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
© James Buckwalter 13
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.
© James Buckwalter 22
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
© James Buckwalter 23
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
© James Buckwalter 32
Mismatch causes Matrix
© James Buckwalter 33
Understanding Mismatch Matrix
© James Buckwalter 34
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
© James Buckwalter 35
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
© James Buckwalter 36
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