technique over MIMO Keyhole Channels . . . 85 5.4 Proposed AMC Method . . . 88 5.5 Simulation Results and Discussion . . . 94 5.6 Summary . . . 102
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5. Automatic Modulation Classification over Spatially Correlated Keyhole MIMO Channels
5.1 Introduction
The MIMO spatial channel alters the statistical properties of the modulated signal, which makes AMC a challenging task. It is, therefore, necessary to cancel the channel mixing effect prior to AMC. It is observed in the literature that the most of the existing MIMO blind channel estimation techniques such as the JADE [31] and the constant modulus algorithm (CMA) [29, 30], require the channel matrix to be of full rank. The majority of the AMC algorithms for MIMO systems have considered the rich scattering environment that offers a full rank channel matrix [1, 15, 17–28]. In practice, MIMO systems can also operate in an insufficient scattering environment, which can lead to a rank deficient channel [14, 46, 47, 49, 50, 52, 56, 74]. For example, one may consider a scenario where a user device located indoor, transmits signals to a cell tower. The opaqueness of building walls to the signal transmitted by the user device allows the most of the signal energy to pass through keyholes like windows, doors etc [14].
The presence of a keyhole is also possible in outdoor scenarios, even under a rich scattering environment [50]. It can happen when the ring of scatterers around the transmitter and the receiver is small compared to the distance between the transmitter and the receiver. Therefore, rich scattering is only a necessary but not sufficient condition for full rank MIMO channels. A rank deficient channel, such as a keyhole channel can severely degrade the AMC performance.
The entries of such a spatial channel matrix are no longer single Rayleigh distributed. It is shown in [46] that these channel entries are double Rayleigh distributed. Such low rank channel models have poor fading statistics because a double Rayleigh channel fades more than a single Rayleigh channel [75]. In addition to the keyhole effect, correlated transmitting and receiving antennas also affect the AMC performance [14]. The antenna correlation is attributed to the inadequate physical separation between antenna elements.
This chapter addresses the problem of AMC over a spatially-correlated keyhole MIMO channels. We propose a novel algorithm, namely the direct modulation recognition (DMR) algorithm, which is capable of classifying the lower order PSK constellations, namely BPSK, QPSK and OQPSK. The proposed algorithm employs the ratios of HOCs derived from the received signal and its backward difference as AMC features. These features are chosen so that
5. Automatic Modulation Classification over Spatially Correlated Keyhole MIMO Channels
. . . .
. . . .
R
NRR
2R
1T
1T
2T
NTg
1g
2g
NTd
2d
NRd
1Figure 5.1: A MIMO system with keyhole
whered= [d1...dNR]T is theNR×1 channel gain vector between the keyhole and the receiver and g= [g1...gNT]T is theNT ×1 channel gain vector between the transmitter and the keyhole.
The rank ofH1is one. The amplitude of each element ofH1is double-Rayleigh distributed [46].
The probability density function (PDF) of double-Rayleigh variable is given by [75].
f(r)= r
σ4K0(2r
σ2), (5.3)
whereK0(.) denotes the zeroth-order modified Bessel function of the second kind andσ2repre- sents the average power of each channel path. A double Rayleigh channel fades more than the standard Rayleigh channel. The PDFs of Rayleigh and double-Rayleigh random variables are plotted in Fig. 5.2.
After incorporating the spatial correlation, the channel matrix H of a spatially correlated MIMO channel is given by
H =θ1/2R H1θ1/2T , (5.4)
where θ and θ are the receiver and the transmitter correlation matrices. The exponential
5.3 Investigation of the Fourth-Order Zero-Conjugate Cumulant based AMC technique over MIMO Keyhole Channels
0 1 2 3 4
0 0.2 0.4 0.6 0.8 1
fading amplitudes
Rayleigh
double Rayleigh
Figure 5.2:PDFs of Rayleigh and Double-Rayleigh Channel Amplitude antenna correlation model is considered for this investigation.
5.3 Investigation of the Fourth-Order Zero-Conjugate Cumulant based AMC technique over MIMO Keyhole Channels
Swami et al., in [4], reported that different digitally modulated signals exhibit distinct fourth-order cumulant values. The fact that the fourth-order cumulants offer resistance to the Gaussian noise makes them attractive AMC features. In [15], Muhlhaus et al. proposed a fourth- order zero-conjugate cumulant (FZC) based AMC algorithm over an uncorrelated Rayleigh MIMO channel. Here, we investigate the performance of this algorithm over a spatially corre- lated MIMO keyhole channel. For this investigation, we consider two modulation types, namely BPSK and QPSK. These two modulation types are indexed by M = 1and2 respectively. The algorithm involves following steps [15]:
Step 1: Equalize the received signal (using ZF or JADE) y(k) to obtain NT transmitted streams ˆxj, j= 1, ...,NT .
step 2: Estimate the 4th-order zero-conjugate cumualnt from ˆxj, j=1, ...,NT. Step 3: Combine the cumulant estimates as
fˆ= PNT
j=1(snrj)2C42(ˆxj) PNT
j=1(snrj)2 (5.5)
TH-2361_126102009
5. Automatic Modulation Classification over Spatially Correlated Keyhole MIMO Channels
wheresnrj is the SNR of the jthpath.
Step 3: DecideM∗as
M∗= argmin
M=1,2 (|fˆ− fM|). (5.6)
where fMis the theoretical FZC value for theMthmodulation type.
Although the algorithm performs very well under a rich scattering environment, it performs poorly in the presence of a keyhole. This is because under the keyhole condition both the ZF and JADE equalizers fail to equalize the received signal. This is validated by the simulation results presented in Fig. 5.3 and Fig. 5.4. We refer the above technique as ZF-FZC if the employed equalization technique is ZF. If the employed equalization technique is JADE, we refer it as JADE-FZC. For the simulation, the number of transmitting and receiving antennas were set at NT = 2 andNR = 4, respectively. For each trial, we generated a random channel matrix and a random source message. For each SNR value, 3000 Monte Carlo simulations were performed to calculate the average probability of correct classificationPC.
Table 5.1: Fourth-Order Zero-Conjugate Cumulant Values for BPSK and QPSK
M fM
1 - 2
2 1
5.3.1 Effect of a Keyhole on Channel Equalization
The equalized signal at the ZF equalizer output can be written as ˆ
y(k)= (HHH)−1HHHx(k)+(HHH)−1HHη(k) (5.7)
= x(k)+z(k),
wherez(k)=(HHH)−1HHη(k).
The total noise variance at the ZF equalizer output can be written as [14]
σ2z = NRσ2ηT r[(HHH)−1]. (5.8) As stated earlier, the rank of the keyhole channel is one. It is clear from Eqn. 5.8 that the noise amplification takes place at the ZF output whenever the rank ofHHHis low [14]. WhenN =1
5.3 Investigation of the Fourth-Order Zero-Conjugate Cumulant based AMC technique over MIMO Keyhole Channels
1 2 3 4 5 6 7 8 9 10
SNR (dB) 0.5
0.6 0.7 0.8 0.9 1
P C Rayleigh Channel
Keyhole Channel
Figure 5.3: PC versus SNR for ZF-FZC withNT =2,NR=4,N=2000 and|ρR|=|ρT|=0
1 2 3 4 5 6 7 8 9 10
SNR (dB) 0.5
0.6 0.7 0.8 0.9 1
P C Rayleigh Channel
Keyhole Channel
Figure 5.4: PC versus SNR for JADE-FZC withNT =2,NR=4,N=2000 and|ρR|=|ρT|=0
, the channel model becomes akin to the keyhole or pinhole MIMO channel. Therefore, the ZF equalizer suffers from noise amplification under the keyhole condition. The JADE technique too fails to equalize the received signal as it requires the channel matrix to be of rank one for TH-2361_126102009
5. Automatic Modulation Classification over Spatially Correlated Keyhole MIMO Channels
the separation of independent components [31] .