If Nyquist tells you how many symbols the bandwidth allows, the Shannon capacity tells you how many bits the noise allows — and it is the deeper, harder limit. Picture talking across a noisy room: the wider the range of pitches your voice can use (bandwidth) and the louder you are over the din (SNR), the more words per second you can reliably get across. Shannon's formula puts a precise, provable ceiling on that: C = B times log2(1 + S/N), where C is capacity in bits per second, B is bandwidth in hertz, and S/N is the signal-to-noise ratio as a plain ratio (not decibels here).
Two levers raise capacity, and they are not equal. Add bandwidth and capacity scales right along with it — double B, double C. But improve S/N and you only gain bits logarithmically: doubling the signal-to-noise ratio adds roughly one extra bit per symbol, no more. So beyond a point, throwing power at noise gives sharply diminishing returns, which is why engineers fight harder for cleaner channels and wider bands than for raw wattage. Shannon's result is also a little eerie: it proves a ceiling no coding scheme can ever beat, and proves that codes exist which sneak right up to it with vanishing error — without telling you how to build them. (Modern turbo and LDPC codes get remarkably close.)
- Find your channel's bandwidth B (in Hz) — say a 3,000 Hz phone line.
- Measure the signal-to-noise ratio as a plain ratio S/N — say 1,000 (which is 30 dB).
- Shannon's noise limit: C = 3,000 times log2(1 + 1,000) = 3,000 times log2(1,001), about 3,000 times 9.97, roughly 29,900 bps.
- Nyquist's bandwidth limit picks the symbol count; the real maximum data rate is whichever of the two is smaller. Noise, via Shannon, is usually the one that bites.
Seeing the limit in your own Wi-Fi
These two formulas are not museum pieces — they run your home network in real time. Modern radios use QAM, which packs bits into a symbol by combining several amplitude levels and several phase angles, drawn as a constellation of dots where each dot is a unique bit pattern. 16-QAM has 16 dots (4 bits per symbol); 256-QAM has 256 dots (8 bits); top-end Wi-Fi reaches 1024-QAM (10 bits). More dots in the same space means more bits per symbol means more speed — exactly the log2(M) lever.
Now watch Shannon enforce the law. Stand next to the router and your SNR is high, the dots are far apart relative to the noise, so the radio safely switches to dense 1024-QAM and you see headline speeds. Walk into the next room and attenuation drops the signal, SNR falls, the dots effectively crowd together until noise starts flipping them — so the radio downshifts to 256-QAM, then 16-QAM, trading bits per symbol for reliability. You did not change the bandwidth; you changed the SNR, and the constellation you are allowed to use is Shannon's ceiling made visible. This is also why the glossy box quoting "1024-QAM!" rarely matches real life: in a house with walls and interference, your devices spend most of their time at far lower QAM.
Step back and the whole picture clicks. The physical layer's job is to push bits across a medium; bandwidth and noise together decide how many bits per second it can manage; Nyquist counts the symbols the bandwidth allows; Shannon counts the bits the noise allows; and the real link rate is the smaller of the two. Everything above this layer — the framing of the next guide, the routing far above it — simply inherits whatever speed and error rate physics hands up. The cleverness in the upper layers is largely about coping gracefully with the limits set right here.