You have been using a noise floor since Module 2 without ever being told where it comes from. The link-budget line that ended at −73 dBm in M2-L4 was only meaningful because something worse was assumed to be underneath it. Shannon’s capacity in M6-L1 is a function of SNR, and the BER curves of M5-L2 and M6-L4 are drawn against Eb/N₀. All of that machinery has been standing on a quantity we borrowed on credit. This lesson pays the debt: it derives the floor from one constant and one temperature, shows how much of it your own receiver adds, and ends with the single formula that turns all of it into the sensitivity figure on a datasheet.
Module 8 spent four lessons making the received signal weaker: free-space loss, obstacles, multipath, walls. None of that would matter if the receiver could simply amplify what arrives. Amplification is cheap; the reason a link fails is that the amplifier raises the noise along with the signal, and the ratio between them is fixed the moment the signal reaches the antenna terminals. A signal is not weak or strong in absolute terms. It is weak or strong relative to the noise it shares a bandwidth with.
So the useful question is not “how small can a signal be?” but “how small is the noise?” — and unlike interference, which is somebody else’s transmitter and therefore negotiable, part of the noise is a consequence of temperature and cannot be negotiated with at all. Any resistor above absolute zero produces a fluctuating voltage across its own terminals, because the charge carriers inside it are in thermal motion. Put that resistor at the input of a receiver and you have injected noise into it without transmitting anything. That is the floor, and it is what we now put a number on.
The effect was measured by John B. Johnson at Bell Labs in 1928 and explained theoretically by Harry Nyquist in the same year and the same journal — which is why it carries both names. Nyquist’s result is remarkable for what it does not contain: the available noise power from a resistor does not depend on the resistance, nor on the material, nor on the shape of the component. It depends only on absolute temperature and on the bandwidth over which you measure.
That gives the most quoted equation in receiver design, and one of the shortest:
Two things about this deserve emphasis before we put numbers in. First, noise power is proportional to bandwidth: doubling the bandwidth doubles the noise, which is 3 dB, and that is the price you pay for every kind of speed. Second, kTB is the noise the source delivers into a matched load, so it is the noise available at the antenna terminals regardless of what the receiver does next. The receiver can only add to it.
Nobody computes 4×10⁻²¹ watts twice. Instead you convert kT once, at a reference temperature, into decibels relative to a milliwatt (M2-L4), and then add bandwidth in decibels. The IEEE reference temperature for noise work is T₀ = 290 K, which is 16.85 °C — chosen so that the arithmetic lands on a round number, and low enough to be a fair standard-room figure. So:
It is worth cross-checking this against a number you have already used. The satellite budget in M8-L1 worked in dBW and quoted Boltzmann’s constant directly as 10 log₁₀(k) = −228.6 dBW/K/Hz. Add the temperature: 10 log₁₀(290) = 24.62 dB, giving −228.6 + 24.62 = −203.98 dBW/Hz. Convert watts to milliwatts by adding 30 dB and you land on −173.98 dBm/Hz. The two conventions are the same statement; satellite engineers keep T separate because their antennas do not look at 290 K, as we will see.
Which temperature a text uses is a real source of small discrepancies, and it is better to name it than to let the reader wonder. This lesson uses 290 K throughout, giving −174 dBm/Hz. Many textbooks use 300 K instead, for which kT = 1.380649×10⁻²³ × 300 = 4.142×10⁻²¹ W/Hz and 10 log₁₀(4.142×10⁻¹⁸) = −173.83, usually quoted as −173.8 dBm/Hz. The gap is 0.15 dB — far below the uncertainty in any real measurement, but if you mix the two mid-calculation you will spend an afternoon looking for a mistake that is not there.
kTB has a ceiling, and you will never meet it. Nyquist’s formula is the low-frequency limit of a quantum expression, and it holds while hf « kT. At 290 K that boundary sits around 6 THz, some two hundred times above the highest millimetre-wave band in this course, so for every wireless system you will ever design, kTB is exact and flat — noise power per hertz is the same at 900 MHz as at 28 GHz. This is why thermal noise is called white.
With −174 dBm/Hz in hand, the thermal floor of any channel is one addition. Note that these are two-sided RF bandwidths — the width of the channel as a spectrum analyser would show it, which is the same B used for occupied bandwidth in M3-L4 and M6-L2:
| Channel bandwidth B | 10 log₁₀(B in Hz) | Thermal floor at 290 K | Where you meet it |
|---|---|---|---|
| 1 kHz | 30.0 dB | −174 + 30.0 = −144 dBm | Narrowband telemetry, CW beacons, GPS tracking loops |
| 200 kHz | 53.0 dB | −174 + 53.0 = −121 dBm | A GSM carrier (M10-L2’s TDMA channel) |
| 1 MHz | 60.0 dB | −174 + 60.0 = −114 dBm | Bluetooth, and the number most engineers memorise |
| 20 MHz | 73.0 dB | −174 + 73.0 = −101 dBm | A Wi-Fi or LTE channel |
Check the awkward one by hand: 10 log₁₀(20×10⁶) = 10 log₁₀(2) + 10 log₁₀(10⁷) = 3.01 + 70 = 73.01 dB, so the floor is −174 + 73.01 = −100.99, i.e. −101 dBm. And one more for scale: a 100 MHz 5G channel has 10 log₁₀(10⁸) = 80 dB of bandwidth and therefore a floor of −94 dBm. Widening from 20 MHz to 100 MHz is a factor of five, 7 dB, and that is exactly the 7 dB by which the floor rose. Wide channels are not free: they buy capacity with noise, which is precisely the trade Shannon priced in M6-L1.
SNR is the plainest definition in this lesson — signal power divided by noise power, both measured in the same bandwidth at the same point in the chain — and it is also the quantity most often quoted ambiguously. The trouble is that four closely related ratios all get called “the signal-to-noise” in conversation, and they differ by terms of 70 dB or more, so a mismatch is not a rounding error.
The bridge you need most often is the one to Eb/N₀, because BER curves are plotted against Eb/N₀ while receivers measure SNR. Signal power is energy per bit times bits per second, Ps = EbRb, and noise power is N₀B, so:
The other three appear on datasheets and in standards documents, usually without being defined:
A worked instance ties them together. Take a receiver holding 20 dB of SNR in a 20 MHz channel with N₀ = −169 dBm/Hz (we will justify that value shortly). Then C/N₀ = 20 + 73.0 = 93 dB-Hz, and at 4 bit/s/Hz the same link has Eb/N₀ = 20 − 6.0 = 14 dB. Three numbers — 20, 93 and 14 — describing one radio. Quote the wrong one and your link budget is out by 73 dB.
kTB is what arrives. Real hardware is worse, because every amplifier, mixer and filter contributes its own thermal and shot noise on top of what it was given. The measure of that damage is the noise factor F: the ratio of the SNR at the input to the SNR at the output, with the input driven from a source at the reference temperature T₀.
Written as a ratio F is called the noise factor; expressed in decibels, NF = 10 log₁₀F, it is the noise figure. The two words are frequently swapped, so read the units rather than the label. A perfect noiseless stage has F = 1, NF = 0 dB — it degrades nothing. Note carefully what F is not: it says nothing about gain. A 30 dB amplifier with NF = 3 dB and a 3 dB attenuator with NF = 3 dB damage the SNR by exactly the same amount.
The conversion is worth doing twice, because the pairing is not intuitive: a 3 dB noise figure sounds like a small imperfection and corresponds to nearly a full 290 K of added noise, while the 1 dB figure of a good LNA corresponds to only 75 K.
The practical consequence is that noise figure adds directly to the floor. A receiver with NF = 5 dB in a 20 MHz channel has an effective noise floor of −101 + 5 = −96 dBm, and a density of N₀ = −174 + 5 = −169 dBm/Hz — the value used in the SNR example above. Every decibel of noise figure is a decibel of range, or a decibel of margin, given away for free.
A receiver is not one stage but a chain: low-noise amplifier, filter, mixer, IF amplifier, detector. Each has a gain and a noise figure, and they do not simply add. Harald Friis published the accounting in 1944, and it is the most consequential design formula in the whole subject — not because it is hard, but because of what it implies about ordering:
Look at the shape of it before the arithmetic. The first stage’s noise factor enters undivided. Every later stage is divided by the total gain ahead of it, so its contribution is suppressed in proportion to how much the signal has already been amplified. The formula is telling you that a receiver’s noise performance is decided almost entirely at its front end.
Take a conventional chain: an LNA with NF = 1 dB and gain 20 dB, a mixer with NF = 8 dB and gain 10 dB, then an IF amplifier with NF = 15 dB whose own gain never enters the sum because nothing follows it.
A chain containing a stage with an 8880 K noise temperature has a total noise figure of 1.28 dB, only 0.28 dB worse than the LNA alone. The mixer contributed 4% of the total and the IF amplifier 2%. That is the whole reason a satellite dish has an amplifier bolted to the feed rather than at the far end of the cable run, and the reason a phone puts its LNA millimetres from the antenna port.
Claims about ordering should be demonstrated rather than asserted, so reorder the same three components — mixer, then LNA, then IF amplifier — and recompute. Nothing about the hardware changes; only the sequence does.
Same three parts, same total gain of 30 dB before the IF stage, and the noise figure went from 1.28 dB to 8.04 dB — a loss of 6.76 dB, which on a range-limited link is slightly more than a factor of two in distance under free-space spreading, where 6.02 dB is exactly a doubling (M8-L1). The first stage dominates because it is the only one whose noise is not divided by anything, and that single fact fixes the physical layout of every receiver ever built.
A corollary about cable. A passive loss ahead of the LNA is a stage with G < 1 and F equal to that same loss, so 2 dB of feeder before the amplifier adds very nearly 2 dB to the system noise figure. This is why a mast-head amplifier and an identical amplifier at the bottom of the mast are not the same product, and why the receive path of a base station is planned around where the loss sits, not merely how much of it there is.
Everything above assembles into one line. A receiver works when the signal is enough above its own noise floor to satisfy the demodulator, so the minimum usable signal is the thermal floor, plus the bandwidth, plus what the receiver adds, plus what the modulation demands:
Read the formula as a list of the only four ways to hear a weaker signal, and notice how few of them are available: cool the front end, narrow the channel, buy a better LNA, or ask less of the demodulator by dropping to a more robust modulation. There is no fifth option, and no amount of gain is one.
Take a Wi-Fi-like receiver on a 20 MHz channel with a noise figure of 5 dB, demodulating 64-QAM at rate 3/4 — roughly 4 bit/s/Hz, needing about 20 dB of SNR to hold the target error rate.
Now compare that to reality, because the gap is instructive. IEEE 802.11 requires only about −65 dBm for that mode; a good commercial radio might claim −72 dBm. Our −76 dBm is the theoretical floor, and real receivers fall short of it by an implementation margin covering phase noise, quantisation, imperfect synchronisation and filter shape. The calculation is still the right one to do first: it tells you what is possible, so you know whether a disappointing measurement is a design fault or a law of physics.
One last honest caveat. Sensitivity is a thermal-noise figure, and it is the correct limit only in a quiet band. Drop the same radio into a crowded office and the floor it actually sees may be 10 or 20 dB higher, because it is now hearing other transmitters rather than the thermal motion of its own front end. That is not a failure of kTB; it is a different problem, and it belongs to M9-L3.
The noise floor has now stopped being an assumption. It is −174 dBm/Hz at 290 K, plus 10 log₁₀B for the channel you chose, plus the noise figure your front end is honest enough to admit to — and the “required SNR” term is the only one left undefined. Filling it in means asking how error probability depends on SNR for each modulation scheme, which is exactly what M9-L2 does with BER and the Eb/N₀ curves you last saw in M6-L4.
After that the lesson order follows the physics. M9-L3 takes up interference — co-channel, adjacent-channel and man-made — and makes the case that in most modern deployments it is interference, not thermal noise, that sets the real limit; a floor lifted by neighbours is the normal condition of an urban network. Then M9-L4 assembles Modules 8 and 9 into a complete link budget: transmit power, antenna gains, path loss, the noise floor from this lesson, the required SNR from the next, and the fade margin that M8-L3 and M8-L4 showed you cannot do without.
What kTB is not. It is not all the noise in a receiver. Flicker (1/f) noise dominates at low frequencies, shot noise appears wherever carriers cross a junction, phase noise from the local oscillator smears the constellation, quantisation noise arrives with the ADC, and the antenna delivers sky and ground noise that has nothing to do with the temperature of your circuit board. Thermal noise earns its central place because it is the one contribution that is unavoidable, calculable and frequency-flat — the floor beneath the floor.