January 2004
doc.: IEEE 802.11-04/0015r2
Comments on Ergodic and
Outage Capacity
Yang-Seok Choi, [email protected]
Siavash M. Alamouti, [email protected]
Submission
Slide 1
Yang-Seok Choi et al., ViVATO
January 2004
doc.: IEEE 802.11-04/0015r2
Questions?
 From IEEE802.11-03/940r1, TGn-channel-models
Model (NLOS)
Mean capacity in
b/s/Hz
% of iid mean
capacity
A (optional)
9.1
83
B
8.9
81
C
8.6
78
D
10.0
92
E
9.3
85
F
10.4
95
iid
10.9
100
Table III: 4x4 MIMO channel mean capacity for the NLOS conditions at 10 dB SNR.
– Can we achieve 10 b/s/Hz at 10 dB SNR?
– If not, how much spectral efficiency can we get at 10 dB
SNR?
– Model B provides better capacity than C?
Submission
Slide 2
Yang-Seok Choi et al., ViVATO
January 2004
doc.: IEEE 802.11-04/0015r2
Comments
 The table may mislead 802.11n participants
regarding practical interpretation of capacity.
– The numbers are neither a lower nor an upper
bound for 802.11n performance criteria (FER as a
function of SNR for a given bandwidth efficiency)
– Relative “theoretical” performance for the different
channels (compared to iid) does not correspond to
the relative practical difference for known
techniques.
 “Outage Capacity” is a more useful metric
than Ergodic “Average Capacity”.
Submission
Slide 3
Yang-Seok Choi et al., ViVATO
January 2004
doc.: IEEE 802.11-04/0015r2
Assumptions
 Block Fading Channel
– Channel is invariant over a frame
– Channel is independent from frame to
frame
 CSI is available to Rx only
– Perfect CSI at RX
– No feedback channel
 Gaussian codebook
Submission
Slide 4
Yang-Seok Choi et al., ViVATO
January 2004
doc.: IEEE 802.11-04/0015r2
System Models
w
d
y
H
y (n)  Hd (n)  w (n)

where y (n) : M 1 received vector,

H : M  N channel Matrix wit h E H (k , l )

2

 1,
d (n) : N 1 data vector wi th E d (n)d (n) H  PI N ,


w (n) : M 1 noise vector wi th E w (n) w (n) H   2 I M ,
SNR :   P /  2 , Total Tx. Power  NP.
Submission
Slide 5
Yang-Seok Choi et al., ViVATO
January 2004
doc.: IEEE 802.11-04/0015r2
“Instantaneous” Capacity
 Capacity under given realization of channel
matrix with perfect knowledge of channel at Rx
C  max I (d ; y | H  H )
 log 2 I M  HH H  log 2 I N  H H H
 If transmitted frames have spectral efficiency
less than above capacity, with arbitrarily large
codeword, FER will be arbitrarily small
 If transmitted frames have spectral efficiency
greater than above capacity, with arbitrarily
large codeword, FER will approach 100%.
Submission
Slide 6
Yang-Seok Choi et al., ViVATO
January 2004
doc.: IEEE 802.11-04/0015r2
Ergodic Capacity
 Ergodic Capacity : Ensemble average of
“instantaneous” capacity over all possible channel
matrices
I (d ; y,H )  I (d ; H )  I (d ; y | H )  I (d ; y | H )
 EI (d ; y | H  H )  EC
 If EC  10 b/s/Hz , does this mean that in average we
achieve 10 b/s/Hz spectral efficiency?
– No in the sense of practical implementation!
– But if CSI is available at Tx, by using adaptive modulation it
can be true when the adaptive modulation can handle spectral
efficiency from 0 to infinity. But if CSI is known to Tx, you can
achieve better capacity.
 Is it possible to achieve Ergodic Capacity?
Submission
Slide 7
Yang-Seok Choi et al., ViVATO
January 2004
doc.: IEEE 802.11-04/0015r2
Ergodic Capacity (cont’d)
 How to achieve Ergodic capacity when CSI is
not available to Tx?
– At least, Your codeword should be spanned over all
possible channel matrices. Otherwise there is no way
to achieve Ergodic Capacity.
– The codeword may have to be spread over all
possible locations.
– Or the frame duration should be much longer than
coherence time.
– And your coding structure should be able to achieve
Ergodic capacity.
 Ergodic capacity is not a useful metric
Submission
Slide 8
Yang-Seok Choi et al., ViVATO
January 2004
doc.: IEEE 802.11-04/0015r2
Outage Capacity
 In fading channel, the capacity is a random
variable.
 Due to delay limitation, outage capacity is more
meaningful than ergodic capacity
 Outage capacity C0 at outage probability r0
Pr(C  C0 )  r0
 When Pr(C  10 b/s/Hz )  0.5
– The above does not mean that in average we achieve
10 bps/Hz spectral efficiency
– But it means that FER is 0.5 even with ideal code if
your frame has 10 b/s/Hz spectral efficiency
Submission
Slide 9
Yang-Seok Choi et al., ViVATO
January 2004
doc.: IEEE 802.11-04/0015r2
Outage Capacity (cont’d)
 CDF in Log scale : Low outage probability is of interest
(some consider zero-outage probability)
– Recall definition of “Capacity” – Maximum rate without error
– Linear scale may not reveal behaviors at low outage probabilities
 Outage Probability
– With ideal code, outage probability is equal to FER of which
spectral efficiency is C0
Pr(C  C0 )  FER
– With non-ideal code, outage probability is lower bound of FER
– Slope of Log outage probability vs. Log SNR plot : Diversity Order
– Slope of non-ideal code FER  Diversity order
Submission
Slide 10
Yang-Seok Choi et al., ViVATO
January 2004
doc.: IEEE 802.11-04/0015r2
Outage Capacity (cont’d)
 100Mbps MAC SAP
– 150 Mbps PHY SAP : required spectral efficiency for
OFDM systems = 4.0 150Mbps
3.2
 12.5 b/s/Hz
48
20MHz
64
(PHY Overhead such as Preamble is excluded)
 Capacity in 11n
– SNR=Received Signal Power per Rx antenna/Noise
Power (at each subcarrier)
1
SNR
C   log 2 I 
H k H kH b/s/Hz
48 k
nT
where H k is a channel matrix at subcarrier k assuming
2
E H k (i, j )  1 and Max delay<GI

Submission

Slide 11
Yang-Seok Choi et al., ViVATO
January 2004
doc.: IEEE 802.11-04/0015r2
Outage Capacity (cont’d)
 Outage Prob. 4-by-4 MIMO OFDM(NLOS, No
shadow fading)
– 0.5 l spacing
Submission
– 1 l spacing
Slide 12
Yang-Seok Choi et al., ViVATO
January 2004
doc.: IEEE 802.11-04/0015r2
Outage Capacity (cont’d)
 Loss due to
–
–
–
–
Non-Ideal code (Space and Frequency diversity)
Non-Ideal Channel Estimation
Implementation Loss
NF
Submission
Slide 13
Yang-Seok Choi et al., ViVATO
January 2004
doc.: IEEE 802.11-04/0015r2
Comparison Table (4x4)
 Required SNR at 10% FER for 12.5 b/s/Hz with
ideal coding
Submission
Model (NLOS)
0.5 l spacing
1 l spacing
B
15.6 dB
13.6 dB
C
15.4 dB
13.4 dB
D
13.2 dB
12.2 dB
E
14 dB
13.1 dB
iid (flat)
13.4 dB
13.4 dB
iid (200 nsec rms)
12.1 dB
12.1 dB
Slide 14
Yang-Seok Choi et al., ViVATO
January 2004
doc.: IEEE 802.11-04/0015r2
Comparison at PHY
 Compare Proposals with ideal coding case
– Slope
– Required SNR at 10% FER
Submission
Slide 15
Yang-Seok Choi et al., ViVATO
January 2004
doc.: IEEE 802.11-04/0015r2
Thank you for your attention!!
Questions?
Submission
Slide 16
Yang-Seok Choi et al., ViVATO
January 2004
doc.: IEEE 802.11-04/0015r2
Back-up
 100Mbps MAC SAP
– 150 Mbps PHY SAP :
required spectral efficiency =
(PHY Overhead such as Preamble is excluded)
150Mbps
 10 b/s/Hz
48
20MHz
64
 Capacity in 11n
– SNR=Total Received Signal Power per Rx
antenna/Noise Power (at each subcarrier)
3.2 1
SNR
H
C
log
I

H
H

2
k
k b/s/Hz
nT
Capacity in OFDM 4 48 k
where H k is a channel matrix at subcarrier k assuming
2
E H k (i, j )  1 and Max delay<GI

Submission

Slide 17
Yang-Seok Choi et al., ViVATO