Signals and Noise

Sept 5, 2002

Announcements

• Homework-Chapter 2, Problems 2, 6, 8, 12, 16, 18• Recommended Problems: 13, 15, 23• Email: ebonilla@gmu.edu• TA: Mahdu Rangarajan, mrangara@gmu.edu• Course URL

http://teal.gmu.edu/ececourses/tcom500_2/lectures.html

• Reading Assignment for next week: • Chapter 3

• Chapter 4, Sections 4.1-4.4

Class Objectives

• Review– Fundamentals of Signals– Mathematical concepts

• Signal-to-Noise Ratio

• Noise Figure

• Bit Error Rate

• Channel Capacity

Fundamentals of Electric Signals

• Electrical signals are created by the flow of electrons

• Electrons flow from high charge potential to lower charge potential (EMF)

• Circuits are conductive paths that direct the flow of electrons

Frequency

• The number of complete cycles of sinusoidal variation per unit time

• 1 Cycle per second = 1 Hertz = 1 Hz

• 1000 cycles per second = 1000 Hz = 1kHz

• 1,000,000 cycles per second = 1 MHz

• 1,000,000,000 Hz = 1 GHz

Frequencies 1Hz, 2, 10, 20 Hz

Sine Wave

• Peak Amplitude (A)– maximum strength of signal– volts

• Frequency (f)– Rate of change of signal– Hertz (Hz) or cycles per second– Period = time for one repetition (T)– T = 1/f

• Phase ()– Relative position in time

PeriodicSignals

Frequencies

• Acoustic frequencies:– human speech: 100 Hz to 7 kHz– Music: up to 20 kHz– ultrasounds: above 20 KHz to 1 MHz

• Electromagnetic carrier frequencies:– AM radio broadcast (example) 710 kHz– FM broadcast 89 MHz- 108 MHz– TV broadcasting 150 MHz- 900 MHz– Cellular telephony ~ 1 GHz

Acoustic Spectrum

MusicSpeech

TelephoneChannel

10 Hz 100 Hz 1k Hz 10 kHz 100 kHz

0 dB

-20 dB

-40 dB

-60 dB

Frequency

Po

we

r R

ati

o

300 Hz 3k Hz

Continuous & Discrete Signals

Frequency Domain Concepts

• Signal can be made up of many frequencies

• Components are sine waves

• Frequency domain functions can be plotted

• Changes in the time domain affects the signal in the frequency domain

Addition of Frequency Components

Time-Frequency Domain

Spectrum & Bandwidth

• Spectrum– range of frequencies contained in signal

• Absolute bandwidth– width of spectrum

• Effective bandwidth– Often just bandwidth– Narrow band of frequencies containing most of the

energy

• DC Component– Component of zero frequency

Signal with DC Component

Data Rate and Bandwidth

• All transmission systems have a limited band of frequencies

• Data Rate: the amount of data that is transmitted in a unit time (one second)

• Limited bandwidth results in limited data rate

Analog and Digital Data Transmission

• Data – Entities that convey meaning

• Signals– Electric or electromagnetic representations of

data

• Transmission– Communication of data by propagation and

processing of signals

Data

• Analog– Continuous values within some interval– e.g. sound, video

• Digital– Discrete values– e.g. text, integers

Signals

• Means by which data are propagated• Analog

– Continuously variable– Various media

• wire, fiber optic, space

– Speech bandwidth 100Hz to 7kHz– Telephone bandwidth 300Hz to 3400Hz– Video bandwidth 4MHz

• Digital– Use discrete components (mostly two DC components)

Analog Transmission Characteristics

• Analog signal transmitted without regard to content

• Vulnerable to noise

• Attenuated over distance

• Use amplifiers to boost signal (& noise)

Digital Transmission Characteristics

• Content sensitive• Integrity endangered by noise, attenuation etc.• Repeaters are used to regenerate signal

– Extracts bit pattern

– Retransmits

– Attenuation is overcome

– Noise is not amplified

Advantages of Digital Transmission• Digital technology

– Low cost LSI/VLSI technology

• Data integrity– Longer distances are possible– Error correction

• Capacity utilization– High bandwidth links economical due to efficiency– High degree of multiplexing easier with digital techniques

• Security & Privacy– Encryption

• Integration– Can treat analog and digital data similarly

Transmission Impairments

• Transmitted signal must be compatible with receiver

• Analog - degradation of signal quality• Digital - bit errors• Signal degradation causes include:

– Attenuation and attenuation distortion– Delay distortion– Noise– Interference

Attenuation

• Signal strength falls off with distance• Depends on medium• Attenuation is an increasing function of

frequency• Received signal strength:

– must be enough to be detected– must be sufficiently higher than noise to be

received without error

Attenuation & Delay Distortion

• Attenuation varies with frequency– Certain frequencies are attenuated more than

others

• Propagation velocity varies with frequency– Some frequencies arrive earlier than others,

results in modifying the phase

Noise & Interference

• Unwanted signals inserted between transmitter and receiver

• Thermal– Due to thermal agitation of electrons– Uniformly distributed (White noise)

• Intermodulation– Signals that are the sum and difference of

original frequencies sharing a medium

• Interference– Identifiable, man-made noise

Noise & Interference (cont)

• Crosstalk– A signal from one line is picked up by another

• Impulse– Irregular pulses or spikes– e.g. External electromagnetic interference– Short duration– High amplitude

Effects of Noise on a Signal

Gaussian Noise Distribution

Crosstalk

Break

Topics – 2nd Half

• Decibels

• Signals and Noise

• Signal-to-noise ratio

• BER, Channel capacity

• Noise types

Electric circuits

http://www.fclabs.com.au/onlinesamples/ec/elect01/els01dx.htm

Animation of Ohm's law

http://www.phy.ntnu.edu.tw/java/rc/rc.html

Animation of an RLC circuit with AC currents http://www.phy.ntnu.edu.tw/java/rlc/rlc.html

Animation of a clipping circuit

http://www.phy.ntnu.edu.tw/java/electronics/clip_e.html

Animation of Fourier synthesis of oscillatory signals http://www.phy.ntnu.edu.tw/java/sound/sound.html

Propagation of Electromagnetic Wave http://www.phy.ntnu.edu.tw/java/emWave/emWave.html

On-line resources for basic concepts of ElectricityCourtesy of Dr. Manitius

Electric PowerPower can be expressed in several ways:

P = VI = I2R = V2/R

If an electric current flows through a resistance R then the power, expressed as I2R, is dissipated (loss) as heat.

rms = root-mean-square value

If voltage V varies in time, then the average power dissipated is proportional to its rms value.

1 2 3 4 5 6 7 8 9 10-5

0

5Voltage V

1 2 3 4 5 6 7 8 9 10-10

-5

0

5

10

15

20V squared

mean-square V

rms value V(rms)

Quantity of Interest: Power Gain/Loss = Ap

Ap = (Output Signal Power)/(Input signal power)

Signal Power Gain/Loss

Signal Input Signal Output

Amplifier

Relative Power Gain

Pinput = Pi = Input Power

Poutput = P0 = Output Power

Ap = Relative Power Gain = P0/Pi

Decibels

• Decibel, dB, is a measure of a relative amplitude (or power) of a signal, be it acoustic or electric

• The term “relative” means that we are measuring the ratio of the given amplitude to another amplitude, for example the ratio of the amplitude (Volts) at the end of the phone line to the amplitude (Volts) at the beginning of the phone line

• In acoustics, the decibel is a measure of the relative level of sound or noise compared to some standardized level of sound or noise. One decibel (0.1 bel) equals 10 times the logarithm of the power ratio of the given sound to the power of the sound barely perceptible by human ear.

Decibels in Acoustics

Sounds Decibel level

Barely audible 0 dB

Whisper 20 dB

Conversation 30 db-50 dB

County level limit 70 dB

Big truck 110 dB

Log Base 10

10x = yFor a given number y (y>0), x is the exponent of 10 that makes 10x equal to y For example:102 = 100 means that 2 is log10(100)

103 = 1000 means that 3 is log10(1000)

104 = 1000 means that 4 is log10(10000)

10-2 = 0.01 means that -2 is log10(0.01)

10-3 = 0.001 means that -3 is log10(0.001)

Linear vs. Logarithmic

Decibels – Power Gain/Loss

A AP

P

P

P

sim plified exam ple

Amplifier has ou tpu t pow er W

and inpu t pow er mW

AP

PdB

P dB Po

i

o

i

P dBo

i

( )

( )

lo g lo g lo g

:

lo g lo g.

lo g

1 0 1 0 1 0

3

3

1 0 1 03

0 0 0 31 0 1 0 0 0 3 0

1 0 1 0

Power Gain/Loss Examples

Let’s do Example 2.1 and others.

Decibels - Voltage

A AV

V

exam ple

repea ter has ou tpu t vo ltage V

and inpu t pow er mV

AV

VdB

V dB Vo

i

V dBo

i

( )

( )

lo g lo g

.

lo g lo g.

lo g . .

2 0 2 0

2 5

2

1 9 5

2 0 2 02

0 1 9 52 0 1 0 2 6 2 0 2

Network element

Signal InputSignalOutput

dB value = 10 log (Output power/Input Power)

dB value = 20 log (Output voltage/ Input Voltage)

e.g. power gain = 2 implies dB value = 10 log 2 = +3.01 dB

e.g. Attenuators can have a fixed values (10, 20, 30 dB)

e.g. Wire AWG 24 produces loss of 2.127 dB/km or 2.16 dB/mile

Decibels

.

P P inpu t pow er mW one m iliW att

P P outpu t pow er in Watts

abso lu te pow er ga inP

mW

AP

mWmeasured in dBm

inpu t i

ou tpu t o

o

P dBmo

1

1

1

( )

,

,( )

dBm - Absolute Power Gain

dBm Examples

Lets do Examples 2.7 and 2.8 with different values.

Key dB Values

Ratio dB1000 30100 2010 102 31 0

0.5 -30.1 -10

0.01 -200.001 -30

Miliwatts and dBm

Miliwatts dBm1000 30100 2010 102 31 0

0.5 -30.1 -100.01 -20

0.001 -30

Linear vs. Logarithmic

SNRSignal-to-Noise Ratio

•Every receiver has a SNR requirement that must be met for error free reception

•Quantifies the “quality” of the received signal

•SNR = 10 log(Signal Power/Noise Power)

•SNR = 20 log(Signal Voltage/Noise Voltage)

SNR Example

A sinusoid signal has a peak-to-peak amplitude of 3 VThere is noise which has a rms value of 640 mVWhat is the SNR?

Let us try to visualize what is going on with the signal and noise.(solve problem and view next 5 slides)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-3

-2

-1

0

1

2

3

time in miliseconds

Vol

ts

Signal

peak 1.5 V

negative peak 1.5V

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-3

-2

-1

0

1

2

3

time in miliseconds

Vol

tsNoise

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-3

-2

-1

0

1

2

3

time in miliseconds

Vo

lts

Signal plus noise

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-3

-2

-1

0

1

2

3

square of the signal

originalsignal

the mean-square value of the signal = 1.125rms = 1.0607

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-3

-2

-1

0

1

2

3

time in miliseconds

Vol

ts (

squ

ared

)

Noise squared and its mean-square value

Mean-square value of the noise =0.41 rms = square root of 0.41 = 0.64

•If SNR is positive, it means the desired signal has greater power than noise. This is good.

•If SNR is negative, it means the desired signal has smaller power than noise. This is bad.

•SNR = 0 means signal and noise have equal power. Not good.

•The higher the SNR, the better for communications. Different systems have different minimum SNR requirements in order to work properly. For example a typical residential phone line has SNR 24 dB to 30dB

Important SNR Concepts

Noise Factor

•Noise Factor: a measure of how noisy a device is

•Ratio of SNR at the input to SNR at the output

•If the device does not add noise to the signal, then the ratio is 1

•Noise Figure is Noise Factor in dB

Noise Sources

• Thermal– Due to thermal agitation of electrons– Uniformly distributed, white noise

• From other signals (Interference)– Signals that are combinations of original

frequencies sharing a medium

Noise Sources (cont)

• Cross-talk– A signal from one line is picked up by another

• Impulse disturbance or impulse noise– Irregular pulses or spikes of short duration and

high amplitude– electromagnetic interference

BER = Bit Error Rate

BER = fraction of bits that are in error (incorrectly transmitted)

E X A M P L E

B E R

1 0

1

1 0

1

6

6

means tha t

m illionone ou t o f a m illion b its

is transm itted w ith error on average( )

BER & Reliability

BER = 1 – Reliability

BER Reliability10-3 99.9%10-4 99.99%10-5 99.999%10-6 99.9999%

Characterizations of Channel Capacity

• Data rate– Bps, In bits per second– Sps - In bauds: symbols per second

• Bandwidth– In cycles per second or Hertz– Bandwidth is limited by electronics and

medium

Shannon’s Law

C = BW log2(1+ S/N) (bps)

BW = bandwidth (Hz)

S/N = Signal-to-Noise Ratio (SNR) (not in dB)

C = channel capacity (bps)

The underlying assumption is that the noise is Gaussian.

Example

Textbook Example 2.13 plus additional exercise

Note: Most calculators work with log10 instead of log2

which is used in Shannon’s Law. To get the correct answer you can multiply your calculator’s answer by a conversion factor of 3.32 and this will give you the log2 answer.

Discussion Exercise

"How many pairs of people can hold conversations simultaneously in a closed room before the background noise becomes too great"?

Explain the key variables that are involved in determining a solution. How would you estimate a solution based on these variables?

Suggestion: Use Signal-to-Noise as your starting point.