Throughput/Delay Measurements of Limited Feedback Beamforming in Indoor Wireless Networks

Abstract

This paper investigates the tradeoff between throughput and feedback delay of limited feedback beamforming in indoor wireless channels with a practical MIMO-OFDM prototype. Past descriptions of this tradeoff are largely based on simplified models of the wireless channel. Unfortunately, wireless channel models may not accurately represent the complexities of a real wireless channel. Furthermore, system impairments, including channel estimation error, only exacerbate the problem of modeling a real wireless system. One such analytical result predicts the performance of limited feedback beamforming as an exponential function of feedback delay. This analytical result has been confirmed through Monte Carlo simulation under Rayleigh fading channel models. Through rigorous measurements and experimentation this paper both evaluates the performance of limited feedback beamforming under feedback delay and confirms the accuracy of the analytical results.

Full text

Throughput/Delay Measurements of Limited Feedback
Beamforming in Indoor Wireless Networks
Robert C. Daniels, Ketan Mandke, Kien T. Truong, Scott M. Nettles, and Robert W. Heath, Jr.
Wireless Networking and Communications Group
The University of Texas at Austin
1 University Station, Mail Code: C0806, Austin, TX 78712
Abstract—This paper investigates the tradeoff between
throughput and feedback delay of limited feedback beamforming
in indoor wireless channels with a practical MIMO-OFDM
prototype. Past descriptions of this tradeoff are largely based on
simplified models of the wireless channel. Unfortunately, wireless
channel models may not accurately represent the complexities
of a real wireless channel. Furthermore, system impairments,
including channel estimation error, only exacerbate the problem
of modeling a real wireless system. One such analytical result
predicts the performance of limited feedback beamforming as
an exponential function of feedback delay. This analytical result
has been confirmed through Monte Carlo simulation under
Rayleigh fading channel models. Through rigorous measurements
and experimentation this paper both evaluates the performance
of limited feedback beamforming under feedback delay and
confirms the accuracy of the analytical results.
I. INTRODUCTION
Multiple-input multiple-output (MIMO) wireless systems
employ multiple antennas at both the transmitter and the
receiver to offer higher data throughput and reliability than
single-antenna links. By aligning the transmitted signal with
the dominant eigenmode of the MIMO channel matrix, beam-
forming techniques allow MIMO systems to obtain full diver-
sity order and improve the bit-error-rate (BER) [1]. Beamform-
ing requires the availability of channel state information at the
transmitter. In the absence of perfect channel knowledge at the
transmitter, the CSI can be quantized at the receiver and sent
back to the transmitter using a low-rate feedback channel [2].
This technique is known as limited feedback. Different types
of limited feedback for beamforming have been developed
[3]–[5]. It was also shown that limited feedback beamforming
systems with only 5 or 6 feedback bits exhibit performance
very close to unquantized beamforming systems [3], [6].
Most prior work on limited feedback beamforming assumes
that the feedback channel is zero-delay such that instantaneous
CSI can be delivered to the transmitter. In practical systems,
however, the existence of feedback delay is inevitable due
to signal processing and transmission delays. Because of the
time-varying nature of the wireless channel, feedback delay
can cause a mismatch between CSI at the transmitter and the
actual channel state. In other words, feedback delay decreases
the performance of limited feedback beamforming systems.
This material is based in part upon work supported by the National
Science Foundation under grants CCF-514194 and CNS-626797, the
Air Force Research Lab under grant numbers FA8750-06-1-0091
and FA8750-05-1-0246, and the DARPA IT-MANET program, Grant
W911NF-07-1-0028.
Analytical results and measurements have characterized the
performance degradation of feedback delay in beamforming
[7]–[10]. Unfortunately these results make specific assump-
tions about the channel model. Recent work has shown a
more general Markov model to characterize capacity loss due
to feedback delay with limited feedback beamforming [11].
This paper, through exhaustive measurements of throughput
(goodput) to approximate capacity, confirms this analytical
result for predicting the observed effects of feedback delay. We
not only verify, but also show that simple measurements can
characterize the feedback delay tradeoff, thus avoiding exten-
sive measurement campaigns as conducted in this paper. The
use of Hydra, a flexible, software-defined wireless prototype,
is a valuable new measurement apparatus that considerably re-
duced the complexity of the substantial experiments performed
in this paper through a programmable physical layer.
Notation: A random variable X∈[a, b]with uniform
distribution is denoted X∼ U[a, b]. The transpose of matrix A
is labeled ATand the Hermitian transpose is given by AH.
Row xor column yof matrix Ais specified by [A]x,:or
[A]:,y, respectively. Similarly, the entry of matrix Aat row x
and column yis stated as [A]x,y. Finally, adopting engineering
notation, j=√−1.
II. SY ST EM DESCRIPTION
All limited feedback beamforming experiments in this paper
were conducted on the Hydra MIMO-OFDM multihop net-
work prototype at the University of Texas at Austin. The reader
is encouraged to see [12]–[14] for a more detailed discussion
of Hydra.
A. RF/Baseband Hardware Specification
Table I lists the relevant characteristics of the Hydra system
hardware for the experiments in this paper. The operating fre-
quency was placed at the upper edge of the 2.45 GHz indus-
trial, scientific, and medical (ISM) band, above all interference
observed by the spectrum analyzer during the experiment [12].
The choice of this frequency is relevant to IEEE 802.11n
wireless LANs that will soon occupy this spectrum with
beamforming-capable physical layers. For this experiment, the
bandwidth is not of considerable importance since, with the
exception of ultrawideband (UWB) channels, coherence time
statistics are not dependent on the bandwidth of operation [15].
The maximum transmit power is listed as 10 mW , but in
order to maintain linearity at the transmitter, typical operation

TABLE I
RF/BASEBAND HARDWARE SPECIFICATION FOR EXPERIMENTS
Specification Value
Operating Frequency 2.5GHz
Maximum Transmit Power 10 mW
Symbol Rate 1M Hz †
Antenna Type L-shaped Microstrip
Antenna Reflection Coefficient <−20 dB
Number of Receive Antennas 2
Number of Transmit Antennas 2
†Although the USRP supports 20 M Hz sampling rates,
the bandwidth is constrained by the USB 2.0 interface.
was below 7.5mW . Custom L-shaped copper microstrip
antennas were chosen over off-the-shelf commercial WLAN
antenna products because of their desirable S11 reflection
coefficient characteristics, which fell below −20 dB for the
entire spectrum of operation.
B. Software Architecture
Hydra features a completely software-defined protocol ar-
chitecture that runs on a general purpose processor. The
physical layer code is implemented using GNU Radio. This
open-source software allows developers to implement modular
signal processing blocks in C++ and flexibly connect them
together using python as a glue language.
C. Physical Layer Algorithms
Hydra follows the IEEE 802.11 MIMO-OFDM PHY as
provided by Task Group N in its Draft 2.0 Standard [16]. The
orthogonal frequency division multiplexing (OFDM) modula-
tion in this physical layer provides efficient equalization of
frequency selective fading. Moreover, the MIMO algorithms
in the standard allow for a framework to perform space-
time processing techniques along with feedback functionality.
Both of these features along with all the components found
in traditional PHYs, make its implementation desirable for
commercial products and experimentation alike. This physi-
cal layer simplified the design process considerably for the
experiments in this paper.
Conforming with the 802.11n physical layer, blocks of
binary data, that we shall denote packets, are transmitted inside
aframe. This frame format, as illustrated in Figure 1 (a), serves
many purposes. First, the training sequences in the packet
allow for asynchronous frame detection. Second, the training
sequences allow for measurement of impairments, such as
frequency offset, and relevant parameters, such as the wireless
channel impulse response for channel equalization. The data
payload is a waveform expression of the binary packet. The
parameters used to map the binary packet to a continuous
waveform are encoded in the header of the frame.
The only non-standard component used in this experiment
is the Extended Training. As mentioned earlier, the training
sequences in the frame can be used for channel impulse
response estimation. This is necessary to perform proper
equalization of the distortion effects of the wireless channel.
Unfortunately, transmission schemes, such as beamforming,
Training + Header Extended
Training Data Payload
(a)
Binary
Input
Bit
Scrambling
Bit
Encoding
Inter-
leaving
Bit Parsing
QAM
Modulation
QAM
Modulation
Spatial
Mapping
IFFT +
Add CP
IFFT +
Add CP
Baseband
Outputs
Inter-
leaving
(b)
Fig. 1. IEEE 802.11n frame format (a) and transmit processing (b)
become part of the effective wireless channel. That means that
CSI feedback from channel estimates of this training no longer
represent the channel of interest, but a beamformed version
of that channel. The 802.11n standard provides a method for
extracting the true channel from the beamformed version of
the channel. Preferring simplicity over efficiency, Hydra was
designed so that extra training information was sent in each
frame without any beamforming, or more generally (as we
shall see later) without spatial mapping. In the experiments,
the Extended Training field provided the CSI feedback as
well as the non-beamformed SNR estimates for comparing
beamforming and cyclic delay diversity.
The structure of the 802.11n PHY transmitter, as imple-
mented in Hydra, is shown in Figure 1 (b). For the purposes of
constructing a mathematical system model of the experiments,
the key component of the transmission structure is the spatial
mapping block. Let xl,m
k=hxl,m
1,k , xl,m
2,k , . . . , xl,m
NST S,k iT
rep-
resent the input to the spatial mapping block where xl,m
i,k ∈C,
for the kth subcarrier of the lth OFDM symbol in packet m,
and NST S is the number of space-time-streams (or distinct
streams) in the MIMO signal. Since the main function of
the system model is for representation of the transmission
strategies, the actual OFDM symbol and packet specified are
irrelevant. In other words, the system model holds equally for
all OFDM symbols in all packets. Therefore, the l,m notation
is dropped and xl,m
i,k =xi,k for the remainder of the paper.
The spatial mapping function s(•)maps the complex vector
of dimension NST S to a complex vector of dimension NTX ,
s:CNST S →CNT X , where NT X is the number of transmit
antennas. Given that the feedback transmission strategies will
not include spatial multiplexing or space-time block coding,
it can be assumed throughout the remainder of this paper that
NST S = 1 ⇒xk=x1,k
∆
=xk. Furthermore, the spatial
mapping function is restricted to the space of linear functions
such that s(xk)represents a matrix transformation. In other
words, s(xk) = Qkxk=qkxk=˜
xk, where ˜
xkis the spatially
mapped output and Qk∈CNT X ×NST S is the spatial mapping
matrix. Several transmit diversity strategies can be packaged
into this matrix framework, including digital beamforming
(BF) and cyclic delay diversity (CDD) [17], both of which
will be revisited after the Hydra receiver is discussed.
The receiver structure is displayed in Figure 2. Like the
transmitter, the key component of the receiver involves spatial
mapping, in this case extracting an estimate ˆ
xkof the kth

Frame
Synch
Packet
Detector
Remove
Freq Offset
Decode
Header
CRC
OK?
yes
no
Decode Payload
Return to Packet Detector
Baseband
Input
(a)
Payload
+
Training
Phase
Tracking
Channel
Estimation
Spatial
Equalization
FFT
Phase
Tracking
Soft Bits
Mapping
Soft Bits
Mapping
Deinter-
leaving
Deinter-
leaving
Soft Bits
Deparsing
Binary
Output
Bit
Scrambling
Viterbi
Decoder
FFT
Delete CP
+ Training
Processed OFDM Symbol by OFDM Symbol
(b)
Fig. 2. Hydra receiver synchronization (a) and data processing (b)
subcarrier in each OFDM symbol. This estimate of xkfollows
the combination of the spatial equalization and FFT blocks in
Figure 1. If frequency domain equalization is performed, this
obviates the need for an explicit FFT operation after the spatial
equalization. In a sense, the Spatial Equalization block is the
inversion of the combination of the wireless channel and the
spatial mapping block. It is assumed that, by using OFDM, the
wireless channel impulse response experienced by each sub-
carrier is frequency flat. Additionally, the model assumes that
the coherence time of the wireless channel impulse response
is large enough such that each packet observes a constant
channel over the duration of waveform transmission. Both of
these assumptions are often accepted for indoor channels [15].
Hence, let Hk∈CNRX ×NT X represent the complex baseband
frequency flat impulse response of the wireless channel for
OFDM subcarrier k. Effectively, the spatial equalization block
receives ykand maps it with function r(•)to ˆ
xk. If we also as-
sume that the receiver is linear, then r(yk) = GH
kyk=gHyk,
where G∈CNRX ×NST S . Note again that ˆ
xk= ˆx1,k = ˆxk
since NST S = 1. It is now possible to create the complex
baseband system model representing the input and output of
the effective channel (combined spatial mapping and wireless
channel). Thus,
ˆxk= (gH
kHkqk)xk(1)
may be used to represent cyclic delay diversity, digital
beamforming, limited-feedback beamforming and equalization
methods.
Cyclic delay diversity is a simple, yet effective method
for achieving full diversity order without knowledge of the
wireless channel impulse response at the transmitter. By apply-
ing a discrete-time, per-transmit-antenna circular shift to each
OFDM symbol it is possible to translate the spatial diversity
experienced by multiple transmit antennas into frequency
diversity. For bit-interleaved and coded OFDM systems, see
Figures 1 and 2, the forward error correction over the subcar-
riers captures the frequency diversity created in the wireless
channel. OFDM symbols preserve the properties of the discrete
Fourier transform (DFT) by adding a cyclic prefix. It follows
that the spatial mapping vector of cyclic delay diversity, for
each scalar element of qCDD
k,
qCDD
i,k
∆
= exp −j2πkNi,C S
NDF T (2)
where NDF T is the block size of the DFT and Ni,C S is the
number of discrete-time symbols to be circularly shifted on
transmit antenna i[18].
Digital beamforming, using channel state information feed-
back, attempts to transmit all of the energy over the maximum
eigenmode of the effective channel gkHHkqk. The optimal
beamforming solution, in terms of maximizing the effective
channel total energy, follows from the singular value decom-
position of the channel matrix, Hk=UkSkVH
k, where Uk
and Vkare unitary matrices and Skis a diagonal matrix of
dimension min{NT X , NRX }. Hence, assuming the diagonal
elements of Skare sorted by descending amplitude, the
optimal beamforming vector is
qBF
k
∆
= [Vk]:,1(3)
and the corresponding optimal combining vector
(gBF
k)H∆
=UH
k1,:(4)
provides the maximum energy effective channel response.
Note that vectors (3) and (4) are not unique and remain optimal
for any scalars ejφ and e−jφ , respectively, for φ∈R.
Limited feedback beamforming reduces the overhead of
transmitting the complete CSI over the feedback channel. By
creating finite-size codebooks with a specified bit-precision,
feedback can drastically be reduced using the beamforming
codeword index instead of the codeword itself. There exist
several ways to construct the codebook including vector quan-
tization [6] and Grassmanian subspace packing [3]. In this
paper the latter codebook is chosen with 32 elements (5-bit
precision). Assigning W32 the notation for the Grassmanian
codebook with 32 elements,
qLF BF
k
∆
= arg min
qk∈W32 kqk−qBF
kk2(5)
and
gLF BF
k
∆
=Hkqk
kHkqkk2
(6)
using (4) above assuming that qLF BF
kapproximates the dom-
inant right singular vector of Hk.
III. EXP ER IM EN T DESCRIPTION
The experiments conducted in this paper were designed to
measure a variety of statistics and metrics of our system that
provide insight into the feedback delay problem.
A. Prototype Setup and Indoor Wireless Channel
The experiments in this paper were performed in the indoor
office environment depicted in Figure 3. The transmitter and
receiver, separated by a distance of 10 m, are located in two
cubicles as might be the case in typical indoor office usage.
At a carrier frequency of 2.5GHz, mobility is primarily
determined by the motion of objects larger than 1×10−2m.
In the absence of movement in the environment, the wireless
channel will remain static as the electromagnetic propagation
in the environment does not change. The wireless channel in

WIRELESS LABORATORY
WIRELESS LABORATORY
WNCG OFFICE
TX
RX
Fig. 3. Floor plan of indoor office setting for feedback beamforming
experiment. The office has rows of cubicles, made with metal frames, and
a large metal support column.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
Channel Correlation
Channel Delay (seconds)
50 % Correlation ~ 250 msec
Fig. 4. Measured channel correlation statistics for indoor wireless channel.
Coherence time is approximately 250 msec.
such a static environment remains highly correlated for hun-
dreds of seconds, as verified through measurement in an empty
office. To produce meaningful results, it is necessary to create
a fading channel with stationary (or at least nearly stationary)
correlation statistics. In order to create such a fading channel
the antennas of each node were mounted on oscilating fans (on
the body of the fan, not on the rotating blades). The two fans
used in this experiment osilated with periods of 13.75 seconds
and 11.25 seconds respectively. This approach was necessary
since moving large objects around the office to create the
desired Doppler effect was not feasible.
Measuring correlation statistics using the fading channel
mentioned required gathering channel measurements. Channel
sounding was performed using repeated transmission of null
packets (i.e. packets containing only the IEEE 802.11n training
sequence and header). The correlation statistics computed from
this data, shown in Figure 4, indicates that the coherence time
of our indoor office setting is approximately 250 msec. This
corresponds to other indoor wireless channel measurements in
the literature [13].
B. Experiment Procedure
This experiment was conducted using a low-latency
and error-free wired feedback channel implemented over
a TCP/IP connection between the transmitter and receiver.
Special care was taken to ensure that feedback delay
was accurately produced and that CSI contained minimal
estimation error. The procedure used in conducting the
feedback experiments for this paper is described below.
P0: Pick a transmit power PT X as a random number with
uniform distribution PT X ∼ U [PT Xmin , PT X max ].
P1: Transmit a packet using CDD, i.e. without feedback at
transmit power PT X for baseline comparison.
P2: Wait for 50 msec to allow receive processing to complete.
P3: Transmit a sounding packet at maximum transmit power
to reduce estimation error in the estimate of the full
MIMO channel.
P4: Wait for desired delay minus some time to account for
processing delay and properly age feedback information.
P5: Transmit a packet using beamforming at transmit power
PT X . The beamforming vector is generated using CSI
obtained over the wired feedback channel.
P6: Repeat steps P4 and P5 for additional feedback delays.
P7: Repeat steps P0-P6 as needed to achieve desired precision
in BER statistics over SNR range of interest.
The procedure above was performed for a variety of coding
and modulation schemes. For each data rate, the impact
of feedback delay can be measured effectively only over a
specific SNR range. Capacity vanishes below this range, and
BER becomes very small above this range. As such, PTX min
and PT Xmax are tuned appropriately for each data rate to
ensure operation in the desired SNR range. It is important
to note that even though beamforming packets have variable
transmit power, sounding packets are always transmitted at the
maximum transmit power in the linear region of the transmit
amplifier. This reduces the chance that channel estimation error
adversely impacts beamforming.
After this procedure (i.e. sounding, wired feedback over
TCP/IP, and feedback aging), the transmitter can use the
uncompressed MIMO CSI to compute optimal beamforming
vectors, or quantize it and then compute limited feedback
beamforming vectors which are then used in the transmission
of step P5. In the case where feedback is not available due to a
dropped sounding packet (usually because of synchronization
error), all subsequent packets will be sent using CDD until
new feedback becomes available.
IV. RES ULT S AN D ANALYSIS
This section presents both the results and analysis of limited
feedback beamforming and CSI delay experiments. These
experiments use the physical layer algorithms of Section II
and the setup procedure in Section III.

A. Throughput Gain Results
Previous theoretical analysis has shown that, under certain
channel assumptions, there exists an upper bound on the capac-
ity gain, ∆C(D), that limited-feedback beamforming provides
(versus capacity of transmitter without CSI). Explicity,
∆C(D)∆
=CBF (D)− CU I (7)
for uniformed transmitter capacity CUI and limited feedback
beamforming capacity CBF as a function of delay D. Unfor-
tunately, system capacity is not necessarily easily translated
into realizable system performance. Past work has shown
that, for a fixed bit-error-rate, QAM constellations exhibit
a constant SNR (in dB) gap from capacity [19]. Therefore,
by transmitting QAM constellations and selecting the proper
constellation order and convolutional coding rate, it is possible
to maintain a relatively static bit error rate and thus, approx-
imate the capacity trend. Since the capacity of the wireless
channel in this experiment is unknown, we measure the impact
of feedback delay by observing the throughput of limited
feedback beamforming and cyclic delay diversity. For a single
spatial stream, the modulation and coding schemes (MCS) 0-
7 in IEEE 802.11n provide rates of 0.5to 5.0Mbps in the
Hydra physical layer. For the IEEE 802.11n physical layer we
define throughput as
TMC S,i(SNR)∆
=Ri(1 −BERi(SNR)) (8)
where Ri, BERiare the total rate and bit error rate for
MCS i, respectively. Using this definition of throughput we
further define throughput gain,∆T(D), for limited-feedback
beamforming as,
∆T(D)∆
=TBF (D)− TC DD (9)
for limited-feedback beamforming throughput TBF and cyclic
delay diversity throughput TCDD . Cyclic delay diversity is
used as a baseline comparison for uninformed transmission.
Cyclic delay diversity provides full diversity order without any
additional receiver processing. Using the frequency domain
spatial mapping framework from Section II it is shown that
cyclic delay diversity is a special case of transmit beamform-
ing.
The bit-errors in each MCS were measured for 1000 packets
over a range of SNR that covered the transition regions of
the throughput curve. This results in a scatter plot of bit-
errors per packet versus SNR measured from the extended
training. An example for MCS 4(16-QAM) can be observed
in Figure 5. The SNR values were binned to small intervals and
the BER was averaged over each bin. This BER versus SNR
curve for MCS iwas translated into throughput using equation
(8). Through visual inspection, the correct SNR regions were
observed for each MCS. Finally, to make the first and second
order derivatives of the throughput function continuous, a
cubic splines interpolation procedure was processed over the
data. Figure 6 demonstrates the throughput calculation for
limited feedback beamforming in a 5−bit codebook with
50 msec feedback delay and cyclic delay diversity.
0
0.1
0.2
0.3
0.4
0.5
0.6
6 8 10 12 14 16 18 20 22
Bit Error Rate
SNR (dB)
Fig. 5. Scatter plot of MCS 4 of experimental data
8 10 12 14 16 18 20
1
1.5
2
2.5
3
3.5
4
4.5
Cyclic Delay Diversity
Limited Feedback (50 msec)
Average Throughput Gain
0.27 bps/Hz
SNR (dB)
Throughput (bps/Hz)
Fig. 6. Throughput versus SNR for limited feedback beamforming with
50 msec feedback delay and cyclic delay diversity
Throughput curves were generated for 100 −1000 msec
feedback delay over all the MCS. In order to compare with
theoretical feedback delay effects from [11], the throughput
gain was calculated using (9) as an approximation of (7). The
resulting throughput calculations are displayed in Figure 7.
Also included in this graph, and as discussed in detail in the
next subsection, is the theoretical upper bound from [11].
B. Performance Analysis
Figure 6 observes the throughput gain of limited feedback
with 50 msec delay versus cyclic delay diversity for a baseline
comparison. As expected, limited feedback beamforming im-
0 0.2 0.4 0.6 0.8 1
0.05
0.1
0.15
0.2
0.25
0.3
Feedback Delay (seconds)
Throughput/Capacity Gain (bps/Hz)
Theoretical Upper Bound (β=0.886)
Measured Data
Best Fit (β=0.877)
Fig. 7. Throughput gain versus feedback delay for limiting feedback
beamforming and 5−bit codebooks

proves the performance of the system over all SNR. Since
the coherence time of the wireless channel was observed
to be approximately 250 msec, a 50 msec delay should
be very close to the maximum achievable throughput gain.
The throughput gain is averaged over 7−22 dB SNR to
yield 0.27 bps/Hz. Experimental results display degraded
performance likely as a result of impairments not observed in
typical wireless channel models. This can include frequency
offset, synchronization error, phase noise, channel estimation
error, and nonlinearities in hardware components.
The principle contribution of this experiment is shown in
Figure 7. This figure displays the loss in throughput versus
feedback delay. Analytical results from [11] show that the
capacity gain as a function of feedback delay can be bounded
by an exponential function. Specifically,
0≤∆C(D)≤αβD(10)
for scalar constants α, β ∈Rand D∈Nthe normalized delay.
Dis the number of state transitions taken in a Markov process
that models the temporal correlation of the wireless channel.
αrefers to the capacity gain of limited feedback beamforming
with zero feedback delay. βis the second largest eigenvalue of
the transition probability matrix which describes the evolution
of the wireless channel through quantized codebook space. For
example in our limited feedback beamforming experiment, we
measure the state of the channel every 50 msec. Each 50 msec
step corresponds to an increment of D. The transition prob-
ability matrix corresponds to the conditional probability that
the quantized limited feedback beamforming vector switches
between any two codebook indices.
In Figure 7, the measured data represents the observed
throughput gain as a function of feedback delay. Using an
exponential fit to the measured data with α= 0.3, the least
squares solution for the exponential term yields β= 0.877.
Additionally, by collecting statistics on the transitions of the
indices of the quantized codebook we were able to reconstruct
the transition probability matrix. This transition probability
matrix provides β= 0.886 from the 2nd largest absolute
eigenvalue. Therefore, we have two results that characterize
capacity (throughput) gain as an exponential function of the
feedback delay. The first result was obtained through ex-
haustive throughput measurements over various SNR, feed-
back delay, and MCS. The second result, however, only
required collecting statistics at a single SNR and feedback
delay (50 msec) to compute the transition probability matrix.
Although the second result theoretically represents an upper
bound on the capacity (throughput) gain as a function of
delay, this upper bound accurately approximates the capacity
(throughput) gain that we observed.
V. CONCLUSION
In this work we have characterized the impact of feedback
delay on the performance of limited feedback beamforming
through measurements in real wireless channels. These mea-
surements were obtained on a IEEE 802.11n draft 2.0 standard
MIMO-OFDM prototype. The measurement of throughput
gain versus feedback delay in this experiment verifies the
accuracy of analytically derived performance bounds [11].
Furthermore, verifying these results suggests a simple pro-
cedure to characterize the performance of feedback delay in
limited feedback beamforming without exhaustive throughput
measurements.
REFERENCES
[1] T. K. Y. Lo, “Maximum Ratio Transmission,” IEEE Trans. on Commu-
nications, vol. 47, pp. 1458–1461, October 1999.
[2] D. J. Love, R. W. Heath, Jr., W. Santipach, and M. L. Honig, “What Is
the Value of Limited Feedback for MIMO Channels?” IEEE Commu-
nications Magazine, vol. 42, pp. 54–59, October 2004.
[3] D. J. Love, R. W. Heath, Jr., and T. Strohmer, “Grassmannian Beam-
forming for Multiple-Input Multiple-Output Wireless Systems,” IEEE
Trans. on Info. Theory, vol. 49, pp. 2735–2747, October 2003.
[4] K. K. Mukkavilli, A. Sabharwal, E. Erkip, and B. Aazhang, “On
Beamforming With Finite Rate Feedback in Multiple-Antenna Systems,”
IEEE Trans. on Info. Theory, vol. 10, pp. 2562–2579, October 2003.
[5] V. K. N. Lau, Y. Liu, and T.-A. Chen, “Optimal Partial Feedback Design
for MIMO Block Fading Channels with Feedback Capacity Constraint,”
in IEEE International Symposium on Information Theory, June 2003,
p. 65.
[6] B. Mondal and R. W. Heath, Jr., “Performance Analysis of Quantized
Beamforming MIMO Systems,” IEEE Transactions on Signal Process-
ing, vol. 54, pp. 4753–4766, December 2006.
[7] A. Hottinen, R. Wichman, and O. Tirkkonen, Multi-Antenna Transceiver
Techniques for 3g and Beyond. New York, NY, USA: John Wiley &
Sons, Inc., 2003.
[8] J. Du, Y. Li, D. Gu, A. F. Molisch, and J. Zhang, “Estimation of Per-
formance Loss Due to Delay in Channel Feedback in MIMO Systems,”
in Proceedings of IEEE Vehicular Technology Conference 2004, vol. 3,
September 2004, pp. 1619–1622.
[9] D. Li and X. Dai, “On the performance of MIMO-OFDM Beamforming
Systems with Feedback Delay,” in International Conference on Wireless
Communications, Networking and Mobile Computing, September 2006,
pp. 1–4.
[10] E. Au, S. Jin, M. McKay, W. H. Mow, X. Gao, and I. Collings, “Ana-
lytical performance of mimo-svd systems in ricean fading channels with
channel estimation error and feedback delay,” Wireless Communications,
IEEE Transactions on, vol. 7, no. 4, pp. 1315–1325, April 2008.
[11] K. Huang, B. Mondal, R. W. Heath, Jr., and J. G. Andrews, “Effect of
Feedback Delay on Multiple-Antenna Limited Feedback for Temporally-
Correlated Channels,” in Proceedings of IEEE Global Telecommunica-
tions Conference, October 2006, pp. 1–5.
[12] K. Mandke, S.-H. Choi, G. Kim, R. Grant, R. C. Daniels, W. Kim,
S. M. Nettles, and R. W. Heath, Jr., “Early Results on Hydra: A
Flexible MAC/PHY Multihop Testbed,” in Proceedings of the 65th IEEE
Vehicular Technology Conference, Apr. 2007, pp. 1896–1900.
[13] K. Mandke, R. C. Daniels, S.-H. Choi, S. M. Nettles, and R. W. Heath,
Jr., “Physical Concerns for Cross-Layer Prototyping and Wireless Net-
work Experimentation,” in Proceedings of the Second ACM International
Workshop on Wireless Network Testbeds, Experimental Evaluation and
Characterization, 2007.
[14] “Hydra: A mimo-ofdm multihop wireless network prototype.” [Online].
Available: http://hydra.ece.utexas.edu
[15] A. Paulraj, R. Nabar, and D. Gore, Introduction to Space-Time Wireless
Communications. Cambridge University Press, 2003.
[16] Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY)
Specifications - Draft 2.0: Enhancements for Higher Throughput, Part
11 standard ed., IEEE 802.11n Working Group, February 2007.
[17] A. Dammann and S. Kaiser, “Standard conformable antenna diversity
techniques for OFDM and its application to the DVB-T system,” in
IEEE Global Telecommunications Conference, vol. 5, 2001, pp. 3100–
3105 vol.5.
[18] G. Bauch, “Capacity optimization of cyclic delay diversity,” in IEEE
Vehicular Technology Conference, vol. 3, 2004, pp. 1820–1824 Vol. 3.
[19] J. Cioffi, Digital Communications. Stanford University Course Reader,
2006.