6. Noise in Communication Systems

We have alluded to "noise" repeatedly. Now we open the box.

6.1 Sources of noise

Thermal (Johnson) noise. Every resistor at temperature $T$ has tiny voltage fluctuations from random thermal motion of charge carriers. The mean-square open-circuit noise voltage in bandwidth $\Delta f$ is

$\overline{v_n^2} = 4 k_B T R\, \Delta f$

where $k_B = 1.38\times 10^{-23}$ J/K. For a $50\,\Omega$ resistor at room temperature ($T = 290$ K) in 1 MHz of bandwidth: $v_n \approx 0.9\,\mu$V RMS. This is the noise floor every receiver must contend with. You cannot beat it with cleverness; it is set by the second law of thermodynamics.

Shot noise. Discrete quantization of charge: a current $I$ is really individual electrons crossing a junction at random instants, and the integer counting statistics produce fluctuations of size $\overline{i_n^2} = 2qI\,\Delta f$. Shot noise dominates in semiconductor junctions and photodiodes. The signal photons' Poisson statistics show up as shot noise in optical receivers.

Flicker (1/f) noise. Slow random fluctuations in transistor parameters whose power spectral density rises as $1/f$ at low frequencies. Dominates below tens of Hz to kHz, depending on the device. Crucial in DC amplifiers, sensor frontends, and oscillator phase noise close to the carrier.

Atmospheric and cosmic noise. Below ~30 MHz, atmospheric noise from lightning and ionospheric activity dominates. Above ~100 MHz, cosmic background noise from galactic synchrotron radiation takes over. This is why radio astronomy and deep-space comm receivers operate at "quiet" frequencies and have to design around the 2.7 K cosmic microwave background as a fundamental floor.

Man-made noise. Spark-gap transmitters (illegal but real), motor brushes, switch-mode power supplies, fluorescent lamps, plasma TVs, switching regulators on every digital board. In urban environments, the man-made noise floor often exceeds thermal by 30 to 60 dB at HF.

6.2 Noise figure and noise temperature

Define the noise factor of a 2-port system as

$F = \frac{\text{SNR}_\text{in}}{\text{SNR}_\text{out}}$

with the input source assumed at the standard temperature $T_0 = 290$ K. The system can only degrade SNR (an active system adds its own noise, a passive lossy system attenuates signal but the ambient resistor noise stays put), so $F \geq 1$ always. The noise figure is just $F$ in decibels:

$\text{NF} = 10\log_{10} F\ \text{dB}$

A noise figure of 0 dB means a noiseless system; 3 dB means it doubles the noise. Modern low-noise amplifiers for cellular radios achieve 0.5 to 1.5 dB. Satellite L-band LNAs (GPS receivers) hit 0.7 dB. Cooled receivers in radio astronomy reach below 0.1 dB.

The equivalent noise temperature is an alternative way of saying the same thing:

$T_e = (F - 1)\,T_0$

It is the temperature you would add to the input source to produce the same noise as the system itself contributes. Useful for satellite receivers because the input noise is dominated by sky temperature (a few tens to hundreds of kelvin) and adding $T_e$ to it directly tells you the operating noise level.

6.3 Friis formula for cascaded amplifiers, derived

Suppose you have a chain of stages with gains $G_1, G_2, G_3, \ldots$ and noise factors $F_1, F_2, F_3, \ldots$. The first stage adds noise referred to its input of $(F_1 - 1) k_B T_0$ per Hz. After amplification by $G_1$ this noise is $G_1(F_1 - 1)k_B T_0$ per Hz at stage 1's output. Stage 2 adds its own input-referred noise $(F_2 - 1) k_B T_0$, which becomes $G_1 G_2 (F_2 - 1)k_B T_0$ at the chain output... no wait, stage 2's added noise enters at its input, not stage 1's. Let me re-derive carefully.

Refer all noises to the input of stage 1.

• Stage 1 contributes $(F_1 - 1)k_B T_0$ Hz⁻¹ of noise at its own input (in addition to the $k_B T_0$ from the source).
• Stage 2 contributes $(F_2 - 1)k_B T_0$ Hz⁻¹ at its own input. Referring this back to stage 1's input means dividing by $G_1$, giving $(F_2 - 1)k_B T_0 / G_1$.
• Stage 3 contributes $(F_3 - 1)k_B T_0/(G_1 G_2)$ when referred to the front.

The total input-referred noise (in addition to the source noise) is:

$N_\text{added} = k_B T_0\left[(F_1 - 1) + \frac{F_2 - 1}{G_1} + \frac{F_3 - 1}{G_1 G_2} + \cdots\right]$

The total noise factor is $F_\text{total} = 1 + N_\text{added}/(k_B T_0)$, so:

$\boxed{F_\text{total} = F_1 + \frac{F_2 - 1}{G_1} + \frac{F_3 - 1}{G_1 G_2} + \frac{F_4 - 1}{G_1 G_2 G_3} + \cdots}$

This is Friis's formula, and it has one critical implication: the first stage dominates. If $G_1$ is large (say 20 dB, or 100x), then stages 2 onward have their noise contributions divided by 100. Even a stage 2 with $F_2 = 10$ contributes only $(10-1)/100 = 0.09$ to the total noise factor, less than 0.4 dB.

The lesson: the first thing your signal touches after the antenna must be a low-noise amplifier (LNA) with high gain. Once the LNA has boosted the signal well above kT, subsequent stages can be noisy and it does not matter. This is why every receiver in existence puts an LNA right after the antenna, often inside the same package as the antenna feed (a "block downconverter" on a satellite dish, an "active antenna" in a phone). It is also why deep-space and radio astronomy receivers go to extraordinary lengths cooling their LNAs in liquid nitrogen or helium: the second-stage noise penalty is unavoidable, but the first-stage temperature you can drive arbitrarily low.

6.4 SNR in AM

For DSB-LC AM with envelope detection at high SNR:

$\text{SNR}_o = \frac{\mu^2}{2 + \mu^2}\cdot\text{SNR}_c$

where SNR_c is the carrier-to-noise ratio at the input. For 100% modulation, the prefactor is $1/3$, so envelope-detected AM is 4.8 dB worse than coherent detection of the same signal. The carrier eats two-thirds of the power; only the sidebands carry information; the envelope detector can only see the message in the sidebands.

DSB-SC with coherent detection, and SSB with coherent detection, are equivalent in noise performance to each other and superior to envelope-detected AM by the same factor.

6.5 SNR in FM, the threshold, and FM advantage

For wideband FM well above threshold, with a single-tone message:

$\text{SNR}_o = 3\beta^2(\beta + 1)\cdot\text{SNR}_i$

The $\beta^2$ scaling is what makes FM the high-fidelity broadcast choice. Doubling $\beta$ (with corresponding bandwidth increase by a factor of $\sim 2$) gains 6 dB of audio SNR while only paying 3 dB in increased noise bandwidth. Tripling $\beta$ gains 9.5 dB. There is a clear payoff to using more bandwidth, up to the threshold.

The FM threshold is the input SNR (typically around 10 to 12 dB) below which the FM demodulator's output SNR plummets. The mechanism: the limiter and demodulator track instantaneous zero crossings of the input. When noise occasionally pushes the signal hard enough that two zero crossings collapse into one, or one crossing splits into three, the demodulator outputs a "click" — a sharp impulse equal to the inverse of the carrier period. Below threshold, click rate explodes and the output is unintelligible. Above threshold, clicks are rare and the audio is clean.

The threshold is one reason narrowband FM is preferred for two-way radio over broadcast FM despite the SNR penalty: with a smaller $\beta$, the threshold is easier to stay above with modest transmit power. A 5-watt handheld pushing narrowband FM to a repeater 30 km away can work fine; the same transmitter trying to broadcast wideband FM (which it cannot legally do anyway) would be below threshold and unintelligible.

Above threshold, FM beats AM by a factor of $3\beta^2$. For $\beta = 5$, that is a 27.5 dB advantage. For $\beta = 1$ (NBFM voice radio), it is 4.8 dB. Pre-emphasis adds a few more dB on top. FM's noise advantage is the entire reason high-fidelity broadcasting moved from AM to FM in the 1940s.