Introduction to Digital Communications pp 193202  Cite as
Principles of Nonlinear MIMO Receivers
Abstract
19.1 Maximum Likelihood MIMO Receiver
Principle
 statistically independent with covariance matrix$$\begin{aligned} \mathbf {R}_{nn}=\sigma _{n}^{2}\mathbf {I}_{N} \end{aligned}$$(19.4)

all noise components \(n_{i}\) possess the same mean power \(\sigma _{n}^{2}\) and zero mean \({\mathbf {E}}\left[ n_{i}\right] =0\,\,;\,\,i=1,2, \ldots ,N\).
 the real part \(n_{R,i}\) and the imaginary part \(n_{I,i}\) of the noise \(n_{i}=n_{R,i}+\mathrm {j}n_{I,i}\) are statistically independent, have the same mean power \(\frac{\sigma _{n}^{2}}{2}\), and the same Gaussian probability density functionwhere x stands for \(n_{R,i}\) and \(n_{I,i}\), \(i=1,2, \ldots ,N\). Consequently, the density function of the noise \(n_{i}\) is given by the product$$\begin{aligned} p_{x}(x)=\frac{1}{\sqrt{2\pi }\sigma _{x}}\mathrm {e}^{\frac{x^{2}}{2\sigma _{x}^{2}}}\,\,;\,\,\sigma _{x}^{2}=\frac{\sigma _{n}^{2}}{2} \end{aligned}$$(19.5)$$\begin{aligned} p_{n_{i}}(n_{i})=p_{n_{R,i}}(n_{R,i})p_{n_{I,i}}(n_{I,i})=\frac{1}{\pi \sigma _{n}^{2}}\mathrm {e}^{\frac{\left n_{i}\right ^{2}}{\sigma _{n}^{2}}}\,\,;\,\,i=1,2, \ldots ,N \end{aligned}$$(19.6)
 the multivariate probability density function of the noise vector \(\mathbf {n}\) then follows as the productor with shorthand notation$$\begin{aligned} p_{\mathbf {n}}\left( n_{1},n_{2}, \ldots ,n_{N}\right) =\left( \frac{1}{\pi \sigma _{n}^{2}}\right) ^{N}\mathrm {e}^{\frac{\left n_{1}\right ^{2}+\left n_{2}\right ^{2}+\cdots +\left n_{N}\right ^{2}}{\sigma _{n}^{2}}} \end{aligned}$$(19.7)$$\begin{aligned} p_{\mathbf {n}}\left( \mathbf {n}\right) =\left( \frac{1}{\pi \sigma _{n}^{2}}\right) ^{N}\mathrm {e}^{\frac{\left\ \mathbf {n}\right\ ^{2}}{\sigma _{n}^{2}}} \end{aligned}$$(19.8)
Just a few words about the computational complexity. As already mentioned, if the transmitter is equipped with \({\text {M}}\) antennas and each antenna output signal can take on \(L_{Q}\) different values, then there are \(L_{Q}^{M}\) different vectors \(\mathbf {s}\), for which the detector has to execute (19.13). We conclude that the number of operations in the maximum likelihood detector grows exponentially with the number M of transmit antennas.
Example 1
Example, calculation steps for maximum likelihood detection
\(\begin{array}{c} \\ \begin{array}{c} \mathbf {s}\end{array}\\ \\ \end{array}\)  \(\begin{array}{c} \\ \left( \begin{array}{c} 1\\ 1 \end{array}\right) \\ \\ \end{array}\)  \(\begin{array}{c} \\ \left( \begin{array}{c} \,\,1\\ 1 \end{array}\right) \\ \\ \end{array}\)  \(\begin{array}{c} \\ \left( \begin{array}{c} 1\\ \,\,1 \end{array}\right) \\ \\ \end{array}\)  \(\begin{array}{c} \\ \left( \begin{array}{c} 1\\ 1 \end{array}\right) \\ \\ \end{array}\) 
\(\begin{array}{c} \\ \mathbf {H}\mathbf {s}\\ \\ \end{array}\)  \(\begin{array}{c} \\ \left( \begin{array}{c} 1.5\\ 1.0\\ 2.0 \end{array}\right) \\ \\ \end{array}\)  \(\begin{array}{c} \\ \left( \begin{array}{c} \,\,\,0.5\\ 1.0\\ 0 \end{array}\right) \\ \\ \end{array}\)  \(\begin{array}{c} \\ \left( \begin{array}{c} 0.5\\ \,\,\,1.0\\ 0 \end{array}\right) \\ \\ \end{array}\)  \(\begin{array}{c} \\ \left( \begin{array}{c} 1.5\\ 1.0\\ 2.0 \end{array}\right) \\ \\ \end{array}\) 
\(\begin{array}{c} \\ \begin{array}{c} \mathbf {r}\mathbf {H}\mathbf {s}\end{array}\\ \\ \end{array}\)  \(\begin{array}{c} \\ \left( \begin{array}{c} 0.4\\ 2.1\\ 1.1 \end{array}\right) \\ \\ \end{array}\)  \(\begin{array}{c} \\ \left( \begin{array}{c} \,\,\,0.6\\ 0.1\\ \,\,\,0.9 \end{array}\right) \\ \\ \end{array}\)  \(\begin{array}{c} \\ \left( \begin{array}{c} \,\,1.6\\ 2.1\\ \,\,\,0.9 \end{array}\right) \\ \\ \end{array}\)  \(\begin{array}{c} \\ \left( \begin{array}{c} \,\,2.6\\ 0.1\\ \,\,\,2.9 \end{array}\right) \\ \\ \end{array}\) 
\(\begin{array}{c} \\ \left\ \mathbf {r}\mathbf {H}\mathbf {s}\right\ ^{2}\\ \\ \end{array}\)  \(\begin{array}{c} \\ 5.78\\ \\ \end{array}\)  \(\begin{array}{c} \\ 1.18\\ \\ \end{array}\)  \(\begin{array}{c} \\ 7.78\\ \\ \end{array}\)  \(\begin{array}{c} \\ 15.81\\ \\ \end{array}\) 
19.2 Receiver with Ordered Successive Interference Cancellation
Prerequisites
Ordered Successive Interference Cancellation
The advantage of this algorithm is its low computational complexity and the feature that it reduces interchannel interference in every decision step. However, decision errors, which may occur at low signaltonoise ratios, are critical, because they can impact the next decision and thus may cause an error propagation for the following steps. We notice that the algorithm is in principle based on the triangulation of a matrix into a lower or an upper triangular matrix also called LU decomposition [1], which is continuously applied from one step to the next. This is in principle a linear operation. However, the described OSIC algorithm gets nonlinear owing to the decision made in each step. The algorithm has been practically used in several systems, such as the layered spacetime architecture.
19.3 Comparison of Different Receivers
Comparison of design criteria for various receivers
\(\begin{array}{c} \\ \mathrm {Receiver}\\ \\ \end{array}\)  \(\begin{array}{c} \\ \mathrm {Target \ Function}\\ \\ \end{array}\)  \(\begin{array}{c} \\ \mathrm {Noise}\\ \\ \end{array}\)  \(\begin{array}{c} \\ \mathrm {Result}\\ \\ \end{array}\)  \(\begin{array}{c} \\ \mathrm {Output}\\ \\ \end{array}\)  \(\begin{array}{c} \\ \mathrm {Method}\\ \\ \end{array}\) 
\(\begin{array}{c} \mathrm {ZeroForcing~(ZF)}\\ \end{array}\)  \(\begin{array}{c} \\ \left\ \mathbf {r}\mathbf {H}\mathbf {s}\right\ ^{2}\\ =\min _{\mathbf {s}\,\epsilon \mathbb {\,C}^{M\mathrm {x}1}}\\ \\ \end{array}\)  \(\begin{array}{c} \\ \mathrm {not}\\ \mathrm {included}\\ \\ \end{array}\)  \(\begin{array}{c} \\ \mathrm {matrix}\\ \mathbf {W}\\ \\ \end{array}\)  \(\begin{array}{c} \\ \mathbf {y=}\\ \mathbf {W}\mathbf {r}\\ \\ \end{array}\)  \(\begin{array}{c}\\ \mathrm {linear}\\ \\ \\ \end{array}\) 
\(\begin{array}{c} \\ \mathrm {MMSE}\\ \\ \end{array}\)  \(\begin{array}{c} \\ \begin{array}{c} {\mathbf {E}}\left[ \left\ \mathbf {W}\mathbf {r}\mathbf {s}\right\ ^{2}\right] \end{array}\\ =\min _{\mathbf {s}\,\epsilon \mathbb {\,C}^{M\mathrm {x}1}}\\ \\ \end{array}\)  \(\begin{array}{c} \\ \mathrm {included}\\ \\ \\ \end{array}\)  \(\begin{array}{c} \\ \mathrm {matrix}\\ \mathbf {W}\\ \\ \end{array}\)  \(\begin{array}{c} \\ \mathbf {y=}\\ \mathbf {W}\mathbf {r}\\ \\ \end{array}\)  \(\begin{array}{c} \\ \mathrm {linear}\\ \\ \\ \end{array}\) 
\(\begin{array}{c} \\ \mathrm {Maximum~Likelihood}\\ \\ \end{array}\)  \(\begin{array}{c} \\ \begin{array}{c} \left\ \mathbf {r}\mathbf {H}\mathbf {s}\right\ ^{2}\\ =\min _{\mathbf {s}\,\epsilon \,\mathcal {A}} \end{array}\\ \\ \end{array}\)  \(\begin{array}{c} \\ \mathrm {included}\\ \\ \\ \end{array}\)  \(\begin{array}{c} \\ \mathrm {symbol}\\ \varvec{\hat{\mathrm{s}}}\\ \\ \end{array}\)  \(\begin{array}{c} \\ \varvec{\hat{\mathrm{s}}}\,\epsilon \,\mathcal {A}\\ \\ \\ \end{array}\)  \(\begin{array}{c} \\ \mathrm {non}\\ \mathrm {linear}\\ \\ \end{array}\) 
\(\begin{array}{c} \\ \mathrm {OSIC}\\ \mathrm {ZF}\\ \\ \end{array}\)  \(\begin{array}{c} \\ \mathrm {as}\\ \mathrm {ZF}\\ \\ \end{array}\)  \(\begin{array}{c} \\ \mathrm {not}\\ \mathrm {included}\\ \\ \end{array}\)  \(\begin{array}{c} \\ \mathscr {\mathrm {\mathrm {symbol}}}\\ \varvec{\hat{\mathrm{s}}}\\ \\ \end{array}\)  \(\begin{array}{c} \\ \varvec{\hat{\mathrm{s}}}\,\epsilon \,\mathcal {A}\\ \\ \\ \end{array}\)  \(\begin{array}{c} \\ \mathscr {\mathrm {non}}\\ \mathrm {linear}\\ \\ \end{array}\) 
\(\begin{array}{c} \\ \mathrm {OSIC}\\ \mathrm {MMSE}\\ \\ \end{array}\)  \(\begin{array}{c} \\ \mathrm {as}\\ \mathrm {MMSE}\\ \\ \end{array}\)  \(\begin{array}{c} \\ \mathrm {included}\\ \\ \\ \end{array}\)  \(\begin{array}{c} \\ \mathrm {\mathrm {symbol}}\\ \varvec{\hat{\mathrm{s}}}\\ \\ \end{array}\)  \(\begin{array}{c} \\ \varvec{\hat{\mathrm{s}}}\,\epsilon \,\mathcal {A}\\ \\ \\ \end{array}\)  \(\begin{array}{c} \\ \mathrm {non}\\ \mathrm {linear}\\ \\ \end{array}\) 
The design of the zeroforcing receiver with and without OSIC does not include the noise at the receiver. Computation of the receiver matrix \(\mathbf {W}\) for the MMSE receiver requires the knowledge of the signaltonoise ratio \(\frac{1}{\alpha }\), which is not needed for the maximum likelihood detection. This method operates without any receiver matrix. Moreover, on the first glance the target functions of the zeroforcing algorithm using the pseudoinverse matrix and the maximum likelihood receiver look the same. Both receivers minimize the squared error \(\left\ \mathbf {r}\mathbf {H}\mathbf {s}\right\ ^{2}\). However, the zeroforcing receiver provides a “soft” output signal \(\mathbf {y\,\epsilon \,\mathbb {C}}^{M\mathrm {x}1}\) with continuous amplitude and phase compared to the output of the maximum likelihood receiver, which is a discrete vector \(\varvec{\hat{\mathrm{s}}}\,\epsilon \,\mathcal {A}\). Hence, the maximum likelihood scheme minimizes the same target function as the zeroforcing receiver, however, as the result of a discrete minimization problem with the constraint \(\varvec{\hat{\mathrm{s}}}\,\epsilon \,\mathcal {A}\). This can be formulated as an integer least squares problem for which several mathematical algorithms from the area of integer programming are known [2, 3]. Such methods have been used for lattice decoding and are summarized as sphere decoding algorithm, because they search in a limited hypersphere of the complex vector space rather than performing an overall brute search [4, 5, 6] and thus do not always provide the global optimum. In principle, the hypersphere is centered around the receive vector \(\mathbf {r}\) and for an efficient search the sphere should cover the lattice points given by the vectors (\(\mathbf {H}\mathbf {s}\,\,;\,\,\mathbf {s}\,\epsilon \,\mathbf {\mathcal {A}}\)) located in the vicinity of \(\mathbf {r}\). As a result, the complexity of the maximum likelihood algorithm can be significantly reduced and sphere decoding became an important alternative to the much simpler but suboptimal linear receivers.
Figure 19.1 shows a rough comparison of the symbol error rate for various receivers as the result of a computer simulation using the platform “webdemo” [7]. According to our expectations, the maximum likelihood detector (ML) demonstrates the best performance followed by the nonlinear receiver with OSIC. Compared to the zeroforcing receiver (ZF), the minimum mean squared error approach (MMSE) takes the noise into account and thus outperforms the ZF receiver in general.
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