Beamforming Techniques for OFDM

Beamforming overview

The objective of beamforming is to separate the desired signal from interfering signals given that all the signals share the same frequency band but have unique spatial signatures associated with them. In addition to interference rejection and multipath fading mitigation, beamforming increases the antenna gain in the direction of the desired user. A beamformer essentially weights and sums the signals from the different antenna elements to optimize the quality of the desired signal.

Beamforming weight update criteria

Minimum Mean Squared Error (MMSE)
Maximum Signal to Interference and Noise Ratio (MSINR)
Maximum Signal to Noise Ratio (MSNR)

Adaptive weight update algorithms

MMSE criteria

• Direct Matrix Inversion (DMI): Directly computes the inverse of the received signal covariance matrix. The optimal weight vector is calculated using the Weiner solution to minimize the MSE between the beamformer output and the desired signal.
   

• where Rxx = E[xxH], rxd = E[xd*], x is the received signal vector from each antenna element, and d is the desired signal vector. Usually, matrix inversion operation leads to high complexity and should to be avoided. However, if 2 or 4 receive antennas are employed, then the size of the covariance matrix is small and calculating the inverse does not pose a computational problem.
   

• Least Mean Square (LMS): Recursively computes and updates the weights to minimize the cost function E[|e|^2]

• where e(n) = d(n) – w^H(n)x(n) and 0≤μ≤1 is convergence rate.  LMS algorithm has low computational complexity, but slowly converges and has potential for instability if μ is too large.

MSINR criteria

• Generalized power method

• Generalized Lagrange multiplier (GLM) method

• Adaptive matrix inversion (AMI)

• Linearized AMI

MSNR criteria

• Power method

• Lagrange multiplier method

• Conjugate gradient method

Beamforming techniques

Time domain beamforming

In time domain beamforming, the beamformer weights are calculated in the time domain, i.e. before the FFT block. The FFT is performed after the signal is combined. Figure 1 shows a typical M-element time domain beamformer.

Figure 1: Time domain narrowband beamformer

Frequency domain beamforming

The main challenge of an OFDM beamformer is to correctly estimate the transmitted data symbols on each of the OFDM sub-carriers. The most intuitive way is to combine the signals in frequency domain, (that is, after the FFT operation at the receiver), since pilot symbols and training sequence are usually inserted in the frequency domain at the transmitter (before the IFFT block). Figure 2 shows a typical Frequency Domain M-element beamformer.

Figure 2: Frequency domain beamformer

The frequency domain beamformer’s task is to minimize the effect of the multiplicative distortion induced by fading channels. Note that the multiplicative distortion can be eliminated by using a one-tap channel equalizer. However, equalization removes the AOA information needed by the beamformer. Therefore, equalization is not performed, and the multiplicative factor is passed to the beamformer which steers the antenna array towards the desired direction.

Sub-band beamforming

Although a channel may appear relatively flat from the view point of an individual sub-carrier, it can be highly frequency selective when considering the whole OFDM bandwidth. Hence, multiplicative distortion may be fairly distinctive across the spectrum. In that case, a single set of beamforming weights is not sufficient for all sub-carriers. A more efficient approach is to have multiple beamformers across the sub-carriers. The optimal solution is to have one beamformer per sub-carrier. However, if the channel does not vary significantly over a specific set of sub-carriers, then a single beamformer can be assigned to these sub-carriers.

Figure 3: Sub-band beamformer

Performance of beamforming for OFDM

Performance of OFDM beamformer for flat channel

Beamforming provides substantial gains over the single antenna case (about 5 dB for a target BER of 10^-2). However, sub-band beamforming is not beneficial, since the channel is flat over all sub-carriers i.e., the multiplicative distortion is constant for all sub-carriers (See figures below).

Figure 4: DMI Beamforming in Flat Rayleigh Fading

Figure 5: LMS Beamforming in Flat Rayleigh Fading

Performance of OFDM beamformer for frequency selective channel

Time-domain and frequency-domain beamforming fail and exhibit flat BER curves for frequency selective channel. The reason for this is that a single beamformer is not sufficient to keep track of fast channel variations on all sub-carriers. However, using a sub-band beamformer with 4 sub-bands, performance is improved drastically and a BER of 10^-2 is obtained at Eb/No = 10 dB (See below figures). Furthermore, increasing the number of sub-bands may further improve the performance, since the channel becomes flatter from the perspective of each group of sub-carriers.

Figure 6: DMI Sub-band Beamforming in Frequency Selective Channel

Figure 7: LMS Sub-band Beamforming in Frequency Selective Channel