Beamforming in MISO Systems: Empirical Results and EVM-based Analysis
11 Beamforming in MISO Systems: EmpiricalResults and EVM-based Analysis
Melissa Duarte, Ashutosh Sabharwal, Chris Dick, and Raghu Rao
Abstract
We present an analytical, simulation, and experimental-based study of beamforming Multiple InputSingle Output (MISO) systems. We analyze the performance of beamforming MISO systems taking intoaccount implementation complexity and effects of imperfect channel estimate, delayed feedback, realRadio Frequency (RF) hardware, and imperfect timing synchronization. Our results show that efficientimplementation of codebook-based beamforming MISO systems with good performance is feasible inthe presence of channel and implementation-induced imperfections. As part of our study we developa framework for Average Error Vector Magnitude Squared (AEVMS)-based analysis of beamformingMISO systems which facilitates comparison of analytical, simulation, and experimental results on thesame scale. In addition, AEVMS allows fair comparison of experimental results obtained from differentwireless testbeds. We derive novel expressions for the AEVMS of beamforming MISO systems andshow how the AEVMS relates to important system characteristics like the diversity gain, coding gain,and error floor.
Index Terms.
Beamforming, MISO systems, EVM, delayed feedback, noisy channel estimate,diversity gain, coding gain.
This work of first two authors was partially supported by NSF Grants CNS-0551692 and CNS-0619767. The first author wasalso supported by a Xilinx Fellowship and a Roberto Rocca Fellowship. The authors also thank Azimuth Systems for providingthe channel emulator used in this work.M. Duarte and A. Sabharwal are with the Department of Electrical and Computer Engineering, Rice University, Houston, TX,77005 USA, e-mail: { mduarte, ashu } @rice.edu.C. Dick and R. Rao are with Xilinx Inc., San Jose, CA, 95124 USA, e-mail: { chris.dick, raghu.rao } @xilinx.com.This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after whichthis version may no longer be accessible. a r X i v : . [ c s . I T ] D ec I. I
NTRODUCTION
Standards for next generation wireless communications have considered the use beamformingMultiple Input Single Output (MISO) systems with codebook-based feedback because thesesystems can potentially achieve same diversity order and larger coding gain compared to non-feedback systems like space-time codes [1–4]. Recently, the performance of beamforming MISOsystems has been analyzed taking into account errors in the channel estimate, and/or feedbackdelay [5–9], and noise in the feedback channel [10]. However, these results do not take intoaccount effects of non-ideal RF processing, imperfect timing synchronization or consider imple-mentation complexity.In this paper we evaluate the performance of codebook based beamforming MISO systemstaking into account implementation complexity and the presence of channel and implementation-induced imperfections. Specifically, we consider channel-induced imperfections which are dueto channel estimation errors and feedback delay and we consider implementation-induced im-perfections which are a result of imperfect timing synchronization and non-ideal RF process-ing, Automatic Gain Control (AGC), Analog to Digital Converters (ADCs), and Digital toAnalog Converters (DACs). Since not all imperfections can be modeled tractably, especiallyimplementation-induced imperfections, we adopt a mixed approach of analytical, simulation,and experimental evaluation. Analytical and simulation results presented in this paper take intoaccount channel-induced imperfections but do not take into account implementation-inducedimperfections because these imperfections are difficult to model in a tractable way. Thus, wecomplement these results with experimental results which do take into account both channel andimplementation-induced imperfections. This mixed approach provides a more complete pictureof expected performance.Inclusion of experimental evaluation poses a unique challenge in the choice of evaluationmetric. Common metrics like Bit Error Rate (BER) or Symbol Error Rate (SER) are usuallyanalyzed as a function of the average Energy per Symbol to Noise ratio ( E s /N o ) or averageEnergy per Bit to Noise ratio ( E b /N ). However, when real hardware is used for evaluation ofwireless systems, getting an accurate measurement of the noise or the E s /N o or E b /N o provesproblematic because the noise can be non-linear, both multiplicative and additive, and maydepend on radio settings and characteristics of the received signal. In contrast, the Average Error Vector Magnitude Squared (AEVMS), a metric commonly used in test equipment, can be easilymeasured since it is computed at the input of the demodulator. As a result, we propose to usethe AEVMS as a metric for performance analysis. This leads to the natural question regardingthe relationship between AEVMS and E s /N o or E b /N o .Our first contribution is a framework for AEVMS-based analysis of beamforming MISOsystems. Although the Error Vector Magnitude (EVM) and EVM-based metrics are heavily usedin industry for testing of wireless devices [11, 12], there is very little theory behind the use ofEVM for performance analysis. Some previous work can be found in [11–16] but no previouswork has analyzed the performance of beamforming MISO systems using an EVM relatedmetric. We present simulation, analytical, and experimental results that show how the AEVMSrelates to the E s /N o , BER, diversity gain, coding gain, and error floor. Since BER and AEVMSare quantities that can be directly measured, using these two metrics allows a straightforwardcomparison of analytical, simulation, and experimental results on the same scale. Furthermore,using metrics like BER and AEVMS facilitates comparison of results obtained with differentwireless testbeds because these metrics are usually easy to measure in any testbed. We showthat BER vs. (1/AEVMS) results can be used to analyze the diversity gain of a system. We alsoshow that coding gain and error floors can be analyzed by looking at the AEVMS performanceas a function of the E s /N o or an E s /N o related metric like the signal power.Our second contribution is the performance analysis of beamforming MISO systems as afunction of the amount of training used for channel estimation. In particular, we consider twodifferent beamforming systems: a 1 round (1R) system which uses only 1 round of training anda 1.5 round (1.5R) system which uses 1.5 rounds of training (we use the terminology for multiround training defined in [17]). We present novel results on the AVEMS vs. E s /N o performanceof the 1R and 1.5R systems in the presence of channel estimation errors and feedback delay.These results show that in the presence of feedback delay, 1.5 rounds of training eliminate theerror floor that is present when only one round of training is used. Taking into account noisychannel estimate and feedback delay, work in [5] analyzed the BER and SER for a 1R systemand work in [8] analyzed the capacity of a 1.5R system. However, previous work does notinclude comparison and AEVMS-based analysis of error floor of 1R and 1.5R systems in thepresence of imperfect channel estimate and feedback delay.Our third contribution is an experimental evaluation which demonstrates that efficient imple- mentation of codebook based beamforming MISO systems with good performance is feasible inthe presence of channel and implementation-induced imperfections. We show that beamformingcodebooks proposed in [18, 19], which are known to facilitate efficient implementation andstorage, can achieve performance close to infinite feedback (infinite codebook size) using onlyfew feedback bits (small codebook size) and have better performance than a space-time codesystem like Alamouti. This result had not been demonstrated in the presence of channel andimplementation-induced imperfections. Experimental results for beamforming systems have beenreported in [20, 21] but these works have not considered codebook based feedback. We alsoconsider the tradeoff between implementation complexity and performance in WiMAX compliantsystems. Our experimental results demonstrate that the Mixed Codebook scheme for WiMAXcompliant systems proposed in [19, 22] has good performance and simplifies implementation ofbeamforming in WiMAX compliant systems.The rest of the paper is organized as follows. Section II describes the channel model andimplementation requirements for the beamforming systems that are considered in this paper. Theframework for AEVMS-based analysis of beamforming MISO systems is presented in SectionIII, this section also presents error floor analysis of 1R and 1.5R systems. Section IV describesthe experimental setup and presents experiment results. Conclusions are presented in Section V.II. B EAMFORMING S YSTEM : M
ODEL AND C ODEBOOKS
A. Channel Model, Channel Estimation and Feedback Delay
We consider a MISO system with T transmit antennas and one receive antenna. The receivedsignal at time k is equal to r [ k ] = h [ k ] x [ k ] + n [ k ] , where the T × vector x [ k ] represents thetransmitted signal at time k , h [ k ] is the × T MISO channel at time k , and n [ k ] represents theadditive white Gaussian noise (AWGN) at the receiver, which is distributed as n ∼ CN (0 , N o ) .The channel vector h [ k ] is given by h [ k ] = [ h [ k ] , h [ k ] , ..., h T [ k ]] , where h i ∼ CN (0 , Ω) andthe entries of h [ k ] are i.i.d. Thus, h ∼ CN ( , Ω I ) .In this paper, we consider closed-loop beamforming based on receiver feedback. Using aunit norm × T beamforming vector w [ k ] , the vector input to the channel is determined as x [ k ] = √ E s w † [ k ] s [ k ] , where s [ k ] denotes the normalized constellation symbol transmitted attime k ( E [ | s [ k ] | ] = 1 ), E s is the average energy of the transmitted signal x [ k ] ( E [ (cid:107) x [ k ] (cid:107) ] = E s ),and () † denotes matrix transpose. Beamforming vectors are part of a predetermined codebook, known to both the transmitter and receiver prior to communication. Furthermore, the codebookis considered to be fixed throughout the communication.Since the channel is time-varying and unknown a priori, the receiver has to estimate thechannel based on training signals. Using training signals sent orthogonal in time with energy E p and assuming AWGN at the receiver, the channel estimate at the receiver is given by (cid:98) h [ k ] = h [ k ] + ∆h [ k ] , (1)where ∆h [ k ] represents the noise in the channel estimate distributed as ∆h ∼ CN ( , σ e I ) and σ e depends on the training signal energy E p to noise energy N o ratio [23]. Thus, the channelestimate is distributed as (cid:98) h ∼ CN ( , Λ I ) , with Λ = Ω+ σ e . The channel estimate in (1) applies toboth Minimum Mean Squared Error estimator and Maximum Likelihood estimator. In general,the training signal energy E p is not exactly equal to the signal energy E s . For example, inWiMAX systems the training energy is 2.5 dB higher than E s . Hence, we assume E p ∝ E s and σ e ∝ ( E s /N o ) − . (2)To account for errors in channel estimation and delay in the feedback channel, we use themodel presented in [5]. In the presence of a feedback delay of D seconds and noisy channelestimate as given in (1) we can write h [ k ] = ρ (cid:115) ΩΛ (cid:98) h [ k − D ] + (cid:113) (1 − | ρ | )Ω v [ k − D ] , (3)where v ∼ CN ( , I ) and ρ is the complex correlation coefficient given by ρ = E [ h i [ k ] (cid:98) h i [ k − D ] ∗ ] √ ΩΛ and we use () ∗ to denote conjugate transpose. As was shown in [5], the correlation coefficient ρ is related to the delay-only correlation coefficient ρ d and the estimation-error-only correlationcoefficient ρ e as ρ = ρ d ρ e where ρ d = E [ h i [ k ] h i [ k − D ] ∗ ]Ω and ρ e = E [ h i [ k ] (cid:98) h i [ k ] ∗ ] √ ΩΛ . Notice that ρ e canbe written in terms of Ω and σ e as [24] ρ e = (cid:113) Ω / (Ω + σ e ) and ρ d does not depend on E s /N o but ρ e does. Using (2) we have that lim EsNo →∞ ρ e = 1 . B. Beamforming with Imprecise Information
We consider a beamforming MISO system with B bits of feedback. The beamforming vector w [ k ] is chosen from a codebook of cardinality N = 2 B . We use an N × T matrix W torepresent a codebook for a system with T transmit antennas and codebook size N , and we use w i to represent the i -th row of matrix W . The beamforming codebook W is known to both thetransmitter and the receiver. The channel estimate at the receiver at time k − D is quantized intoone of the codewords in the codebook, quantization is performed via an exhaustive search overthe codewords in the codebook [3], b = arg max ≤ i ≤ N (cid:12)(cid:12)(cid:12)(cid:98) h [ k − D ] w † i (cid:12)(cid:12)(cid:12) . (4)Index b output by the channel quantizer is feedback to the transmitter and the transmitter choosesvector w b for beamforming (we assume error-free feedback channel). Hence, with a feedbackdelay of D , the beamforming vector used at time k is w [ k ] = w b and the received signal at time k is equal to r [ k ] = h [ k ] w † b (cid:113) E s s [ k ] + n [ k ] . (5)In the case of infinite feedback ( N = ∞ ) the beamforming vector is given by w [ k ] = (cid:98) h [ k − D ] ∗ || (cid:98) h [ k − D ] || . C. Codebooks
We consider four different types of beamforming codebooks which represent a tradeoff be-tween performance and implementation complexity, and represent the best known methodsspanning the two metrics. Specifically, we choose Maximum Welch Bound Equality (MWBE),WiMAX , Equal Gain Bipolar (EGB), and Tripolar codebooks. Description of these codebooks isshown in Table I. The EGB and Tripolar codebooks can be generated using the vector mappingtechniques in [19]. Also, EGB codebooks can be designed via a Kerdock code construction [18].Our previous work [19] showed that implementation of the channel quantization operation in(4) can require a large amount of complex multiplications depending on the codebook structure.Table I shows the amount of resources required for channel quantization for the four types ofcodebooks considered. Using an MWBE or a WiMAX codebook requires a large amount ofcomplex multipliers. In contrast, using an EGB or a Tripolar codebook does not require anycomplex multipliers. EGB and Tripolar codebooks allow implementation of complex multipli-cations for channel quantization using simple multiplexers as was shown in [19]. The EGBcodebook requires the least amount of resources among the four types of codebooks considered.For the Rayleigh i.i.d channel described in Section II-A and assuming an ideal scenariowhere there are no channel or implementation-induced imperfections, it is known that all thecodebooks considered in this paper have similar performance [2–4, 18, 19, 25–27]. WiMAX and
MWBE codebooks result in slightly better performance than EGB or Tripolar codebooks, but theperformance difference is usually less than 0.5 dB. Hence, in this ideal scenario, EGB codebooksare a good design choice because of their good performance and efficient implementation. Inthis paper we will investigate how MWBE, WiMAX, EGB, and Tripolar codebooks perform inthe presence of channel and implementation-induced imperfections.We will also present experimental evaluation of the WiMAX Mixed Codebook scheme pro-posed in our previous work in [19, 22]. In a WiMAX Mixed Codebook scheme, the WiMAXcodebook is used at the transmitter for beamforming while channel quantization at the receiver isimplemented using an EGB or a Tripolar codebook which is obtained by mapping the WiMAXcodebook. The mapping is performed as proposed in [19]. The Mixed Codebook scheme remainsWiMAX compliant because the mapped WiMAX codebook is only used for channel quantization.For more details on the WiMAX Mixed Codebook scheme please refer to [19, 22].III. E
RROR V ECTOR M AGNITUDE A NALYSIS OF B EAMFORMING
MISO S
YSTEMS
For a receiver demodulator using a normalized constellation, the AEVMS is given byAEVMS = E [ | s [ k ] − (cid:98) s [ k ] | ] , (6)where (cid:98) s [ k ] is the decision variable which is input to the demodulator. In this section we analyzethe AEVMS to understand the error floor, BER, diversity gain, and coding gain of a beamformingMISO system. The framework for AEVMS-based analysis developed in this section will be usedfor analysis of experimental results presented in Section IV. A. Training
The received signal r [ k ] is equalized to obtain the decision variable (cid:98) s [ k ] which is input to thedemodulator. The value used for equalization depends on the channel estimate obtained fromtraining. We consider a 1R system and a 1.5R system which differ in the amount of trainingthat is used for channel estimation. The two systems are explained below.
1) 1R system:
Fig. 1(a) shows a frame for a 1R system. The 1R system uses only one trainingsequence and the channel is estimated only once per frame. The channel estimate is computed atthe receiver upon reception of the training sequence. The channel estimate is used for computationof the feedback information and for equalization. The training sequence consists of transmission of a training signal from each transmitter antenna and training signals from different antennasare sent orthogonal in time. Two preambles are transmitted, one before the training sequenceand one before the payload, the preambles are used for AGC and timing synchronization. Thesecond preamble and the payload are beamformed.
2) 1.5R system:
This system uses two training sequences, as shown in Fig. 1(b). The firsttraining sequence is exactly the same as the training sequence used in the 1R system. The secondtraining sequence is the beamformed version of the first training sequence. Two preambles, usedfor AGC and timing synchronization, are transmitted before each training sequence. The secondpreamble and the payload are beamformed. In the 1.5R system the channel is estimated twice.The first estimate is computed from the first training sequence and this estimate is used forcomputation of the feedback information. The second channel estimate is computed after thesecond training sequence, since the second training sequence is beamformed, the second channelestimate corresponds to the equivalent beamformed channel. The estimate of the equivalentbeamformed channel is used for equalization.The second training sequence is beamformed because of two reasons. First, beamforming thesecond training sequence simplifies the implementation of the equalizer. Second, although weassume noiseless feedback, estimating the equivalent beamformed channel may be useful in asystem with noisy feedback where the codeword chosen by the receiver is not the codewordbeing used by the transmitter. In a noisy feedback system it may be better to get an estimateof the equivalent beamformed channel and use this channel estimate for equalization, insteadof estimating the channel and then computing the equalization signal using this estimate andthe codeword chosen by the receiver. This intuition is based on results presented in [17] forfeedback based power control schemes, where it is shown that using power controlled trainingimproves performance in a noisy feedback system.We use q [ k ] to denote the beamformed channel at time k . The × T vector q [ k ] is given by q [ k ] = [ q [ k ] , q [ k ] , ..., q T [ k ]] where q i [ k ] = h i [ k ] w b,i and w b,i denotes the i -th entry of vector w b .The additive Gaussian noise in the estimation of q [ k ] is same as in the estimation of h [ k ] andthe estimator used is also the same. Hence, the estimate of the equivalent beamformed channelis given by (cid:98) q [ k ] = q [ k ] + ∆q [ k ] , where ∆q ∼ CN ( , σ e I ) . B. Equalization and Decision Variable
We now compute the decision variable for the 1R system and for the 1.5R system. Resultspresented in this section and in Sections III-C to III-E take into account channel-iduced imper-fections and assume there are no implementation-induced imperfections.
1) 1R system:
The receiver knows (cid:98) h [ k − D ] w † b √ E s and the decision variable is equal to (cid:98) s R [ k ] = r [ k ] ( (cid:98) h [ k − D ] w † b √ E s ) ∗ | (cid:98) h [ k − D ] w † b √ E s | . Substituting r [ k ] and h [ k ] using (5) and (3) respectively we obtain. (cid:98) s R [ k ] = ρ (cid:115) ΩΛ s [ k ] + (cid:113) (1 − | ρ | )Ω v [ k − D ] w † b (cid:16)(cid:98) h [ k − D ] w † b (cid:17) ∗ (cid:12)(cid:12)(cid:12)(cid:98) h [ k − D ] w † b (cid:12)(cid:12)(cid:12) s [ k ]+ n [ k ] √ E s (cid:16)(cid:98) h [ k − D ] w † b (cid:17) ∗ (cid:12)(cid:12)(cid:12)(cid:98) h [ k − D ] w † b (cid:12)(cid:12)(cid:12) , (7)
2) 1.5R system:
The receiver has knowledge of (cid:98) a [ k ] √ E s where (cid:98) a [ k ] = (cid:80) Ti =1 (cid:98) q i [ k ] = (cid:80) Ti =1 q i [ k ]+ (cid:80) Ti =1 ∆ q i [ k ] = a [ k ] + ∆ a [ k ] . We use (cid:98) q i [ k ] and ∆ q i [ k ] to denote the i -th entry of vectors (cid:98) q [ k ] and ∆q [ k ] respectively and we define a [ k ] = (cid:80) Ti =1 q i [ k ] and ∆ a [ k ] = (cid:80) Ti =1 ∆ q i [ k ] . Using (5)and the expressions for a [ k ] and (cid:98) a [ k ] above, we obtain the decision variable for the 1.5R system (cid:98) s . R [ k ] = r [ k ] ( (cid:98) a [ k ] √ E s ) ∗ | (cid:98) a [ k ] √ E s | = a [ k ] (cid:98) a [ k ] s [ k ] + 1 (cid:98) a [ k ] √ E s n [ k ] . (8) C. AEVMS of a beamforming MISO System
In this section we compute the AEVMS for the 1R and 1.5R systems.
1) 1R system:
Using (6) with (cid:98) s [ k ] substituted with (7) we obtainAEVMS R = E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:32) − ρ (cid:115) ΩΛ − (cid:113) (1 − | ρ | )Ω v [ k − D ] w † b (cid:16)(cid:98) h [ k − D ] w † b (cid:17) ∗ (cid:12)(cid:12)(cid:12)(cid:98) h [ k − D ] w † b (cid:12)(cid:12)(cid:12) (cid:33) s [ k ] − n [ k ] √ E s (cid:16)(cid:98) h [ k − D ] w † b (cid:17) ∗ (cid:12)(cid:12)(cid:12)(cid:98) h [ k − D ] w † b (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (9)Since s [ k ] , n [ k ] , v [ k − D ] , and (cid:98) h [ k − D ] are independent and E [ | s [ k ] | ] = 1 , E [ | n [ k ] | ] = N o ,and E [ n [ k ]] = 0 , we can simplify the expression above to obtainAEVMS R = 1 − Re { ρ } (cid:115) ΩΛ + | ρ | ΩΛ + (1 − | ρ | )Ω E (cid:12)(cid:12)(cid:12) v [ k − D ] w † b (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:98) h [ k − D ] w † b (cid:12)(cid:12)(cid:12) + E (cid:12)(cid:12)(cid:12)(cid:98) h [ k − D ] w † b (cid:12)(cid:12)(cid:12) N o E s . (10) In the Appendix we show that E (cid:12)(cid:12)(cid:12) v [ k − D ] w † b (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:98) h [ k − D ] w † b (cid:12)(cid:12)(cid:12) = E (cid:12)(cid:12)(cid:12)(cid:98) h [ k − D ] w † b (cid:12)(cid:12)(cid:12) (11)and E (cid:12)(cid:12)(cid:12)(cid:98) h [ k − D ] w † b (cid:12)(cid:12)(cid:12) = 1Λ( T −
1) Ψ (12)where
Ψ = T − T N − T − F , T, T, (cid:18) N (cid:19) T − . (13)Substituting (11) and (12) in (10) we obtain AEVMS R in closed formAEVMS R = 1 − Re { ρ } (cid:115) ΩΛ + | ρ | ΩΛ + (1 − | ρ | )Ω 1Λ( T −
1) Ψ + 1Λ( T −
1) Ψ 1 E s /N o . (14)In the case of infinite feedback we have that N = ∞ hence Ψ = 1 .
2) 1.5R system:
For a 1.5R system the AEVMS, computed using (6) and (8), is equal toAEVMS . R = E (cid:34)(cid:12)(cid:12)(cid:12)(cid:12) − a [ k ] (cid:98) a [ k ] (cid:12)(cid:12)(cid:12)(cid:12) (cid:35) + E (cid:34) | (cid:98) a [ k ] | (cid:35) E s /N o . (15)We do not have a closed form expression for AEVMS . R because we do not have a closed formexpression for the expectations in (15). Computing these expectations is complicated becausethey depend on the channel estimate at time k and the quantized channel estimate at time k − D .However, (15) allows us to do an asymptotic (large E s /N o ) analysis of a 1.5R system and wealso provide simulation results and experiment results that show how this system performs. D. Relation Between AEVMS and Error Floor
At infinite E s /N o we expect the AEVMS to be equal to zero. If this is not the case then thesystem has an error floor. If the system has an error floor then it will not be possible to decreasethe AEVMS below a certain value greater than zero no matter how large the E s /N o . We nextshow that in the ideal case of no feedback delay and no channel estimation error, the 1R and1.5R systems do not have an error floor. However, when feedback delay and channel estimationerrors are taken into account, the 1R system has an error floor while the 1.5R system does not. Definition 1:
An error floor exists if lim
EsNo →∞ AEVMS > . An error floor does not exist if lim
EsNo →∞ AEVMS = 0 . Proposition 1:
For the 1R system, the following is true. (a)
In the ideal case of no feedback delay ( D = 0 hence ρ d = 1 ) and no channel estimationerror ( σ e = 0 ), the 1R system does not have an error floor and the AEVMS given byAEVMS R | σ e =0 ,ρ d =1 = 1Ω( T −
1) Ψ 1 E s /N o . (16) (b) In the case of feedback delay ( | ρ d | < ) and channel estimation error such that σ e satisfies(2), the 1R system has an error floor. Proof:
Part (a) follows from Definition 1 and simplification of (14) using ρ d = 1 and σ e = 0 .Part (b) follows from Definintion 1 and use of (14), | ρ d | < , lim EsNo →∞ ρ e = 1 , and lim EsNo →∞ Λ =Ω to compute lim
EsNo →∞ AEVMS R | σ e ∝ ( E s /N o ) − , | ρ d | < = | ρ d − | + (1 − | ρ d | ) T − Ψ > . Proposition 2:
For the 1.5R system, the following is true. (a)
In the ideal case of no feedback delay ( D = 0 hence ρ d = 1 ) and no channel estimationerror ( σ e = 0 ), the 1.5R system does not have an error floor and the AEVMS given byAEVMS . R | σ e =0 ,ρ d =1 = 1Ω( T −
1) Ψ 1 E s /N o . (17) (b) In the case of feedback delay ( | ρ d | < ) and channel estimation error such that σ e satisfies(2), the 1.5R system does not have an error floor. (c) In the case of feedback delay ( | ρ d | < ) and channel estimation error such that σ e satisfies(2), the AEVMS of the 1.5R system at high E s /N o can be approximated asAEVMS . R | σ e ∝ ( E s /N o ) − , | ρ d | < ≈ E (cid:34) | a [ k ] | (cid:35) E s /N o . (18) Proof:
Part (a) follows from Definition 1 and simplification of (15) using ρ d = 1 and σ e = 0 . Part (b) follows from Definition 1 and use of (15), | ρ d | < , lim EsNo →∞ (cid:98) a [ k ] = a [ k ] and lim EsNo →∞ E [1 / | (cid:98) a [ k ] | ] = E [1 / | a [ k ] | ] to compute lim EsNo →∞ AEVMS . R | σ e ∝ ( E s /N o ) − , | ρ d | < = 0 .Proof of part (c) is as follows. At high E s /N o we can approximate σ e ≈ , E [ | − a [ k ] / (cid:98) a [ k ] | ] ≈ ,and E [1 / | (cid:98) a [ k ] | ] ≈ E [1 / | a [ k ] | ] . Using these approximations in (15) we obtain (18).Proposition 1(b) and Proposition 2(b) are verified via simulation results shown in Fig. 2(a) andFig. 2(b). Fig. 2 shows results for five different scenarios labeled as SN1, SN2, ... , SN5. Fig.2(d) specifies the legend for Fig. 2(a), Fig. 2(b), and Fig. 2(c). All results in Fig. 2 correspondto 16-QAM modulation and Ω = 1 . The legend in Fig. 2(d) specifies if results were obtainedvia simulation or analytically and if results correspond to a 1R or a 1.5R system. Results shown in Fig. 2(a) and Fig. 2(b) show that for ρ d = 0 . and σ e = ( E s /N o ) − / the 1R system has anerror floor and the 1.5R system does not have an error floor. Error floors are usually identifiedby observing the BER vs. E s /N o performance. Equivalently, errors floors and their causes canbe identified by analyzing the performance of the AEVMS as a function of the E s /N o .Intuitively, results in Proposition 1(b) and Proposition 2(b) can be explained as follows. Forequalization, the 1R system uses (cid:98) h [ k − D ] w † b √ E s and the 1.5R system uses (cid:98) a [ k ] √ E s . Ideally, h [ k ] w † b √ E s should be used for equalization. Since σ e → as E s /N o → ∞ , then for high E s /N o the signal used for equalization in the 1.5R system approximates to the ideal value, while forthe 1R system the signal used for equalization approximates to h [ k − D ] w † b √ E s .Results asymptotic in E s /N o for the AEVMS of the 1.5R system in Proposition 2(c) areverified in Fig. 2(b). Notice that at high E s /N o the AEVMS of the 1.5R system decays pro-portionally with the increase in E s /N o , as stated in (18). Fig. 2 also shows that analytical andsimulation results for the 1R system match accurately. Analytical results presented in Fig. 2(a)were computed using the equations for BER derived in [5], these BER equations were also usedto obtain the analytical results in Fig. 2(c) by writing E s /N o in terms of the AEVMS using(14). Analytical results in results in Fig. 2(c) were computed using (14). In Fig. 2 we have onlyincluded analytical results for SN1 in order to avoid cluttering the graphs. We note that analyticalresults for finite N were obtained assuming an optimum codebook (see Appendix) and we haveverified that this assumption yields analytical results which match accurately simulation resultsfor MWBE, WiMAX, EGB and Tripolar codebooks and different values of N , T , σ e , ρ d and Ω . E. Relation Between AEVMS, BER, Diversity Gain and Coding Gain
Based on preceding analysis, the BER vs. E s /N o plot can be decomposed into two plots: BERvs. (1/AEVMS) and AEVMS vs. E s /N o . This decomposition is shown in Fig. 2. For example,from Fig. 2(a) we can observe that for SN4 a BER of − is obtained at E s /N o ≈ dB.Equivalently, one can first observe from Fig. 2(c) that for SN4 a BER of − is obtained atAEVMS ≈ − dB (i.e an 1/AEVMS value of 18 dB) and then observe from Fig. 2(b) that anAEVMS of − dB is obtained at E s /N o ≈ dB. In order to decompose the BER vs. E s /N o plot into BER vs. (1/AEVMS) and AEVMS vs. E s /N o , one computes (analytically) or measures(in simulation) the BER and AEVMS corresponding to a given E s /N o value. A system has a diversity gain G d and coding gain G c if at high E s /N o the BER scales as [28]BER ≈ (cid:18) G c E s N o (cid:19) − G d . (19)Hence, when (19) is a valid approximation, the BER vs. E s /N o plot captures the coding anddiversity gain of a system and in a log-log scale this plot is well approximated by a straight linethat decays with slope G d and has a horizontal shift of G c dB relative to the benchmark curveof ( E s /N o ) − G d [29]. In this section we show that when (19) is a valid approximation and theAEVMS can be approximated as AEVMS ≈ β E s /N o , (20)where β is a positive and finite constant, we have the following. When the BER vs. E s /N o plot isdecomposed into BER vs. (1/AEVMS) and AEVMS vs. E s /N o plots, we have that the diversitygain and the part of the coding gain that depends on the modulation scheme (constellation) arecaptured by the BER vs. (1/AEVMS) plot and the part of the coding gain that does not depend ofthe modulation scheme is captured by the AEVMS vs. E s /N o plot. The following result showsthe relation between AEVMS and diversity gain. Proposition 3:
If at high E s /N o the AEVMS can be approximated as in (20) and the BERcan be approximated as in (19), then at high E s /N o , or equivalently at high values of 1/AEVMS,the BER as a function of the AEVMS is given byBER ≈ (cid:18) G c β AEVMS (cid:19) − G d (21)and the the diversity gain G d can be computed as G d = − lim AEVMS →∞ log BER log AEVMS . (22)Consequently, the BER vs. (1/AEVMS) curve plotted in a log-log scale decays with slope G d and the BER vs. (1/AEVMS) plot captures the diversity gain. Proof:
Solving for E s /N o from (20) and substituting the result in (19) we obtain (21). (22)can be readily verified by substituting (21) in (22) and computing the limit.As an example, consider the case of no feedback delay and no channel estimation errors. In thiscase the SER (hence the BER) can be approximated as in (19) at high E s /N o [29] and the 1Rand 1.5R systems have the same AEVMS which satisfies (20) for all values of E s /N o (as can be seen from (16) and (17)). Hence, by solving for E s /N o from (16) or (17) and substituting in(19) we obtain that at high 1/AEVMS the BER for ρ d = 1 and σ e = 0 can be approximated asBER ≈ (cid:32) G c T −
1) Ψ 1
AEVMS (cid:33) − G d . (23)In the presence of feedback delay and channel estimation errors, the AEVMS of the 1Rsystem cannot be approximated as in (20) because the system has an error floor, as was shownin Propostion 1(b). Because of the error floor the BER of the 1R system does not decay to zero,as can be seen Fig. 2(a) and Fig. 2(c), and (19) is not a valid approximation. For the 1.5R systemand taking into account feedback delay and channel estimation errors, Proposition 2(c) showsthat the AEVMS at high E s /N o can be approximated as in (20) with β = E [1 / | a [ k ] | ] . TheBER for the 1.5R system taking into account feedback delay and noisy channel estimate is notknown. However, we note that it was shown in [6] that for a 1.5R system with feedback delayand perfect channel estimation (19) is a valid approximation and the diversity gain reduces toone when | ρ d | < . This is consistent with results in Fig. 2(a) which suggest that there is a lossin diversity gain (diversity gain less than T ) as can be observed from the slope of the curvesfor the 1.5R system at high E s /N o . Notice that for the 1.5R system, the slope of decay of theBER vs. E s /N o curves is the same as for the BER vs. (1/AEVMS) curves, hence, the diversitygain is captured by the BER vs. (1/AEVMS) curves, this is consistent with Proposition 3.We now analyze the relation between AEVMS and coding gain. Part of the coding gain ofa beamforming MISO system depends on the modulation (e.g. MPSK or MQAM) being used.We use G c,m to label the part of the coding gain that depends on the modulation being used andwe use G c,p to label the part of the coding gain that does not depend on the modulation beingused. G c,p is mainly due to power gain or effective increase in received signal power as definedin [28]. The following conjecture relates the AEVMS and the coding gain. Conjecture 1:
For a beamforming MISO system with channel estimation errors and feedbackdelay the following is true. If at high E s /N o the AEVMS can be approximated as in (20) andthe BER can be approximated as in (19), then at high E s /N o , or equivalently at high values of1/AEVMS, G c,m and G c,p satisfy the following. (a) G c,m , which is the part of the coding gain that depends on the modulation being used, iscaptured by the BER vs. (1/AEVMS) curve. (b) G c,p , which is the part of the coding gain that does not depend on the modulation being used, is captured by the AEVMS vs. E s /N o curve.Results in (14) and (15) do show that the AEVMS vs. E s /N o curve is independent of themodulation. A formal proof for Conjecture 1 would require a general expression for the codinggain of the 1R and 1.5R systems taking into account channel estimation errors and feedback delay,this general expression is not known. We have performed extensive simulations that indicate thatConjecture 1 will most likely hold. Below we give an example.As can be observed from Fig. 2(a), at high E s /N o the difference in performance betweenSN4 and SN5 is due to coding gain (at high E s /N o the BER vs. E s /N o curves differ only by ahorizontal shift). Observe from Fig. 2(c) that the BER vs. (1/AEVMS) plots for SN4 and SN5lie on top of each other. Hence, from Proposition 3 we have that SN4 and SN5 have the same G d and from Conjecture 1(a) we have that SN4 and SN5 have the same G c,m , which is consistentwith the fact that results for SN4 and SN5 are both for 16 QAM. Since G d and G c,m for SN4and SN5 are the same, then the difference in coding gain between SN4 and SN5 is only dueto a difference in G c,p . Hence, from Conjecture 1(b), the horizontal shift between the BER vs. E s /N o plots for SN4 and SN5 must be equal to the horizontal shift between the AEVMS vs. E s /N o plots for SN4 and SN5 and this can be verified from Fig. 2(a) and Fig. 2(b).IV. E XPERIMENT S ETUP AND R ESULTS
In Section III we presented an AEVMS-based analysis of beamforming MISO systems whichaccounted for effects of delay and channel estimation errors. In order to simplify analysis,we did not take into account implementation-induced imperfections. In this section we presentan empirical evaluation of beamforming MISO systems which was conducted using WARP[30] and a wireless channel emulator [31, 32]. Using a channel emulator allowed us to controlchannel related parameters like Ω and ρ d . In addition, using real hardware for transmission andreception of RF signals allowed us to obtain results which account for real-world hardwareeffects. Hence, our experimental results take into account both channel and implementation-induced imperfections. We present experiment results using the AEVMS-based framework wepresented in Section III. A. Experiment Setup and Scenarios Considered
Experiments were implemented using the WARPLab framework [33] which allows rapidprototyping of physical layer algorithms by combining the ease of MATLAB with the capabilitiesof WARP. The WARPLab framework provides the software necessary for easy interaction withthe WARP nodes directly from the MATLAB workspace, the software consists on FPGA codeand MATLAB m-code functions, which are all available in the WARP repository [34]. TwoWARP nodes were used, one as a transmitter node and the other one as the receiver node. Themain component of the WARP node hardware is a Xilinx Virtex-II Pro FPGA. Each node alsohas four daughter card slots, each slot is connected to a dedicated bank of I/O pins on the FPGA,these daughter card slots were used to connect the FPGA to up to four different radio boards.For our experiments, we used two and four radios at the transmitter to build a × and a × MISO system respectively. At the receiver, only one radio board was used.The experiments were implemented using the basic WARPLab setup [33] where two WARPnodes are connected to a host PC via an Ethernet switch. The baseband waveforms (samples)were constructed in MATLAB and the samples were stored in buffers on the FPGA on thetransmitter node, download of samples from the MATLAB workspace to the FPGA buffers wasdone using the software provided in the WARPLab framework. A trigger signal sent from thehost PC to the WARP nodes started transmission of samples from the transmitter node andstorage of received samples on buffers on the receiver node. The radio boards at the transmitternode upconverted the baseband samples to RF waveforms and the radios at the receiver nodedownconverted the received RF signal to baseband samples that were stored on the buffers on thereceiver FPGA. The samples in the receive buffers were loaded to the MATLAB workspace onthe host PC using functions from the WARPLab framework. Processing of the received basebandsamples was done in MATLAB. The error-free feedback channel was implemented in the hostPC.Experiments were performed for a × and a × MISO system. We only implementedthe 1.5R system in Fig. 1(b). The 1R system was not considered for experimental evaluationbecause, as was shown in Section III, the 1.5R system outperforms the 1R system for a largerange of E s /N o values. In order to compare the performance of a feedback-based system likebeamforming with a non-feedback-based system like Alamouti, we also implemented and tested a × Alamouti scheme [35] using the WARPLab framework. For the Alamouti implementationonly one training sequence was sent and payload was sent immediately after the training sequencewas transmitted. The rest of the experiment conditions in the Alamouti implementation wereequal to the experiment conditions in the beamforming implementation. Experiment conditionsare shown in Table II. We note that the number of payload symbols per frame was limited to110 due to the characteristics of the transmitted signal (128 samples per symbol plus samplesused for training and preamble) and the maximum number of samples that can be stored perreceiver radio in a WARP node ( samples). The clock was shared between the transmitterand the receiver to avoid carrier frequency offset effects. The wireless channel emulator wasset so that an RF link was enabled from each transmitter radio to the receiver radio in order toemulate a MISO system and each RF link consisted of three paths. Since the delay spread wasmuch smaller than the symbol period the transmitted signal went through a flat fading channel.The emulated channel corresponds to the channel model described in Section II-A. B. Empirical Results Using a Wireless Channel Emulator
Results obtained using the channel emulator are presented in Fig. 3. Fig. 3(d) specifies thelegend for Fig. 3(a), Fig. 3(b) and Fig. 3(c), the seven different scenarios that were evaluatedvia experiments are labeled as EXP1, EXP2, ..., EXP7. Fig. 3(c) also includes simulation resultsfor a × and a × MISO system with σ e = ( E s /N o ) − / . (noise variance of the channelestimate is 3.6 dB lower than ( E s /N o ) − to match the fact that in experiments the total trainingsignal energy per antenna was 3.6 dB larger than the total energy per symbol), ρ d = 0 . (asin experiments), and Ω = 1 . Including the effect of the channel, the average energy per symbolis equal to Ω E s and the average energy per symbol to noise ratio is equal to Ω E s /N o . In theexperiments, the emulator output power is equal to Ω E s .Results in Fig. 3(a) verify that a feedback system like beamforming has better performancethan a non feedback system like Alamouti. From results in Fig. 3(a) we also observe that theperformance of MWBE, EGB, and WiMAX codebooks is approximately the same. For T = 2 and N = 4 , the curves corresponding to the EGB codebook and the MWBE codebook areapproximately the same, for parts of the curves it may seem that one codebook has better per-formance than the other but the curves cross each other several times indicating the performancefor the two codebooks is approximately the same. Similarly, we observe that for T = 4 and N = 64 the EGB and the WiMAX codebook have similar performance.Results for T = 4 in Fig. 3(a) show that the diversity gain with infinite feedback is thesame as the diversity gain with finite feedback, since the BER curves appear to decay with thesame slope (EXP4, EXP5, EXP6 and EXP7 decay with approximately same slope). The onlydifference between infinite and finite feedback is the coding gain, as can be observed from thehorizontal shift for T = 4 curves in Fig. 3 (a) (shift of EXP7 with respect to EXP4, EXP5 andEXP6). The difference in performance between infinite feedback and finite feedback is between1 dB and 2 dB for most of the average received signal powers considered.Results in Fig. 3(a) show that the WiMAX Mixed Codebook scheme (EXP4) has worseperformance than the WiMAX scheme (EXP5) and the performance loss is approximately 1dB.In the WiMAX Mixed Codebook scheme used to obtain EXP4 result, a Tripolar codebook wasused for channel quantization. Using an EGB instead of a Tripolar codebook would allow a moreefficient implementation but results in [19] showed that this would result in a worse performanceof the Mixed Codebook scheme. There is a tradeoff between implementation complexity andperformance; using a WiMAX Mixed Codebook scheme simplifies the implementation of thechannel quantizer but results in a small performance degradation.In the presence of channel and implementation-induced imperfections, results in Fig. 3(a)demonstrate that EGB codebooks have good performance. Since EGB codebooks also allowefficient implementation, we conclude that EGB codebooks are the best option out of the fourtypes of codebooks considered. Results in Fig. 3(a) also demonstrate that in a WiMAX compliantsystem, a WiMAX Mixed Codebook scheme using a Tripolar codebook offers a good tradeoffbetween implementation complexity and performance.BER vs. Ω E s results in Fig. 3(a) can be used to compare different experimental results butare not useful to compare experimental results with simulation or analytical results. Translating Ω E s values to E s /N o or vice versa is complicated because measuring N o or E s /N o is notstraightforward since the noise can be non-linear, both multiplicative and additive and maydepend on radio settings and characteristics of the received signal. Hence, translating the BERvs. Ω E s results into BER vs. E s /N o or vice versa proves problematic. To facilitate comparisonbetween simulation and experimental results we decompose results in Fig. 3(a) into BER vs.(1/AEVMS) and AEVMS vs. Ω E s , as show in Fig. 3(c) and Fig. 3(b) respectively. In order to dothis decomposition one measures the BER and the AEVMS for a given Ω E s . This decomposition is analogous to the one done in Section III-E where the BER vs. E s /N o plot was decomposedinto BER vs. (1/AEVMS) and AEVSM vs. E s /N o . BER and AEVMS are metrics that can beeasily measured (the AEVMS is computed before the demodulator and the BER is computed afterthe demodulator) and are commonly measured in testing of wireless devices [11, 12]. Hence,using BER vs. (1/AEVMS) for performance analysis allows a straightforward comparison ofexperimental results with simulation results on the same scale, as shown in Fig. 3(c). Resultsin this figure show that experimental results match closely simulation results, there are somedifferences but these may be due to hardware effects that were not considered in simulations.It is important to keep in mind that, as shown in Section III-E, part of the coding gain of asystem is not captured by the BER vs. 1/AEVMS plots. Differences in coding gain that are notcapture in the BER vs. 1/AEVMS plots can observed by plotting the AEVMS as a function ofthe E s /N o , as shown in Section III-E, or an E s /N o related metric like Ω E s . As an example,consider results in Fig. 3(a) for T = 4 . All curves for T = 4 decay with approximately the sameslope and the main difference between curves is a horizontal shift, hence, the main differencebetween results is due to a difference in coding gain. However, curves for T = 4 in Fig. 3(c)are approximately the same, consequently at least part of the coding gain is not being capturedby the BER vs. (1/AEVMS) plots. As can be seen in Fig. 3(b), part of the coding gain that isnot captured by the BER vs. (1/AEVMS) results is captured by the AEVMS vs. Ω E s results.BER vs. (1/AEVMS) and AEVMS vs. Ω E s plots can also be used to facilitate comparisonof results obtained with different wireless testbeds. As we have mentioned, BER and AEVMScan be directly measured. Hence, BER vs (1/AEVMS) results obtained with different wirelesstestbeds can be directly compared without need for calibration between testbeds. Also, AEVMSvs. Ω E s results can be used to compare how good the testbed is: for a given Ω E s the best testbedis the one that has the lowest AEVMS. Metrics like BER, AEVMS, and Ω E s which facilitatecomparison between results obtained with different wireless testbeds and between experimentaland simulation results are of great value for benchmarking and debugging.V. C ONCLUSION
We presented a comprehensive study of beamforming MISO systems. We presented simulation,analytical, and experimental results, and analyzed implementation requirements and effect ofchannel and implementation-iduced imperfections. Our results show that, using EGB codebooks, it is feasible to efficiently implement codebook-based beamforming MISO systems that have goodperformance. We also showed that the Mixed Codebook scheme simplifies the implementationcomplexity of WiMAX beamforming systems. Finally, we showed that the AEVMS is a relevantmetric for performance analysis of beamforming MISO systems which facilitates comparisonbetween theoretical and experimental results and can also facilitate comparison between exper-imental results obtained with different wireless testbeds.A PPENDIX
We show how to obtain equations (11) and (12). To obtain (11) we rewrite E (cid:12)(cid:12)(cid:12) v [ k − D ] w † b (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:98) h [ k − D ] w † b (cid:12)(cid:12)(cid:12) = E (cid:34) | v [ k − D ] w b, + ... + v T [ k − D ] w b,T | (cid:12)(cid:12)(cid:12)(cid:98) h [ k − D ] w † b (cid:12)(cid:12)(cid:12) (cid:35) , (24)where w b,i and v i [ k − D ] denote the i -th entry of vector w b and v [ k − D ] respectively. Since v i [ k − D ] and w b,i are independent and E [ v i [ k − D ]] = 0 , crossterms in the expectation in (24)cancel. Then, using the fact that E [ | v i [ k − D ] | ] = 1 and || w b || = 1 , (24) reduces to (11).To obtain (12) we rewrite (cid:12)(cid:12)(cid:12)(cid:98) h [ k − D ] w † b (cid:12)(cid:12)(cid:12) = || (cid:98) h [ k − D ] || max ≤ i ≤ N (cid:12)(cid:12)(cid:12)(cid:101) h [ k − D ] w † i (cid:12)(cid:12)(cid:12) , where (cid:101) h [ k − D ] = (cid:98) h [ k − D ] || (cid:98) h [ k − D ] || . Since (cid:101) h [ k − D ] and (cid:98) h [ k − D ] are independent [4], we can write E (cid:12)(cid:12)(cid:12)(cid:98) h [ k − D ] w † b (cid:12)(cid:12)(cid:12) = E (cid:34) || (cid:98) h [ k − D ] || (cid:35) E ≤ i ≤ N (cid:12)(cid:12)(cid:12)(cid:101) h [ k − D ] w † i (cid:12)(cid:12)(cid:12) = 1Λ( T − E ≤ i ≤ N (cid:12)(cid:12)(cid:12)(cid:101) h [ k − D ] w † i (cid:12)(cid:12)(cid:12) , (25)where we have used the fact that (cid:107) (cid:98) h (cid:107) ∼ gamma ( T, Λ) hence (cid:107) (cid:98) h (cid:107) ∼ inverse gamma ( T, ) , and E (cid:20) (cid:107) (cid:98) h (cid:107) (cid:21) = T − . Using the relation between correlation and chordal distance [29] we havethat max ≤ i ≤ N (cid:12)(cid:12)(cid:12)(cid:101) h [ k − D ] w † i (cid:12)(cid:12)(cid:12) = 1 − min ≤ i ≤ N d ( (cid:101) h [ k − D ] , w i ) , where d ( (cid:101) h [ k − D ] , w i ) is thechordal distance between (cid:101) h [ k − D ] and w i . We rewrite the expectation in (25) as E ≤ i ≤ N (cid:12)(cid:12)(cid:12)(cid:101) h [ k − D ] w † i (cid:12)(cid:12)(cid:12) = 1 + E (cid:34) min ≤ i ≤ N d ( (cid:101) h [ k − D ] , w i )1 − min ≤ i ≤ N d ( (cid:101) h [ k − D ] , w i ) (cid:35) . (26)Denote Z = min ≤ i ≤ N d ( (cid:101) h [ k − D ] , w i ) , an approximation to the pdf of Z (assuming an optimumcodebook designed based on the Grassmannian criterion [3]) was found in [29] and is equal to p Z ( z ) = N ( T − z T − for ≤ z ≤ (1 /N ) / ( T − . The expectation in (26) can be computed as E (cid:34) min ≤ i ≤ N d ( (cid:101) h [ k − D ] , w i )1 − min ≤ i ≤ N d ( (cid:101) h [ k − D ] , w i ) (cid:35) = (cid:90) ( N ) T − z − z N ( T − z T − dz = T − T N − T − F , T, T, (cid:18) N (cid:19) T − , (27)where F denotes the Gauss Hypergeometric function and the result of the integration wasfound in [36]. Using (27), (26), and (25), we obtain (12).R EFERENCES [1] A. Hottinen, M. Kuusela, K. Hugl, J. Zhang, and B. Raghothaman, “Industrial embrace of smart antennas and MIMO,”
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IGURES 23
Transmit preamble andtraining sequence Feedback delay(Estimate Channel, quantize channel estimate, and send feedback) Transmit beamformedpreamble Receiver processing(Estimate transmitted bits)Transmit beamformed payload (a) Time diagram of a frame of a beamforming system which uses only one training sequence (1R system).
Transmit preamble andfirst training sequence Feedback delay(Estimate Channel, quantize channel estimate, and send feedback) Transmit beamformedpreamble and second training sequence Receiver processing(Estimate transmitted bits)Transmit beamformedpayload60 ms (b) Time diagram of a frame of a beamforming system which uses two training sequences (1.5R system).Fig. 1. Time diagrams for different beamforming systems.
IGURES 24 − − − − − − Es/No (dB) BE R (a) BER vs. E s /N o − − − − − − AEV M S ( d B ) (b) AEVMS vs. E s /N o − − − − − − − − BE R (c) BER vs. 1/AEVMS − − − − − − − − BE R SN1 (Simulation): T = 2, N = 4, EGB Codebook, ! e2 = (Es/No) − /2, " d = 0.99, 1R system.SN1 (Analytical): T = 2, N = 4, Optimum Codebook, ! e2 = (Es/No) − /2, " d = 0.99, 1R system.SN2 (Simulation): T = 2, N = 4, EGB Codebook, ! e2 = (Es/No) − /2, " d = 0.99, 1.5R system.SN3 (Simulation): T = 4, N = 64, EGB Codebook, ! e2 = (Es/No) − /2, " d = 0.99, 1R system.SN4 (Simulation): T = 4, N = 64, EGB Codebook, ! e2 = (Es/No) − /2, " d = 0.99, 1.5R system.SN5 (Simulation): T = 4, N = , ! e2 = (Es/No) − /2, " d = 0.99, 1.5R system. (d) Legend for Fig. 2(a), Fig. 2(b), and Fig. 2(c).Fig. 2. Simulation and analytical results showing the performance of a beamforming MISO system using BER and AEVMSfor performance analysis. All results correspond to 16QAM modulation. IGURES 25 − − − − − − − − − − − − − − ! Es (dBm) BE R (a) BER vs. Ω E s − − − − − − − − − − − − − − − − − ! Es (dBm)
AEV M S ( d B ) (b) AEVMS vs. Ω E s
10 12 14 16 18 20 22 24 26 2810 − − − − − − − BE R Simulation, T = 2, N = 4, EGB Codebook, ! e2 = (Es/No) − /2.3, " d = 0.9996, 1.5R system.Simulation, T = 4, N = 64, EGB Codebook, ! e2 = (Es/No) − /2.3, " d = 0.9996, 1.5R system. (c) BER vs. 1/AEVMS
10 12 14 16 18 20 22 24 26 2810 − − − − − − − BE R EXP1, T = 2, Alamouti.EXP2, T = 2, N = 4, EGB Codebook.EXP3, T = 2, N = 4, MWBE Codebook.EXP4, T = 4, N = 64, WiMAX Mixed Codebook.EXP5, T = 4, N = 64, WiMAX Codebook.EXP6, T = 4, N = 64, EGB Codebook.EXP7, T = 4, N = ! . (d) Legend for Fig. 4(a), Fig. 4(b), and Fig. 4(c).Fig. 3. Emulator and simulation results showing the performance of a beamforming MISO system using BER and AEVMSfor performance analysis. All results correspond to 16QAM modulation. ABLES 26
TABLE IC
ODEBOOK D ESCRIPTION AND C OMPARISON OF RESOURCES REQUIRED FOR CHANNEL QUANTIZATION FOR DIFFERENTCODEBOOKS .Description Resource RequirementsResource Total Total for N = 64 T = 4 .Complex Mults. NT NT − N N N N − NT NT − N N N N − NT − N W = GC , where Real Mults. N G is an N × N diagonal matrix Real Adds. N NT C is an N × T matrix whose Mux 4 Inputs NT { , − , j, − j } . Mux 9 Inputs 0 0Relational N − NT − N W = GC , where Real Mults. N + N G is an N × N diagonal matrix Real Adds. NT + N NT C is an N × T matrix whose Mux 4 Inputs 0 0entries belong to { , , j, − , − j, Mux 9 Inputs NT j, − j, − − j, − j } . Relational N − ABLES 27
TABLE IIE
XPERIMENT C ONDITIONS
Parameter ValueNumber of transmitter antennas T = 2 and T = 4 Number of receiver antennas 1Carrier frequency 2.4 GHzNumber of subcarriers 1Bandwidth 625 kHzADC/DAC sampling frequency 40 MHzPulse shaping filter Squared Root Raised CosineSRRC roll-off factor 1Symbol time 3.2 µ sPayload symbols per frame 110Modulation 16 QAMCoding Rate 1 (No error correction code)Training signal energy per antenna E p = E s + 3 . dBFeedback delay D = 60 msPaths per emulated RF link 3Model per path Jake’s model for all 3 pathsFading Doppler per path 0.1 Hz in all 3 pathsDelay per path Path 1 = 0 µ s , Path 2 = 0.05 µ s,Path 3 = 0.1 µ sRelative path loss per path Path 1 = 0 dB , Path 2 = 3.6 dBPath 3 = 7.2 dBDelay correlation coefficient For Jake’s model is computed as ρ d = J o (2 π · . Hz · ms ) = 0 . where J o ( x ))