---
description: >-
  The following is a calculation and measurement of the noise temperature of the
  receiver.
---

# Receiver Noise Temperature

Every component in the receiver chain contributes thermal noise. The radiometer output power is the sum of soil emission and receiver noise ($$T_\mathrm{rec}$$​). The figure below shows the main components contributing to$$T_\mathrm{rec}$$​.

<figure><img src="https://732569122-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FpjLxyzuWzCEm73ytLKrU%2Fuploads%2Fy6bXuqGMp0fW5rRoED3q%2FScheme-it-export-Receiver-Noise-Calc-2023-12-29-11-32.png?alt=media&#x26;token=9e2d0035-d4c0-4b6c-929a-9a730955ca3c" alt="" width="452"><figcaption><p><em>Receiver components that are primarily contributing to the noise figure of the sensor. Not shown are some cable losses and contributions from the mixer and A/D converter components.</em></p></figcaption></figure>

### Theoretical Calculation

The combined noise figure of the cascading system is:

$$\mathrm{NF}_\mathrm{system} = \mathrm{L}_\mathrm{antenna} + \mathrm{L}_\mathrm{switch} + \mathrm{NF}_\mathrm{LNA}+ \dfrac{\mathrm{L}_\mathrm{bandpass} + \mathrm{NF}_\mathrm{amp}-1}{\mathrm{G}_\mathrm{LNA}} + \dfrac{L_\mathrm{Cable} + \mathrm{NF}_\mathrm{ADC} - 1}{\mathrm{G}_\mathrm{LNA} \cdot \mathrm{G}_\mathrm{amp}}$$

$$T_\mathrm{rec}$$ is calculated based on the following assumptions for Loss $$\mathrm{L}$$, noise figure $$\mathrm{NF}$$, and gain $$\mathrm{G}$$:

$$\begin{aligned} \mathrm{L}_\mathrm{antenna} &= 1.0\,\mathrm{dB}\,\,\mathrm{(estimated)}\\ \mathrm{L}_\mathrm{switch} &= 0.3\,\mathrm{dB}\\ \mathrm{NF}_\mathrm{LNA} &= 0.5\,\mathrm{dB}\\ \mathrm{G}_\mathrm{LNA} &= 20.5\,\mathrm{dB}\\ \mathrm{L}_\mathrm{bandpass} &= 1.5\,\mathrm{dB}\\ \mathrm{NF}_\mathrm{amp} &= 2.5\,\mathrm{dB}\\ \mathrm{G}_\mathrm{amp} &= 20.0\,\mathrm{dB}\\\mathrm{L}_\mathrm{Cable} &= 1.0\,\mathrm{dB}\,\,\mathrm{(estimated)}\\\mathrm{NF}_\mathrm{ADC} &= 3.0\,\mathrm{dB}\,\,\mathrm{(estimated)}\\\end{aligned}$$​

The noise figures of the amplifiers, as specified in their data sheets, and the noise temperatures of the resistors, which are affected by their losses, are dependent on their physical temperatures. Thus, we can calculate the overall noise temperature of the system for two realistic operating temperatures.

| Receiver Phys. Temp. | Receiver Noise Temp. | Remark                                                      |
| -------------------- | -------------------- | ----------------------------------------------------------- |
| 10 deg C             | 151 K                | After sensor startup on a cool day.                         |
| 50 deg C             | 168 K                | After about 20-30 minutes of data collection on a warm day. |

For this table, we assume the physical temperature of the antenna to be constant at 20 degrees Celsius and only change the physical temperature of the electronic receiver components.

The temperature of the PCB is measured by a digital sensor and recorded as "lna\_temperature\_degC" in the raw data file.

In summary, the noise temperature of the receiver varies from 150 to 170 Kelvin, depending on the temperature of the receiver components.

### Measurement

To actually measure the noise temperature of the receiver, the following three calibration measurements were made:

1.  The antenna was replaced with a pair of 75 cm long RF cables, each with a measured loss of 0.6 dB. 50-Ohm terminations were attached to one side of the cable, and the other was connected to the LNA inputs. The terminations were then placed in temperature-controlled environments at around -20 deg C and +110 deg C.

    1. Measurement 1 is the cold load at around -16 deg C.
    2. Measurement 2 is the hot load at around +110 deg C.



    <figure><img src="https://732569122-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FpjLxyzuWzCEm73ytLKrU%2Fuploads%2F3I9PYigRrKOjkH1NrBzi%2FScheme-it-export-hot-cold-measurement2-2024-12-10-13-47.png?alt=media&#x26;token=501d8500-52c1-4a63-a8e1-d8743603b648" alt="" width="563"><figcaption><p>Shown here is one of the two input channels with the RF cable connected to a 50-ohm load in a temperature controlled environment. The same is repeated for the second channel.</p></figcaption></figure>
2. The next measurement was made with the antenna connected directly to the LNA (without the pair of RF cables). The sensor was placed inside a shallow metal can for this measurement, with the antenna pointing up at the sky. The microwave brightness temperature of the sky is around 4 Kelvin. If we add a few Kelvin contributions from the antenna's side and back lobe, the expected noise temperature of the sky is around 10 K.
   1. Measurement 3 is the brightness temperature of the sky.

The following sensor outputs were recorded for the three measurements: a) cold load: 262 K, b) hot load: 355 K, and c) sky: 131 K. It is important to keep in mind that these temperatures do not represent absolute noise temperatures but have undergone the internal calibration process and are therefore expressed in terms relative to the internal noise source (see section "Internal Calibration").

<div data-full-width="true"><figure><img src="https://732569122-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FpjLxyzuWzCEm73ytLKrU%2Fuploads%2FzMtmaHfE1xZ3WUxoLnlv%2Flinreg_tsys.png?alt=media&#x26;token=5355fa5e-b280-4f7b-9d56-a4241b117af2" alt="" width="320"><figcaption><p>Plot of sensor output in units of Kelvin versus actual, physical brightness temperature of the source.</p></figcaption></figure></div>

In the case of measurements 1 and 2, we have to include the loss of the RF cable in the calculation of the system noise temperature:

$$
L_\text{cable} = 10^{\frac{\text{Loss (dB)}}{10}}
 \approx 1.148
$$

And the gain of the cable:

$$
G_\text{cable} = \frac{1}{L_\text{cable}}
 \approx 0.87
$$

The cable noise temperature is:

$$
T_\mathrm{cable}​=(L_\mathrm{cable}​−1)⋅T_\mathrm{phys}
$$

Assuming a physical temperature of the cable of 290K:

$$
T_\mathrm{cable}​=(1.148−1)⋅290 \approx 43.3K
$$

Thus, the noise temperature of the system used for measurements 1 and 2 is:

$$
T_\mathrm{sys}​=T_\mathrm{ant​}⋅G_\mathrm{cable}​+T_\mathrm{cable​}+T_\mathrm{LNA,eff}
$$

This results in the noise temperature of the sensor (including the antenna and all internal components) being 146 Kelvin and the loss of the antenna is 1.3 dB.

### Script to Calculate the Theoretical Receiver Noise Temperature

Below is a Python script for calculating the receiver noise temperature at different antenna and component temperatures.&#x20;

```python
import numpy as np

T_phys_Antenna_degC = 20
T_phys_PCB_degC = 10

Antenna = {"Loss": {"dB": 1.0}}
Switch = {"Loss": {"dB": 0.3}}
LNA = {"NF": {"degC": [-40, 85], "dB": [0.5, 0.68]}, "Gain": {"degC": [-40, 85], "dB": [21.05, 20.6]}}
BP = {"Loss": {"dB": 1.5}}
Amp = {"NF": {"degC": [-40, 85], "dB": [2, 3]}, "Gain": {"degC": [-40, 85], "dB": [20, 19.2]}}
Cable = {"Loss": {"dB": 1}}
LMS = {"NF": {"degC": [-40, 85], "dB": [3, 3]}, "Gain": {"degC": [-40, 85], "dB": [35, 35]}}

NF_LNA = np.interp(T_phys_PCB_degC, LNA["NF"]["degC"], LNA["NF"]["dB"])
NF_Amp = np.interp(T_phys_PCB_degC, Amp["NF"]["degC"], Amp["NF"]["dB"])
NF_LMS = np.interp(T_phys_PCB_degC, LMS["NF"]["degC"], LMS["NF"]["dB"])
Gain_LNA = np.interp(T_phys_PCB_degC, LNA["Gain"]["degC"], LNA["Gain"]["dB"])
Gain_Amp = np.interp(T_phys_PCB_degC, Amp["Gain"]["degC"], Amp["Gain"]["dB"])

NF_Chain = Switch["Loss"]["dB"] + NF_LNA + (BP["Loss"]["dB"] + NF_Amp - 1)/Gain_LNA + (Cable["Loss"]["dB"] + NF_LMS - 1)/(Gain_LNA*Gain_Amp)

NT_Chain = (T_phys_Antenna_degC + 273.15) * (pow(10, (Antenna["Loss"]["dB"]/10))-1) + (T_phys_PCB_degC + 273.15) * (pow(10, (NF_Chain/10))-1)

print(NT_Chain)
```

### Script to Calculate the Receiver Noise Temperature from Hot/Cold and Sky Measurements

```python
Temp_Termination_Cold_degC = -16
Temp_Termination_Hot_degC = 110
Temp_Termination_Sky_K = 10

Temp_Calsource_degC = 30
Noisetemp_Receiver_K = 45

Cable_Loss_dB = 0.6
L_cable = pow(10, Cable_Loss_dB / 10)
G_cable = 1/L_cable

Antenna_Loss_dB = 1.30
L_antenna = pow(10, Antenna_Loss_dB / 10)
G_antenna = 1/L_antenna

Temp_Termination_Cold_K = Temp_Termination_Cold_degC + 273.15
Temp_Termination_Hot_K = Temp_Termination_Hot_degC + 273.15
Temp_Calsource_K = Temp_Calsource_degC + 273.15
Temp_Cable_K = (L_cable-1) * 290
Temp_Antenna_K = (L_antenna-1) * 290

T_sys_source_Cold_K = Temp_Termination_Cold_K * G_cable + Temp_Cable_K + Noisetemp_Receiver_K
T_sys_source_Hot_K = Temp_Termination_Hot_K * G_cable + Temp_Cable_K + Noisetemp_Receiver_K
T_sys_source_Sky_K = Temp_Termination_Sky_K * G_antenna + Temp_Antenna_K + Noisetemp_Receiver_K
T_sys_cal_K = Temp_Calsource_K + Noisetemp_Receiver_K + 12
T_output_Cold_K = Temp_Calsource_K * T_sys_source_Cold_K / T_sys_cal_K
T_output_Hot_K = Temp_Calsource_K * T_sys_source_Hot_K / T_sys_cal_K
T_output_Sky_K = Temp_Calsource_K * T_sys_source_Sky_K / T_sys_cal_K

print("Tsys: {:.1f} K".format(Noisetemp_Receiver_K + Temp_Antenna_K))
print("Cold: {:.1f} K, Hot: {:.1f} K, Sky: {:.1f} K".format(T_output_Cold_K, T_output_Hot_K, T_output_Sky_K))
```
