*4.1. Transmitter*

The transmitted data in each of the *D* modes are modulated using a PAM with a square-root raised cosine pulse *p*(*t*) with roll-off factor equal to *α* and normalized power, which can be expressed as [38]:

$$p(t) = \frac{4\alpha}{\pi\sqrt{T}} \cdot \frac{\cos\left(\left[1+\alpha\right]\frac{\pi t}{T}\right) + \frac{T\cdot\sin\left(\left[1-\alpha\right]\frac{\pi t}{T}\right)}{4\alpha t}}{1 - \left(\frac{4at}{T}\right)^2}.\tag{12}$$

Hence, we can write the sequence of PAM pulses for a given mode *i* as:

$$x\_i(t) = \sum\_{n = -\infty}^{\infty} s\_i[n] p(t - nT),\tag{13}$$

where *si*[*n*] is a random variable with values taken from the set defined by the PAM modulation scheme. Let us define the global impulse response *q*(*t*) as the convolution of the transmitting pulse *p*(*t*) and the optical channel impulse response matrix *htot*(*t*) as:

$$
\underline{q}(t) = \underline{\underline{h}}\_{tot}(t) \* p(t), \tag{14}
$$

so that *q*(*t*) is a *D* × *D* matrix of impulse responses. This way, *qij*(*t*) describes the impulse response between the transmitter mode *i* and the receiver mode *j*. Therefore, we can write the relationship between the transmitted symbols *sj*[*n*] and the received signal in mode *i*, *yi*(*t*), as:

$$y\_i(t) = \sum\_{n} \sum\_{j=1}^{D} s\_j[n] q\_{ij}(t - nT) + n\_i(t) \tag{15}$$

with *ni*(*t*) as the noise in the *i*-th receiver mode.

#### *4.2. Linear MMSE MIMO Receiver*

The most widely used linear MIMO receiver for SDM systems is based on the design of filter *<sup>O</sup>*(*ω*) in Figure 1 to minimize the mean squared error (MSE), which is called a linear MMSE MIMO receiver [28,37,39]. Mathematically, the MSE for the linear MIMO receiver under the MMSE criterion is defined as:

$$
\sigma^2\_{\text{MMSE-LE}} = \mathbb{E}\left[\underline{\mathfrak{e}}^H[n]\underline{\mathfrak{e}}[n]\right] \tag{16}
$$

where

$$\underline{\underline{e}}[n] = \underline{\underline{s}}[n] - \underline{\underline{s}}[n] \tag{17}$$

with *s*<sup>ˆ</sup>[*n*] as the output vector of the linear MIMO receiver *<sup>O</sup>*(*ω*).

It is well known that the structure of a linear MMSE MIMO receiver can be divided into a matched filter *Q<sup>H</sup>*(*ω*) operating in continuous time, a sampler operating at the symbol rate, and a discrete-time equalizer of response *<sup>W</sup>*(Ω), as presented in Figure 2.

**Figure 2.** SDM communication system model with linear minimum mean square error (MMSE) MIMO receiver.

We denote as *y*[*n*] the vector of samples after the match filter *Q<sup>H</sup>*(*ω*) = F {*q<sup>H</sup>*(−*<sup>t</sup>*)} in the receiver and a symbol rate sampling, being *q*(*t*) = F <sup>−</sup><sup>1</sup>{*Q*(*ω*)} defined in (14). Now, we define the sampled impulse response at *t* = *nT* of the convolution of *q*(*t*) and its matched filter *<sup>q</sup><sup>H</sup>*(−*<sup>t</sup>*), which represents the equivalent discrete channel response as:

$$
\underline{\underline{\mathbf{q}}}[n] = \underline{\underline{q}}(t) \* \underline{\underline{q}}^H(-t)\Big|\_{t=nT'} \tag{18}
$$

and its discrete Fourier transform pair as:

$$\underline{\underline{G}}(\Omega) = \sum\_{n=-\infty}^{\infty} \underline{g}[n] \cdot e^{-j\Omega n}.\tag{19}$$

Hence, the optimal discrete-time MIMO equalizer *Wopt*(Ω) according to the MMSE criterion becomes:

$$
\underline{\underline{W}}\_{\gamma\text{pt}}(\Omega) = \left[ \underline{\underline{G}}(\Omega) + \frac{I}{\underline{n}} \cdot \left( \frac{N\_0}{2} \right) \right]^{-1} \tag{20}
$$

where we are considering a normalized transmission power equally distributed in each of the *D* modes.

When *G*(Ω) satisfies the Nyquist criterion for MIMO systems *G*(Ω) = *I*, there is neither ISI nor cross-talk at the matched filter output, further equalization would not be needed, and the optimum linear receiver consists only in the matched filter. However, if such a criterion is not fulfilled, the equalizer *W*(Ω) is essential and some SNDR loss at the output will be unavoidable w.r.t. the ideal case.

#### *4.3. Matched Filter-Based Receiver for SDM*

In this subsection we will explore the optical channel requirements to reduce the linear MIMO receiver *<sup>O</sup>*(*ω*) in Figure 1 to a simple matched filter-based receiver. Furthermore, we will show that the resulting receiver is optimal in the sense that the discrete-time system response of the SDM communication system is the identity matrix, followed by the addition of the AWGN noise.

Let us first write out:

$$
\underline{Q}(\omega) = P(\omega) \cdot \underline{H}\_{tot}(\omega),
\tag{21}
$$

where *Htot*(*ω*) and *<sup>P</sup>*(*ω*) are the Fourier transforms of *htot*(*t*) and *p*(*t*), respectively and according to what is plotted in Figure 2. It follows that:

$$
\underline{\underline{Q}}(\omega)^H = P^\*(\omega) \cdot \underline{\underline{H}}\_{tot}(\omega)^H. \tag{22}
$$

The signal at each of the *D* branches *yi*(*t*), defined in (15), is processed before sampling by the continuous-time filter *Q<sup>H</sup>*(*ω*). The equivalent scheme for this matched filter-based MIMO receiver is shown in Figure 3a. By using the linearity of the system we can rearrange Figure 3a to obtain Figure 3b. Then, elaborating the expression *Q*(*ω*)*Q<sup>H</sup>*(*ω*) we obtain that:

$$\underbrace{Q}\_{\text{III}}(\omega)\underbrace{Q^{H}}\_{\text{III}}(\omega) = P(\omega)\cdot\underbrace{H}\_{\text{left}}(\omega)\cdot\underbrace{H^{H}}\_{\text{left}}(\omega)\cdot P^{\*}(\omega) = P(\omega)\cdot\underbrace{H}\_{\text{left}}(\omega)\cdot\underbrace{H^{H}}\_{\text{left}}(\omega)\cdot P^{\*}(\omega), \tag{23}$$

where we have used that:

$$H\_{\rm CD}(\omega) \cdot H\_{\rm CD}^\*(\omega) = 1. \tag{24}$$

**Figure 3.** SDM communication system model with matched filter-based receiver (**a**) and its reordered version (**b**).

From (10) we have that:

*<sup>H</sup>*(*ω*) · *<sup>H</sup>*(*ω*)*<sup>H</sup>* = *Kamp*−<sup>1</sup> ∏*k*=1 *<sup>V</sup>*(*k*)Λ(*k*)(*ω*)*U*(*k*)*<sup>H</sup>* · *<sup>V</sup>*(*Kamp*)Λ(*Kamp*)(*ω*)*U*(*Kamp*)*<sup>H</sup>* · *U*(*Kamp*)Λ(*Kamp*)*<sup>H</sup>* (*ω*)*V*(*Kamp*)*<sup>H</sup>* · *Kamp* ∏*k*=2 *<sup>U</sup>*(*Kamp*−*k*+<sup>1</sup>)Λ(*Kamp*−*k*+<sup>1</sup>)*<sup>H</sup>* (*ω*)*V*(*Kamp*−*k*+<sup>1</sup>)*<sup>H</sup>* = *Kamp*−<sup>1</sup> ∏*k*=1 *<sup>V</sup>*(*k*)Λ(*k*)(*ω*)*U*(*k*)*<sup>H</sup>* · *V*(*Kamp*) · |Λ(*Kamp*)(*ω*)|<sup>2</sup> · *V*(*Kamp*)*<sup>H</sup>* · *Kamp* ∏*k*=2 *<sup>U</sup>*(*Kamp*−*k*+<sup>1</sup>)Λ(*Kamp*−*k*+<sup>1</sup>)*<sup>H</sup>* (*ω*)*V*(*Kamp*−*k*+<sup>1</sup>)*<sup>H</sup>*. (25)

And the diagonal matrix:

$$\begin{split} \left| \underline{\Delta}^{(\mathcal{K}\_{\text{amp}})} (\omega) \right|^{2} &= \\ \text{diag} \left( \left[ e^{\frac{1}{2} \mathcal{S}\_{1}^{(\mathcal{K}\_{\text{amp}})} - j\omega \tau\_{1}^{(\mathcal{K}\_{\text{amp}})}}, \dots, e^{\frac{1}{2} \mathcal{S}\_{D}^{(\mathcal{K}\_{\text{amp}})} - j\omega \tau\_{D}^{(\mathcal{K}\_{\text{amp}})}} \right] \right) \\ \text{diag} \left( \left[ e^{\frac{1}{2} \mathcal{S}\_{1}^{(\mathcal{K}\_{\text{amp}})} + j\omega \tau\_{1}^{(\mathcal{K}\_{\text{amp}})}}, \dots, e^{\frac{1}{2} \mathcal{S}\_{D}^{(\mathcal{K}\_{\text{amp}})} + j\omega \tau\_{D}^{(\mathcal{K}\_{\text{amp}})}} \right] \right) \\ = \left| \text{diag} \left( \left[ e^{\frac{1}{2} \mathcal{S}\_{1}^{(\mathcal{K}\_{\text{amp}})}}, \dots, e^{\frac{1}{2} \mathcal{S}\_{D}^{(\mathcal{K}\_{\text{amp}})}} \right] \right) \right|^{2} \end{split} \tag{26}$$

that does not allow simplifying (25) unless the following holds:

$$|\underline{\Lambda}^{(K\_{\text{amp}})}(\omega)|^2 = \mathfrak{e}^{\left(\mathcal{S}^{(K\_{\text{amp}})}\right)} \cdot \underset{\overline{\mathbb{m}}}{I}.\tag{27}$$

This latter condition is equivalent to assuming that:

$$\mathfrak{c}\left(^{\frac{1}{2}\mathfrak{s}^{(K\_{\text{amp}})}}\right) = \mathfrak{c}\left(^{\frac{1}{2}\mathfrak{s}\_1^{(K\_{\text{amp}})}}\right) = \mathfrak{c}\left(^{\frac{1}{2}\mathfrak{s}\_2^{(K\_{\text{amp}})}}\right) = \dots = \mathfrak{c}\left(^{\frac{1}{2}\mathfrak{s}\_D^{(K\_{\text{amp}})}}\right),\tag{28}$$

or, in other words, that the MDL is negligible for the *Kamp*-th span. When the condition expressed in (27) is satisfied for all the spans of the system, we can commute the terms in (25), and therefore, we can obtain:

$$\underline{\underline{H}}(\omega) \cdot \underline{\underline{H}}(\omega)^{H} = \prod\_{k=1}^{\mathcal{K}amp} |\underline{\underline{\mathcal{H}}}^{(k)}(\omega)|^{2} = \prod\_{k=1}^{\mathcal{K}amp} e^{\left(\mathcal{S}^{(k)}\right)} \cdot \underline{\underline{I}} = e^{\left(\sum\_{k=1}^{\mathcal{K}amp} \mathcal{S}^{(k)}\right)} \cdot \underline{\underline{I}}.\tag{29}$$

Revisiting (23), and plugging in (29) under the assumption of a negligible MDL in the optical channel, we can write:

$$\underline{\underline{\mathbf{G}}}(\omega) = \underline{\underline{\mathbf{Q}}}(\omega) \cdot \underline{\underline{\mathbf{Q}}}^H(\omega) = |P(\omega)|^2 \cdot \varepsilon^{\binom{\sum\_{k=1}^{\text{Carup}} \underline{\mathbf{s}}^{(k)}}} \cdot \underline{\underline{\mathbf{I}}} \cdot \tag{30}$$

Therefore, without loss of generality, *e* <sup>∑</sup>*Kamp k*=1 *g*(*k*) = 1 can be assumed and, after sampling at the symbol rate, (30) can be written as:

$$\underline{G}(\Omega) = \frac{1}{T} \cdot \sum\_{l=-\infty}^{\infty} \left| P\left(\frac{\Omega + 2\pi l}{T}\right) \right|^2 \cdot \underline{I} = \underline{I} \tag{31}$$

where we have used that *p*(*t*) defined in Equation (12) is a square-root raised cosine pulse satisfying the Nyquist criterion.

Regarding the filtered noise waveforms *<sup>z</sup>*1(*t*) to *zD*(*t*) in Figure 4a, they have an autocorrelation function matrix *R zz*(*t*), whose Fourier transform pair *Sz*(*ω*) can be expressed as:

$$
\underline{\underline{S}}\_{\varpi}(\omega) = \underline{\underline{Q}}(\omega) \cdot \underline{\underline{S}}\_{\varpi}(\omega) \cdot \underline{\underline{Q}}^{H}(\omega),
\tag{32}
$$

where *S n*(*ω*) is the Fourier transform of the autocorrelation function matrix *Rnn*(*t*) = *I* · *N*02 · *δ*(*t*) of the received noise vector *n*(*t*) = [*<sup>n</sup>*1(*t*), *<sup>n</sup>*2(*t*),..., *nD*(*t*)]*<sup>T</sup>*. We remind that the noise components of the noise vector *n*(*t*) were assumed uncorrelated with identical power in each mode equal to *N*0/2. Using (30) and (32) leads to:

$$\underline{\underline{S}}\_{\omega}(\omega) = \frac{N\_0}{2} \cdot \underline{\underline{Q}}(\omega) \cdot \underline{\underline{Q}}^H(\omega) = \frac{N\_0}{2} \cdot |P(\omega)|^2 \cdot e^{\left(\sum\_{k=1}^{\mathbb{K}amp} \underline{\mathbb{S}}^{(k)}\right)} \cdot \underline{\underline{I}}\tag{33}$$

Using the previous assumptions about gains *g*(*k*) and *<sup>P</sup>*(*ω*) made before, we obtain that the sampled noise vector *<sup>z</sup>*[*n*]=[*<sup>z</sup>*1[*n*], ... , *zD*[*n*]]*<sup>T</sup>* has an autocorrelation matrix function *R zz*[*n*] = *I* · *N*02 · *<sup>δ</sup>*[*n*].

Therefore, we can conclude that a *D* × *D* MIMO coherent optical communication system using a continuous-time matched filter as a receiver completely eliminates channel ISI and crosstalk when the MDL in the channel is negligible. Moreover, the equivalent discrete-time system model reduces to *D* discrete parallel AWGN channels as shown in Figure 4b. Hence, there would be no loss of performance w.r.t. the AWGN channel without distortion.

#### **5. Numerical Simulation of Linear MIMO FSE Receiver for MDL-Impaired Optical Channel**

In this section, we assess the performance of the ideal MMSE linear receiver when MDL is present. Specifically, we study the SNDR degradation at the receiver output w.r.t. the case when the MDL is negligible. To carry out this study, we will use a FSE-based receiver, as shown in Figure 5, which is the most common implementation of the ideal linear filter in discrete-time systems (see *<sup>O</sup>*(*ω*)in Figure 1). Note that this scheme is only valid for integer oversampling rates *rov*.

**Figure 5.** SDM communication system model with linear fractionally-spaced equalizer (FSE) MIMO receiver and integer oversampling rate *rov*.

The FSE oversampling rate *rov* has been set to two [1,12] and the discrete-time equalizer, with a *WFSE*(Ω) response, has been designed with a number of taps *Ntaps* large enough so that any further increase does not lead to a significantly better SNDR at the receiver output. The decimated output of *WFSE*(Ω), by a *rov* factor, are the estimated symbol *<sup>s</sup>*<sup>ˆ</sup>[*n*].

We define the receiver performance metric for each mode *i*, *<sup>L</sup>*(*i*), as the difference in dB between the output SNR of an ISI and crosstalk-free system with *D* parallel AWGN channels (see Figure 4b), and the output SNDR of the FSE-based receiver, denoted as *SNDRout*.

## *5.1. Channel Model*

We decide to carry out the numerical simulations of the *<sup>H</sup>*(*ω*) channel model described in (6) in the time domain, so that the relative delays of the different modes can be easily described as a time shift between them. The different mode amplitudes can also be handled simply by a diagonal matrix. The chromatic dispersion, which have a SISO frequency response *HCD*(*ω*) that does not depend on the mode *i* ∈ {0, . . . , *<sup>D</sup>*}, is represented as [7]:

$$H\_{CD}(\omega) = e^{-j\mathcal{G}\frac{\omega^2}{2}},\tag{34}$$

where *β* = *β*¯ 2*tot*, and *tot* = *Kampspan* when all spans are considered of equal length.

The MDL effect is modeled with an amplification factor for each mode and each optical amplifier (located at the end of each span). These factors are considered time-invariant for a given channel realization in the form of a vector for the *k*-th span *g*(*k*) = [*g* (*k*) 1 , *g* (*k*) 2 , ... , *g* (*k*) *D* ], where *g* (*k*) *i* for *i* ∈ {0, ... , *D*} is expressed in dB and taken from a Gaussian distribution with zero mean and standard deviation (STD) *<sup>σ</sup>g*. The sum of all factors ∑*<sup>D</sup> i*=1 *g* (*k*) *i* is set to 0 for normalization purposes. Hence, the amplitudes matrix of the *k*-th span, frequency independent, is given by:

$$\underline{\mathbf{A}}^{(k)} = \text{diag}\left( \left[ e^{\left(\frac{1}{2}\mathbf{x}\_1^{(k)}\right)}, \dots, e^{\left(\frac{1}{2}\mathbf{x}\_D^{(k)}\right)} \right] \right). \tag{35}$$

Alternatively, for each span *k* of the communication link we have the delays matrix:

$$\underline{\mathbf{A}}^{(k)}(\omega) = \underline{\mathbf{A}}^{(k)} \cdot \text{diag}\left(\left[e^{-j\omega \tau\_1^{(k)}}, \dots, e^{-j\omega \tau\_D^{(k)}}\right]\right),\tag{36}$$

being *τ*(*k*) = [*τ*(*k*) 1 , *τ*(*k*) 2 , ... , *τ*(*k*) *D* ] the vector that models the MD with group delays for each mode of the *k*-th span.

To obtain the delays, we generate the first *D*/2 values of *τ*(*k*) from a Gaussian distribution with STD *<sup>σ</sup>gd*, and the second *D*/2 values are taken as the opposite of these, which satisfies that ∑*<sup>D</sup> i*=1 *τ*(*k*) *i* = 0, since we consider that the system uses polarization multiplexing as part of the SDM [40].

The time-domain impulse response for each of the spans *k* is calculated by applying the inverse Fourier transform to (7) and can be expressed as:

$$
\underline{\underline{h}}^{(k)}(t) = \underline{\underline{V}}^{(k)} \underline{\underline{A}}^{(k)} \underline{\underline{d}}^{(k)}(t) \underline{\underline{V}}^{(k)},\tag{37}
$$

where

$$\underline{d}^{(k)}(t) = \text{diag}\left(\left[\delta(t - \tau\_1^{(k)}), \dots, \delta(t - \tau\_D^{(k)})\right]\right),\tag{38}$$

and we have used that the matrices *A*(*k*) , *<sup>U</sup>*(*k*), and *V*(*k*) are constant.

Equation (37) describes that incoming signal at the *k*th span is multiplied by the unitary matrix *<sup>U</sup>*(*k*), then each modal impulse response is delayed by *τ*(*k*) *i* , the amplification factor is set by the diagonal matrix *A*(*k*) and the mode-mixing unitary matrix *V*(*k*) is applied. Finally, the impulse response of the complete channel is given by:

$$
\underline{\underline{h}}\_{tot}(t) = \underline{\underline{h}}^{(K\_{amp})}(t) \* \underline{\underline{h}}^{(K\_{amp}-1)}(t) \* \cdots \* \underline{\underline{h}}^{(1)}(t) \* h\_{CD}(t), \tag{39}
$$

where *hCD*(*t*) = F <sup>−</sup><sup>1</sup>{*HCD*(*ω*)}.

Note that, due to the random nature of *g*(*k*) and *τ*(*k*) in each *k* span, we can generate an arbitrary number *Nch* of channel realizations of *Htot*(*ω*) = F {*htot*(*t*)} for a given value of *<sup>σ</sup>g* and *<sup>σ</sup>gd*.

Since we consider all the modes to be strongly coupled, the *U*(*k*) and *V*(*k*) matrices of each span *k* are modeled as unitary Gaussian random matrices obtained from a QR factorization of a complex random matrix whose elements have a zero mean and STD equal to 1. The two orthogonal matrices after QR factorization of two independent realizations of the random matrix are used as *U*(*k*) and *<sup>V</sup>*(*k*), respectively.

We consider a total number of *Kamp* = 100 spans, each *span* = 50 km long. For the fiber parameters, we used the multi-core fiber data reported in [41], considering the number of modes *D* = 6 and the central wavelength *λc* = 1469 nm. The selection of this multi-core fiber allow us to compare the results of this work with those presented in [28], and to obtain the fiber parameters needed for the numerical simulations from [41].

We take 2% as the underestimation dispersion factor that is applied to the dispersion coefficient *DCD* to obtain the residual CD experienced by the receiver. For the gain STD *<sup>σ</sup>g*, we considered several values in the range of the systems referenced in [27]. For the numerical simulations, we compute a total of *Nch* = 10,000 realizations of the channel frequency response *Htot*(*ω*) defined in (5).

#### *5.2. Transmitter and Linear MIMO FSE Receiver Parameters*

As described in Section 4, the transmitter uses a generalized PAM modulation and an square-root raised cosine for pulse shaping with a roll-off factor equal to *α*. We will show results of the numerical simulation for several values of *α*. The symbol period *T* has been set for a symbol rate *Rs* = 64 GBaud. The FSE-based MIMO receiver has an oversampling factor *rov* = 2, and a number of equalizer taps *Ntaps* = 1000 has been selected to ensure that it does not limit the receiver performance for the considered channel MDL.

#### *5.3. Signal-to-Noise at the Input of the Receiver*

The signal-to-noise ratio at the input of each mode *S N in*(*i*) for *i* ∈ {0, ... , *D*} is defined as: 

$$\left(\frac{S}{N}\right)\_{\rm in}(i) = \frac{P\_{\rm in}(i)}{N\_0/2}.\tag{40}$$

The signal-to-noise at the input of the receiver *SNRin* in dB can be written as:

$$\overline{\text{SNR}}\_{\text{in}} = 10 \cdot \log\_{10} \left( \frac{\frac{1}{D} \cdot \sum\_{i=1}^{D} P\_{\text{in}}(i)}{N\_0/2} \right) = 10 \cdot \log\_{10} \left( \frac{1}{D} \cdot \sum\_{i=1}^{D} \left( \frac{S}{N} \right)\_{\text{in}}(i) \right), \tag{41}$$

and it is taken from the set of values in Table 1. *Pin*(*i*) is the receiver input power in the mode *i* for the current channel realization.

#### *5.4. Performance Loss Metric for FSE-Based MIMO Receiver*

We define the performance loss metric (in dB) of the FSE-based MIMO receiver in MDL-impaired channels for certain mode *i* as:

$$L(i) = SNR\_{in}(i) - SNRR\_{out}(i) \tag{42}$$

where *SNR*(*in*)(*i*) = 10 · log10 *S N in*(*i*), and *SNDRout* = [*SNDRout*(1), *SNDRout*(2), ... , *SNDRout*(*D*)]*<sup>T</sup>* is calculated as defined in [42] and Equation (28) in [43] for MIMO implementation of the FSE. Given a set of system model parameters, the numerical simulation will generate a total of *D* · *Nch* values of *<sup>L</sup>*(*i*). The average loss *AL* is calculated for each channel realization of among the available *Nch* as:

$$AL = \frac{1}{D} \sum\_{i=1}^{D} L(i). \tag{43}$$

Two FSE-based MIMO receiver performance metrics can be extracted from the *D* · *Nch* calculated values of *<sup>L</sup>*(*i*):

•*ML*95 is defined as the 95th percentile of the *L*(*i*) distribution obtained for any optical channel realization and mode;

• *AML*95 is defined as the 95th percentile of the *AL* distribution obtained for any optical channel realization.

The values for the parameters used in the simulation are summarized in Table 1.

**Table 1.** Simulation parameters.


#### *5.5. Numerical Simulation Results*

The first result is focused on the impact of PAM pulses roll-off factor *α* and MDL level, represented by *<sup>σ</sup>g*, on the SDM optical system performance. Figure 6 shows that for systems with transmitters using a higher *α*, the degradation is a bit lower. The effect is higher with increasing *<sup>σ</sup>g* for systems working at a high regime of *SNRin*, as seen in Figure 6b.

**Figure 6.** *Cont*.

**Figure 6.** *ML*95 (up) and *AML*95 (down) as defined in Section 5.4 for *SNRin* = 5, 6.2, and 10 dB (**a**) and *SNRin* = 15, 30, and 60 dB (**b**) for different values of the transmitter roll-off factor *α*. Note that *<sup>σ</sup>g* = 0 corresponds to a channel without MDL.

In practical systems, the allowable loss of *SNDRout* in a channel with elements introducing MDL w.r.t. an ideal channel without MDL is around 1–2 dB. We can observe that, assuming a maximum degradation of 2 dB in the system with a 95% confidence, *<sup>σ</sup>g* values of the amplifiers should not exceed 0.2 dB. These results are in agreemen<sup>t</sup> with the capacity limits of a MIMO MMSE receiver and MDL channel calculated in [27].

A second result is presented in Figure 7, where the probability distribution of *AL* and *SNRin* estimated from the analysis of all channel realizations is plotted. We are comparing different levels of SNR regimes, with *SNRin* = 5, 6.2, and 10 dB (Figure 7a) and *SNRin* = 15, 30, and 60 dB (Figure 7b), for a roll-off factor of *α* = 0.9. The upper and lower limits of the blue boxes represent the 25th and 75th percentiles respectively. The red line inside the box indicates the median of the metric. In case the distribution of values was Gaussian, the whisker bounds correspond to 2.7 times the STD of the metric or, in other words, the number of values between the upper and lower bounds of the whiskers contains 99.3% of the values. Values outside these limits are considered outliers and are individually represented by red crosses.

We make the following observations from Figure 7:


**Figure 7.** Probability distribution for *AL* as defined in (43) (up) and *L*(*i*) as defined in (42) (down) for *SNRin* = 5, 6.2, and 10 dB (**a**) and *SNRin* = 15, 30, and 60 dB (**b**). *α* = 0.9 for all graphs. Note that *<sup>σ</sup>g* = 0 corresponds to a channel without MDL.
