Noise and dB

Covered topics

• AWGN

• complex noise

• SNR/SINR

• dB

Gaussian noise

noise (signal processing)

Unwanted modifications that a signal may suffer.

Fig. 47 A noise signal. The average is zero and higher fluctuations are more rare than the lower ones.

Source M. Lichtman | License: CC BY-NC-SA 4.0

If the average would not be zero, then we have an easy solution for the average part: Just subtracting the average value. Then we would be left out with the rest again.

When many random things add up, then we get Gaussian noise.

Fig. 48 Many kinds of random processes added up leads to a Gaussian distribution.

Source M. Lichtman | License: CC BY-NC-SA 4.0

Also called normal distribution or bell curve.

Has two parameters:

• mean

• variance

mean \(\mu\)

average value of the distribution.

variance \(\sigma^2\)

standard deviation \(\sigma\) of the distribution squared. It is a metric that represents how much the values deviate from the average.

For example, if both the mean and variance are 1, approximately 68% of the values will be between 0 and 2 (mean ± 1 standard deviation).

noise power

represents how strong the noise is. Defined by the variance \(\sigma^2\). We use a square, because power is proportional to the amplitude of a signal: \(P = U^2 R\) (power equals to the square of the voltage multiplied by the resistance).

Exercise 85

Where could be the Gaussian noise helpful for us in context of wireless communication?

Exercise 86

You want to add Gaussian noise to your radio signal in GNU Radio to model system noise. You choose a mean value of 2 and a variance of 4. Your signal has the following values:

signal = [-5, -4, -3, -2, -1, 0, 1, 2, 3, 4]

Give an example output. For simplicity, stick to integer numbers.

Energy & power & duty cycle

energy

a quantitative property that is transferred, recognizable as work, heat, and light. Unit: J

In small batteries, we use mAh, i.e., milliampere hours. E.g., if a battery has a capacity of 1000 mAh, then it can deliver 1 A for one hour.

power

the amount of energy transferred or converted per unit time. Unit: W

Power corresponds to voltage times current: \(P = V \cdot I\)

duty cycle

the fraction of one period in which a signal or system is active.

Fig. 49 Duty cycle \(D\) varied in 10% increments. The signal is active, when it is high.
Public domain. By Eighthave, modified by Teslaton. Source: Wikimedia Commons

Exercise 87

How long can a typical rechargeable AAA battery feed the Bluetooth sensor with the following properties:

• sensor consists of two main components: processor and transceiver.

• transmission power of 1 mW.

• device gets activated every second and transmits data, but not receives any.

• the transmitter has a duty cycle of 5 %.

• the processor on the sensor has an average power consumption of 5 mW when it is active.- outside the duty cycle, the device is in sleep mode and consumes 1 μW.

Decibel (dB)

Imagine you have to compare a signal at 2W with the noise floor at 0.0000002 W. The ratio contains many zeroes, but a logarithmic metric can squeeze these zeroes.

decibel (dB)

relative unit of measurement equal to one tenth of a bel (B). The comparison is done on a logarithmic scale and the compared values are power or root-power quantities.

Fig. 50 Waterfall plot of three signals in linear and logarithmic scale. There are more details on the logarithmic scale.

Source M. Lichtman | License: CC BY-NC-SA 4.0

Decibel conversion:

• a signal power ratio: \(x_{\mathrm{dB}} = 10 \log_{10}x\)

  • 10 comes from deci

• a signal amplitude ratio: we use 20 instead of 10

  • reason: decibel compares in the power domain. If we compare amplitudes, we have to square additionally the ratio we get: \(x^2 = 20 \log_{10}x\).

In the decibel world, multiplications become additions:

Fig. 51 Three amplification values get multiplied together, but their decibel counterparts get added.

Source M. Lichtman | License: CC BY-NC-SA 4.0

dB is not a unit, it is just a factor represented in a domain we don’t usually use, namely logarithmic domain. Sometimes we append a unit, then it becomes relative to the lowest integer value of this unit:

• dBW is relative to 1 W. 2W becomes \(10 \log_{10} 2 ~= 3 \mathrm{dB}\).

• dBmW or shortly dBm is relative to 1 mW.

Transmitter is set to 3 dBW.

Means to 2 W, even 2x corresponds to ≈3.01 dB.

The signal I received was 10 dB above the noise floor.

Means SNR = 10 dB

Linear	dB
1x	0 dB
2x	3 dB
10x	10 dB
0.5x	-3 dB
0.1x	-10 dB
100x	20 dB
1000x	30 dB
10000x	40 dB

dBm examples from Wikipedia:

dBm	Notes
526	an example black hole collision
296	power output of the sun
80	Typical transmission power of FM radio station with 50 km range
60	Typical combined radiated RF power of microwave oven elements
27	Typical cellular phone transmission power
23	WLAN 5 GHz
15	Typical wireless LAN transmission power in laptops
4	Bluetooth Class 2 radio, 10 m range
0	Bluetooth Class 3 radio, 1 m range
-10	Maximal received signal power of wireless network (802.11 variants)
-100	Minimal received signal power of wireless network (802.11 variants)
-111	Thermal noise floor for commercial GPS single-channel signal bandwidth (2 MHz)
-127.5	Typical received signal power from a GPS satellite
-174	Thermal noise floor for 1 Hz bandwidth at room temperature (20 °C)

Decibel allows us to describe such a vast range from a thermal noise to a black hole collision! ↕️

Exercise 88

In the following you have transmission power data for various devices:

1. A Bluetooth class 3 based sensor with 1 m range: -10 dBm.

2. A mobile phone radio: 0.5 W.

3. You laptop Wi-Fi: 32 mW.

Calculate for each case transmission power in dBW and W. You don’t have to use a calculator – approximate values are sufficient.

Noise in the frequency domain

Gaussian noise in the time domain is also Gaussian noise in the frequency domain.

Fig. 52 Top: Gaussian noise in the time domain. Bottom: Gaussian noise in the frequency domain.

Source M. Lichtman | License: CC BY-NC-SA 4.0

In the frequency domain we use dB, so top and bottom do not look the same.

Complex noise

noise power is split between the real and imaginary portions, i.e., amplitude and phase.

If we apply Gaussian noise on the real and imaginary parts:

Fig. 53 IQ-plot of Gaussian noise applied on I and Q.

Source M. Lichtman | License: CC BY-NC-SA 4.0

If we apply on a QPSK constellation:

Fig. 54 a QPSK constellation on the left. With noise applied on the right. AWGN stands for additive white Gaussian noise

Source M. Lichtman | License: CC BY-NC-SA 4.0

Increasing the power of the noise:

Fig. 55 QPSK constellations with different noise powers.

Source M. Lichtman | License: CC BY-NC-SA 4.0

At noise power

• 0.3: we are able to differentiate in general

• 0.6: there is no clear separation, so decoding is more difficult. Error rate will be higher compared to the case with 0.3 noise power.

AWGN

additive white Gaussian noise

• additive: because we add the noise

• white: spectrum is flat, i.e., noise applies to all frequencies equally

We can typically assume AWGN when we are talking about noise

SNR

signal-to-noise ratio (SNR)

measure that compares the level of a desired signal to the level of background noise.

\(\mathrm{SNR} = \frac{P_\mathrm{signal}}{P_\mathrm{noise}} \)

In the dB domain, we just subtract the values.

Examples:

• 0 dB: signal and noise power are the same

• positive: signal has a higher power than noise

• negative: it is difficult to detect, however not impossible. Refer to the GPS signal levels on the previous table.

Exercise 89

Your laptop is connected to a Wi-Fi station and the signal you receive has the following power:

$ iw dev wlp0s20f3 station dump | grep 'signal avg'
	signal avg:	-45 dBm
	beacon signal avg:	-45 dBm

1. What is the SNR assuming a noise power of -90 dBm?

2. Is the received signal power acceptable?

Appendix

SINR

Noise comes from two sources:

1. thermal noise in the receiver

2. interference from other radio signal sources

Signal-to-interference-plus-noise ratio

Compared to SNR, also interference or disturbance from other signal sources are accounted

Fig. 56 The base station wants to receive the green signal S, but is interfered by the red signals.
CC BY-SA 4.0. By Kirlf. Source: Wikimedia Commons