---
# System prepended metadata

title: Noise
tags: [' ss', Communication System Design, ' ncu']

---

---
tags: Communication System Design, ss, ncu
author: N0-Ball
title: Noise
GA: UA-208228992-1
---

[ToC]

# Preknowledges

- All electrical circuites wiith T > 0 K **->** have noise
- Random motion of electrons causes fluctuation **->** AC current = thermal noise
- Thermal noise sets a lower limit on signal power in any receiver
- Often, a **10-dB margin** is needed to make demodulators work in receivers
- Noise power needs to be estimated for a receiver
- **Carrier-to-noise ratio** defines system performance
- C/N is assessed at the input of demodulator
- C/N determines
    - S/N (SNR) for analog output
    - BER for digital output

# Link Budget with Noise

![](https://i.imgur.com/ZvFAC2b.png)

- C = P~t~

$$
C/N = \frac{P_r}{kT_sB_n}
$$
where T~s~ is the **System Noise Temperature** for the receiver

$$
T_s = T_A + T_R
$$

where 
:    T~A~ is antenna noise temperature
:    T~R~ is receiver noise temperature

in dB, we write:
$$
C/N = P_t + G_t + G_r - L_p - losses -k -T_s - B_n
$$
where **k** is Boltzmann's constant, $1.38 \times 10 ^ {-32}$ or -228.6 dB

## Noise Temperature & Noise Power

System Noise Temperature
:    Model receiver with a noiseless receiver and a single noise source at the antenna terminals

- A block body withg physical temperature, T~p~[K], generates electrical noise over a wide bandwidth

$$
P_n = kT_PB_n \equiv kT_sB_n
$$
- k: Boltzmann's constant
- T~s~: System Noise Temperature
- B~n~: Noise Bandwidth in which the noise power is mearused, in Hz

:::warning
Noise temperature is not real temperature
:::

# Noise Sources

- Individual blocks in the receiver generate noise
- Thermal noise is present whenever electrons move in a circuit
- Need a noise model for receivers

```graphviz
digraph {
  rankdir=LR;
  
  Antenna [shape="point", xlabel="Antenna"]
  LNA [shape="triangle", fixedsize=true, orientation=30]
  BPF1 [shape="square"]
  Mixer [shape="square"]
  BPF2 [shape="square"]
  IF_amp [shape="triangle", fixedsize=true, orientation=30, label="IF\n amp"]
  Demodulator [shape="square"]
  
  Antenna -> LNA -> BPF1 -> Mixer -> BPF2 -> IF_amp -> Demodulator
}
```

## Noise model

![](https://i.imgur.com/grUVYGS.png)

- Receiver is modeled as a noiseless receiver with system noise temperature, T~s~
- Output of receiver is P~out~
    - $P_{out} = P_{in} \times G = P_{in} \times G_{RF} \times G_{m} \times G_{IF}$
- Signal power, P~out~ coexists with the present of noise power P~n~
    - $P_n = G_{RF}G{m}G_{IF}\left[\frac{kT_{IF}B_{n}}{G_{RF}G_{m}} + \frac{kT_mB_n}{G_{RF}} + k\left(T_{RF} + T_{in}\right)B_n\right]$
    - $P_n = G_{RF}G{m}G_{IF}k \left[ T_{RF} + T_{in} + \frac{T_m}{G_{RF}} + \frac{T_{IF}}{G_{RF}G_{m}} \right]B_n = GkT_sB_n$
        - where $T_s = T_{in} + T_{RF} + \frac{T_m}{G_{RF}} + \frac{T_{IF}}{G_{RF}G_m}$
    - Low Noise Amplifyer
        - Low T~RF~
        - High G~RF~
- End-to-end gain of a receiver is $G_{RF} + G_{m} + G_{IF} = G$
- Each noise source contributes noise power $N = kT_nB_n$
- Signal power out of receiver is P~out~
- Signal power in to receiver is P~in~

## Noise model with a Lossy device

- A lossy device not only attenuates the gain but also radiates noise
- Equivalent output noise source T~l~
    - $T_l = T_p(1 - G_l) = T_p(1 - \frac{1}{L})$
- A lossy device also attenuates incomming noise

Where 

G~l~ 
:    linear gain (less than 1, not in dB) of the attenuating device

T~p~ 
:    physical temperature of the lossy device
        

![](https://i.imgur.com/edDBy5n.png)


# Noise Bandwidth

- Noise bandwidth of a filter is the equivalent rectangular bandwidth that passes the same power as the actual filter
- All practical bandpass filters have B~n~ = 3 dB bandwidth
- We use 3 dB bandwidth and call it noise bandwidth

# G/T

- Receiving stations can be characterized by their G/T ratio
- G/T = receiving antenas's gain divided by system noise

$$
C/N \propto G_r/T_s \\[1em]
\frac{C}{N} = \frac{P_r}{P_n} = \frac{P_tG_tG_r}{kT_sB} \propto \frac{G_r}{T_s}
$$

- Target
    - hight gain
    - low noise temperature

# Multipath

![](https://i.imgur.com/CjjLohY.png)

:::info
occurs when indirect ray is a multiple number of half wavelengths longer than the direct 
ray
:::

![](https://i.imgur.com/BV4noTG.png)

- Typical LOS link
    - 50 km
    - Antenna beamwidth is ~1–2° at 100 km. 
    - Reflection Point is at 25 km
    - Angle α = 100 / 25000 rads = 0.23°

## Deep fade
![](https://i.imgur.com/vTg01yC.png)

## Mitigating Cancellation

**If**
Signal is below -30 dB when indirect ray has phase of 180° $\pm$ 1.8°

### Solution 1
Second Antenna 3 m below

- will have different indirect ray phase

![](https://i.imgur.com/JEgo92i.png)

### Solution 2
If upper antenna fades out, signal switched to lower antenna

- called switched antenna diversity

![](https://i.imgur.com/CL5XJTJ.png)

### Solution 3
Frequency Diversity

- Band is divided into 12 channels
    - 36 MHz / channel
    - frequency spacing: 40 MHz
- Switching channel changes phase of direct and indirec rays

# Propagation 

Different weather will effect propagation attenuation

- Outage occurs
    - S/N at baseband
    - BER falls below accptable
- express outage as percent of average year 

## Clear air

- oxygen and water vapor
    - depends on relative humidity
## Rain 

- Effects significantly at 6 GHz or above
- slightly higher with HP (horizontal polariztion) than VP (vertical polarization)
- Types
    - Stratiform
        - near uniform
        - widespread
        - typically associated with warm fronts
        - rate < 10 mm/hr
    - Convective
        - Non-uniform
        - Typically associated with warm fronts
        - Rain rate > 10 mm/hr

# Rain Attenuation

:::info
Not only attenuates signals but also increases effective antenna noise temperature by $\Delta T$
:::

$$
\Delta T \cong 290 \times \left( 1 - \frac{1}{L_{ntl}} \right) K
$$

**Where**

L~ntl~
:    natural scale as loss
$$
A_{dB} = 10 \cdot \log_{10} \left(L_{ntl}\right) 
$$

## Solutions

- More complex model is needed to find path length in rain
- Models give effective path length L~e~

## Example
:::info
- Reciever's T~s~ = 600K
- Rain attenuation, A = 10.0 dB
:::

$$
\Delta T \cong 290 \times \left( 1 - 0.1 \right) = 261\ K
$$

**new Noise Temperature**
$$
T_s' = 600 + 261 = 861\ K
$$

**Therefore**

- Noise increase
    - 1.6 dB ($10 \log_{10} (\frac{861}{600})$)
- C/N decreases
    - 1.6 + 10 = 11.6 dB


# Noise Figure

- Manufactures of microwave device characterize them with a **Noise Figure**
- Measured with 290 input termination
- usually given in dB

$$
\begin{split}
T_n &= 290 \times (\text{NF} - 1) \\[1em] 
&= 290 \times \left(10^{\frac{10 \log(\text{NF})}{10}} - 1 \right) \\[1em]
&= 290 \times \left(10^{\frac{\text{NF (dB)}}{10}} - 1 \right)
\end{split}
$$

## Example #1

:::info
- $T_{in} = 25 K,\ T_{RF} = 50 K,\ T_m = 500 K,\ T_{IF} = 1000K$
- $G_{RF} = 23 dB = 200$
- $G_m = 1 = 0dB$
:::

$$
T_s = 50 + 25 + \frac{500}{200} + \frac{1000}{200} = 82.5 K
$$

## Example #2

:::info
- $T_{in} = 25 K,\ T_{RF} = 50 K,\ T_m = 500 K,\ T_{IF} = 1000K$
- **$G_{RF} = 50 dB = 10^5$**
- $G_m = 1 = 0dB$
:::

$$
T_s = 50 + 25 + \frac{500}{10^5} + \frac{1000}{10^5} \approx 75.0 K
$$

## Example #3

:::info
- $T_{in} = 25 K,\ T_{RF} = 50 K,\ T_m = 500 K,\ T_{IF} = 2000K,\ T_p = 300 K$
- **$G_{RF} = 50 dB = 10^5$**
- $G_m = 1 = 0dB$
- $G_l = 10 ^ (-0.2) \approx 0.631$
:::

$$
T_{wg} = T_p(1 - G_l) = 300 \times (1-0.631) = 110.7 K \\[1em]
\begin{split}
T'_s &= \frac{T_{in}}{L} + T_{wg} + T_{RF} + \frac{T_m}{G_{RF}} + \frac{T_{IF}}{G_{RF}G_m} \\[1em]
&\approx \frac{T_{in}}{L} + T_{wg} + T_{RF} = 176.5 K
\end{split}
$$

- wthout a lossy waveguide
$$
T_s = T_{in} + T_{RF} + \frac{T_m}{G_{RF}} + \frac{T_{IF}}{G_{RF}G_m} \approx T_{in} + T_{RF} = 75 K
$$


## Summary 

Low noise RF amplifiers (**LNA**) should have a high signal
- low noise component
- RF amplifier
    - can be up to x20 and over
