5.3. noise characteristics

1

5.3. Noise characteristics

Reference: [4]

The signal-to-noise ratio is the measure for the extent to which

a signal can be distinguished from the background noise:

SNR SN

where Sin is the signal power, and Nin is the noise power.

5.3.1. Signal-to-noise ratio, SNR

5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.1. Signal-to-noise ratio, SNR

2

SNRin Sin

Nin


A. Signal-to-noise ratio at the input of the system, SNRin

It is usually assumed that the signal power, Sin, and the noise

power, Nin, are dissipated in the noiseless input impedance of

the measurement system.

Measurement object Measurement system

vS

ZS=RS + jXS

Zin=Rin + jXin

Noiseless

RL

SNRin


Example: Calculation of SNRin

1) Sin ,VS

2 Zin

(ZS + Zin)2 2) Nin

,

Vn 2 Zin

(ZS + Zin)2

3) SNRin VS

2

Vn 2

VS 2

4 k T RS


vS

ZS=RS + jXS

Zin=Rin + jXin

Noiseless

RL

SNRin

Note that SNRin is not a function of Zin.

4

SNRo srcSNRin

SNRo src SNRin



vS Power gain, Ap

Noiseless

SNRo src So

No src

ZS=RS + jXS

1) The measurement system is noiseless.

Sin Ap

Nin Ap

Sin

Nin

RL

B. Signal-to-noise ratio at the output of the system, SNRo

5

SNRo

SNRo SNRin



vS Power gain, Ap

Noisy

SNRo So

No

ZS=RS + jXS

2) The measurement system is noisy.

SNRin

Sin Ap

(Nin+Nin msr) Ap

Sin

Nin

RL

6

SNRo

Noise factor is used to evaluate the signal-to-noise degradation

caused by the measurement system (H. T. Friis, 1944).

5.3.2. Noise factor, F, and noise figure, NF

5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF

F SNRin

SNRo


vS Power gain, Ap

NoisyZS=RS + jXS SNRin

RL

7

SNRoSNRin

The signal-to-noise degradation is due to the additional noise,

No msr , which the measurement system contributes to the load.



vS Power gain, Ap

NoisyZS=RS + jXS

F SNRin

SNRo

SNRo src

SNRo

So /No src

So /No

No

No src

No src + No msr

No src

RL

No msr

No src


RS


F No

No src

Vno2/RL

4 kTRS B )GV AV(2 /RL

Vno2

4 kTRS B )GV AV(2

vovin

Example: Calculation of noise factor

Voltage gain, AVRL

enS GV

Here and below, we assume that the reactance in the source

output impedance is compensated by the properly chosen input

impedance of the measurement system (noise tuning).


F Vno

2

4 kTRS B )G AV(2

The following three characteristics of noise factor can be seen

by examining the obtained equation:

1. It is independent of the load resistance RL,

2. It does depend on the source resistance RS,

3. If the measurement system were completely noiseless,

the noise factor would equal one.

Reference: [2]

Conclusions:


Noise factor expressed in decibels is called noise figure )NF(:

NF 10 log F

Due to the bandwidth term in the denominator

there are two ways to specify the noise factor: (1) a spot noise,

measured at specified frequency over a 1Hz bandwidth, or (2)

an integrated, or average noise measured over a specified

bandwidth.

C. Noise figure

F Vno

2

4 kTRS B )G AV(2

Reference: [2]


Reference: [2]

We will consider the following methods for the measurement of

noise factor: (1) the single-frequency method, and (2) the white

noise method.

E. Measurement of noise factor

1) Single-frequency method. According to this method, a

sinusoidal test signal vS is increased until the output power

doubles. Under this condition the following equation is satisfied:

RS


vS

vovin

Voltage gain, AvGv

RL


Reference: [2]

RS


vS

vovin

1) (VS GV AV)2 + Vno

2 2 Vno

2

VS 0 VS 0

2) Vno2

)VS GV AV(2

VS 0

3) F No src

Vno2

VS 0 (VS GV AV)2

4 kTRS B )GV AV(2

VS

2

4 k T RS B

Voltage gain, AVGV

RL


Reference: [2]

F

The disadvantage of the single-frequency method is that the

noise bandwidth of the measurement system must be known.

A better method of measuring noise factor is to use a white

noise source.

2) White noise method. This method is similar to the previous

one. The only difference is that the sinusoidal signal generator

is now replaced with a white noise source:

VS2

4 k T RS B



in ) f (

vovin

1) (in RS GR AV)2 B + Vno2

2 Vno2

in 0 in 0

2) Vno2

)in RS GR AV(2 B in 0

3) F No src

Vno2

in 0 (in RS GR AV)2 B

4 kTRS B )GR AV(2

in

2 RS

4 k T

RS

Voltage gain, AvGR

RL

15


F

The noise factor is now a function of only the test noise signal,

the value of the source resistance, and temperature. All of

these quantities are easily measured.

Neither the gain nor the noise bandwidth of the measurement

system need be known.

in2

RS

4 k T

The standard reference temperature is T0 = 290 K for that

k T0= 4.001021. (H. T. Friis: NF, Pa, and T0.)

16

Reference: [2]

5.3.3. VnIn noise model

The actual network can be modeled as a noise-free network

with two noise generators, en and in, connected to its input

(Rothe and Dahlke, 1956):

RS


vS

vo

Rin

Noiseless

AV RL

In a general case, the en and in noise generators are correlated.

en

in

5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.3. VnIn noise model

17

Reference: [2]

The en source represents the network noise that exists when RS

equals zero, and the in source represents the additional noise

that occurs when RS does not equal zero,

The use of these two noise generators plus a complex

correlation coefficient completely characterizes the noise

performance of a linear network.

RS


vS

vo

Rin

Noiseless

AV RL

en

in


18

Reference: www.analog.com

Example: Input voltage and current noise spectra (ultralow noise, high speed, BiFET op-amp AD745)

en

in


19

The total equivalent noise voltage reflected to the source

location can easily be found if we apply the following

modifications to the input circuit:

A. Total input noise as a function of the source impedance

RS


vS

vo

Rin

Noiseless

AV RL

en


in

20

en at S = 4 kT RS + en 2 + 2 Vn In + )in RS(2

RS


vs

vo

en

in Rin

Noiseless

AV

RS


vS

vo

in Rs

Noiseless

AV

RL

RL

en


21

RS


vS

vo

Voltage gain, AV

We now can connect an equivalent noise generator in series

with the input signal source to model the total input voltage of

the whole system.

We assume that the correlation coefficient in the previous

equation 0. (For the case 0, it is often simpler to

analyze the original circuit with its internal noise sources.)

en at S

en at S = 4 kT RS + en 2 + )in RS(2

RL

Reference: [7]



B. Measurement of en and in

Measurement system

Noiseless

AV

en = )Vn o / B( / AV

1) en o >> )4 kT Rt + en2(0.5

2) in Rt = )Vn o / B( / AV

3) in = [)Vn o / B( / AV ] / Rt

Measurement system

Noiseless

AVRt

vn o

RL

vn o

RL

en

in

en

in

235. SOURCES OF ERRORS. 5.4. Noise matching: maximization of SNR

5.4. Noise matching: maximizing SNR

The purpose of noise matching is to let the measurement

system add as little noise as possible to the measurand.

Influence

Measurement System

Measurement Object

Mat

chin

g

+ xx

245. SOURCES OF ERRORS. 5.4. Noise matching: maximization of SNR

where, Nat S and SNRat S are the noise power and the signal-to-

noise ratio at the source location.

We then will try and maximize the SNRo at the output of the

measurement system by matching the source resistance.

VS2

Nat SSNRo = SNRat S

VS2

4kTRS+ en2 + )in RS(2 f ) RS

(

Let us first find the noise factor F and the signal-to-noise ratio

SNRo of the measurement system as a function of the source

resistance: F = f ) RS ( and SNRo = f ) RS (.

4kTRS + en2 + )in RS(2

4kTRS

No

No srcF

Nat S

NR f ) RS

(S

25

F 0.5, dB

SNR 0.5, dB

5.4.1. Optimum source resistance

1

10

100

0.1101 102 103 104100

e n at

S ,

nV/H

z0.5

en = 2 nV/Hz0.5, in = 20 pA /Hz0.5

in RS

RS min F

RS max SNR

5. SOURCES OF ERRORS. 5.4. Noise matching: maximization of SNR. 5.4.1. Optimum source resistance

RS , 101 102 103 104100

-30

-20

-10

0

10

20

Measurement

system noise

en

Source noise 4kTRS

en = in Rn

RS opt =en

in

RS opt is called

the optimum source resistance

)also noise resistance.)

VS2

4kTRS+ en2 + )in RS(2SNR


4kTRSF

vS = en1 Hz0.5

26

SNR 0.5, dB

F 0.5, dB

RS , 101 102 103 104100

-30

-20

-10

0

10

20

It is important to note that the source resistance that maximizes

SNR is RS max SNR 0, whereas the source resistance that

minimizes F is RS min F RS opt .

We can conclude therefore, that for a given RS, SNR cannot be

increased by connecting a resistor to RS.

Vs2



4kTRSF


27

RS

Measurement object

vS

SNR 0.5, dB

F 0.5, dB

RS , 101 102 103 104100

-30

-20

-10

0

10

20

VS2



4kTRSF


Adding a series resistor, R, increases the total source

resistance up to RS opt = RS + R and (!) decreases SNR.

RS RS opt

28

VS2



4kTRSF

SNR 0.5, dB

RS

Measurement object

vS

+R


F 0.5, dB

RS , 101 102 103 104100

-30

-20

-10

0

10

20

4kTRS opt + en2 + )in RS opt (2

4kTRS opt F

VS2

4kTRS+ en2 + )in RS(2SNR VS

2


RS R RS opt

Adding a series resistor, R, increases the total source

resistance up to RS opt = RS + R and (!) decreases SNR.

29

Adding a parallel resistor, R, decreases by the same factor both

the input signal and the source resistance seen by the

measurement network, and therefore (!) decreases SNR.

RS

vS


VS2



4kTRSF

Measurement object

SNR 0.5, dB

F 0.5, dB

SNR 0.5, dB

RS , 101 102 103 104100

-30

-20

-10

0

10

20

RS RS opt

30

SNR 0.5, dB

F 0.5, dB

SNR 0.5, dB

RS , 101 102 103 104100

-30

-20

-10

0

10

20

VS2



4kTRSF

RS

Measurement object

vS

R RS / [)RS R(/R] RS opt

VS / [)RS R(/R]VS


VS2


VS 2

4kTRS+ en2 + )in RS (2SNR

4kTRS opt + en2 + )in RS opt (2

4kTRS opt F

Adding a parallel resistor, R, decreases by the same factor both

the input signal and the source resistance seen by the

measurement network, and therefore (!) decreases SNR.

F 0.5, dB

SNR 0.5, dB

RS , 101 102 103 104100

-30

-20

-10

0

10

20

31

Conclusions.

The noise factor can be very misleading: the minimization of F

does not necessarily leads to the maximization of the SNR.

This is referred to as the noise factor fallacy (erroneous belief).


325. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.2. Methods for the increasing of SNR

Methods for the increasing of SNR are based on the following

relationship:

SNRo = SNRin

1

F

The strategy is simple: to increase SNRo, keep SNRin constant

while decreasing the noise figure:

SNRo = SNRin

1

F

The SNR at the output will increase because the relative noise

power contributed by the measurement system will decrease.

5.4.2. Methods for the increasing of SNR

33

Reference: [7]

io scvin

k

Measurement system

Rinro

Rinro

gm vin

gm vin

A. Noise reduction with parallel input devices

This method is commonly used in low-noise OpAmps:

to increase the SNR, several active devices are connected

in-parallel:


en

in

en

in

34

Reference: [7]

Home exercise: Prove that the following network is equivalent to the

previous one.

io sc

Equivalent measurement system

vin

Rin

kro

k gm vin


en/k 0.5

in k 0.5

35

io sc

Reference: [7]

Measurement object

vS

k= en / in

RS

Equivalent measurement system

vin

Rin

k

RS

Thanks to parallel connection of input devices, it is possible to

decrease the ratio, (en / in (p (en / in (single / k , with no change in

vS and RS, and hence in the SNRin.

Note that SNRo cannot be improved if the RS is too large.

ro

SNRo = SNRo max and F = Fmin at RS =en

k in

k gm vin


en/k 0.5

in k 0.5

36

Reference: [7]Reference: [7]

Home exercise: Prove that

SNRo p = k SNRo single at F min


37

SNRin (n VS )2

4 kT n2 Rin

const,

SNRo = SNRin .1

F

RS

Measurement object

vin1: n

n2 RS

n vS

vo

Measurement system

AV

vS

F SNRin

SNRo

RL

B. Noise reduction with an input transformer


en

in

38

Example: Noise reduction with an ideal input transformer

1

10

100

0.1101 102 103 104100

e n at

S ,

nV/H

z0.5

B = 1 Hz, en = 2 nV/Hz0.5, in = 20 pA /Hz0.5

en

in Rs

RS n2

vS n

RS

vS

1: n

SNRo )1: n( = SNRo

FFmin

F 0.5, dB

RS, 101 102 103 104100

-30

-20

-10

0

10

20

SNR 0.5, dB SNRo )1: n(

0.5

SNRo = SNRin

1

F

Measurement

system noise

Source noise

RS for minimum F

SNRo )1: n( = n2 SNRo F min

n2= RS opt

RS

4kTRsB


vS = en1 Hz0.5

F 0.5Fmin 0.5

SNRo 0.5 SNRo )1: n(

0.5

SNRo F min0.5


Home exercise: Prove that

SNRo )1: n( = n2 SNRo F min

40

(RS + R1 )n2 + R2

vS n

RS

vS

1: n R1 R2

Example: Noise reduction with a non-ideal input transformer


41

Reference: [4]

Our aim in this section is to maximize the SNR of a three-stage

amplifier.

RS

vS

AV 1 AV 2 AV 3

vO

enS1 enS2 enS3

5.4.3. SNR of cascaded noisy amplifiers

5. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.3. SNR of cascaded noisy amplifiers

42

Reference: [4]

2) Vno 2 = [enS1

2 AV1

2 AV22 AV3

2 + enS22 AV2

2 AV32 + enS3

2 AV32 ] B

1) SNRo SNRat S

VS 2

Vno2

/) AV12 AV2

2 AV32(

3) SNRo VS

2 / B

enS12

+ enS22

/AV12 + enS3

2 /AV1

2 AV2

2

Conclusion: keep AV1 >> 1 to neglect the noise contribution of

the second and third stages.

5. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.3. SNR of cascaded noisy amplifiers

RS

vS

AV 1 AV 2 AV 3

vO

enS1 enS2 enS3

43Next lecture

Next lecture:

5.3. noise characteristics

Documents