reliability in rf communications - ucsb · evm and impairments •we have a good idea how white...

Reliability in RF Communications

© James Buckwalter 1

Signal-to-Noise, SNR

• Ratio of power and noise in band

• There are many ways to describe SNR.

• Ebno - Energy per bit 2

21

2

bnoE

e

bno

P eE

s s

bno

o

PTE

N

© James Buckwalter

How does EVM relate to SNR?2

Error Vector Magnitude

© James Buckwalter

• EVM is another metric of BER. EVM can be related to a symbol error rate and therefore to a BER.

2

, ,

1

2

,

1

1

100% 100%1

N

ref k meas k

error k

N

referenceref k

k

S SP N

EVMP

SN

3

Creating a Reference Constellation

• How would one choose G? • This is complicated but a straightforward answer would

be to use an LMS technique that compares measured constellation to the reference constellation and adapts gain according to

© James Buckwalter

S

ref= G

refm

I+ jm

Q( ) = mI+ jm

Q

Serror ,k

2

k=1

N

å = Smeas,k

- Sref ,k

2

k=1

N

å = Smeas,k

- Gref

mI+ jm

Q( )2

k=1

N

å

® choose Gref

that minimizes the function

4

In the presence of white noise (I)

© James Buckwalter

S

ref ,k= m

I ,k+ jm

Q,k

Each ideal symbol can be represented by

White noise can be represented by

n = n

I+ jn

Q

5

In the presence of white noise (II)

© James Buckwalter

Smeas,k

= mI ,k

+ nI+ j m

Q ,k+ n

Q( )S

error ,k= S

ref ,k- S

meas,k

Serror ,k

= mI ,k

+ nI( ) - m

I ,k+ j m

Q ,k+ n

Q( ) - mQ ,k( )

Serror ,k

= nI+ jn

Q

6

In the presence of white noise (III)

• The addition of white noise with a variance of n2

means that the error vector is

• Expectation of error over many samples is

© James Buckwalter

Serror ,k

2

k=1

N

å = nI+ jn

Q

2

k=1

N

å = Nn2

After many

samples

E Serror ,k

néë ùû2

k=1

N

åé

ëê

ù

ûú = Ns

n

2

7

In the presence of white noise (IV)

• Define average symbol power from estimation of constellation

• Now the EVM is rewritten in terms of signal power and noise

© James Buckwalter

2

,

1

1 N

S ref k

k

P SN

Serror ,k

2

k=1

N

å

Sref ,k

2

k=1

N

å=

Nsn

2

NPS

=s

n

2

PS

8

In the presence of white noise (V)

• Now the EVM is rewritten in terms of signal power and noise

© James Buckwalter

EVM =s

n

2

PS

=1

SNRS

=1

SNRblog

2N

Sometimes called Ebno

9

EVM versus BER

© James Buckwalter

10

(%) 100%

( ) 10log

error

reference

error

reference

PEVM

P

PEVM dB

P

10

Simulation of EVM

• Noise due to white noise sources spreads ideal digital points.

• Here SNR per symbol is 20dB

© James Buckwalter

Random cluster of points

11

Signal-to-Noise, SNR

• SNR is related to BER or SER so EVM is also related to BER/SER

© James Buckwalter

, 2

2

, ,

1 3 11

2 1

1 1

e s

e b e s

P QM EVMM

P P

12

Probability of correct symbol

BER vs EVM


EVM and Impairments

• We have a good idea how white noise degrades our receiver behavior. Remember that we can now use the noise figure of the system to express the receiver as

• How about other impairments? Phase noise, LO leakage, image rejection, AM-AM compression, AM-PM compression and so on!

© James Buckwalter

s

n

2 = No= kTFB

14

EVM due to Phase Noise (I)

• Received signal is represented as in-phase and quadrature message with respect to the carrier.

• Sometimes this is better described in polar coordinates

© James Buckwalter

cos sinI RF Q RFx t m t t m t t

2 2 1

cos

where and tan

RF

Q

I Q

I

x t A t t t

m tA t m t m t t

m t

15

EVM due to Phase Noise (II)

• Add a phase noise contribution to

• This is mixed with the in-phase and quadrature LOs

• Here we assume that the receiver is homodyne

© James Buckwalter

V = A t( )cos w

RFt +f t( )( )

mI= 2A t( )cos w

RFt +f t( )( )cos w

LOt +q( ) » A t( )cos f t( ) -q( )

mQ

= -2A t( )cos wRF

t +f t( )( )sin wLO

t +q( ) » A t( )sin f t( ) -q( )

16

EVM due to Phase Noise (III)

© James Buckwalter

mI+ jm

Q=

A t( ) cos f t( ) -q( ) + jsin f t( ) -q( )éë

ùû

= A t( )ej f t( )-q( )

• Phase noise causes shift in detected phase

17

EVM due to Phase Noise (IV)

• We have found the constellation in the presence of phase noise. Now what would the ideal constellation be?

© James Buckwalter

S

ref= A t( ) cosf t( ) + jsinf t( )( ) = A t( )e

jf t( )

18

EVM due to Phase Noise (V)

• Determine the error due to phase noise

© James Buckwalter

S

error= A t( )e

jf t( )e- jq -1( )

Serror

» mI+ jm

Q( ) cosq -1- jsinq( )

Serror

2

= mI

cosq -1( ) + mQ

sinq( )2

+ mQ

cosq -1( ) - mIsinq( )

2

Serror

2

= 2A2 t( ) 1- cosq( )

S

error

2

= 2A2 t( ) 1- cosq( ) »q 2 A2 t( )

19

Note that the error increases with the power of the constellation symbol

EVM due to Phase Noise (V)

• Remember that phase noise

© James Buckwalter

EVM =

Serror ,k

2

k=1

N

å

Sref ,k

2

k=1

N

å=

q 2 A2 t( )k=1

N

å

A2 t( )k=1

N

å= q

EVM = sq

After many

samples

20

sq

2 t( ) = 4 Sqq

f( )sin2 p ft( )-¥

¥

ò df s

q

2 t( ) =wo

2ctFrom last time

EVM with SNR and Phase Noise

• Noise sources are considered independent.

• Impact on EVM is therefore independent and powers add.

© James Buckwalter

21

S

EVMSNR

21

Simulation of Phase Noise

• Rms phase jitter of 10 degrees.


Carrier Leakage/Image Rejection

• EVM shows up in homodyne and heterodyne transmitters and receivers due to LO leakage and image rejection.

– LO leakage: particular problem of homodyne

– IRR: particular problem of heterodyne


EVM due to Carrier Leakage (I)

• Define RX voltage with LO leakage signal

• Assume homodyne conversion

© James Buckwalter

cos cosRF LO LOV A t t t V t

Desired Signal LO Leakage with phase noise

24

EVM due to Carrier Leakage (II)

© James Buckwalter

mI= 2 A t( )cos w

RFt +f( ) +V

LOcos w

LOt +q( )( )cos w

LOt +q( )

mI» A t( )cos f -q( ) +V

LO

mQ

= -2 A t( )cos wRF

t +f( ) +VLO

cos wLO

t +q( )( )sin wLO

t +q( )

mQ

» A t( )sin f -q( )

25

EVM due to Carrier Leakage (III)

• The error contains a voltage shift

• Ignore phase error for now…

© James Buckwalter

m

I+ jm

Q= A t( )e

j f t( )-q( )+V

LO

S

error ,k= A t( )e

jf t( )e- jq -1( )+V

LO

Serror ,k

2

k=1

N

å = VLO

2

k=1

N

å = NVLO

2

26

EVM due to Carrier Leakage (III)

• EVM is reduced with lower leakage.

• EVM due to LO leakage is introduced as a d.c. offset.

© James Buckwalter

LO Leakage Suppression (LLS) is ratio of LO power to signal power.

2

,

1

2

,

1

N

error k

k LO

N

Sref k

k

SP

EVM LLSP

S

27

EVM due to Carrier Leakage and Phase Noise (I)

• Return to expression for LO leakage and phase noise

• Find error in presence of LO leakage and phase noise

© James Buckwalter

S

error ,k= A t( )e

jf t( )e- jq -1( )+V

LO

Serror

» mI+ jm

Q( ) cosq -1- jsinq( ) +VLO

Serror

2

= mI

cosq -1( ) + mQ

sinq +VLO( )

2

+ mQ

cosq -1( ) - mIsinq( )

2

Serror

2

= 2A2 1- cosq( ) + 2VLO

A 1- cosq( ) +VLO

2

28

EVM due to Carrier Leakage and Phase Noise (II)

• Making a small angle approximation

© James Buckwalter

Serror ,k

2

k=1

N

å = A2q 2 + Aq 2VLO

+VLO

2( )k=1

N

å

Serror ,k

2

k=1

N

å = NPSq 2 + NP

LO+q 2V

LOA

k=1

N

å

S

error

2

= A2q 2 + Aq 2VLO

+VLO

2

29

EVM due to Carrier Leakage and Phase Noise (III)

• Impact of phase noise and LO leakage. LO leakage shows up as mean for phase noise.

© James Buckwalter

EVM =

Serror ,k

2

k=1

N

å

Sref ,k

2

k=1

N

å=

NPSq 2 + NP

LO+q 2V

LOA

k=1

N

å

NPS

= q 2 1+ LLSE Aéë ùû

VLO

æ

èç

ö

ø÷ + LLS

EVM =1

SNRS

+sq

2 1+ LLS × X( ) + LLS

where X= E Aéë ùû

VLO

* You might argue that there might be better “references” to use with LLS 30

Determining the IRR

• Remember that our IRR was contributed through the gain and phase mismatch between the quadrature paths of our receiver (or transmitter).

© James Buckwalter

IRR =1

DG

2G

æ

èç

ö

ø÷

2

+q

2

æ

èç

ö

ø÷

2

IRR =1

a

2

æ

èç

ö

ø÷

2

+q

2

æ

èç

ö

ø÷

2

31

IRR in terms of Circuit

• Alpha is the gain mismatch

• Theta is the phase mismatch


Mismatch causes Matrix


Understanding Mismatch Matrix


M =

1+a

2

æ

èç

ö

ø÷cos

q

2

æ

èç

ö

ø÷ - 1-

a

2

æ

èç

ö

ø÷sin

q

2

æ

èç

ö

ø÷

- 1+a

2

æ

èç

ö

ø÷sin

q

2

æ

èç

ö

ø÷ 1-

a

2

æ

èç

ö

ø÷cos

q

2

æ

èç

ö

ø÷

é

ë

êêêêê

ù

û

úúúúú

EVM impact for Mismatch


Smeas

= G 1+a

2

æ

èç

ö

ø÷e

jq

2 mI+ j 1-

a

2

æ

èç

ö

ø÷e

- jq

2mQ

é

ëê

ù

ûú

Smeas

- Sref

= G 1+a

2

æ

èç

ö

ø÷e

jq

2 mI+ j 1-

a

2

æ

èç

ö

ø÷e

- jq

2 mQ

é

ëê

ù

ûú

Smeas

- Sref

2

Sref

2=

G2 1+a

2

æ

èç

ö

ø÷e

jq

2 -1æ

èçç

ö

ø÷÷m

I+ j 1-

a

2

æ

èç

ö

ø÷e

- jq

2 -1æ

èçç

ö

ø÷÷m

Q

2

G2 mI

2 + mQ

2( )

EVM impact for Mismatch


Smeas

- Sref

2

Sref

2=

G2 1+a

2

æ

èç

ö

ø÷e

jq

2 -1æ

èçç

ö

ø÷÷m

I+ j 1-

a

2

æ

èç

ö

ø÷e

- jq

2 -1æ

èçç

ö

ø÷÷m

Q

2

G2 mI

2 + mQ

2( )

= 1+a

2

æ

èç

ö

ø÷e

jq

2 -1æ

èçç

ö

ø÷÷

2

mI

2 + 1-a

2

æ

èç

ö

ø÷e

- jq

2 -1æ

èçç

ö

ø÷÷

2

mQ

2

= 1+a

2

æ

èç

ö

ø÷

2

e jq - 2+a( )ejq

2 +1

æ

è

çç

ö

ø

÷÷m

I

2 + 1-a

2

æ

èç

ö

ø÷

2

e- jq - 2 -a( )e- j

q

2 +1

æ

è

çç

ö

ø

÷÷m

Q

2

» 1+a +a

2

æ

èç

ö

ø÷

2æ

è

çç

ö

ø

÷÷

1+ jq -q 2

2

æ

èç

ö

ø÷- 2+a( ) 1+ j

q

2

æ

èç

ö

ø÷+1

æ

è

çç

ö

ø

÷÷m

I

2 + 1-a( ) 1- jq( ) - 2- jq( )+1( ) mQ

2

»a

2

æ

èç

ö

ø÷

2

-q 2

2+

a jq

2

æ

è

çç

ö

ø

÷÷

EVM due to Images (I)

• This is mixed with the in-phase and quadrature heterodyne LOs

© James Buckwalter

cos cosRF IMV A t t t A t IRR t

IFI= 2A cos w

RFt +f( ) + IRRcos w

IMt( )( )cos w

LOt( )

IFI» A cos w

IFt +f( ) + IRRcos w

IFt( )( )

IFQ

= -2A cos wRF

t +f( ) + IRRcos wIM

t( )( )sin wLO

t( )

IFQ

» A sin wIF

t +f( ) - IRRsin wIF

t( )( )37

EVM due to Images (II)

• Images appear at both I

© James Buckwalter

mI= A cos w

IFt +f( ) + IRRcos w

IFt( )( )2cos w

IFt( )

mI= Acos f( ) + A × IRR

mQ

= A sin wIF

t +f( ) - IRRsin wIF

t( )( ) -2cos wIF

t( )( )m

Q= Asin f( )

S

error

2

= A× IRR( )2

= A2 × IRR2

38

EVM due to Images (III)

© James Buckwalter

2

,21

2

,

1

N

error k

k

N

ref k

k

S

EVM IRR IRR

S

Serror ,k

2

k=1

N

å = N × PS× IRR2

39

• For example, 5% EVM

• 64-QAM gives 10^-6

• 256-QAM gives 10^-2

EVM Summary

• All contributions of error in the receiver can be combined to find the overall EVM.

• We add each contribution as an independent source of error.

© James Buckwalter

EVM =1

SNRS

+sq

2 1+ LLS × X( ) + LLS + IRR2

40

reliability in rf communications - ucsb · evm and impairments •we have a good idea how white...

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