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2. Next Topic: NOISE

Noise determines the minimum signal power (minimum detectable signal or MDS) at the input of the system required to obtain a signal to noise ratio of 1. A S/N = 1 is usually considered to be the lower acceptable limit except in systems where signal averaging or processing gain is used. Noise figure is a figure of merit used to describe the amount of degradation in S/N ratio that the system introduces as the signal passes through.

For some applications, the minimum signal power that is detectable is

important.

o Satellite receiver

o Terrestrial microwave links

o 802.

Noise limits the minimum signal that can be detected for a given signal input

power from the source or antenna.

We will identify sources of noise, and define related quantities of interest:

o S/N = Signal to noise ratio

o MDS = Minimum Detectable Signal

o F = Noise factor

o NF = 10 * log(F) = Noise figure

Noise Basics:

What is noise? How is it evident to us? Why is it important?

What:

1. Any unwanted random disturbance

2. Random carrier motion produces a current. Frequency and phase

are not predictable at any instant in time

3. The noise amplitude is often represented by a Gaussian probability

density function.

The cumulative area under the curve represents the probability of the event

occurring. Total area is normalized to 1.

Because of the random process, the average value is zero:

(^1) [ ()] 0 1 1

= lim ∫ =

+

→∞

v t dt

T

v

t T t

n T

n

We cannot predict vn(t), but the variance (standard deviation) is finite:

v n

t P

v n vn

t P

vn

[ ] 2

1

= lim^1 ∫ () = σ

+

→∞

v t dt

T

v

t T t

n T

n

Often we refer to the rms value of the noise voltage or current:

vn , rms = vn

Sources of Noise in Circuits:

o Shot noise forward-biased junctions

o Thermal Noise any resistor

o Flicker (1/f) noise trapping effects

Shot noise: This is due to the random carrier flow across a pn junction.

Electrons and holes randomly diffuse across the junction producing noise

current pulses that occur randomly in time. The steady state current

measured across a forward biased diode junction is really a large number of

discrete current pulses.

p

I D I

The variance of this current:

( ) 2

0

= lim ∫ − = σ →∞

i T I I dt

T

D

T

It can be shown that this mean square noise current can be predicted by

i^2 = 2 qID B

where

q = charge of an electron = 1.6 x 10 -

I D = diode current

B = bandwidth in Hertz (sometimes called Δf)

The noise current spectral density: i^2 / B = 2 qID

o Independent of frequency (white noise)

o Independent of temperature for a fixed current

o Proportional to the forward bias current

o Gaussian probability distribution

1 mA of current corresponds to a noise current spectral density of

18 pA/√Hz

read: 18 picoamp per root Hertz

Thermal Noise: Thermal noise, sometimes called Johnson noise, is due to

random motion of electrons in conductors. It is unaffected by DC current

and exists in all conductors. Its spectral density is also frequency

independent, but is directly proportional to temperature. The noise

probability density is Gaussian.

v^2 = 4 kTRB

i^2 = 4 kTB / R

4kT = 1.66 x 10 –20^ V-C

A 50 ohm resistor produces a noise voltage spectral density of

0.9 nV/√Hz

or a Norton equivalent noise current spectral density of

18 pA/√Hz

Flicker or 1/f noise. This noise source is most evident at very low

frequencies. It is hard to localize its physical mechanisms in most devices.

There is usually some 1/f noise contribution due to charge traps with long

time constants. The trap charge then is randomly released after some

relatively long period of time. 1/f noise is modeled by:

f

i^2 / B = K I

™ K is a fudge factor. It can vary wildly from one type of transistor to

the next or even from one fabrication lot to the next.

™ I is the current flowing through the device.

™ B is the bandwidth.

™ 1/f noise can be described by a corner frequency.

™ Carbon resistors exhibit 1/f noise; metal film resistors do not.

Log (i^2 /B)

Log f

Corner frequency

Noise can be modeled as a Thevenin equivalent voltage source or a Norton

equivalent current source. The noise contributed by the resistor is modeled

by the source, thus the resistor is considered noiseless.

It is important to note that noise sources:

o Do not have polarity (the arrow is just to distinguish current

from voltage)

o Do not add algebraically, but as RMS sums

v n , total = vn + vn = 4 kTBR + 4 kTBR

If the sources are correlated (derived from the same physical noise source),

then there is an additional term:

2 , 21 22 2

v n total = vn + vn + Cvn vn

C can vary between –1 and 1.

R

vn in^ R

R

vn in^ R

vn1 R

v R

vn1 R1 n

v R

n

The available noise power can be calculated from the RMS noise voltage or

current:

i R kTB

R

P vn n

av =^4 = 4 =

That is, the available noise power from the source is

o independent of resistance

o proportional to temperature

o proportional to bandwidth

o has no frequency dependence

P av = 4 x 10 -21^ watt

in a 1 Hz bandwidth at the standard noise room temperature of 290 K. If

converted to dBm = 10 log(P/10 -3^ ), this power becomes

• 174 dBm/Hz

We are generally interested in the noise power in other bandwidths than 1 Hz. It’s easy to calculate: P = kTB where kT = -174 dBm To convert bandwidth in Hertz to dB: 10 log B

EX: Suppose your B = 1000 Hz. P = kTB. In dBm, P = -174 + 10 log (1000) = -174 + 30 = -144 dBm

Can a resistor produce infinite noise voltage? Vn^2 = 4 kTBR

Equivalent circuit for noisy resistor. Always some shunt capacitance.

to find total noise power: ∫ 0 ∞ Vno^2 df = kTC = Vno^2

total noise power is independent of R

Low Pass

Vno = Vn 1 +ω^12 C 2 R 2

9 R

4 R

R

f

log 10 Vno

Vn

R

C^ Vno

Noise Equivalent Bandwidth

An amplifier or filter has a nonideal frequency response. Noise power transmitted through is determined by the bandwidth.

Noise power ∝ V^2 (mean square voltage) – white noise

vi^2 A f ( )^2 = vo^2 / Hz in a 1 Hz interval

Summation over entire frequency band

∫ o ∞^ vo^2 ( ) f df^ = v^ i^2 ∫ o^ ∞ A f (^ )^2^ df

We choose an equivalent BW, B, with rectangular profile whose area is the same.

Am^2 B = ∫ o^ ∞ A f ( )^2 df

B = A^1

m^2

∫ o^ ∞ A f ( )^2^ df

This is the definition of bandwidth that we will assume in subsequent calculations.

A f ( )^2

AM^2

B^ f

vi^2 A f ( ) vo^2

Signal-to-noise ratio

Several definitions SNR = (^) PPS N

= (^) NS generally use available power

S + N

N and^

S + N + D

N or^ SINAD^ are alternate definitions. Why is S/N important? Affects the error rate when receiving information.

Pav = VS

2 4 R

rms voltage VS

V^ R

S^ +-

Ref: S. Haykin,Communication Systems, 4th ed., Wiley, 2001

Noise Factor, F :

is a measure of how much noise is added by a component such as an amplifier.

F = (^) SSi^ /^ Ni o /^ No

> 1

because S/N at input will always be greater than S/N at output, F > 1. Noise factor represents the extent that S/N is degraded by the system.

F = (^) noise power available at output due to source @ 290total noise power available at output k

= (^) N Navo avi ⋅^ Gav

Gav = SS^ avo avi

F = (( SS^ // NN )) avi

avo

Noise Figure: NF = 10 log 10 F

Si , Ni Ga So , No

source at 290 K

The higher the noise factor (or noise figure), the larger the degradation of S/N by the amplifier.

Ex.

Gav = 10 dB NF = 3 dB B = 10 6 Hz

amplifier specification

signal available power

W dBm

R

S v

s

avi s^510113

= 2 = × −^12 = × −^15 = −

signal av. pwr. = Savi = vs

2 4 RS^ =^

10 −^12

200 =^5 ×^10

(^15) N ⇒ − 113 dBm

noise av. pwr. = N (^) avi = kTB = − 174 + 60 = − 114 dBm Since noise power increases with B 10 log 10 B = 60 dB (in this example)

10 log (^) NSavo avo

⎛

⎝⎜^

⎞

⎠⎟ =^ 10 log^

Savi Navi

⎛

⎝⎜^

⎞

⎠⎟ −^ NF

= − 113 − −( 114 ) − 3

= 1 dB − 3 dB = − 2 dB (not very good)

How can S (^) o / No be improved?

1. Reduce^ F.^ Slight room for improvement
2. Reduce B. Major improvement if application can tolerate reduced B.
3. Increase antenna gain. Lots of room for improving Si/Ni

VS = 1 μ V

RS = 50 Ω

Gav RL

( S (^) o / N (^) o ) = Si^ F /^ N^ i

+-

W

v S = 1.4 μV

say B = 105 Navi = − 174 + 50 = − 124 dBm Savi N (^) avi^ =^11 dB^ and^

Savo Navo^ =^8 dB

Ex. Noise Floor of Spectrum Analyzer typical NF ≅ 25 dB for SA. NAVO = NAVI ⋅ F ⋅ GAV

N AVI = (− 174 dBm / Hz ) + 10 log B

NF = 25 dB

G AV = 1 0( dB )

RBW NAVO

1 kHz 10 kHz 100 kHz etc.

− 119 dBm − 109 − 99

We will see later how this can be improved.

resolution bandwidth (RBW)

The excess noise added by an active circuit such as an amplifier can also be modeled by an extra resistor at an effective input noise temperature, Te.

is equivalent to:

In terms of noise factor:

F = noise out due to DUT + noise out due to sourceNoise out due to source

= kTe^ BGkT^ +^ kTo^ BG o BG^

= 1 + TTe o

or Te = 290 ( F − 1 )

(where F is a number, not dB )

G POUT

noisy amp

Pav = kTo B

G POUT =^ k T (^ o +^ Te ) B^ ⋅^ G Pav kTe B

Σ

noiseless

(useful when To ≠ 290 k )

Significance of Te : excess noise.

N (^) avo ( total ) = kBG ( T (^) o + Te )

Example: NF = 1 dB ⇒ F = 1. = 1 + Te To = 1 + Te 290 so Te = 75 K total output noise ⇒ 290 + 75 = 365 K equivalent source temp So what? Not major increase in noise power. Further reduction in F may not be justified.

But, for space application: To = 20 K is possible. Then T = To + Te = 20 + 75 = 95 K major degradation in noise temp. F or NF at room temperature doesn’t reveal this so clearly.

F = 1 + 75/20 = 4.5 (NF = 7 dB)

due to source resistor

due to amplifier

Noise Figure of Cascaded Stages. Use Available gain. Why available gain? Noise power defined as available power. Cascading of noise is more convenient when GA is used. Second Stage Noise Contribution

No 1 = k T ( (^) o + T 1 ) BG 1 N (^) o 2 = k T ( (^) o + T 1 ) BG 1 G 2 + kT 2 BG 2 To get total input referred noise power: No 2 G 1 G 2 =^ Ni^ (equivalent)^ =^ k T (^ o^ +^ T^1 ) B^ +^ kT^2 B^ /^ G^1 excess noise at input: kT 1 B + kT 2 B / G 1

Recall that F = 1 + TTe o Te = T 1 + T 2 G 1 FTOTAL = 1 + T T^1 o F 1



+ TT^2

o G 1 F 2 − 1 G 1

N

Third Stage:

• F G^3 −^1 1 G 2

G 1 G 2

Ni = kTo B (^) No 1 No 2

F 1 F 2

T 1 = eff. noise temp @ input

T 2

+

RS

Noise Figure of Cascaded Stages

Fi = Noise Factor G (^) i = Available Gain

⎫⎬

⎭

not in dB

FTOTAL = F 1 + F^2 G^ −^1

1

+ F G^3 −^1

1 G 2

+ ...

= Input Total Noise Factor

(^ S N ) IN ( S N ) OUT^ =^ FTOTAL

Or: ( S N ) OUT dB = ( S N ) IN dB − NFTOTAL

( S N ) IN (^) F 1

G 1

F 2

G 2

F 3

G 3

( S N ) OUT

Additional stages in the cascade treated the same way. Total available gain of cascade = Ga 1 Ga 2 Ga 3 ...

1. If noise figure is important in a receiver, it is standard procedure to design so that the first stage sets the noise performance.

FTOTAL = F 1 + F^2 G^ −^1

1

+ F G^3 −^1

1 G 2

This will require a large enough G 1 to diminish the noise contribution of the second stage.

2. How is the minimum detectable signal or MDS defined?
   • at a given B (very important) PMDS ⇒ S^ + N^ N = 3 dB or S = N S N =^ O^ dB PMDS = 10log( kTB )+ NF dB ( )

OR

PMDS = − 174 dBm Hz / + 10log B + NF dB ( )

Noise figure of Passive Networks

ex. attenuator filter matching network No active components. Only resistors and reactances.

no excess noise is generated by network Savo Savi^ =^ Gav so, ( S N ) i = (^) kTPS o B ( S N ) o = GkT^ ⋅^ PS o B F = (( S NS N^ )) i o

= (^) G^1 Noise factor is just the inverse of gain.

or, NF = − G dB ( )

ex. 10 dB attenuator Gav = − 10 dB NF = 10 dB

passive network Gav F

ZS

PS

Navi = kTo B Navo = kTo B

Measure noise figure of amplifier.

Method #1: Use the spectrum analyzer as a noise receiver.

B = 106 Hz G 1 = 10 dB F 1 = 2 dB

( 1.58 )

B very wide F 2 = 30 dB

( 1000 )

FTOTAL = 1.58 + 99910 = 101.5 ( 20 dB )

completely dominated by second stage.

Now add preamp ahead of SA.

( )

1 1

10 2

G dB F dB

=

( )

2 2

30 1

G dB F dB

= = F 3 (^) = 30 dB

( ) ( )( ) ( )

1.26 (^1 ) 1.58 1. 10 10 1000

F TOTAL

dB

− = + + =

With preamp, SA noise contribution can be kept small enough that front end noise figure can be determined with accuracy. Otherwise, rather hopeless.

ohms

DUT

AMP

Spect. Analyzer

ohms

DUT

AMP

Pre Amp

Spect. Analyzer

Measuring NF. Method #2 Use a calibrated signal source, matched correctly to amp under test.

can measure Vrms across known RL

• B 2 << B 1 Noise equiv. BW set by BPF Two measurements

1. Generator inactive, but still properly terminating amp. Must have correct source impedance.

PS = Pavs = N avi = kT Bo 1

P 1 = output noise power from chain ← measurement 1 = FkTo At B 2

total F transducer gain = (^) Power available from sourcePower delivered to load

2. Generator on. PS = VS

2 8 RS

available power from generator in excess of kTB

P 2 = FkTo At B 2 + PS At ← measurement 2 Eliminate At : F^ =^ P^1 2 P 1 −^1

⋅ (^) kTPS o B 2 Y ≡ P P^2 1

SIG

GEN DUT

Power PS Meter F 1 , G 1 , B 1^ B^2

BPF G 2

Method #3: HOT-COLD NF You can also use a calibrated noise source for measuring NF.

Bn >> B 1 (^) < B 2 The advantage here is that we don’t need to know noise equivalent BW accurately. Noise source has very wide BW compared with system under test. PH = noise power with source on = kTH B TH = effective noise temp. of source Po = kTo B = noise power with source off. To = 290 k Excess Noise Ratio = ENR = PH^ P −^ Po o

= TTH

o

− 1

ENR dB ( ) = 10 log 10^ TTH

o

⎛⎝⎜ − 1 ⎞⎠⎟

Y factor for noise source:

YS = PPH o

= TTH

o

LNA

50 Ω

noise source

Power Meter

B 1 B 2^ B^3 >^ B^2

DUT preamp

So, we can use the noise source instead of the signal generator.

1. Source off. Noise power at meter: P 1 = F kTo B AT

total noise factor transducer gain

2. Source on.

P 2 = P 1 + Ys kT 0 BA T

Divide: P P^2 1

= Y = 1 + YFS

again, the transducer gain cancels, and now B cancels too. We can solve for F from the measured P 2 P 1. F = (^) YY − S 1 Noise factor – numerical ratios, not dB. and

NF = 10 log F ( dB )

The tunable noise figure meter is a receiver. The mixer block upconverts the input noise

signal to a 2 GHz power meter. fin^ =^2 GHz^ − fLO

Thus, by choosing the local oscillator frequency fLO, we measure the noise power within the bandwidth of the IF filter. The noise figure meter also applies a square wave to turn the noise source on and off, obtaining the HOT/COLD input noise condition needed to determine F. As an added bonus, the meter also measures the gain of the device under test.