Considering an *N* × *N* transceiving beam formation system, there are also lots of imperfect elements even if the beams between transmitting and receiving ends are completely in the right direction. Some of these unsatisfactory factors such as narrowband or wideband amplitude and phase error, the amplitude and phase error in in-phase (I) channel, quadrature (Q) channel, and phase noise, are focused in this article. Each of them is added into the ideal model and discussed, respectively (Figure 1).

### 2.1. Narrowband amplitude and phase error

In this study, we consider the case where the amplitude and phase error in the signal path are narrowband. Denote the output signal of filter in each transmitting path by *s*(*t*), and the output of the *m*_{
th
} (*m* = 1,..., *N*) transmitting channel with error by {g}_{m}{e}^{j{\varphi}_{m}}. Then the output of every transmitting antenna can be written as {g}_{m}{e}^{j{\varphi}_{m}}\cdot s\left(t\right), where *g*_{
m
} = 1-*σ*_{
m
}*, σ*_{
m
} is random and exponentially distributed, and *ϕ*_{
m
} is uniformly distributed in the range of [0,*θ*]. The *n*_{
th
} (*n* = 1,...,*N*) channel noise is denoted by *n*_{
n
} (*t*). If the amplitude and phase error exists in each signal path at both the transmitting and receiving ends, which are expressed as {g}_{m}{e}^{j{\varphi}_{m}} and {h}_{n}{e}^{j{\phi}_{n}} respectively, the total signal received will be

\sum _{n=1}^{N}\left[{h}_{n}{e}^{j{\phi}_{n}}\cdot \left(\left(\sum _{m=1}^{N}{g}_{m}{e}^{j{\varphi}_{m}}\cdot s\left(t\right)\right)+{n}_{n}\left(t\right)\right)\right]

(1)

In the multi-antenna transceiving systems, the counterparts of signal-to-noise ratio in single-antenna systems (SNR) will be updated to

SN{R}^{\prime}=\frac{E\left[{\left|\sum _{n=1}^{N}\left[{h}_{n}{e}^{j{\phi}_{n}}\cdot \left(\sum _{m=1}^{N}{g}_{m}{e}^{j{\varphi}_{m}}s\left(t\right)\right)\right]\right|}^{2}\right]}{E\left[{\left|\sum _{n=1}^{N}{h}_{n}{e}^{j{\phi}_{n}}\cdot {n}_{n}\left(t\right)\right|}^{2}\right]}=\frac{{\left|\sum _{n=1}^{N}{h}_{n}{e}^{j{\phi}_{n}}\right|}^{2}\cdot {\left|\sum _{m=1}^{N}{g}_{m}{e}^{j{\varphi}_{m}}\right|}^{2}\cdot {P}_{signal}}{\left(\sum _{n=1}^{N}{h}_{n}^{2}\right)\cdot {P}_{noise}}=\frac{{\left|\sum _{n=1}^{N}{h}_{n}{e}^{j{\phi}_{n}}\right|}^{2}\cdot {\left|\sum _{m=1}^{N}{g}_{m}{e}^{j{\varphi}_{m}}\right|}^{2}}{\left(\sum _{n=1}^{N}{h}_{n}^{2}\right)}\cdot SNR

(2)

that is,

SN{R}^{\prime}\left(\mathsf{\text{dB}}\right)=10{log}_{10}\left({\left|\sum _{n=1}^{N}{h}_{n}{e}^{j{\phi}_{n}}\right|}^{2}\right)+10{log}_{10}\left({\left|\sum _{m=1}^{N}{g}_{m}{e}^{j{\varphi}_{m}}\right|}^{2}\right)-10{log}_{10}\left(\sum _{n=1}^{N}{h}_{n}^{2}\right)+SNR\left(\mathsf{\text{dB}}\right)

(3)

### 2.2 Wideband amplitude and phase error

When wideband amplitude and phase error is considered, the signal propogation channel is considered as a matrix [6]:

\mathbf{h}=\left[\begin{array}{ccc}\hfill {h}_{11}\hfill & \hfill \cdots \phantom{\rule{0.3em}{0ex}}\hfill & \hfill {h}_{1N}\hfill \\ \hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \vdots \hfill \\ \hfill {h}_{N1}\hfill & \hfill \cdots \phantom{\rule{0.3em}{0ex}}\hfill & \hfill {h}_{NN}\hfill \end{array}\right]

(4)

where *h*_{
mn
} represents the amplitude fading and phase difference between the *m*_{
th
} transmitting antenna and the *n* th receiving antenna (Figure 2).

After considering the multi-path transmission, *h*_{
mn
} varies with time and can be modeled as [7]

{h}_{mn}\left(t\right)=\delta \left(t\right){e}^{jk{d}_{mn}}+b\delta \left(t-\tau \right){e}^{j{\varphi}_{mn}}

(5)

where b is the relative amplitude fading, *τ* is the counter-shifting, *ϕ* is the relative phase, k=\frac{2\pi}{\lambda} is the wave number and *d*_{
mn
} is the distance between the *m* th transmitting antenna and the *n* th receiving end. The Fourier transform of the channel matrix is

H\left(\omega \right)=\left[\begin{array}{ccc}\hfill {e}^{jk{d}_{11}}\hfill & \hfill \cdots \phantom{\rule{0.3em}{0ex}}\hfill & \hfill {e}^{jk{d}_{1N}}\hfill \\ \hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \vdots \hfill \\ \hfill {e}^{jk{d}_{N1}}\hfill & \hfill \cdots \phantom{\rule{0.3em}{0ex}}\hfill & \hfill {e}^{jk{d}_{NN}}\hfill \end{array}\right]+b{e}^{-j\omega \tau}\left[\begin{array}{ccc}\hfill {e}^{j\varphi {d}_{11}}\hfill & \hfill \cdots \phantom{\rule{0.3em}{0ex}}\hfill & \hfill {e}^{j\varphi {d}_{1N}}\hfill \\ \hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \vdots \hfill \\ \hfill {e}^{j\varphi {d}_{N1}}\hfill & \hfill \cdots \phantom{\rule{0.3em}{0ex}}\hfill & \hfill {e}^{j\varphi {d}_{NN}}\hfill \end{array}\right]

(6)

Thus, the channel's total power can be computed as

P=\sum _{m=1}^{N}\sum _{n=1}^{N}{\left|{e}^{jk{d}_{mn}}+b{e}^{-j\omega \tau}{e}^{j{d}_{mn}}\right|}^{2}

The channel gain is *G* = 10 · log_{10}*P*.

The received signal vector can be expressed as

\left[\begin{array}{c}\hfill {r}_{1}\hfill \\ \hfill {r}_{2}\hfill \\ \hfill \vdots \hfill \\ \hfill {r}_{N}\hfill \end{array}\right]=H\cdot \left[\begin{array}{c}\hfill {s}_{1}\hfill \\ \hfill {s}_{2}\hfill \\ \hfill \vdots \hfill \\ \hfill {s}_{N}\hfill \end{array}\right]+\left[\begin{array}{c}\hfill {n}_{1}\hfill \\ \hfill {n}_{2}\hfill \\ \hfill \vdots \hfill \\ \hfill {n}_{N}\hfill \end{array}\right]

(7)

### 2.3. Amplitude and phase error in in-phase and quadrature channels

The procedure of extracting the in-phase and quadrature parts and up-converting brings amplitude and phase error to the receiving signal. The errors in the transmitting and receiving ends have similar expressions, which are modeled [8] as: *T*_{
m
} (*t*) = 2[*k*_{
m
} cos(*ωt*), sin(*ωt*+*φ*_{
m
} )] and *R*_{
n
} (*t*) = [*l*_{
n
} cos (*ωt*), sin(*ωt*+*γ*_{
n
} )]. The *k*_{
m
} and *l*_{
n
} (*m*,*n* = 1,...,*N*) represent the amplitude errors and they are both in the range of [1-*ξ*,1+*ξ*], where *ξ* is a tiny number. The phase errors *φ* and *γ* are assumed to be much less than \frac{\pi}{2}. After the low-pass filtering, the received signal of the *n* th path is

\begin{array}{cc}\hfill {r}_{n}\left(t\right)& =LPF\u27e8{R}_{n}\left(t\right)\cdot \left[\left(\sum _{m=1}^{N}{T}_{m}\left(t\right)\cdot s\left(t\right)\right)+{n}_{n}\left(t\right)\right]\u27e9\hfill \\ =LPF\u27e8{R}_{n}\left(t\right)\cdot \left(\sum _{m=1}^{N}{T}_{m}\left(t\right)\right)\cdot s\left(t\right)+{R}_{n}\left(t\right)\cdot {n}_{n}\left(t\right)\u27e9\hfill \\ \triangleq {H}_{n}\cdot s\left(t\right)+{n}_{rn}\left(t\right)\hfill \end{array}

(8)

where *H*_{
n
} represents the *n* th channel matrix of I/Q channel and is defined as

\begin{array}{cc}\hfill {H}_{n}& =LPF\u27e8{R}_{n}\left(t\right)\cdot \left(\sum _{m=1}^{N}{T}_{m}\left(t\right)\right)\u27e9\hfill \\ =LPF\u27e82{\left[{l}_{n}cos\left(\omega t\right),sin\left(\omega t+{\gamma}_{n}\right)\right]}^{T}\cdot \left[\left(\sum _{m=1}^{N}{k}_{m}\right)cos\left(\omega t\right),\sum _{m=1}^{N}sin\left(\omega t+{\phi}_{m}\right)\right]\u27e9\hfill \\ =\left[\begin{array}{cc}\hfill {l}_{n}\cdot \left(\sum _{m=1}^{N}{k}_{m}\right)\hfill & \hfill {l}_{n}\cdot \left(\sum _{m=1}^{N}sin\left({\phi}_{m}\right)\right)\hfill \\ \hfill \left(\sum _{m=1}^{N}{k}_{m}\right)\cdot sin\left({\gamma}_{n}\right)\hfill & \hfill \sum _{m=1}^{N}cos\left({\gamma}_{n}-{\phi}_{m}\right)\hfill \end{array}\right]\hfill \end{array}

The noise form in (8) is n_{
nr
}(*t*) = *LPF*〈R_{
n
}(*t*)*n*_{
n
} (*t*)〉.

The sum of the signal from all *N* channels is

r\left(t\right)=\sum _{n=1}^{N}{r}_{n}\left(t\right)=\sum _{n=1}^{N}\left[{H}_{n}\cdot s\left(t\right)+{n}_{nr}\left(t\right)\right]

### 2.4. The model and correction of phase noise

For both the single-antenna and multi-antenna systems, local oscillator (LO) will bring phase noise inevitably. Usually, the phase noise is assumed to obey the white Gaussian noise model, colored Gaussian noise model, or Wiener model [7, 9–11]. Here we consider the Wiener model and denotes the phase noise for the *k* th symbol by *e*^{jφk} . The discrete form of phase noise can be written as

\phi \left(k\right)=\alpha \cdot \phi \left(k-1\right)+\Delta \phi

(9)

where Δ*φ* obeys the normal distribution and is random generated, that is, Δ*φ*~*N*(0,*σ*^{2}), {\sigma}^{2}=\frac{4\pi {f}_{\Delta \mathsf{\text{3dB}}}}{{F}_{s}}, *f*_{Δ3dB} is the 3 dB line width, *F*_{
s
} is the sampling frequency, *α* is a constant near but less than 1. The relational expression displayed in (9) indicates that the current phase noise is related with its previous moment, which is different to the two other counterparts. Assumed that *s*(*t*) is the output signal in each transmitting path and all the transmitted signals can be received by every receiving antenna. The sampled received signal in the *n* th path can be written as

{r}_{n}\left(k\right)=\left[N\cdot s\left(k\right)+{n}_{n}\left(k\right)\right]{e}^{j{\phi}_{n}\left(k\right)}

(11)

where *N* denotes the number of antennas which is the same for both the transmitting and receiving ends. The sum of received signals is equivalent to *N*^{2} times the transmitting signal in the ideal case. Then the error can be expressed as

\begin{array}{cc}\hfill e\left(k\right)& =\left(\frac{1}{{N}^{2}}\sum _{n=1}^{N}{r}_{n}\left(k\right)\right)-s\left(k\right)\hfill \\ =\left(\frac{1}{{N}^{2}}\sum _{n=1}^{N}\left[\left(N\cdot s\left(k\right)+{n}_{n}\left(k\right)\right)\cdot {e}^{j{\phi}_{n}\left(k\right)}\right]\right)-s\left(k\right)\hfill \\ =s\left(k\right)\cdot \left[\left(\frac{1}{N}\sum _{n=1}^{N}{e}^{j{\phi}_{n}\left(k\right)}\right)-1\right]+\frac{1}{{N}^{2}}\sum _{n=1}^{N}{n}_{n}\left(k\right)\cdot {e}^{j{\phi}_{n}\left(k\right)}\hfill \end{array}

(12)

Thanks to the especial character, a few approaches are derived to eliminate the error *e*(*n*). Gitlin [12] modeled the phase noise as sinusoidal signal and presented self-adaptation and compensation method named finite impulse response-adaptive line enhancer (FIR-ALE). However, the ability of compensation is limited. Another method, the infinite impulse response-adaptive line enhancer (IIR-ALE) [13] has obvious effect provided many prior information being known. extended Kalman filter (EKF) [14] is used to estimate phase noise by linearizing first order digital phase locked loop (DPLL), in which, however, the non-convergent condition happens sometimes.

In this article a more accurate method is adopted which is called angle-offset least mean square (AO-LMS). As shown in Figure 3, the angle difference between input signal and its hard-decision results is used as the input of LMS to estimate the real phase noise.

As an improvement approach to FIR-ALE, the input of LMS algorithm is a more accurate input, which is closer to the real phase offset caused by phase noise. It means a clearer aim and a better compensation ability.

### 2.5 Simulation results

The four errors introduced above are simulated. We consider a 64-ary QAM multi-antenna system. The transmitting and receiving ends each has 4 antennas. The sampling rate is *f*_{
s
} = 4 MHz. Rooted raised cosine (RRC) filters are adopted in both the transmitting and receiving ends. The roll-off factor is *α* = 0.02. The rate of sampling period and symbol period is 2, and the order of RRC is 16. The effects of different factors on the transceiving performance are shown in Figures 4, 5, and 6. As is shown in Figure 4, symbol error rate (SER) gets a sharply increase when the narrowband phase error starts rising. Comparatively, the amplitude error puts a much less influence on it. Figure 5 displays the effect of wideband amplitude and phase error. When the channel fading is small (i.e., b1 = 0.01), the situation is very similar to the ideal case. However, just a little increase of the fading can lead to the rapid rise of SER much less a large value near 1. The amplitude and phase error caused in I/Q channels are measured by another criteria error vector magnitude (EVM) as it is shown in Figure 6. Yet the amplitude error and phase error both have a drastic influence on the transceiver's performance.

The following simulation results verify the compensation ability of AO-LMS. Constellation before and behind the block of phase noise compensation are shown in Figures 7 and 8. In the view of constellation, the signal with phase noise brings a circular change along the amplitude circles. It is distinctly presented when SNR is large enough. Figure 8 shows that the algorithm adopted can take an apparent clustering even if large Gaussian channel noise exists. On the other hand, as is shown in Figure 9, the symbol error number can decrease sharply to less than one-tenth compared to the number before correction and at least one order of the SNR (dB) is gotten. Furthermore, the algorithm is adapted to engineering application due to its less tap number. The tap number used in the simulation is 16.