S4-5.4 Noise
Standard probability and random-processes theory — written August 2026
What this is and why it exists
Noise is the adversary every communication system is engineered against — the reason links have ranges, receivers have specifications and satellite dishes have their shapes. This unit characterises the enemy (thermal, white, colored, AWGN) and prices every component's contribution to it in two interchangeable currencies: noise figure and noise temperature. Its cascade formula delivers one of the most practical sentences in radio engineering: the first stage decides almost everything.
The vocabulary
- Thermal noise — the voltage generated by thermal agitation of charge in any resistance; power proportional to temperature and bandwidth (kTB).
- White noise — flat power spectral density over all frequencies of interest; "white" as in containing every frequency equally.
- Colored noise — any noise whose spectrum is not flat; white noise after any real filter.
- AWGN — additive white Gaussian noise: adds to the signal, flat in spectrum, Gaussian in amplitude — the standard channel model.
- Noise figure (F) — the factor by which a component degrades signal-to-noise ratio from input to output.
- Equivalent noise temperature — the same degradation expressed as the temperature of a fictitious input resistor producing that much added noise.
- Noise bandwidth — the width of the ideal rectangular filter passing the same noise power as the real filter.
- Friis cascade formula — total noise factor: F1 plus (F2 minus 1) over G1, plus (F3 minus 1) over G1 G2, and so on.
The mental model
Every warm resistor hums. Random thermal motion of charge makes a fluctuating voltage across any resistance above absolute zero — no current drawn, no fault present. Its available power is kTB: Boltzmann's constant, absolute temperature, bandwidth. At room temperature that is minus 174 dBm per hertz — the noise floor, the number below which no signal at room temperature can be heard unaided, and the starting line of every link budget. The model stack is honest about idealisation: white noise (flat spectrum) is a fiction with infinite total power, but a harmless one, because every receiver filters — and filtered white noise is colored, its spectrum shaped by the filter. AWGN completes the standard adversary: added to the signal, white across the band, Gaussian by the Central Limit Theorem's verdict on summed microscopic sources.
Noise figure prices a component's guilt. An amplifier amplifies the incoming noise (forgivable — the ratio survives) and adds noise of its own manufacture (the crime). F is the factor by which signal-to-noise worsens through the device; its decibel form is the datasheet number, and an ideal noiseless component scores F of one, zero dB. Noise temperature says the same thing in kelvin — as if the device were an added warm resistor at the input — and shines when noise figures crowd near one: satellite receiver front-ends advertise 40 kelvin, a cleaner statement than a noise figure of 1.06. Converting between the currencies is one line of arithmetic and a routine professional task.
The cascade formula is the strategy lesson. Each later stage's added noise arrives divided by all the gain before it: F total is F1 plus (F2 minus 1)/G1 plus onward. With decent first-stage gain, the denominators bury every later contribution — the first stage's noise figure is nearly the system's. Hence the low-noise amplifier bolted directly at the antenna, ahead of the lossy cable: put the quietest, highest-gain stage first, and the rest of the chain can be ordinary. One formula, and it dictates the physical layout of every receiving installation on earth.
What you should now be able to explain or do
Compute thermal noise power from temperature and bandwidth. Distinguish white, colored and AWGN and say which idealisation each carries. Convert between noise figure and noise temperature. Run the cascade formula on a three-stage front end and defend the LNA-first architecture with it.
Check yourself
Why does a resistor generate noise with no current flowing through it?
Thermal agitation — charge in any resistance above absolute zero moves randomly, producing a fluctuating voltage with available power kTB. Only cooling reduces it.
White noise has infinite total power, yet the model survives. Why?
Because no receiver sees all frequencies: every system filters, and within any finite bandwidth the flat-spectrum model is accurate and the integrated power is finite. The fiction never faces the test that would break it.
A device has a noise figure of 2 (3 dB). What has it done to the signal?
Halved the signal-to-noise ratio: whatever quality margin arrived at its input, half remains at its output — the signal may be bigger, but the ratio is twice worse.
In a cascade, stage two is fairly noisy. Under what condition does the system barely notice?
When stage one has high gain: stage two's excess noise enters the total divided by G1, so a quiet, high-gain first stage suppresses every later sinner.
Why is the LNA mounted at the antenna rather than beside the receiver indoors?
The cable's loss would otherwise come FIRST in the cascade — attenuating signal before any gain exists to protect the ratio. Amplifying at the antenna puts high gain and low noise ahead of the loss, where the Friis formula wants them.
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