Appearance
7.1 — The Communication Model, Noise and SNR
Every communication system, from a shout across a field to a satellite link, has the same five parts: a source, a transmitter, a channel, a receiver, and a destination. And every one of them fights the same enemy — noise, which is added by the channel and by the receiver itself, and which cannot be removed.
This chapter is about that enemy: where it comes from, how much of it there is, and what it costs.
Everything in this Part is a fight against the quantity introduced here, and the tools for the fight are the frequency-domain methods of Part 4 and the probability of Volume II Part 7 — because noise is random, and the only honest way to describe it is statistically.
1. The model
\text{source}\to\text{transmitter}\to\underbrace{\text{channel}}_{+\text{ noise}}\to\text{receiver}\to\text{destination}
The transmitter converts the message into a form the channel can carry — which usually means shifting it to a much higher frequency, for reasons section 6 explains.
The channel is whatever lies between: a copper pair, a coaxial cable, an optical fibre, or free space. It does three things to the signal, all of them unwanted:
- Attenuates it — reduces the amplitude.
- Distorts it — different frequencies are attenuated and delayed differently.
- Adds noise.
The receiver undoes as much of that as it can.
The asymmetry worth noticing: attenuation and distortion are in principle reversible, since you can amplify and equalise. Noise is not. Once it is added, it is part of the signal forever, and amplifying the signal amplifies the noise equally. That is why noise, and not attenuation, sets the fundamental limit.
2. Where noise comes from
Thermal noise
Every resistor generates noise because its charge carriers are in random thermal motion. John Johnson measured it at Bell Labs in 1926 and Harry Nyquist explained it theoretically the following year, which is why it carries both their names.
\boxed{v_n=\sqrt{4kTRB}}
with k = 1.38\times10^{-23} J/K, T in kelvin, R in ohms, B the bandwidth in hertz.
Read the structure. Noise voltage grows as the square root of resistance, of temperature, and of bandwidth. Halving the bandwidth reduces the noise by 30%; halving the resistance does the same.
Worked example. A 1 kΩ resistor at 290 K over a 1 MHz bandwidth:
v_n=\sqrt{4\times1.38\times10^{-23}\times290\times1000\times10^6}=\sqrt{1.6\times10^{-11}}=4.0\ \mu\text{V}
Four microvolts, from a resistor doing nothing. That is the floor beneath which no measurement can go.
The available noise power — the most a source can deliver into a matched load — is remarkably simple:
P_n=kTB
The resistance has cancelled out. At 290 K:
P_n = 1.38\times10^{-23}\times290\times B=4\times10^{-21}B \ \text{watts}
In decibels relative to a milliwatt:
P_n(\text{dBm})=-174+10\log_{10}B
That -174 dBm per hertz is worth memorising. It is the thermal noise floor at room temperature, and every receiver specification is measured against it.
| Bandwidth | Noise floor |
|---|---|
| 1 Hz | -174 dBm |
| 1 kHz | -144 dBm |
| 1 MHz | -114 dBm |
| 20 MHz | -101 dBm |
Thermal noise is white, meaning its power is spread equally across all frequencies, and its amplitude distribution is Gaussian. That combination is called AWGN — additive white Gaussian noise — and it is the standard channel model for essentially all of communication theory.
Shot noise
Current is made of discrete electrons arriving at random times, so the current itself fluctuates:
i_n=\sqrt{2qI B}
Note it depends on the current, not the resistance. Shot noise dominates in photodiodes and in low-current semiconductor circuits, and it is why an optical receiver's noise gets worse as the received light gets stronger — which sounds backwards until you realise the signal grows faster than the noise, so the ratio still improves.
Flicker noise
Also called 1/f noise, because its power density is inversely proportional to frequency. Its origin is trapping and release of carriers at defects in semiconductors, and it dominates below a corner frequency of anywhere from a few hertz to a few kilohertz depending on the device.
Why it matters: it is the reason DC-coupled precision measurements drift, and the reason a chopper amplifier (Chapter 2.5) exists — moving the signal up to a higher frequency, amplifying it there, and bringing it back down puts the amplification above the flicker corner.
Interference
Not really noise, since it is somebody's signal, but it has the same effect. Mains hum, switching supplies, other radio users, digital clocks radiating from a nearby board. Unlike thermal noise it usually has structure, which means it can sometimes be filtered, notched or cancelled — which is why identifying the mechanism (Chapter 5.P) matters more than filtering blindly.
What the noise is doing to your signal
The formulas below are ratios, and the ratio is easier to trust once you have seen what changing it does.
Three things to take from the panels before the definition arrives.
Noise adds, it does not replace. The signal is still fully present in the right-hand panel — it has not been damaged or removed. It has been added to. That is why the fix is never "make the signal louder" alone; raising both signal and noise together changes nothing, and only the ratio matters.
The failure is sudden, not gradual. The middle panel is imperfect but entirely usable, and the right one is hopeless. Digital links show this sharply: a few decibels of extra noise takes a channel from a handful of correctable errors to no connection at all, which is why streaming video is either fine or gone and rarely in between.
The ratio is of powers, not amplitudes, which is why the decibel figures look large. That is the definition the next section builds on.
3. Signal-to-noise ratio
\text{SNR}=\frac{P_{signal}}{P_{noise}}, \qquad \text{SNR}_{dB}=10\log_{10}\frac{P_s}{P_n}
In voltage terms it is 20 log, since power goes as voltage squared:
\text{SNR}_{dB}=20\log_{10}\frac{V_s}{V_n}
The decibel scale is used because SNR spans an enormous range and because gains and losses along a chain then add rather than multiply.
What different values mean in practice:
| SNR | Consequence |
|---|---|
| 0 dB | signal equals noise |
| 10 dB | speech intelligible with effort |
| 20 dB | acceptable voice quality |
| 40 dB | good analog audio |
| 60 dB | high-quality analog audio |
| 96 dB | 16-bit digital audio's theoretical best |
And a key fact about digital systems: they do not degrade gradually with SNR. They work perfectly until a threshold, then fail completely. That cliff is the reason digital television either looks perfect or freezes, with no snowy in-between — and it is a direct consequence of error correction, which either succeeds or does not.
4. Noise figure
An amplifier adds its own noise, so the SNR at its output is always worse than at its input. The noise figure measures how much worse:
F=\frac{\text{SNR}_{in}}{\text{SNR}_{out}}, \qquad NF=10\log_{10}F \ \text{dB}
A perfect amplifier has F=1, or 0 dB. A good low-noise amplifier at radio frequencies achieves 0.5 to 1 dB; a general-purpose one, 3 to 6 dB.
The Friis formula, and why the first stage decides everything
For amplifiers in cascade:
\boxed{F_{total}=F_1+\frac{F_2-1}{G_1}+\frac{F_3-1}{G_1G_2}+\cdots}
Read the denominators. The second stage's noise contribution is divided by the first stage's gain; the third stage's by the product of the first two.
Worked example. Stage 1: NF=1 dB (F=1.26), gain 20 dB (G=100). Stage 2: NF=10 dB (F=10), gain 20 dB. Stage 3: NF=15 dB (F=31.6).
F_{total}=1.26+\frac{10-1}{100}+\frac{31.6-1}{10{,}000}=1.26+0.09+0.003=1.353
NF_{total}=10\log(1.353)=1.31\ \text{dB}
A 15 dB noisy stage contributed 0.003 to the total. The answer is dominated almost entirely by the first stage.
Now swap stages 1 and 2:
F_{total}=10+\frac{1.26-1}{100}+\cdots=10.003 \;\Rightarrow\; NF=10\ \text{dB}
From 1.31 dB to 10 dB, purely by changing the order. Nine decibels — a factor of eight in effective sensitivity — thrown away.
The design rules this produces, and they are followed universally:
- Put the lowest-noise, highest-gain amplifier first, immediately at the antenna.
- Never put a lossy component before the first amplifier. A lossy component's noise figure equals its loss, so 3 dB of cable before the amplifier costs 3 dB of noise figure directly.
- This is why a satellite dish has the amplifier mounted at the feed rather than in the house, and why a mobile base station puts its low-noise amplifier at the top of the mast.
Noise temperature
An alternative and often more convenient measure:
T_e=(F-1)T_0, \qquad T_0=290\ \text{K}
Read it as: the temperature a resistor at the input would have to be at, to produce the same noise the amplifier adds.
Used in satellite and radio astronomy because the numbers are more meaningful there. A 0.5 dB noise figure is T_e = 35 K; a 3 dB one is 290 K. When the antenna itself sees a cold sky at 10 K, a 290 K receiver is throwing away most of the advantage, which is why radio telescope receivers are cryogenically cooled.
5. The link budget
Adding everything up to see whether a link works. The calculation is entirely in decibels, so everything adds.
P_{rx}=P_{tx}+G_{tx}-L_{path}-L_{misc}+G_{rx}
The free space path loss, for distance d and wavelength \lambda:
L_{fs}=20\log_{10}\left(\frac{4\pi d}{\lambda}\right)
or in practical units:
L_{fs}(\text{dB})=32.4+20\log_{10}d_{km}+20\log_{10}f_{MHz}
Two things fall out of that formula and both matter.
Doubling the distance costs 6 dB. Not 3 — the power spreads over a sphere whose area grows as d^2.
Doubling the frequency also costs 6 dB. This surprises people, since free space does not absorb anything. The loss is not in the space; it is in the receiving antenna. A fixed-gain antenna has an effective area proportional to \lambda^2, so at higher frequency the same antenna captures less of the passing wave. Use a fixed physical size of antenna instead and the higher frequency actually wins.
Worked link budget — a Wi-Fi link
- Transmit power: 20 dBm (100 mW)
- Transmit antenna gain: 2 dBi
- Distance: 50 m
- Frequency: 2400 MHz
- Receive antenna gain: 2 dBi
- Receiver bandwidth: 20 MHz
- Receiver noise figure: 6 dB
- Required SNR for the chosen data rate: 20 dB
Path loss:
L=32.4+20\log_{10}(0.05)+20\log_{10}(2400)=32.4-26.0+67.6=74.0\ \text{dB}
Received power:
P_{rx}=20+2-74+2=-50\ \text{dBm}
Noise floor:
P_n=-174+10\log_{10}(20\times10^6)+6=-174+73+6=-95\ \text{dBm}
SNR:
\text{SNR}=-50-(-95)=45\ \text{dB}
Required 20 dB, available 45 dB — a margin of 25 dB. The link works comfortably.
Now add two internal walls at 6 dB each and a body blocking at 4 dB:
\text{SNR}=45-16=29\ \text{dB}
Still 9 dB of margin. And that is why home Wi-Fi mostly works, and why it stops working at three walls rather than two.
Push the distance to 500 m: the path loss rises by 20 dB, leaving 9 dB of SNR — below the requirement, so the link would drop to a lower data rate that needs less SNR. That automatic rate adaptation is what every Wi-Fi device does, and it is why the throughput falls as you walk away rather than the connection simply dying.
6. Why modulation exists
The message is at baseband — audio is 20 Hz to 20 kHz. Why not transmit it directly?
Three reasons, and the first is decisive.
1. Antenna size. An efficient antenna is comparable to a quarter of a wavelength. At 3 kHz:
\lambda=\frac{3\times10^8}{3000}=100\ \text{km}, \qquad \frac{\lambda}{4}=25\ \text{km}
A twenty-five kilometre antenna. At 100 MHz the quarter wave is 0.75 m, which is the whip on a car.
2. Sharing. Every transmitter using baseband would occupy the same frequencies and interfere with every other. Shifting each to a different carrier lets thousands share the spectrum, which is the frequency division of Chapter 7.4.
3. Propagation. Different frequencies behave completely differently — some follow the ground, some bounce off the ionosphere, some go straight through and require line of sight. Choosing a carrier is choosing a propagation behaviour.
Modulation is therefore the operation of putting a low-frequency message onto a high-frequency carrier, and Chapter 4.4's frequency-shift property is the mathematics of it:
m(t)\cos(\omega_ct) \;\longleftrightarrow\; \tfrac12M(\omega-\omega_c)+\tfrac12M(\omega+\omega_c)
Multiplying by a carrier shifts the whole spectrum up to the carrier frequency. That is all modulation is, and Chapters 7.2 and 7.3 are the variations.
7. Bandwidth, and why it is the scarce resource
Bandwidth is the range of frequencies a signal occupies. It is the resource everything competes for, because the useful spectrum is finite and physically shared.
B=f_{high}-f_{low}
Typical allocations:
| Service | Bandwidth per channel |
|---|---|
| Telephone voice | 4 kHz |
| AM broadcast | 10 kHz |
| FM broadcast | 200 kHz |
| Analog TV | 6–8 MHz |
| Wi-Fi channel | 20–160 MHz |
| 5G channel | up to 400 MHz |
Spectrum is allocated by regulators and sold at auction for billions, which is a direct economic consequence of the fact that it cannot be manufactured.
Chapter 7.5 gives the exact relationship between bandwidth, noise and how much information can flow. Everything in between — every modulation scheme, every coding technique — is an attempt to get closer to that limit.
8. Distortion, and how it differs from noise
Three kinds, and the distinction matters because the remedies differ.
Amplitude distortion. The channel attenuates different frequencies differently. Correctable by an equaliser that applies the inverse response.
Phase distortion. Different frequencies are delayed differently, so the waveform's shape changes even though its spectrum is intact. Measured by group delay (Chapter 4.4), and correctable by an all-pass filter (Chapter 5.2).
Nonlinear distortion. The channel or an amplifier is not linear, so new frequencies appear that were not in the input. This one is not correctable, because information has been mixed irreversibly.
The measure is total harmonic distortion:
\text{THD}=\frac{\sqrt{V_2^2+V_3^2+\cdots}}{V_1}
Where nonlinearity bites hardest in radio: intermodulation. Two signals at f_1 and f_2 passing through a nonlinear amplifier produce sums and differences, and the third-order products at 2f_1-f_2 and 2f_2-f_1 land very close to the original signals, where no filter can remove them.
Worked example. Two signals at 100.0 and 100.2 MHz. Third-order products at 2(100.0)-100.2 = 99.8 MHz and 2(100.2)-100.0=100.4 MHz. Both inside the band you are trying to receive.
This is why an amplifier's linearity, quoted as its third-order intercept point, is as important as its noise figure, and why a receiver in a crowded band needs both. It is also why an overloaded receiver hears stations that are not there.
Chapter 7.2 puts the message onto the carrier the oldest way — by varying the carrier's amplitude or frequency — and explains why FM sounds so much better than AM.
Every formula above, built from scratch
None of the results in this chapter are worth memorising, because each one can be rebuilt in under a minute from something simpler. What follows is that rebuilding, one result at a time, so the formula and the reason for it sit on the same page as the explanation that needed them.
Noise
v_n=\sqrt{4kTRB} \qquad\text{(Johnson-Nyquist thermal noise)}
P_n=kTB \qquad\text{(available noise power — the resistance cancels)}
P_n(\text{dBm})=-174+10\log_{10}B \qquad\text{at } T=290\ \text{K}
The -174 dBm/Hz is worth memorising. Every receiver specification is measured against it.
i_n=\sqrt{2qIB} \qquad\text{(shot noise — depends on current, not resistance)}
\text{SNR}_{dB}=10\log_{10}\frac{P_s}{P_n}=20\log_{10}\frac{V_s}{V_n}
Noise figure
F=\frac{\text{SNR}_{in}}{\text{SNR}_{out}}, \qquad NF=10\log_{10}F
T_e=(F-1)\times290\ \text{K}
Friis cascade formula:
F_{total}=F_1+\frac{F_2-1}{G_1}+\frac{F_3-1}{G_1G_2}+\cdots
Read the denominators: every later stage's contribution is divided by all the gain before it. The first stage dominates, which is why the low-noise amplifier goes at the antenna.
A lossy component's noise figure equals its loss in dB, which is why cable before the first amplifier is so expensive.
Distortion
\text{THD}=\frac{\sqrt{V_2^2+V_3^2+\cdots}}{V_1}
Third-order intermodulation products fall at 2f_1-f_2 and 2f_2-f_1 — inside the wanted band, where no filter reaches them.
What the next chapter fixes
Noise sets what a channel can do. It does not tell you how to get a signal onto it, and putting audio straight onto a radio wave fails for reasons of antenna size alone. Chapter 7.2 introduces modulation, deriving AM and FM and showing why one is fragile in a thunderstorm and the other is not.