# Signals and Noise

AI and Robotics

Nov 24, 2013 (4 years and 5 months ago)

54 views

Chapter 5

Signals and Noise

Signal

carries

information

the

analyte

that

is

of

interest

to

us
.

Noise

is

up

of

extraneous

information

that

is

unwanted

because

it

the

accuracy

and

precision

of

an

analysis

Signal
-
to
-
Noise Ratio

S/N = (mean)/(Standard deviation) =

Signal
-
to
-
noise

(S/N)

is

much

more

useful

figure

of

merit

than

noise

alone

for

describing

the

quality

of

an

analytical

method
.

The

magnitude

of

the

noise

is

defined

as

the

standard

deviation

s

of

numerous

measurements

and

signal

is

given

by

the

mean

x

of

the

measurements
.

S/N

is

the

reciprocal

of

the

relative

standard

deviation
.

S/N

<

2

or

3

impossible

to

detect

a

signal
.

x
s
RSD

1

Figure 5
-
1

Figure 5
-
2

Sources of Noise

Analysis are affected by two types of noise:

1. Chemical noise

2. Instrumental noise

Chemical

noise
:

Arises

from

an

uncontrollable

variables

that

effect

the

chemistry

of

the

system

being

analyzed
.

Examples

are

undetected

variations

in

temperature,

pressure,

chemical

equilibria,

humidity,

light

intensity

etc
.

Instrumental

Noise
:

Noise

is

associated

with

each

component

of

an

instrument

i
.
e
.
,

with

the

source,

the

input

transducer,

signal

processing

elements

and

output

transducer
.

Noise

is

a

complex

composite

that

usually

cannot

be

fully

characterized
.

Certain

kinds

of

instrumental

noise

are

recognizable,

such

as
:

1
.

Thermal

or

Johnson

noise

2
.

Shot

noise

3
.

Flicker

or

1
/f

noise

4
.

Environmental

noise

Instrumental Noise

1
.

Thermal

Noise

or

Johnson

Noise
:

Thermal

noise

is

caused

by

the

thermal

agitation

of

electrons

or

other

charge

carriers

in

resistors,

capacitors,

transducers,

electrochemical

cells

and

other

resistive

elements

in

an

instruments
.

The

magnitude

of

thermal

noise

is

given

by

where,

rms

=

root

mean

square

noise,

f

=

frequency

band

width

(Hz),

k

=

Boltzmann

constant

(
1
.
38

x

10
-
23

J/K),

T

=

temperature

in

Kelvin,

R

=

resistance

in

ohms

of

the

resistive

element
.

Thermal

noise

can

be

decreased

by

narrowing

the

bandwidth,

by

lowering

the

electrical

resistance

and

by

lowering

the

temperature

of

instrument

components
.

rms
=
4kTR
f

Instrumental Noise

2
.

Shot

Noise
:

Shot

noise

is

encountered

wherever

electrons

or

other

charged

particles

cross

a

junction
.

Where,

i
rms

=

root
-
mean
-
square

current

fluctuation,

I

=

average

direct

current,

e

=

charge

on

the

electron

(
1
.
60

x

10
-
19

C),

f

=

band

width

of

frequencies
.

Shot

noise

in

a

current

measurement

can

be

minimized

only

by

reducing

bandwidth
.

i
=
2Ie
f
rms

3
.

Flicker

Noise
:

Flicker

noise

is

characterized

as

having

a

magnitude

that

is

inversely

proportional

to

the

frequency

of

the

signal

being

observed
.

It

is

sometimes

termed

1
/f

(one
-
over
-
f)

noise
.

The

cause

of

flicker

noise

are

not

well

understood

and

is

recognizable

by

its

frequency

dependence
.

Flicker

noise

becomes

significant

at

frequency

lower

than

100

Hz
.

Flicker

noise

can

be

reduced

significantly

by

using

wire
-
wound

or

metallic

film

resistors

rather

than

the

more

common

carbon

composition

type
.

4
.

Environmental

Noise
:

Environmental

noise

is

a

composite

of

different

forms

of

noise

that

arise

from

the

surroundings
.

Much

environmental

noise

occurs

because

each

conductor

in

an

instrument

is

potentially

an

antenna

capable

of

picking

up

electromagnetic

and

converting

it

to

an

electrical

signal
.

Figure 5
-
3

Signal
-
to
-
Noise

Enhancement
:

When

the

need

for

sensitivity

and

accuracy

increased,

the

signal
-
to
-
noise

ratio

often

becomes

the

limiting

factor

in

the

precision

of

a

measurement
.

Both

hardware

and

software

methods

are

available

for

improving

the

signal
-
to
-
noise

ratio

of

an

instrumental

method
.

Hardware

method
:

Hardware

noise

reduction

is

accomplished

by

incorporating

into

the

instrument

design

components

such

as

filters,

choppers,

shields,

modulators,

and

synchronous

detectors
.

These

devices

remove

or

attenuate

the

noise

without

affecting

the

analytical

signal

significantly
.

Hardware

devices

and

techniques

are

as

follows
:

1
.

Grounding

and

Shielding
:

Noise

that

arises

from

environmentally

generated

electromagnetic

can

be

substantially

reduce

by

shielding,

grounding

and

minimizing

the

length

of

conductors

within

the

instrumental

system
.

2
.

Analog

Filtering
:

By

using

low
-
pass

and

high
-
pass

analog

filters

S/N

ratio

can

be

improved
.

Thermal,

shot

and

flicker

noise

can

be

reduced

by

using

analog

filters
.

3
.

Modulation
:

In

this

process,

low

frequency

or

dc

signal

from

transducers

are

often

converted

to

a

higher

frequency,

where

1
/f

noise

is

less

troublesome
.

This

process

is

called

modulation
.

After

amplification

the

modulated

signal

can

be

freed

from

amplifier

1
/f

noise

by

filtering

with

a

high
-
pass

filter,

demodulation

and

filtering

with

a

low
-
pass

filter

then

produce

an

amplified

dc

signal

suitable

for

output
.

4
.

Signal

chopping
:

In

this

device,

the

input

signal

is

converted

to

a

square
-
wave

form

by

an

electronic

or

mechanical

chopper
.

Chopping

can

be

performed

either

on

the

physical

quantity

to

be

measured

or

on

the

electrical

signal

from

the

transducer
.

5
.

Lock
-
in
-
Amplifiers
:

Lock
-
in
-
amplifiers

permit

the

recovery

of

signals

even

when

the

S/N

is

unity

or

less
.

It

requires

a

reference

signal

that

has

the

same

frequency

and

phase

as

the

signal

to

be

amplified
.

A

lock
-
in

amplifier

is

generally

relatively

free

of

noise

because

only

those

signals

that

are

locked
-
in

to

the

reference

signal

are

amplified
.

All

other

frequencies

are

rejected

by

the

system
.

Software

Method
:

Software

methods

are

based

upon

various

computer

algorithms

that

permit

extraction

of

signals

from

noisy

data
.

Hardware

convert

the

signal

from

analog

to

digital

form

which

is

then

collected

by

computer

equipped

with

a

data

acquisition

module
.

Software

programs

are

as

follows
:

1
.

Ensemble

Averaging
:

In

ensemble

averaging,

successive

sets

of

data

stored

in

memory

as

arrays

are

colleted

and

summed

point

by

point
.

After

the

collection

and

summation

are

complete,

the

data

are

averaged

by

dividing

the

sum

for

each

point

by

the

number

of

scans

performed
.

The

signal
-
to
-
noise

ratio

is

proportional

to

the

square

root

of

the

number

of

data

collected
.

Figure 5
-
9

2
.

Boxcar

Averaging
:

Boxcar

averaging

is

a

digital

procedure

for

smoothing

irregularities

and

enhancing

the

signal
-
to
-
noise

ratio
.

It

is

assumed

that

the

analog

analytical

signal

varies

only

slowly

with

time

and

the

average

of

a

small

number

of

points

is

a

better

measure

of

the

signal

than

any

of

the

individual

points
.

In

practice

2

to

50

points

are

averaged

to

generate

a

final

point
.

This

averaging

is

performed

by

a

computer

in

real

time,

i
.
e
.
,

as

the

data

is

being

collected
.

Its

utility

is

limited

for

complex

signals

that

change

rapidly

as

a

function

of

time
.

Figure 5
-
11

3
.

Digital

filtering
:

Digital

filtering

can

be

accomplished

by

number

of

different

well
-
characterized

numerical

procedure

such

as

(a)

Fourier

transformation

and

(b)

Least

squares

polynomial

smoothing
.

(a)

Fourier

transformation
:

In

this

transformation,

a

signal

which

is

acquired

in

the

time

domain,

is

converted

to

a

frequency

domain

signal

in

which

the

independent

variable

is

frequency

rather

than

time
.

This

transformation

is

accomplished

mathematically

on

a

computer

by

a

very

fast

and

efficient

algorithm
.

The

frequency

domain

signal

is

then

multiplied

by

the

frequency

response

of

a

digital

low

pass

filter

which

remove

frequency

components
.

The

inverse

Fourier

transform

then

recovers

the

filtered

time

domain

spectrum
.

(b)

Least

squares

polynomial

data

smoothing
:

This

is

very

similar

to

the

boxcar

averaging
.

In

this

process

first

5

data

points

are

averaged

and

plotted
.

Then

moved

one

point

to

the

right

and

averaged
.

This

process

is

repeated

until

all

of

the

points

except

the

last

two

are

averaged

to

produce

a

new

set

of

data

points
.

The

new

curve

should

be

somewhat

less

noisy

than

the

original

data
.

The

signal
-
to
-
noise

ratio

of

the

data

may

be

enhanced

by

increasing

the

width

of

the

smoothing

function

or

by

smoothing

the

data

multiple

times
.