Reciprocal Mixing: The trouble with oscillators · Reciprocal Mixing: LTE Example •Note the noise...

Reciprocal Mixing:The trouble with oscillators

Tradeoffs in RX

• Noise Figure(Sensitivity)

• Distortion (Linearity)

• Phase Noise (Aliasing)

©James Buckwalter 2

Phase Noise

• Phase noise is the frequency domain representation of rapid, short-term, random fluctuations in the phase of a waveform, caused by time domain instabilities ("jitter").


Note that commercial generators don’t do better than around -160 dBc/Hz

What does phase noise look like?


Simple Phase Noise Model

• Consider an oscillator with some noise

• When that noise is small,

• Our power spectrum indicates

©James Buckwalter

sinlo loV t A t t

sin coslo lo loV t A t t t

2

4VV lo lo lo lo

AS f S S

5

Phase Noise Definition

• We can define the phase noise from the ratio of the noisy signal to the ideal carrier component

• Independent of amplitude. Note that the units of phase noise are dBc per Hz. dBc means a relative level with respect to the carrier.

©James Buckwalter

2

10 102

410log 10log

4

lolo

lo

lolo

AS S

LA

6

Characterizing Phase Noise

©James Buckwalter

10

,110log

sideband LO

carrier

P f f HzL f

P

7

Approximating the Phase Noise

• Phase noise is generally broken into well-defined regions.

©James Buckwalter

1

1

1

log logk

i i i i i

i

L f a f f b U f f U f f

1/f3 1/f2 Constant

8


• Let’s relate this back to a timing parameter

• Phase jitter:

• This generally looks like


sq

2 t( ) = 4 Sqq

f( )sin2 p ft( )-¥

¥

ò df S

qqf( ) =10

L w{ }/10

s

t

2 t( ) = ct +kt2

Sqq

f( ) = cf

osc

f

æ

èç

ö

ø÷

2

sq

2 t( ) = c 2p fosc

t( )2 sin2 p ft( )

p ft( )2

-¥

¥

ò df = c2p f

osct( )

2

pt

sin2 x( )x2

-¥

¥

ò dx = c 2p fosc( )

2

t


• Cycle-to-cycle Jitter:


st

2 1

fosc

æ

èçç

ö

ø÷÷ =

c

fosc

Approximating the Phase Noise

• By parameterizing the phase noise, we can find the expected phase jitter

©James Buckwalter

1

1

1

log logk

i i i i i

i

L f a f f b U f f U f f

1 1 1/10 /102 10 10

1

1

2 10 110

i i

i i

a akb a i

i i i

i

af f f

ai is the slope

Fi

bi

11

Calculating Phase Noise

• In 1/f2 region,

• If treated as constant,

©James Buckwalter

L f{ } = -20 log

10f( )- log

10f

i( )( )+ bi

12

L f{ }fl

fh

ò df =10b

l10

102log

10fl( )

10log

10f -2( )

fl

fh

ò df

=10b

l10

fl

2 f -2

fl

fh

ò df =10b

l10

fl

2 -1

ffl

fh

=10b

l10

fl

2 f -2

fl

fh

ò df =10b

l10

fl

2 -1

ffl

fh

=10b

l fl

fh- f

l

fh

æ

èçç

ö

ø÷÷

L f

x{ } = -20 log10

fx( )- log

10f

i( )( )+bi

L fx{ }

fl

fh

ò df =10L f

x{ } 10f

h- f

l( ) =10b

lf

l

fx

æ

èçç

ö

ø÷÷

2

fh- f

l( )

10b

lf

x

fl

æ

èçç

ö

ø÷÷

2

fh- f

l( ) =10b

l fl

fh- f

l

fh

æ

èçç

ö

ø÷÷®

fl

fx

æ

èçç

ö

ø÷÷

2

=f

l

fh

æ

èçç

ö

ø÷÷® f

x= f

hf

l

Oscillator Conditions

©James Buckwalter

G jw o( )Z jwo( ) = 1

ÐG jwo( )Z jwo( ) = 2pn

13

LC Oscillator Tank

• Look at perturbed behavior near resonance

©James Buckwalter

1 1

1 1

1

1

o

o

o

o

Z R RR j

sCR Q QsL j

Z Rj

Q Qj

1 21 1 1

o

oo o

R RZ

jQjQ jQ

14

LC Oscillator

• Now add negative resistance to impedance of tank…

©James Buckwalter

Z wo+ Dw( ) »

R

1+ 2 jQDw

wo

15

LC Oscillator

• Now add negative resistance to impedance of tank…

©James Buckwalter

1 1

11 1 1 1 2

2

o

m

m

o o o

o

o

RZ

gjQ jQ g jQ

R

RZ

jQ

16

LC Oscillator Phase Noise

©James Buckwalter

L Dw{ } = 10logv

noise

2

vs

2

æ

èçç

ö

ø÷÷

17

LC Oscillator Phase Noise (cont)

©James Buckwalter

vnoise

2 = Z wo+ Dw( )

2 in

2

Df

vnoise

2 =R

2QDw

wo

æ

è

çççç

ö

ø

÷÷÷÷

2

4kTG

R= 4kTGR

wo

2QDw

æ

èçö

ø÷

2

18



vnoise

2

vs

2=

4kTGRw

o

2QDw

æ

èçö

ø÷

2

vs

2

vnoise

2

vs

2=

4kTG

Posc

wo

2QDw

æ

èçö

ø÷

2


©James Buckwalter

• A factor of ½ is included in noise expression to indicate AM-PM / AM-AM distinction.

• Phase noise is related to

– Oscillator Q

– Oscillator power

– Noise in tank circuit

L Dw{ } = 10log2kTG

Posc

wo

2QDw

æ

èçö

ø÷

2æ

èçç

ö

ø÷÷

20

Leeson’s Formula

• This expression can be empirically expanded to describe noise in other regimes.

• At low frequency

• At high frequency

©James Buckwalter

2

1/210log 1 1

2

fo

osc

kTL

P Q

2

1/210log

2

fo

osc

kTL

P Q

2

10logosc

kTL

P

21

Phase Noise Regions

©James Buckwalter

2

1/210log 1 1

2

fo

osc

kTL

P Q

L Dw{ } = 10log2kTG

Posc

wo

2QDw

æ

èçö

ø÷

2æ

èçç

ö

ø÷÷

2

10logosc

kTL

P

22

Classical Phase Noise Theory

• Use a linear time invariant model

• Assume all noise sources are white

• In reality, the oscillator is a large-signal circuit

• Transconductor might be on only part of the time

• Oscillator noise is cyclostationary

• Leeson’s formula while intuitive is empirical!


Why is Phase Noise Important?

• Channels are spaced tightly to provide a large number of users.

• Reciprocal Mixing

• Adjacent channel power might be much higher than desired signal. Mixing smears the strong signal into the desired channel.


Phase Noise Impairment (I)


P

int- P

sig+ SNR

min= SNR

PN

• Compare the adjacent channel power to the desired power to determine the signal level that will fall into the desired band.

Phase Noise Impairment (II)

• For cell standards, in-band jammer can be 40 dB greater than desired signal.

• SNRmin should be > 20 dB

©James Buckwalter

SNRPN

= Pint

- Psig

+ SNRmin

SNR

PN< 40 + 20 = 60dB

26

Phase Noise Impairment

• How do we specify the noise due to the phase noise?

• In 1/f2 region,

• From earlier..

©James Buckwalter

SNRPN

=1

L Df{ }fl

fh

ò dDf

SNRPN

=1

L fh

fl{ } f

h- f

l( )

L f{ } = -20 log

10f( )- log

10f

i( )( )+ bi

27

A Cautionary Note

• Summary

L f

hf

l{ } = Psig

- Pint

- SNRmin

-10log10

fh- f

l( )

GSM Frequency Plan

A Cautionary Note

• Summary

L f

hf

l{ } = Psig

- Pint

- SNRmin

-10log10

fh- f

l( )

Phase Noise Impairment (IV)

• Example: GSM channel spacing (600 kHz)

• We specify that the oscillator should have a phase noise of under -99 dBc at a 100 kHz offset frequency.

©James Buckwalter

L fh

fl{ } = L 519kHz{ } = -99dBm- -43dBm( ) -10log

10600kHz( ) = -113dBc

L 100kHz{ } = -113dBc+ 20log10

519kHz

100kHz

æ

èç

ö

ø÷ = -99dBc

31

L f

hf

l{ } = Psig

- Pint

- SNRmin

-10log10

fh- f

l( )

Basic FDD SystemWhat about our out-of-band Blockers?

Reciprocal Mixing

• Assume RX at 2120 MHz and TX at 1970 MHz.

• Therefore, frequency offset is 50 MHz.

• TX signal is 10 dBm, RX sensitivity is -110 dBm.

• What is the required phase noise if the reciprocal mixing is 10 dB less that RX sensitivity?


Reciprocal Mixing: LTE Example• Calculate the SNR phase noise

• The phase noise is

• Translate to 10 MHz

SNRPN

= Pint

- Psig

+ SNRmin

SNRPN

=10dBm- -110dBm( )+10dB =130dB


L fh

fl{ } = -SNR

PN-10log

10f

h- f

l( )

L 49MHz{ } = -130dBc -10log10

60MHz - 40MHz( )L 49MHz{ } = -130dBc -73dB = -203dBc

L 10MHz{ } = -203dBc+10log

104.9( ) = -197dBc

Reciprocal Mixing: LTE Example

• Note the noise floor for the phase noise is around -174 dBm.

• If the noise floor is -164 dBm, this suggests that the oscillator power must be 33 dBm. This is very high oscillator power (and dc power consumption).


L 10MHz{ } = -197dBc


• Let’s ask the opposite question. Given the noise floor and oscillator power, how much transmit power can we tolerate?

• The phase noise is


SNRPN

= -10log10

fh- f

l( ) - L 50MHz{ }

SNRPN

= -10log10

60MHz - 40MHz( ) - -174dBc( )SNR

PN=174dBc -73dB =101dB

L 50MHz{ } = -164dBm-10dBm = -174dBc


• The phase noise SNR is related to

• This isn’t much blocker power. Consider the out-of-band blocker levels that we have discussed for 800 MHz and 2.4 GHz. We MUST filter signals to avoid reciprocal mixing!

Pint

= SNRPN

+ Psig

- SNRmin

Pint

=101dB -110dBm-10dB = -19dBm


Conclusions

• Oscillator phase noise is a silent killer in interference limited systems.

• Requirements for phase noise are dictated by phase noise in 1/f^2 and thermal noise floor.

Reciprocal Mixing: The trouble with oscillators · Reciprocal Mixing: LTE Example •Note the noise...

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