## Introduction When designing an RF chain, front end, or experiment, noise is a first-order consideration. This note shows that: 1. Noise is a fundamental part of the universe and exists even outside of circuitry. 2. Useful RF performance metrics must include noise (e.g., $C/N_0$ or $E_b/N_0$). Raw power measurements such as dBm are only meaningful if the noise floor is assumed to be far away, and even then, a few active or passive components can quickly invalidate that assumption. 3. Noise has a minimum floor. As a result, $C/N_0$ is *always* degraded by adding components: - **Passive devices** attenuate both $C$ and $N_0$, but $N_0$ has a floor. - **Active devices** amplify both $C$ and $N_0$, while also adding their own noise temperature, further degrading $C/N_0$. The goal is to motivate why noise temperature is often the most natural and robust way to analyze RF systems. ## Background on Noise Temperature ### What is it? Thermal noise is everywhere and represents the minimum power a system will deliver: $ \text{Thermal Noise} = kTB $ This equation will be justified later, but the key idea should be kept in mind. While thermal noise appears to depend on three variables, with standard definitions it effectively reduces to a dependence on temperature alone: - $k = 1.38\times10^{-23}\,\text{J/K}$ is Boltzmann’s constant (recall $\text{W}=\text{J/s}$, so $k=\text{W/(K·Hz)}$) - $T$ is the physical temperature in Kelvin - $B$ is the bandwidth in Hz, typically normalized to 1 Hz With $B=1$, thermal noise depends only on temperature. Lower temperatures yield less noise; higher temperatures yield more. Because the universe has a cosmic microwave background, with an effective temperature of about 2.4 K, there is always nonzero noise in any electrical system. **Table 1: Thermal Noise Values at Different Temperatures** | C | F | K | Thermal Noise / Hz (dBm) | | --- | --- | --- | ------------------------ | | -40 | -40 | 233 | -174.9 | | -20 | -4 | 253 | -174.6 | | 0 | 32 | 273 | -174.2 | | 17 | 63 | 290 | -174.0 | | 50 | 122 | 323 | -173.5 | | 75 | 167 | 348 | -173.2 | In practice, it is common to use **–174 dBm/Hz** as a default reference value. ### Relating Noise Temperature to Noise Factor and Noise Figure Noise Factor $F$ is defined as the ratio of input SNR to output SNR: $ F = \frac{S_{in}/N_{in}}{S_{out}/N_{out}} = \frac{\text{SNR}_{in}}{\text{SNR}_{out}} $ In an ideal device, input and output SNR are identical and $F=1$. In real systems, SNR always degrades, so $F>1$. Noise Figure (NF) is simply Noise Factor expressed in dB: $ \text{NF} = 10\log_{10}(F) $ For active devices, NF is usually given in the datasheet. For passive devices, the noise figure is equal to the insertion loss. ### Proof that Noise Figure of a Passive Device Equals Its Loss Let the gain of a passive device be $-0.30\,\text{dB}$. Assume (and justify later) that the noise entering and leaving the passive device is equal: $ N_{in} = N_{out} $ Inject a signal: $ S_{in} = 1\,\text{mW} = 0\,\text{dBm} $ With a loss of 0.30 dB, the output signal power is: $ S_{out} = -0.3\,\text{dBm} = 10^{-0.3/10} = 0.93\,\text{mW} $ Substituting into the Noise Factor definition: $ F = \frac{1}{0.93} = 1.075 $ $ \text{NF} = 10\log_{10}(1.075) \approx 0.3\,\text{dB} $ Therefore: $ \text{NF} = -(\text{gain in dB}) \quad \text{for a passive device} $ ### Temperature and Noise Figure In many applications, it is more useful to work *backwards* from temperature. Noise Figure and Noise Factor are only meaningful if all devices reference the same ambient temperature. This becomes problematic in space applications, where antenna temperature is rarely 290 K. Noise Factor is simply a transformation of noise temperature *assuming* a reference temperature: $ \text{NF} = 10\log_{10}\!\left(\frac{T_\text{ref}+T_\text{noise}}{T_\text{ref}}\right) $ $ T_\text{noise} = T_\text{ref}\left(10^{\text{NF}/10}-1\right) $ Noise temperature expresses noise power in terms of the equivalent temperature required to generate it: $ T_n = \frac{P_n}{k_B B} $ ## Relating Power to Temperature (Johnson–Nyquist Noise) ### Noise Is a Property of the Universe, Not the Circuitry **Claim:** The noise power delivered by a circuit depends solely on its equivalent noise temperature. To show this, consider three elements: loads, passive devices, and active devices. A generic load (spectrum analyzer, power sensor, modem, etc.) can be modeled as a resistance $R_\text{Load}$. ![[load_resistor.png|200]] A passive component (cable, attenuator, splitter) can be modeled as a resistor. Johnson showed that a resistor $R$ at temperature $T$, over bandwidth $B$, produces a mean-square noise voltage: $ V_n^2 = 4k_BTRB $ All real components therefore add noise, since each term on the right-hand side is strictly positive. A passive component can be modeled as a noiseless resistor in series with a noise voltage source corresponding to its equivalent temperature. Using a Norton equivalent yields: ![[resistor_to_noiseless_resistor.png|300]] For a matched load (typically 50 Ω in RF systems), the noise current divides equally: ![[norton_noiseless_resistor_diagram.png|300]] $ P = \left(\frac{I}{2}\right)^2 R $ $ P = \frac{4k_BTB}{4R}\cdot R = k_BTB $ Thus, the noise power depends *only* on temperature. Thermal noise is a property of the universe—not the circuitry. Any attempt to reduce noise power by mismatching impedance also reduces signal power, leaving $C/N_0$ unchanged or worse. ### Impedance Matching Reducing current by increasing resistance does reduce noise power—but it also reduces signal power by the same mechanism. Since signal quality depends on $C/N_0$, impedance mismatch always degrades performance. Maximum power transfer occurs only under matched conditions. ### Impedance Matching of a Resistor (Exercise) How can a device have a 50 Ω impedance yet introduce, say, 6 dB of loss? Can such a device be subdivided indefinitely while remaining matched? ![[impedance_matching_of_a_resistor_image.png|500]] This circuit represents a matched attenuator. For a matched system, the input and output impedances must both equal $R_o = 50\,\Omega$. Let $r = R_1 = R_2$. The impedance seen from node A is: $ Z_A = R_3\frac{r+R_L}{R_3+(r+R_L)} $ With $Z_{in} = r + Z_A = R_o$, solving yields: $ R_3 = \frac{R_o^2 - r^2}{2r} $ Setting $r = 16.6\,\Omega$ gives $R_3 = 66.9\,\Omega$, producing a 6 dB attenuator that remains matched. ### High-Level View of Multiple Passive Devices Each passive device contributes its own noise source. When cascaded: $ P = k_B (T_1 + T_2) B $ ![[norton_diagram_with_two_sources.png|400]] ### Another Way to Obtain Noise Power Using a Thevenin noise source: ![[thevenin_circuit_with_amp.png|500]] With: $ V_N = \sqrt{4RBk_BT} $ Maximum power transfer occurs when $R_{in}=R$, yielding: $ P_\text{max} = k_BTB $ Again, noise power depends only on temperature. ## System Receiver Noise A receiver chain can be simplified as: > Antenna → Amplifier → Bandpass Filter → DUT - **Antenna**: Sets the initial noise temperature. - **Amplifier**: Adds gain and noise (referenced to its input). - **Bandpass Filter**: Defines bandwidth (often normalized to 1 Hz). The objective is to compute the total noise temperature at the DUT input. ### Cascaded Devices ![[system_noise_frontend.png]] With antenna temperature $T_A$ and amplifier noise temperature $T_G$: $ \text{SNR} = \frac{P}{k_B B (T_A + T_G)} $ Amplifier gain cancels, reinforcing why SNR—not absolute power—is the relevant metric. ### Active Devices (Friis in Temperature Form) For cascaded stages: $ T_{Rx} = T_{G1} + \frac{T_{G2}}{G_1} + \frac{T_{G3}}{G_1 G_2} $ Early stages dominate noise performance, making a low-noise first amplifier critical. Including the antenna: $ T_{tot} = T_A + T_{Rx} $ ## Noise Power Density from Receiver Temperature $ N_0 = k_B T_{tot} \quad [\text{W/Hz}] $ $ N = N_0 \cdot \text{BW} $ An equivalent receiver noise figure can be derived if desired: $ T_{Rx} = T_0(F_{Rx}-1) $ ## Noise Figure Revisited $ F = \frac{T_0 + T_G}{T_0}, \quad T_0 = 290\,\text{K} $ Noise Figure is simply a logarithmic representation of noise temperature. While convenient, it assumes a common reference temperature, which often fails in space and remote-sensing systems. **In practice, noise temperature provides a clearer and more physically meaningful path to final SNR.** ## Conclusions Noise temperature provides a unified and physically grounded framework for RF system analysis. Unlike noise figure, it naturally accommodates components operating at different physical temperatures—such as antennas exposed to space and electronics inside a controlled environment. For accurate SNR prediction, especially in non-terrestrial systems, noise temperature is the preferred metric. ## Resources - Johnson–Nyquist circuit walkthrough: https://www.bing.com/videos/riverview/relatedvideo?q=theiven+circuit+and+the+johnson+nyquist+circuit - In-depth noise analysis notes: https://www.ece.mcmaster.ca/faculty/nikolova/antenna_dload/current_lectures/L07_Noise.pdf - JPL noise calculations paper: https://ipnpr.jpl.nasa.gov/progress_report/42-150/150F.pdf - Equivalent noise temperature overview: https://www.allaboutcircuits.com/technical-articles/characterize-rf-noise-components-using-equivalent-noise-temperature/ - Draw.io sources: [[equivalent circuits.drawio]], [[noise_util.drawio]] ![[Connect#Connect]]