Part 3 set the upper limit on what a receiver can handle. Receiver sensitivity sets the lower limit: it is the weakest signal, in dBm at the antenna, that still gives the required quality. The receiver sensitivity formula has only three parts, thermal noise in the channel, the noise figure of the receiver, and the signal-to-noise ratio the demodulator needs. This part works through those terms, shows how front-end amplifiers set the noise figure, and checks the result against the reference sensitivity of real systems.
The Receiver Sensitivity Equation
In logarithmic form, RX sensitivity is:
Smin (dBm) = 10 log10(kT × 1000) + 10 log10B + NF + SNRmin
The first term is the thermal noise density. Any resistive source at temperature T delivers noise power kTB to a matched load, with k = 1.381 × 10−23 J/K. At the standard reference temperature of 290 K that is −173.98 dBm/Hz, normally rounded to −174 dBm/Hz. At 300 K it is −173.83 dBm/Hz, so the small difference is rarely important. The page on thermal noise and kTB derives the value.
The second term shows why bandwidth matters. Noise power grows with bandwidth, so the narrowest filter ahead of the demodulator sets B. In a superheterodyne receiver this is the final IF filter; in a direct-conversion receiver it is the baseband or digital channel filter. Doubling the bandwidth costs 3 dB of sensitivity, as Figure 1 shows.
The third term is the noise figure. The noise factor F is the input SNR divided by the output SNR with a 290 K source, and NF = 10 log10F. A noiseless block has NF = 0 dB, and any real one is above that. A matched passive device at 290 K has a noise figure equal to its loss, so a 2 dB filter has NF = 2 dB.
The fourth term depends on the modulation, the coding and the required error rate. It can be negative, since coded signals with strong redundancy can work below the noise. The SNR and Eb/N0 page relates it to bit rate and bandwidth.
A term called the minimum detectable signal (MDS) is often used for the noise floor itself, that is, the case with SNRmin = 0 dB. It is a useful common reference, but sensitivity is the figure that describes real operation at a stated error rate.
Cascaded Noise Figure and the Friis Formula
A receiver is a cascade, and the Friis formula gives the total noise factor from the stage noise factors F and linear gains G:
Ftotal = F1 + (F2 − 1)/G1 + (F3 − 1)/(G1G2) + …
Gain ahead of a stage divides that stage’s noise contribution, so the first stages count the most. This is the reverse of the linearity result in Part 3.
Worked example: a receiver noise budget
| Stage | Gain (dB) | NF (dB) | Added noise factor | Running NF (dB) |
|---|---|---|---|---|
| Preselector | −1.5 | 1.5 | 1.413 | 1.50 |
| LNA | 16.0 | 1.3 | 0.493 | 2.80 |
| Image filter | −2.0 | 2.0 | 0.021 | 2.85 |
| Mixer | −8.0 | 9.0 | 0.390 | 3.65 |
| IF amp | 20.0 | 4.0 | 0.536 | 4.55 |
The chain has 24.5 dB of gain and a cascaded noise figure of 4.55 dB. Figure 2 shows the running total. The preselector loss arrives first and counts in full, the LNA adds 1.3 dB of its own, and after that the additions shrink only as far as the gain allows. The mixer and IF amplifier still add 1.71 dB between them here, since the net gain ahead of them is only 12.5 dB at the mixer and 4.5 dB at the IF amplifier, and in a real design more gain ahead of the mixer would be weighed against linearity.
With a 9 MHz noise bandwidth, the noise floor is −174.0 + 69.5 + 4.55 = −99.9 dBm. A coded QPSK signal needing an SNR of about −1 dB then has a sensitivity of −100.9 dBm.
Noise temperature and antenna noise
The same budget can be expressed as an equivalent noise temperature, Te = 290(F − 1). The chain above has F = 2.85, so Te = 537 K. This form is useful when the antenna is not at 290 K. The 290 K assumption suits a terrestrial antenna looking at its surroundings, but a satellite-pointing antenna may see only 50 K or so. Noise is then proportional to Ta + Te, and the noise floor falls by 1.49 dB compared with the 290 K case. Noise temperature covers the conversion.
Noise bandwidth is also slightly different from 3 dB bandwidth. It is the width of an ideal rectangular filter passing the same noise power, and it is a little larger than the 3 dB width for a filter of low order. For the sharp channel filters of modern receivers the two are close, so the signal bandwidth is a fair approximation.
Checking Reference Sensitivity Against Real Systems
Reference sensitivity is the level a standard requires at the antenna connector, under a defined signal and error rate. It can be sanity-checked with the same equation, using a noise figure and required SNR the standard’s authors would have assumed and adding an implementation margin. The table below does this, and the assumed values are typical and approximate. The result lands close to the published requirements, which shows that nothing mysterious lies behind those numbers.
| System | B (MHz) | kTB (dBm) | NF (dB) | SNR (dB) | Margin (dB) | Result (dBm) | Typical requirement |
|---|---|---|---|---|---|---|---|
| GSM voice | 0.2 | −121.0 | 9 | 9 | 0.0 | −103.0 | about −102 dBm (handset) |
| LTE 10 MHz, QPSK, one branch | 9 | −104.4 | 9 | −1 | 2.5 | −93.9 | about −94 dBm (Band 1, single-port categories) |
| LTE 10 MHz, QPSK, two branches | 9 | −104.4 | 9 | −4 | 2.5 | −96.9 | about −97 dBm (Band 1) |
| Wi-Fi 20 MHz, BPSK rate 1/2 | 20 | −101.0 | 10 | 4 | 5.0 | −82.0 | about −82 dBm |
| Wi-Fi 20 MHz, 64-QAM rate 3/4 | 20 | −101.0 | 10 | 20 | 5.0 | −66.0 | about −65 dBm |
The LTE rows show the effect of receive diversity: with two antennas, the same signal needs about 3 dB less SNR per branch, which moves the result from −93.9 to −96.9 dBm. The Wi-Fi rows show how far modulation can move the sensitivity. A 16 dB difference in required SNR gives 16 dB less sensitivity in the same 20 MHz channel, and this is why a Wi-Fi link falls back to lower data rates as range increases. A wider channel costs sensitivity as well, because 40 MHz has 3 dB more noise than 20 MHz.
Measuring Receiver Sensitivity
Sensitivity measurement means reducing the input level until quality just meets the requirement. Analog receivers traditionally used SINAD, the ratio of signal plus noise and distortion to noise and distortion, with 12 dB a common reference for voice. Digital receivers use bit error rate, block error rate or packet error rate, and the receiver sensitivity in dBm is the level at which the agreed rate is met, for example a 10% packet error rate for Wi-Fi. Throughput-based tests are also common for LTE and 5G.
The signal generator must be well shielded, because leakage around a test fixture easily exceeds the very low levels involved. Noise figure measurement gives a separate check on the same budget.
How to Improve Receiver Sensitivity
The equation shows what can be changed, and the Friis formula shows how much each change is worth. Starting from 4.55 dB in the example chain:
| Change | New NF (dB) | Effect (dB) |
|---|---|---|
| Cut preselector loss by 0.5 dB | 4.05 | −0.50 |
| Use an LNA with 0.8 dB NF in place of 1.3 dB | 4.23 | −0.33 |
| Put the LNA ahead of the preselector | 3.63 | −0.92 |
| Reduce LNA gain from 16 to 10 dB | 7.54 | +2.99 |
| Add a 10 dB attenuator ahead of the LNA | 14.55 | +10.00 |
- Reduce loss ahead of the LNA. Each decibel of cable, switch or filter loss adds directly. An antenna-mounted LNA gains the full cable loss.
- Improve LNA noise figure and gain together. The table shows a lower NF giving 0.33 dB here, while losing 6 dB of LNA gain costs 2.99 dB. How to choose an LNA covers the trade.
- Narrow the bandwidth. A channel filter closer to the signal bandwidth removes noise directly, as far as the modulation allows.
- Lower the SNR requirement. Stronger coding, lower-order modulation, diversity and averaging all reduce it, usually at the cost of data rate.
Moving the LNA ahead of the filter wins 0.92 dB on paper, but it exposes the LNA to every out-of-band signal, so it is only sensible where the environment is quiet.
Front-End Amplifiers
The LNA sets the noise of the receiver, provides gain to suppress later stages, and must stay linear when strong signals arrive. It is specified by noise figure, gain, bandwidth, input and output match, IIP3, compression point and supply power. Silicon-germanium HBT and scaled CMOS processes now reach noise figures that once required gallium arsenide at frequencies up to several gigahertz, while GaAs pHEMT and GaN devices remain common where the lowest noise or the highest survivability is wanted. The RF amplifiers section compares these options.
Dynamic Range and Receiver Saturation
Sensitivity is one end of the receiver’s dynamic range. The other end is receiver saturation, where gain compresses and distortion takes over. Using a largest wanted signal of −25 dBm, the example receiver spans 75.9 dB from sensitivity to maximum, as shown in Figure 3. An IIP3 of −8 dBm gives an input P1dB near −17.6 dBm by the relation in Part 3, and an SFDR of 61.3 dB. Sensitivity sits below the noise floor in the figure because the assumed SNR is negative.
Peak levels need attention too. Multicarrier and OFDM signals have a high peak-to-average ratio, so a signal with a safe average level may still reach compression on its peaks. Part 16 quantifies this.
Automatic gain control extends the top of the range by reducing gain when signals are strong. The cost shows in the table above: 10 dB of attenuation ahead of the LNA gives 14.55 dB NF, a full 10 dB loss of sensitivity. That is acceptable, because a strong signal does not need the sensitivity. It would not be acceptable if a strong blocker triggered the AGC while the wanted signal was weak, so a receiver has to decide carefully which signal it is measuring. A related control sits next to the converter, which brings the analysis to the end of the receive chain.
Next in the Series
Part 5, RF ADCs in Receivers, reaches the end of the receive path: sampling and Nyquist zones, converter noise and resolution, how much gain to put in front of an ADC, and how an AGC loop keeps the signal in range.