Noise and SNR

2

Noiseunwanted signals inserted between transmitter and receiver

is the major limiting factor in communications system performance

3

• Noise can broadly be defined as any unknown signal that affects the recovery of the desired signal.

• The received signal is modeled as

s(t) is the transmitted signal

w(t) is the additive noise

)1.9()()()( twtstr

4

Categories of NoiseThermal noise

• due to thermal agitation of electrons

• uniformly distributed across bandwidths

• referred to as white noiseIntermodulation noise

• produced by nonlinearities in the transmitter, receiver, and/or intervening transmission medium

• effect is to produce signals at a frequency that is the sum or difference of the two original frequencies

5

Categories of NoiseCrosstalk:

– a signal from one line is picked up by another

– can occur by electrical coupling between nearby twisted pairs or when microwave antennas pick up unwanted signalsImpulse Noise:

– caused by external electromagnetic interferences

– noncontinuous, consisting of irregular pulses or spikes

– short duration and high amplitude

– minor annoyance for analog signals but a major source of error in digital data

6

Thermal Noise

• Thermal noise known as white noise. Noise is assumed to be independent of frequency, uniformly distributed spectrally from 0 to about 1013 Hz.

• Thermal noise, its energy increase with temperature.• The noise voltage varies in time with a Gaussian probability

distribution function and mean value of zero.

Power spectral density (PSD) of thermal noise

7

Thermal Noise (Cont)• The noise power density (amount of

thermal noise to be found in a bandwidth of 1Hz in any device or conductor) is:

W/Hz k0 TN N0 = noise power density in watts per 1 Hz of bandwidthk = Boltzmann's constant = 1.3803 10-23 J/KT = temperature, in kelvins (absolute temperature)

0oC = 273 Kelvin

8

Thermal Noise (cont)

• Because of the weakness of the signal received by satellite earth stations, thermal noise is particularly significant for satellite communication.

• Thermal noise power present in a bandwidth of B Hertz (in watts):

or, in decibel-watts (dBW),

BTN log10 log 10k log10

BT log10 log 10dBW 6.228

TBN k

9

Other noises• Intermodulation noise – occurs if signals with

different frequencies share the same medium– Interference caused by a signal produced at a

frequency that is the sum or difference of original frequencies

• Crosstalk – unwanted coupling between signal paths

• Impulse noise – irregular pulses or noise spikes– Short duration and of relatively high amplitude– Caused by external electromagnetic

disturbances, or faults and flaws in the communications system

10

Signal-to-Noise Ratios

• The desired signal, s(t), a narrowband noise signal, n(t)

• Signal-to-noise ratio is defined by

• The signal-to-noise ratio is often considered to be a ratio of the average signal power to the average noise power.

)()()( tntstx

)]([E

)]([ESNR

2

2

tn

ts

11

Noise in Digital Communications

• Two strong external reasons for the increased dominance of digital communication The rapid growth of machine-to-machine

communications. Digital communications gave a greater

noise tolerance than analogue.• Broadly speaking, the purpose of detection is

to establish the presence of an information-bearing signal in noise.

Bit Error Rate (BER)• Let n denote the number of bit errors observed in a

sequence of bits of length N; then the relative frequency definition of BER is

• BER and Packet error rate (PER)speech, a BER of 10-2 to 10-3 is sufficient.data transmission over wireless channels, a bit

error rate of 10-5 to 10-6 is often the objective. video transmission, a BER of 10-7 to 10-12 is often

the objective.financial data, a BER of 10-11or better is often the

requirement. 12

N

nNlimBER

SNR in digital systems– The ratio of the modulated energy per information bit to the

one-sided noise spectral density; namely,

1. The analogue definition was a ratio of powers. The digital definition is a ratio of energies.

2. The definition uses the one-sided noise spectral density; that is, it assumes all of the noise occurs on positive frequencies. This assumption is simply a matter of convenience.

3. The reference SNR is independent of transmission rate. Since it is a ratio of energies, it has essentially been normalized by the bit rate.

13

0

digitalref density spectral Noise

bitper energy ModulatedSNR

N

Eb

14

15

Nyquist Bandwidth

In the case of a channel that is noise free:• if rate of signal transmission is 2B then can carry

signal with frequencies no greater than B – given bandwidth B, highest signal rate is 2B

• for binary signals, 2B bps needs bandwidth B Hz• can increase rate by using M signal levels

• Nyquist Formula is: C = 2B log2M

• data rate can be increased by increasing signals– however this increases burden on receiver– noise & other impairments limit the value of M

16

Channel Capacity

Maximum rate at which data can be transmitted over a given communications channel under given conditions

data rate

in bits per second

bandwidth

in cycles per

second or Hertz

noise

average noise level over path

error rate

rate of corrupted

bits

limitations due to

physical properties

main constraint

on achieving efficiency is noise

17

Shannon Capacity Formula

• considering the relation of data rate, noise and error rate:– faster data rate shortens each bit so bursts of noise

corrupts more bits– given noise level, higher rates mean higher errors

• Shannon developed formula relating these to signal to noise ratio (in decibels)

• SNRdb=10 log10 (signal/noise)

• capacity C = B log2(1+SNR)– theoretical maximum capacity– get much lower rates in practice

Signal to Noise Ratio – SNR (1)

• Ratio of the power in a signal to the power contained in the noise present at a particular point in the transmission.

• Normally measured at the receiver with the attempt to eliminate/suppressed the unwanted noise.

• In decibel unit,

where PS = Signal Power, PN = Noise Power

• Higher SNR means better quality of signal.

N

SdB P

P1010logSNR

Signal to Noise Ratio – SNR (2)

• SNR is vital in digital transmission because it can be used to sets the upper bound on the achievable data rate.

• Shannon’s formula states the maximum channel capacity (error-free capacity) as:

– Given the knowledge of the receiver’s SNR and the signal bandwidth, B. C is expressed in bits/sec.

• In practice, however, lower data rate are achieved.• For a fixed level of noise, data rate can be increased by

increasing the signal strength or bandwidth.

SNR1log2 BC

Expression of Eb/N0 (1)

• Another parameter that related to SNR for determine data rates and error rates is the ratio of signal energy per bit, Eb to noise power density per Hertz, N0; → Eb/N0.

• The energy per bit in a signal is given by:– PS = signal power & Tb = time required to send one bit which can be

related to the transmission bit rate, R, as Tb = 1/ R.

• Thus,

• In decibels:

bSb TPE

TR

P

N

RP

N

E SSb

k

/

00

dB

b

N

E

0

TRP dBS 101010)( 10logk10log10log

– 228.6 dBW

Expression of Eb/N0 (2)

• As the bit rate R increases, the signal power PS relative to the noise must also be increased to maintain the required Eb/N0.

• The bit error rate (BER) for the data sent is a function of Eb/N0 (see the BER versus Eb/N0 plot).

• Eb/N0 is related to SNR as:

R

BSNR

R

B

P

P

N

E

N

Sb

0

BER versus Eb/N0 plot

where B = Bandwidth, R = Bit rate

Higher Eb/N0, lower BER

22

Definition of Q(x)

x

dssxQ )2/exp(2

1)( 2

23

Performance comparison

10_16

25This example is printed on your tutorial sheet.

26

𝑆𝑁𝑅= 𝐸[𝑆2ሺ𝑡ሻ]𝐸[𝑛2ሺ𝑡ሻ] (9.6)