Joint Calibration of Transmitter and Receiver Impairments in Direct-Conversion Radio Architecture

全文

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Joint Calibration of

Transmitter and Receiver Impairments in

Direct-Conversion Radio Architecture

Chen-Jui Hsu and Wern-Ho Sheen, Member, IEEE

Abstract—Direct-conversion radio architecture is a low-cost,

low-power and small-size design that has been widely employed in today’s wireless devices. This architecture, however, induces radio impairments such as I-Q imbalance and dc offset that may incur severe degradation in communication performance if left uncompensated. In this paper, a new method is proposed to calibrate simultaneously a transceiver’s own transmitter and receiver radio impairments with no dedicated analog circuit in the feedback loop. Based on a unified time-domain approach, the proposed method is able to calibrate jointly the frequency-independent I-Q imbalance, frequency-dependent I-Q imbalance and dc offset and is applicable to any type of communication systems (single-carrier, multiple-carrier, etc.). The existing meth-ods in the literature either need a dedicated analog circuit in the feedback loop and/or are applicable only to a particular type of systems with some radio impairments present. The issue of training sequence design is also investigated to optimize the calibration performance, and analytical and simulation results show that the performance loss due to radio impairments can be recovered by the proposed method.

Index Terms—Direct-conversion transceiver, self-calibration,

I-Q imbalance, DC offset.

I. INTRODUCTION

D

IRECT-CONVERSION radio architecture is a low-cost, low power and small-size design that has gained popular-ity in today’s wireless devices [1][2]. The radio impairments of this architecture such as I-Q imbalance and dc offset, however, result in a severe degradation in communication performance if left uncompensated [3]-[9]. This is particularly true in the next-generation high data-rate systems where a wide bandwidth and a high-order modulation are deemed to be employed. Removal and/or compensation of the radio im-pairments in the direct-conversion radio architecture have been an area of extensive research. Generally speaking, two types of techniques have been proposed [3]-[19]: one is calibration and the other is estimation/compensation. Calibration is a tech-nique used to remove the effects of a transceiver’s own radio impairments [9]-[19], whereas the estimation/compensation technique is to counteract the cascaded transmitter and receiver Manuscript received May 18, 2011; revised August 31, 2011; accepted September 27, 2011. The associate editor coordinating the review of this paper and approving it for publication was G. Colavolpe.

C. J. Hsu is with the Department of Electrical Engineering, National Chiao Tung University, Hsinchu 300, Taiwan, R.O.C.

W. H. Sheen is with the Department of Information and Communication Engineering, Chaoyang University of Technology, Wufong, Taichung41349, Taiwan (e-mail: linch.ee89@nctu.edu.tw, whsheen@cyut.edu.tw).

Digital Object Identifier 10.1109/TWC.2011.110811.110939

radio impairments at the receiving side [3]-[9]. Both types of techniques find their applications in real systems [3]-[19]. In this work, we focus on the digital calibration technique.

Many interesting works have been devoted to the calibration of the radio impairments in the direct-conversion architec-ture [9]-[19]. In [9]-[11], adaptive methods were proposed to calibrate transmitter frequency-independent I-Q imbalance and dc offset by using an analog envelope-detector (ED) in the feedback loop. Transmitter frequency-dependent I-Q imbalance was investigated for the continuous frequency-shift-keying (CFSK) systems in [12][13], with no consideration on other impairments; frequency-dependent I-Q imbalance is particularly problematic in a wideband system where it is very challenging to keep the I- and Q-branch analog filters perfectly matched over the entire band. The works in [14]-[16] discussed calibration of the transmitter frequency-independent and dependent I-Q imbalances by using the low-IF architecture in the feedback loop.

So far, most of the calibration techniques in the literature have focused on the transmitter radio impairments, either by employing an ED [9]-[11] or the low-IF radio architecture [14]-[16] in the feedback loop so that the receiver impairments can be neglected safely. However, using an ED or the low-IF radio architecture in the feedback loop increases the complex-ity in the analog domain. In addition, calibration of the receiver impairments is important in its own right; for example, if one’s own receiver has been calibrated, only the transmitter-side impairments (of the transmitting device), rather than the cascaded transmitter and receiver impairments, need to be estimated and compensated for at the receiver, and that reduces the receiver complexity [4]-[8].

Very recently, the issue of joint calibration of a transceiver’s own transmitter and receiver radio impairments in the direct-conversion radio architecture was investigated in [17]-[19]. In [17], a two-feedback method was proposed, where the phase of the receiver oscillator is shifted by exact 90 degrees in the second feedback, aiming to separate the transmitter and receiver I-Q imbalances. Unfortunately, it is not practical to have an exact 90 degrees phase rotation in real systems. In [18], a new calibration method was proposed for the OFDM (orthogonal frequency-division multiplexing) type of systems with no dedicated analog circuit in the feedback loop. The method, however, is designed for the OFDM-type of systems and can only calibrate the frequency-independent I-Q imbalance and dc offset.

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Direct-Conversion RF Transmitter cos(2πf tT) sin(2 ) T f tT T α π θ − + ( ) x t 2cos(2πf tR) 2αRsin(2πf tR θR) − + 0Q d 0 I d ( ) I r n ( ) Q r n Direct-Conversion RF Receiver ( )n ρ ( ) r n ( )* ( )* Joint Estimator of RF Impairments ˆ ( )n ρ ˆ d ˆb ˆ ( ) w n

Receiver Calibration Circuit

( ) p s t b ( ) w n ( ) s n complex signal real signal 0( ) v t ( ) c r n

{

2

}

0 Re b ejπf tT

Transmitter Calibration Circuit To Power

Amplifier From LNA 1 S 2 S 3 S ( ) Q T h t ( ) I T h t ( ) T h t ( ) I R h t h tR( ) ( ) Q R h t h tR( ) ( ) T h t d 4 S 

( )

y t

Fig. 1. The system consisting of a direct-conversion RF transceiver, calibration circuits and a joint estimator of the calibration parameters.

In this paper, a new method is proposed to calibrate a transceiver’s own transmitter and receiver impairments si-multaneously, with no dedicated analog circuit in the feed-back loop. Compared to the work in [18], the proposed method is unique in that it is applicable to any type of communication systems and is able to calibrate jointly the frequency-independent Q imbalance, frequency-dependent I-Q imbalance, and dc offset, thanks to the proposed unified time-domain approach. This work is an extension of our work in [19], where parts of the results in this paper were firstly reported.

The rest of this paper is organized as follows. Section II describes the models of radio impairments and digital calibration circuits. Section III develops a joint estimator under the principle of nonlinear least-squares. The proposed method is then analyzed in Section IV, and numerical results and conclusions are given in Section V and VI, respectively.

II. RADIOIMPAIRMENTSANDCALIBRATIONCIRCUITS Figure 1 depicts the considered system that consists of calibration circuits, the direct-conversion radio transceiver, and a joint estimator of the transmitter and receiver radio impairments. The switches 𝑆1, 𝑆2, 𝑆3 and 𝑆4 are used to

control the signal flow paths; during calibration training,𝑆1 ,

𝑆3 and𝑆4 are at the upper positions, and𝑆2 is at the lower

position to form an internal loopback, whereas during normal communication,𝑆1,𝑆3 and𝑆4are at the lower positions, and

𝑆2is at the upper position. Since the transceiver’s own receiver

is used for internal loopback during calibration training, a

dedicated calibration period is assumed in this method prior to normal communication where transmitter and receiver are located in different devices.

A. Radio Impairments

At the transmitter, the radio impairments that are investigated include frequency-dependent I-Q imbalance, frequency-independent I-Q imbalance and carrier feed-through. Frequency-dependent I-Q imbalance is due to mis-match between the in-phase (I) and quadrature-phase (Q) analog filters which are denoted by ℎ𝑇(𝑡) ⊗ ℎ𝐼𝑇(𝑡) and ℎ𝑇(𝑡) ⊗ ℎ𝑄𝑇(𝑡). Here, ℎ𝑇(𝑡) is the common part of the

filters, and ⊗ denotes the operation of linear convolution. Frequency-independent I-Q imbalances are due to the gain and phase mismatches between the I and Q branches of the mixer circuitry and denoted by𝛼𝑇 and𝜃𝑇, respectively. And,

the carrier feed-through induces dc offset at the receiving side [1][2] and is characterized by Re{𝑏0𝑒𝑗2𝜋𝑓𝑇𝑡}, where

𝑏0 = 𝑏𝐼0+ 𝑗𝑏𝑄0 , 𝑓𝑇 is the transmit center frequency, and

𝑗 = √−1. In the following, 𝑏0 is called the transmitter dc

offset. Define 𝑠𝑝(𝑡) .= 𝑠𝐼𝑝(𝑡) + 𝑗𝑠𝑄𝑝 (𝑡) =𝑛 𝑠𝑝(𝑛) 𝛿 (𝑡 − 𝑛𝑇𝑠), (1)

be the signal appearing at the input of the transmitter, where 𝑠𝑝(𝑛) is its discrete-time equivalent, 𝑇𝑠is the symbol duration,

and𝛿(𝑡) is the Dirac delta function. Under the effects of the radio impairments, the pass-band transmit signal is ˜𝑥 (𝑡) =

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Re{𝑥 (𝑡) 𝑒𝑗2𝜋𝑓𝑇𝑡}with its base-band equivalent given by [15] 𝑥 (𝑡) = ℎ𝑇,+(𝑡) ⊗ 𝑠𝑝(𝑡) + ℎ𝑇,−(𝑡) ⊗ 𝑠∗𝑝(𝑡) + 𝑏0, (2) where ℎ𝑇,±(𝑡) = 1/2 ⋅ [ ℎ𝐼 𝑇(𝑡) ± 𝛼𝑇𝑒𝑗𝜃𝑇ℎ𝑄𝑇 (𝑡) ] ⊗ ℎ𝑇(𝑡) , (3) and𝑎∗denote the complex conjugate of𝑎. In (2), 𝑠

𝑝(𝑡) can be

viewed as being transmitted through two paths along with a corruption from dc offset; one is the desired path with impulse responseℎ𝑇,+(𝑡), and the other is the mirror-frequency path

with impulse responseℎ𝑇,−(𝑡). Clearly, I-Q imbalances incur

mirror-frequency interference in the transmitted signal. With no I-Q imbalances and dc offset, i.e.,ℎ𝐼

𝑇(𝑡) = ℎ𝑄𝑇 (𝑡) = 𝛿 (𝑡), 𝛼𝑇 = 1 and 𝜃𝑇 = 𝑏0= 0, 𝑥 (𝑡) = 𝑠𝑝(𝑡)⊗ℎ𝑇(𝑡) as one might

expect.

Likewise, the radio impairments that are investigated at the receiver include frequency-independent I-Q imbalance, char-acterized by𝛼𝑅and𝜃𝑅, frequency-dependent I-Q imbalance,

characterized by the filtersℎ𝑅(𝑡) ⊗ ℎ𝐼𝑅(𝑡) and ℎ𝑅(𝑡) ⊗ ℎ𝑄𝑅(𝑡),

and dc offset, 𝑑0 = 𝑑𝐼0+ 𝑗𝑑𝑄0. ℎ𝑅(𝑡) is the common part

of the filters, and 𝑓𝑅 = 𝑓𝑇 − Δ𝑓 is the receive center

fre-quency. During normal communication,Δ𝑓 is a real frequency offset between transmitter and receiver which are located in different devices, whereas during calibration training, Δ𝑓 is an intentional frequency shift1 that is introduced purposely in

our method to help the radio impairments estimation, as is to be detailed in Section III.

Denote ˜𝑦(𝑡) and ˜𝑣0(𝑡) be the received pass-band signal and

additive white Gaussian noise in Figure 1. Under the effects of radio impairments, the received base-band signal is given by [8]

𝑟 (𝑡) = ℎ𝑅,+(𝑡) ⊗[𝑒𝑗2𝜋Δ𝑓𝑡𝑦 (𝑡) + 𝑣0(𝑡)]

+ ℎ𝑅,−(𝑡) ⊗[𝑒𝑗2𝜋Δ𝑓𝑡𝑦 (𝑡) + 𝑣0(𝑡)]∗+ 𝑑0, (4)

where𝑦(𝑡) and 𝑣0(𝑡) are the low-pass equivalents of ˜𝑦(𝑡) and

˜𝑣0(𝑡), respectively, and ℎ𝑅,±(𝑡) = 1/2 ⋅ [ ℎ𝐼 𝑅(𝑡) ± 𝛼𝑅𝑒∓𝑗𝜃𝑅ℎ𝑄𝑅(𝑡) ] ⊗ ℎ𝑅(𝑡) . (5)

Again, (4) says that the receiver I-Q imbalances induce mirror-frequency interference, and in the absence of I-Q imbalance and dc offset, 𝑟 (𝑡) = ℎ𝑅(𝑡) ⊗[𝑒𝑗2𝜋Δ𝑓𝑡𝑦 (𝑡) + 𝑣0(𝑡)]. It is

worthy to note that during calibration training,𝑦 (𝑡) = 𝑥 (𝑡) which is the signal transmitted from its own transmitter be-cause of the internal loopback. During normal communication, on the other hand,𝑦 (𝑡) = 𝑥 (𝑡)⊗𝑐 (𝑡), where 𝑥(𝑡) is the signal transmitted from the transmitter in other device, and 𝑐(𝑡) is the channel impulse response.

B. Calibration Circuits

At the transmitter, we propose to use a pre-distortion filter, 𝑤(𝑛), and a dc correction term, 𝑏, to calibrate I-Q imbalances and dc offset, respectively as in Figure 12. After calibration,

𝑠𝑝(𝑛) is given by

𝑠𝑝(𝑛) = [𝑠 (𝑛) + 𝑏] + 𝑤 (𝑛) ⊗ [𝑠 (𝑛) + 𝑏]∗, (6) 1In practice, the intentional frequency shift can be implemented precisely with two digital frequency synthesizers from a single reference oscillator [20]. 2The idea of using a pre-distortion filter to remove the mirror-frequency interference was also reported in [15].

where 𝑠(𝑛) is the transmitted symbol. For convenience, the equivalent discrete-time model will be used throughout the rest of the paper. In this way, (2) is rewritten as

𝑥 (𝑛) = 𝑔𝑇,+(𝑛) ⊗ 𝑠 (𝑛) + 𝑔𝑇,−(𝑛) ⊗ 𝑠∗(𝑛) + Δ𝑏, (7) where𝑢 (𝑛) = 𝑢 (𝑡)∣𝑡=𝑛𝑇𝑠 𝑢 ∈ {𝑥, ℎ𝑇,+, ℎ𝑇,−}, 𝑔𝑇,+(𝑛) = ℎ𝑇,+(𝑛) + 𝑤∗(𝑛) ⊗ ℎ𝑇,−(𝑛) , (8) 𝑔𝑇,−(𝑛) = ℎ𝑇,−(𝑛) + 𝑤 (𝑛) ⊗ ℎ𝑇,+(𝑛) , (9) and Δ𝑏 = 𝑔𝑇,+(𝑛) ⊗ 𝑏 + 𝑔𝑇,−(𝑛) ⊗ 𝑏∗+ 𝑏0. (10)

In (8)-(10), 𝑔𝑇,+(𝑛) is regarded as the overall impulse

re-sponse of the desired path after calibration,𝑔𝑇,−(𝑛) is that of

the mirror-frequency path, and Δ𝑏 is the residual dc offset. Ideally,𝑔𝑇,−(𝑛) = 0 and Δ𝑏 = 0 which lead to

𝑤𝑜𝑝𝑡(𝑛) = −(ℎ𝑇,+(𝑛) )⊗ ℎ 𝑇,−(𝑛) , (11) 𝑔𝑇,+,𝑜𝑝𝑡(𝑛) = ℎ𝑇,+(𝑛) + 𝑤𝑜𝑝𝑡∗ (𝑛) ⊗ ℎ𝑇,−(𝑛) , (12) and 𝑏𝑜𝑝𝑡= −𝑏0(𝑔𝑇,+,𝑜𝑝𝑡(𝑛) ), (13)

where the notation (ℎ (𝑛))† is to denote the inverse filter of

ℎ(𝑛). As is expected, 𝑤𝑜𝑝𝑡(𝑛) = 𝑏𝑜𝑝𝑡 = 0 for the case of

no I-Q imbalances and dc offsets. The estimation of𝑤𝑜𝑝𝑡(𝑛)

and 𝑏𝑜𝑝𝑡 is done during calibration training and used during

normal communication.

As in [15][16], the image-rejection-ratio (IRR) will be adopted as the performance measure for the I-Q imbalance calibration, which is defined as

𝐼𝑅𝑅𝑇(𝑓) = 10log10∣𝐺𝑇,+(𝑓)∣ 2

∣𝐺𝑇,−(𝑓)∣2dB,

(14) where 𝑈 (𝑓) .= FT [𝑢 (𝑛)] is the Fourier transform (FT) of 𝑢(𝑛). In addition, the ratio

𝜀𝑇 = 10log10∣Δ𝑏∣ 2

∣𝑏02dB

(15) will be adopted as the performance measure for the dc offset calibration.

At the receiver, a time-domain calibration filter, 𝜌 (𝑛), is employed to remove the receiver mirror-frequency interfer-ence, and a dc correction term, 𝑑, is used to remove the dc offset (see Figure 1). Thus, the received signal after calibration is given by 𝑟𝑐(𝑛) = (𝑟 (𝑛) − 𝑑) − 𝜌 (𝑛) ⊗ (𝑟 (𝑛) − 𝑑)∗ = 𝑔𝑅,+(𝑛) ⊗[𝑒𝑗2𝜋𝜇𝑛𝑦 (𝑛) + 𝑣0(𝑛)] + 𝑔𝑅,−(𝑛) ⊗[𝑒𝑗2𝜋𝜇𝑛𝑦 (𝑛) + 𝑣0(𝑛)]∗+ Δ𝑑, (16) where 𝑟 (𝑛) = ℎ𝑅,+(𝑛) ⊗[𝑒𝑗2𝜋𝜇𝑛𝑦 (𝑛) + 𝑣0(𝑛)] + ℎ𝑅,−(𝑛) ⊗[𝑒𝑗2𝜋𝜇𝑛𝑦 (𝑛) + 𝑣0(𝑛)]∗+ 𝑑0, (17)

𝜇 = Δ𝑓𝑇𝑠 is the normalized frequency offset,

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Δ𝑑 = (𝑑0− 𝑑) − 𝜌 (𝑛) ⊗ (𝑑0− 𝑑)∗, (19)

and {𝑣0(𝑛)} are i.i.d. (independent, and identically

dis-tributed) zero-mean Gaussian noise with variance of 𝜎2 0. In

(16), 𝑦 (𝑛) = 𝑥 (𝑛) ⊗ 𝑐 (𝑛) with 𝑥(𝑛) being the signal transmitted from the other device (normal communication), and𝜇 is the real frequency offset between the transmitter and receiver which are located in different devices. Clearly, it is desirable to have𝑔𝑅,−(𝑛) = 0 and Δ𝑑 = 0; or equivalently,

𝜌𝑜𝑝𝑡(𝑛) =(ℎ∗𝑅+(𝑛)

)⊗ ℎ

𝑅−(𝑛) , (20)

and

𝑑𝑜𝑝𝑡= 𝑑0. (21)

Similar to the transmitter case, the receiver calibration perfor-mance is evaluated by 𝐼𝑅𝑅𝑅(𝑓) = 10log10∣𝐺𝑅,+(𝑓)∣ 2 ∣𝐺𝑅,−(𝑓)∣2dB (22) and 𝜀𝑅= 10log10∣Δ𝑑∣ 2 ∣𝑑02dB, (23) for the I-Q imbalance and dc offset, respectively. Also, 𝜌𝑜𝑝𝑡(𝑛) = 𝑑𝑜𝑝𝑡 = 0 for the case of no receiver I-Q

imbal-ances and dc offset. 𝜌𝑜𝑝𝑡(𝑛) and 𝑑𝑜𝑝𝑡 are estimated during

calibration training. With perfect calibration in the transmitter and receiver,𝑟𝑐(𝑛) in (16) is given by,

𝑟𝑐(𝑛) = 𝑔𝑅,+(𝑛) ⊗ [ 𝑒𝑗2𝜋𝜇𝑛(𝑔 𝑇,+(𝑛) ⊗ 𝑠 (𝑛) ⊗ 𝑐 (𝑛)) + 𝑣0(𝑛) ] . (24) In this case, the frequency offset 𝜇 and channel 𝑐(𝑛) are the radio parameters left to be estimated and compensated at the receiver before making a detection on𝑠(𝑛). Quite a lot of methods have been proposed for channel and frequency-offset estimation, for example in [4]-[5],[7]-[8] and references therein.

III. JOINTESTIMATIONOFCALIBRATIONPARAMETERS In this section, a joint estimation of the calibration pa-rameters, 𝑤𝑜𝑝𝑡(𝑛), 𝑏𝑜𝑝𝑡, 𝜌𝑜𝑝𝑡(𝑛), and 𝑑𝑜𝑝𝑡 is developed first,

followed by the design of training sequence and frequency shift 𝜇 to optimize the calibration performance. Recall that the estimation is done during calibration training (internal loopback), where 𝑠𝑝(𝑛) = 𝑠 (𝑛) is the training sequence, 𝑦 (𝑡) = 𝑥 (𝑡) with 𝑥(𝑡) being the signal transmitted from its own transmitter, and 𝜇 is the intentional frequency shift introduced to help estimation of the calibration parameters. A. Non-linear Least-Squares Estimation

Using (7), the received signal in (17) can be rewritten as 𝑟 (𝑛) = 𝑒𝑗2𝜋𝜇𝑛[𝑓 1,+(𝑛) ⊗ 𝑠 (𝑛) + 𝑓1,−(𝑛) ⊗ 𝑠∗(𝑛) + 𝑏1] + 𝑒−𝑗2𝜋𝜇𝑛[𝑓 2,+(𝑛) ⊗ 𝑠 (𝑛) + 𝑓2,−(𝑛) ⊗ 𝑠∗(𝑛) + 𝑏2] + 𝑑0+ 𝑣 (𝑛) , (25) where 𝑓1,±(𝑛) =(ℎ𝑅,+(𝑛) 𝑒−𝑗2𝜋𝜇𝑛)⊗ ℎ𝑇,±(𝑛) , (26) 𝑓2,±(𝑛) =(ℎ𝑅,−(𝑛) 𝑒𝑗2𝜋𝜇𝑛)⊗ ℎ∗𝑇,∓(𝑛) , (27) 𝑏1=(ℎ𝑅,+(𝑛) 𝑒−𝑗2𝜋𝜇𝑛)⊗ 𝑏0, (28) 𝑏2=(ℎ𝑅,−(𝑛) 𝑒𝑗2𝜋𝜇𝑛)⊗ 𝑏∗0, (29) and 𝑣 (𝑛) = ℎ𝑅,+(𝑛) ⊗ 𝑣0(𝑛) + ℎ𝑅,−(𝑛) ⊗ 𝑣0∗(𝑛) . (30)

Our goal here is to estimate 𝑤𝑜𝑝𝑡(𝑛), 𝑏𝑜𝑝𝑡,𝜌𝑜𝑝𝑡(𝑛) and 𝑑𝑜𝑝𝑡

from𝑟(𝑛), given the training sequence 𝑠(𝑛) and the intentional frequency shift 𝜇. Obviously, one possible way to do this is to estimate ℎ𝑇,±(𝑛), ℎ𝑅,±(𝑛), 𝑏0 and𝑑0 directly from

(25)-(30) and then apply them to (11)-(13), (20) and (21). Direct estimation of ℎ𝑇,±(𝑛), ℎ𝑅,±(𝑛), 𝑏0 and𝑑0, however, is very

complex as can be seen from (25)-(29). Instead, a simpler method is proposed here based on the following observations:

𝑤𝑜𝑝𝑡(𝑛) = −(ℎ𝑇,+(𝑛) )⊗ ℎ 𝑡,−(𝑛) = −(𝑓1,+(𝑛))†(𝑛) ⊗ 𝑓1,−(𝑛) , (31) 𝑏𝑜𝑝𝑡= −𝑏0(𝑔𝑇,+,𝑜𝑝𝑡(𝑛) ) = −𝑏1⊗ [𝑓1,+(𝑛) + 𝑤𝑜𝑝𝑡∗(𝑛) ⊗ 𝑓1,−(𝑛)]†, (32) and 𝜌𝑜𝑝𝑡(𝑛) =(ℎ∗𝑅,+(𝑛) )⊗ ℎ 𝑅,−(𝑛) =(𝑓∗ 1,+(𝑛) 𝑒−𝑗2𝜋𝜇𝑛 )(𝑓 2,−(𝑛) 𝑒−𝑗2𝜋𝜇𝑛). (33)

Therefore, 𝑤𝑜𝑝𝑡(𝑛), 𝑏𝑜𝑝𝑡, and 𝜌𝑜𝑝𝑡(𝑛) can be calculated

through 𝑓1,±(𝑛), 𝑓2,−(𝑛), and 𝑏1 which along with 𝑑0 can

be estimated from (25)-(30) in a much easier way, as to be discussed below. In the proposed method,𝑓1,±(𝑛), 𝑓2,±(𝑛),

𝑏1,𝑏2and𝑑0will all be estimated under the principle of

least-squares with 𝑓2,+(𝑛) and 𝑏2 serving as auxiliary variables

which are not needed in the final evaluation (see (31)-(33)). To this end, firstly let 𝑓1,±(𝑛) and 𝑓2,±(𝑛) abe modeled as

FIR (finite impulse response) filters,

f𝑖,±= [𝑓𝑖,±(0) , 𝑓𝑖,±(1) , . . . , 𝑓𝑖,±(𝐿𝑓− 1)]𝑇, 𝑖 = 1, 2,

(34) where 𝐿𝑓 is the filters’ length and usually not known in

advance. In Section V, it will be shown that the estimation performance is quite insensitive to the value of 𝐿𝑓 if it is of

sufficient length.

Consider a training sequence {𝑠 (𝑛)}𝑁−1𝑛=−𝐾, where 𝐾 ≥ 𝐿𝑓, and 𝑠(𝑛) = 𝑠(𝑛 + 𝑁) 𝑛 = −𝐾, ⋅ ⋅ ⋅, −1 is the

cyclic-prefix3. Define S be the 𝑁 × 𝐿

𝑓 signal matrix with

[S]𝑖,𝑗 = 𝑠 (𝑖 − 𝑗), 0 ≤ 𝑖 ≤ 𝑁 − 1, 0 ≤ 𝑗 ≤ 𝐿𝑓 − 1,

andf =[f𝑇

1,+, f1,−𝑇 , 𝑏1, f2,+𝑇 , f2,−𝑇 , 𝑏2, 𝑑0]𝑇. Then, (25) can be

rearranged into the following vector-matrix form

r = Φf + v, (35)

where r = [𝑟 (0) , 𝑟 (1) , . . . , 𝑟 (𝑁 − 1)]𝑇, Φ = [

Γ𝑁(𝜇) T Γ𝑁(−𝜇) T 1𝑁 ], T = [ S S 1𝑁 ], Γ𝑁(𝜇) = diag{1,𝑒𝑗2𝜋𝜇, . . . , 𝑒𝑗2𝜋𝜇(𝑁−1)} is the diagonal 3Generally,𝑁 is selected based on a tradeoff between calibration perfor-mance and complexity. Given a design of a radio transceiver where the worst values of radio impairments are specified,𝑁 can be selected according to the designer’s own tradeoff on the performance vs. complexity.

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matrix with elements {1,𝑒𝑗2𝜋𝜇, ⋅ ⋅ ⋅ , 𝑒𝑗2𝜋𝜇(𝑁−1)},

1𝑁 is the all 1 vector with dimension 𝑁, and

v = [𝑣 (0) , 𝑣 (1) , . . . , 𝑣 (𝑁 − 1)]𝑇. From (35), the

least-squares estimate ˆf is given by

ˆf = Υr, (36)

whereΥ =(Φ𝐻Φ)−1Φ𝐻 is the pseudo inverse of Φ. Note

thatΦ has to have full-rank in order to assure identifiability.

After obtaining ˆf, 𝑤𝑜𝑝𝑡(𝑛),𝑏𝑜𝑝𝑡,and𝜌𝑜𝑝𝑡(𝑛) can be evaluated

as in (31), (32), and (33), respectively. Substitute (36) into (35), one has

ˆf = f + Υv (37)

which is an unbiased estimate of ˆf with the mean-square error

(MSE) given below, E[ ˆf− f 2

]

= 𝑡𝑟{ΥE[vv𝐻]Υ𝐻}= 𝑡𝑟{ΥCvΥ𝐻},

(38) whereE [⋅] denotes the operation of taking expectation, Cv =

E[vv𝐻] is the noise correlation matrix, and𝑡𝑟 {X} denotes

the trace of the square matrixX. Notice that with no frequency

shift, i.e.,𝜇 = 0, (35) becomes

r = S(f1,++ f2,+)+ S(f

1,−+ f2,−

)

+ (𝑏1+ 𝑏2+𝑑0) 1𝑁.

(39) In such an undesirable case,f1,+andf2,+are not identifiable,

so aref1,−andf2,−, and𝑏1,𝑏2, and𝑑0, and, therefore, it is not

possible to estimate𝑤𝑜𝑝𝑡(𝑛), 𝜌𝑜𝑝𝑡(𝑛) and 𝑏𝑜𝑝𝑡as in (31)-(33),

respectively. This explains the necessity of the introduction of the frequency shift𝜇 during calibration training.

B. Training Sequence Design

Theoretically, the optimal training sequence is the one that minimizes MSEE[ ˆf− f 2

]

= 𝑡𝑟{ΥCvΥ𝐻}. As is seen

in (30), however,Cv is a function ofℎ𝑅,+(𝑛) and ℎ𝑅,−(𝑛),

and therefore the optimal training sequence differs from one transceiver to another and there is no way to design it. In our method, the simplified measure

𝑡𝑟{ΥΥ𝐻}= 𝑡𝑟{(Φ𝐻Φ)−1} (40)

is adopted in search of good training sequences. The measure is optimal only if𝑣 (𝑛) , 𝑛 = 0, ⋅ ⋅ ⋅, 𝑁 − 1 are white Gaussian noises.

Let ∑𝑁−1𝑛=0 ∣𝑠 (𝑛)∣2/𝑁 = 1. It can be shown that

𝑡𝑟{Φ𝐻Φ} = (4𝐿

𝑓+ 3) ⋅ 𝑁, (41)

and from [21] the minimum MSE in (40) is achieved provided that

Φ𝐻Φ = 𝑁 ⋅ I

4𝐿𝑓+3 (42)

which in turns leads to

T𝐻T = 𝑁 ⋅ I 2𝐿𝑓+1, (43) T𝐻Γ (−2𝜇) T = 0 2𝐿𝑓+1×2𝐿𝑓+1, (44) and T𝐻Γ (±𝜇) 1 𝑁 = 02𝐿𝑓+1×1, (45)

whereI𝑚is the identity matrix with dimension𝑚, and 0𝑚×𝑛

is the all zero matrix of size𝑚 × 𝑛. Clearly, from (43)-(45), 𝑠(𝑛) has to be designed jointly with frequency shift 𝜇 in order to have the best performance, but that, unfortunately, com-plicates the design significantly. In the following, a simpler method is proposed.

Consider a periodic training sequence that consists of𝑃 +1 periods with𝐾 samples in each period, i.e., 𝑠(𝑛) = 𝑠(𝑛+𝐾), 𝑛 = −𝐾, ⋅ ⋅ ⋅ , 0, ⋅ ⋅ ⋅ , 𝑁 − 𝐾 − 1, where 𝑁 = 𝐾𝑃 . Define

S1 be the signal matrix for one period (from the second

period), with its 𝑖 − 𝑗 element given by [S1]𝑖,𝑗 = 𝑠 (𝑖 − 𝑗),

0 ≤ 𝑖 ≤ 𝐾 − 1, and 0 ≤ 𝑗 ≤ 𝐿𝑓− 1, T1= [S1, S1, 1𝐾], and

T = [T𝑇

1, . . . , T𝑇1]

  

𝑃

𝑇, then the matrix Φ can be decomposed

as follows. Φ =[ Γ𝑁(𝜇) T Γ𝑁(−𝜇) T 1𝑁 ] = ⎡ ⎢ ⎢ ⎢ ⎣ Γ𝐾(𝜇) T1 Γ𝑁(−𝜇) T1 1𝐾 𝑒𝑗2𝜋𝜇𝐾Γ 𝐾(𝜇) T1 𝑒−𝑗2𝜋𝜇𝐾Γ𝐾(−𝜇) T1 1𝐾 .. . ... ... 𝑒𝑗2𝜋𝜇(𝑃 −1)𝐾Γ 𝐾(𝜇) T1 𝑒−𝑗2𝜋𝜇(𝑃 −1)𝐾Γ𝐾(−𝜇) T1 1𝐾 ⎤ ⎥ ⎥ ⎥ ⎦ .= Φ2Φ1, (46) where Φ1= ⎡ ⎣ Γ0𝐾×2𝐿𝐾(𝜇) T𝑓+11 Γ0𝐾𝐾×2𝐿(−𝜇) T𝑓+11 00𝐾×1𝐾×1 01×2𝐿𝑓+1 01×2𝐿𝑓+1 𝐾⎦ , (47) Φ2= ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ I𝐾 I𝐾 1𝐾 ⋅ 1𝐾 𝑒𝑗2𝜋𝜇𝐾I 𝐾 𝑒−𝑗2𝜋𝜇𝐾I𝐾 1𝐾 ⋅ 1𝐾 .. . ... ... 𝑒𝑗2𝜋𝜇(𝑃 −1)𝐾I 𝐾 𝑒−𝑗2𝜋𝜇(𝑃 −1)𝐾I𝐾 1𝐾 ⋅ 1𝐾 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ . (48) Thus, Φ𝐻 1 Φ1 = 𝐾 ⋅ I4𝐿𝑓+3 and Φ𝐻2Φ2 = 𝑃 ⋅ I2𝐾+1 are to constitute a sufficient condition of (42). Furthermore, the condition Φ𝐻

1 Φ1 = 𝐾 ⋅ I4𝐿𝑓+3, called Condition-A, can be split into the following three sub-conditions:

𝐶𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛 − 𝐴.1 : S𝐻 1S1= 𝐾 ⋅ I𝐿𝑓, (49) 𝐶𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛 − 𝐴.2 : S𝑇 1S1= 0𝐿𝑓×𝐿𝑓, (50) and 𝐶𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛 − 𝐴.3 : S𝑇 11𝑁 = 0𝐿𝑓×1. (51) In [22], methods were given to design sequences that satisfy Condition-A.1 and Condition-A.2, while Condition-A.3 just demands that the designed sequence has a zero mean. As an example, using the frequency-domain nulling (FDN) method in [22], the sequence (𝐾 = 64) 𝑠 (𝑛) =𝑁1 63 ∑ 𝑘=0 𝑆 (𝑘) 𝑒𝑗2𝜋𝑛𝑘64 , 𝑛 = 0, ⋅ ⋅ ⋅, 63, (52) with 𝑆 (𝑘) = ⎧ ⎨ ⎩ 𝑒𝑗𝜙𝑘, arbitrary 𝜙 𝑘, for 𝑘 ∈ 𝐽, and 𝐽 = [1, 5, 9, . . . , 61] 0, for 𝑘 /∈ 𝐽 (53)

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The condition Φ𝐻 2Φ2 = 𝑃 ⋅ I2𝐾+1, called Condition-B, amounts to ⎡ ⎣ 𝛾1𝑃 ⋅ I(𝜇) ⋅ I𝐾𝐾 𝛾1(−𝜇) ⋅ I𝑃 ⋅ I𝐾 𝐾 𝛾𝛾22(−𝜇) ⋅ 1(𝜇) ⋅ 1𝐾𝐾 𝛾2(𝜇) ⋅ 1𝐻𝐾 𝛾2(−𝜇) ⋅ 1𝐻𝐾 𝑃⎦ = 𝑃 ⋅I2𝐾+1. (54) That is, 𝛾1(𝜇) = 1 − ( 𝑒𝑗4𝜋𝜇𝐾)𝑃 1 − 𝑒𝑗4𝜋𝜇𝐾 = 0, (55) and 𝛾2(𝜇) = 1 𝐾⋅ 1 −(𝑒𝑗2𝜋𝜇𝐾)𝑃 1 − 𝑒𝑗2𝜋𝜇𝐾 = 0. (56)

From (55) and (56), it is concluded that

𝜇𝑜𝑝𝑡= 𝑃 𝐾𝑘 , {𝑘 ∈ 𝑍 ∣𝑘 /∈ 𝑖𝑃/2, 𝑖 ∈ 𝑍 } . (57)

In (49)-(51) and (57), we have successfully separated the design of𝑠(𝑛) from that of 𝜇.

IV. PERFORMANCEANALYSIS

In this section, we aim to analyze the calibration perfor-mance with the estimates given in (36). Specifically, we aim to analyze the probability density functions (pdfs) of calibrated 𝐼𝑅𝑅𝑇(𝑓), 𝐼𝑅𝑅𝑅(𝑓), 𝜀𝑇 and𝜀𝑅. For brevity, only𝐼𝑅𝑅𝑇(𝑓)

and𝜀𝑇 will be treated explicitly here; similar procedures can

be applied to analyze𝐼𝑅𝑅𝑅(𝑓) and 𝜀𝑅, as were detailed in

[23]. Numerical results will be given in Section V to verify the accuracy of the analyses.

To begin with, define Υ .=

[

Υ𝑇

f1,+ Υ𝑇f1,− Υ𝑇𝑏1 Υ𝑇f2,+ Υ𝑇f2,− Υ𝑇𝑏2 Υ𝑇𝑑0 ]𝑇

. From (37), then we have ˆf1,+ = f1,+ + Υf1,+v,

ˆf1,−= f1,−+Υf1,−v, ˆ𝑏1= 𝑏1𝑏1v, ˆf2,−= f2,−+Υf2,−v, and ˆ𝑑0 = 𝑑0 + Υ𝑑0v. During the internal loopback, the signal-to-noise ratio (SNR𝑙𝑜𝑜𝑝𝑏𝑎𝑐𝑘) is usually very high,

say SNR𝑙𝑜𝑜𝑝𝑏𝑎𝑐𝑘> 30dB, and, therefore, it is reasonable

to assume that ˆf1,+ ≈ f1,+. Recall that f1,+ is the desired

channel response from transmitter to receiver, as is given in (26). Using this approximation, the estimated calibration filters𝑤(𝑛) in (31) isˆ ˆ 𝑤 (𝑛) = −(𝑓ˆ1,+(𝑛))†⊗ ˆ𝑓1,−(𝑛) ≈ −(𝑓1,+(𝑛) )⊗ ˆ𝑓 1,−(𝑛) . (58) Without loss of generality, 𝑤 (𝑛) will be modeled asˆ an FIR filter in this analysis and denoted by w =ˆ [ ˆ𝑤 (0) , ˆ𝑤 (1) , . . . , ˆ𝑤 (𝐿 − 1)]𝑇, where 𝐿 can be selected as long as one wishes for the desirable analysis accuracy. With this modeling, (58) can be rearranged in the following vector-matrix form ˆ w ≈ −F1,+× ˆf1,−= −F1,+×(f1,−+ Υf1,−v ) = w + v𝑤, (59) where F1,+ is the 𝐿 × 𝐿𝑓 sized convolution matrix of the

truncated inverse filter(𝑓1,+(𝑛)),

w .= [𝑤 (0) , 𝑤 (2) , ⋅ ⋅ ⋅, 𝑤 (𝐿 − 1)]𝑇 = −F1,+f1,−, (60)

(see (31)), and

v𝑤 .= [𝑣𝑤(0) , 𝑣𝑤(1) , . . . , 𝑣𝑤(𝐿 − 1)]𝑇 = −F1,+Υf1,−v. (61)

In addition, 𝑔𝑇,+(𝑛) in (8) and 𝑔𝑇,−(𝑛) in (9) are

approxi-mated as ˆ𝑔𝑇,+(𝑛) .= ℎ𝑇,+(𝑛) + ˆ𝑤∗(𝑛) ⊗ ℎ𝑇,−(𝑛) ≈ ℎ𝑇,+(𝑛) , (62) and (see (59)) ˆ𝑔𝑇,−(𝑛) .= ℎ𝑇,−(𝑛) + ˆ𝑤 (𝑛) ⊗ ℎ𝑇,+(𝑛) = ℎ𝑇,−(𝑛) + 𝑤 (𝑛) ⊗ ℎ 𝑇,+(𝑛) ≈0 +𝑣𝑤(𝑛) ⊗ ℎ𝑇,+(𝑛) ≈ 𝑣𝑤(𝑛) ⊗ ℎ𝑇,+(𝑛) . (63)

The approximation in (62) is good because ∣ℎ𝑇,+(𝑛)∣ ≫ ∣ℎ𝑇,−(𝑛)∣ in real systems (see Section V), and (63) is good

because ideally 𝑤(𝑛) is sought to make ℎ𝑇,−(𝑛) + 𝑤 (𝑛) ⊗ ℎ𝑇,+(𝑛) = 0 in our method. Using (62) and (63), the

calibrated𝐼𝑅𝑅𝑇(𝑓) in (14) is evaluated by 𝐼𝑅𝑅𝑇(𝑓) ≈ 10log10 ∣𝐻𝑇,+(𝑓)∣ 2 ∣𝐻𝑇,+(𝑓)∣2∣𝑉𝑤(𝑓)∣2 = −10log10∣𝑉𝑤(𝑓)∣2(dB). (64) Furthermore, 𝑣(𝑛) in (30) is approximated by ℎ𝑅,+(𝑛) ⊗ 𝑣0(𝑛) because ∣ℎ𝑅,+(𝑛)∣ ≫ ∣ℎ𝑅,−(𝑛)∣. Using this

approxi-mation in (61), we have

𝑉𝑤(𝑓) .= FT [v𝑤] ≈ −Ψ𝐻(𝑓) F1,+Υf1,−Gv0= 𝝌𝐻𝑤(𝑓) v0,

(65) where Ψ (𝑓) = [1, 𝑒𝑗2𝜋𝑓, . . . , 𝑒𝑗2𝜋𝑓(𝐿−1)]𝑇, G is the 𝑁 × 𝑁 sized convolution matrix of

ℎ𝑅,+(𝑛), v0 = [𝑣0(0) , 𝑣0(1) , . . . , 𝑣0(𝑁 − 1)]𝑇, and 𝝌𝐻

𝑤(𝑓) = −Ψ𝐻(𝑓) F1,+Υf1,−G. Since {𝑣0(𝑛)}𝑁−1𝑛=0

are zero mean i.i.d complex circular symmetric Gaussian variables with variance 𝜎2

0, ∣𝑉𝑤(𝑓)∣2 is an exponentially

distributed random variable with the pdf

𝑝(∣𝑉𝑤(𝑓)∣2)=𝝌𝐻 1 𝑤(𝑓) 𝝌𝑤(𝑓) 𝜎02𝑒 1 𝝌𝐻𝑤 (𝑓)𝝌𝑤(𝑓)𝜎02∣𝑉𝑤(𝑓)∣2, ∣𝑉𝑤(𝑓)∣2≥ 0, (66) and the pdf of𝐼𝑅𝑅𝑇(𝑓) is 𝑝 (𝐼𝑅𝑅𝑇(𝑓)) =10 ⋅ 𝝌𝐻log𝑒10 𝑤(𝑓) 𝝌𝑤(𝑓) 𝜎2010 −𝐼𝑅𝑅𝑇 (𝑓) 10 𝑒 10−𝐼𝑅𝑅𝑇 (𝑓)10 𝝌𝐻𝑤 (𝑓)𝝌𝑤(𝑓)𝜎20, − ∞ < 𝐼𝑅𝑅𝑇(𝑓) < ∞, (67)

Finally, it can be shown from (66) and [24] that E {𝐼𝑅𝑅𝑇(𝑓)} = −10 ⋅ E { log10∣𝑉𝑤(𝑓)∣2} = −10log10𝝌𝐻𝑤(𝑓) 𝝌𝑤(𝑓) 𝜎02 𝑒C , (68) E{(𝐼𝑅𝑅𝑇(𝑓))2 } = 100 ⋅ E{(log10∣𝑉𝑤(𝑓)∣2)2 } =(10log10𝑒√𝜋 6 )2 + (E {𝐼𝑅𝑅𝑇(𝑓)})2, (69) and VAR {𝐼𝑅𝑅𝑇(𝑓)} = ( 10log10𝑒√𝜋 6 )2 , (70)

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TABLE I THERF IMPAIRMENTS

RF Impairments Parameter Value

Frequency independent I-Q imbalance (𝛼𝑇= 1.05, 𝜃𝑇= −5𝑜), (𝛼𝑇, 𝜃𝑇), (𝛼𝑅, 𝜃𝑅) (𝛼𝑅= 1.08, 𝜃𝑅= 5𝑜) Frequency dependent I-Q imbalance 𝐼 part : [1 0.2 0.1 0.05] {ℎ𝐼

𝑇(𝑛), ℎ𝑄𝑇(𝑛)}, {ℎ𝐼𝑅(𝑛), ℎ𝑄𝑅(𝑛)} 𝑄 part : [0.9 0.1 0.08 0.12] DC offset𝑏0and𝑑0, with signal 𝑏0= −0.1 × (1 + 𝑗)/√2,

power normalized to 1 𝑑0= 0.1 × (1 + 𝑗)/√2

To analyze 𝜀𝑇, first note that ˆ𝑏 ≈ −ˆ𝑏1(𝑓1,+(𝑛)

) in (32) because𝑓1,+(𝑛) ≫ 𝑓1,−(𝑛) in real systems. Using ˆ𝑏1=

𝑏1+ Υ𝑏1v, then ˆ𝑏 is approximated as ˆ𝑏 ≈ −𝑏1− Υ𝑏1v 𝑛 𝑓1,+(𝑛) ≈ 𝑏 + 𝑣𝑏, (71) where 𝑣𝑏= −Υ𝑏1v 𝑛 𝑓1,+(𝑛) . (72)

Furthermore, from (10), (71) and (62),Δ𝑏 is approximated as Δ𝑏 ≈ ˆ𝑔𝑇,+(𝑛) ⊗ ˆ𝑏 + ˆ𝑔𝑇,−(𝑛) ⊗ ˆ𝑏∗+ 𝑏0 = ˆ𝑔𝑇,+(𝑛) ⊗ 𝑏 + ˆ𝑔𝑇,−(𝑛) ⊗ 𝑏∗+ 𝑏0 ≈0 +ˆ𝑔𝑇,+(𝑛) ⊗ 𝑣𝑏 + ˆ𝑔𝑇,−(𝑛) ⊗ 𝑣∗𝑏 ≈ ˆ𝑔𝑇,+(𝑛) ⊗ 𝑣𝑏 ≈ ℎ𝑇,+(𝑛) ⊗ 𝑣𝑏 = − ( 𝑛 ℎ𝑇,+(𝑛) ) ⋅ Υ𝑏1v𝑛 𝑓1,+(𝑛) ≈ 𝝌𝐻 Δ𝑏v0, (73) where 𝝌𝐻 Δ𝑏= − ( 𝑛 ℎ𝑇,+(𝑛) ) ⋅ Υ𝑏1G𝑛 𝑓1,+(𝑛) (74) Similar to (67)-(70), we have 𝑝 (𝜀𝑇) = ∣𝑏0 2log 𝑒10 10 ⋅ 𝝌𝐻 Δ𝑏𝝌Δ𝑏𝜎02 10𝜀𝑇10𝑒 −∣𝑏0∣210𝜀𝑇10 𝝌𝐻Δ𝑏𝝌Δ𝑏𝜎2 0, − ∞ < 𝜀𝑇 < ∞, (75) E {𝜀𝑇} = E { 10log10∣Δ𝑏∣2 ∣𝑏02 } ≈ 10log10𝝌𝐻Δ𝑏𝝌Δ𝑏𝜎20 ∣𝑏02𝑒C , (76) and VAR {𝜀𝑇} = ( 10log10𝑒√𝜋 6 )2 . (77) −8 −6 −4 −2 0 2 4 6 8 20 25 30 35 40 45 50 55 Simulation, SNRloopback = 35dB Frequency (MHz) E[ IRR T ] (dB) L f = 5 Lf = 6 L f = 7 Lf = 8 L f = 12

Fig. 2. Performance of the calibratedE [𝐼𝑅𝑅𝑇] with different 𝐿𝑓′𝑠.

−8 −6 −4 −2 0 2 4 6 8 20 25 30 35 40 45 50 55 Simulation, SNRloopback = 35dB Frequency (MHz) E[ IRR R ] (dB) Lf = 5 Lf = 6 Lf = 7 Lf = 8 Lf = 12

Fig. 3. Performance of the calibratedE [𝐼𝑅𝑅𝑅] with different 𝐿𝑓′𝑠.

V. NUMERICALRESULTS

In this section, the performance of the proposed cali-bration method is evaluated through analysis and computer simulations. Table 1 summarizes the transmitter and receiver RF impairments which are typical values in real systems [1]-[18],[25]. In all results, 1/𝑇𝑠 = 20MHz, SNR .=

(1/𝑁)𝑁𝑛=0∣𝑠 (𝑛)∣2/𝜎2

0 , and each simulation point is

ob-tained with 106 realizations. In addition, 𝐾 = 64, 𝑃 = 3,

and 𝑁 = 𝐾𝑃 = 192. Note that two types of SNR need to

be differentiated in the proposed method: SNR𝑙𝑜𝑜𝑝𝑏𝑎𝑐𝑘 and

SNR𝑐ℎ𝑎𝑛𝑛𝑒𝑙.SNR𝑙𝑜𝑜𝑝𝑏𝑎𝑐𝑘 is the SNR defined for calibration

training (internal loopback) whereas SNR𝑐ℎ𝑎𝑛𝑛𝑒𝑙 defined for

normal communication. In real systems, SNR𝑙𝑜𝑜𝑝𝑏𝑎𝑐𝑘

SNR𝑐ℎ𝑎𝑛𝑛𝑒𝑙 because there is no propagation loss during the

internal loopback.

Figures 2 and 3 investigate the effects of 𝐿𝑓 on the

per-formance of the calibrated E [𝐼𝑅𝑅𝑇(𝑓)] and E [𝐼𝑅𝑅𝑅(𝑓)],

respectively, by computer simulations. The training sequence is the one in (53), and𝜇 = 23/(3 ⋅ 64). Recall that 𝐿𝑓 is the

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TABLE II

EXAMPLE MEANS AND STANDARD DEVIATIONS OF THE CALIBRATED𝐼𝑅𝑅𝑇,𝐼𝑅𝑅𝑅,𝜀𝑇,AND𝜀𝑅. Parameters SNR𝑙𝑜𝑜𝑝𝑏𝑎𝑐𝑘 SimulationMean (dB)Analysis Standard deviation (dB)Simulation Analysis 𝐼𝑅𝑅𝑇 (4MHz) 35 50.8 50.9 5.59 5.57 45 60.8 60.9 5.57 5.57 55 70.8 70.9 5.56 5.57 𝐼𝑅𝑅𝑅(4MHz) 35 50.8 50.8 5.53 5.57 45 60.8 60.8 5.58 5.57 55 70.8 70.8 5.58 5.57 𝜀𝑇 35 −40 −40.3 5.57 5.57 45 −50 −50.3 5.57 5.57 55 −60 −60.3 5.56 5.57 𝜀𝑅 35 −38 −38 5.6 5.57 45 −48 −48 5.57 5.57 55 −58 −58 5.59 5.57 −8 −6 −4 −2 0 2 4 6 8 15 20 25 30 35 40 45 50 55 SNRloopback = 35dB Frequency (MHz) E[ IRR T ] (dB) Training−1 Training−2 Training−3 Training−4 Simulation No Calibration

Fig. 4. Performance of the calibrated E [𝐼𝑅𝑅𝑇] with different training designs. −8 −6 −4 −2 0 2 4 6 8 15 20 25 30 35 40 45 50 55 SNRloopback = 35dB Frequency (MHz) E[ IRR R ] (dB) Training−1 Training−2 Training−3 Training−4 Simulation No Calibration

Fig. 5. Performance of the calibrated E [𝐼𝑅𝑅𝑅] with different training designs.

calibration performances are quite insensitive to the values of 𝐿𝑓 as long as it is larger than 6 in this case; similar results are

observed for the dc offset calibration. Since𝐿𝑓 may not be

known exactly in advance, it is advisable to use a sufficiently large𝐿𝑓 to avoid performance degradation.𝐿𝑓 = 7 is used

40 60 80 100 120 140 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 min{ IRRT (4MHz) } = 35dB IRRT(4MHz) (dB) Simulation Analysis SNRloopback = 35dB SNRloopback = 55dB SNRloopback = 45dB

Fig. 6. Analytical and simulated pdfs of the calibrated𝐼𝑅𝑅𝑇 at frequency 4MHz. −120 −100 −80 −60 −40 −20 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 εT (dB) max{εT } = −24dB Simulation Analysis SNRloopback = 55dB SNRloopback = 35dB SNRloopback = 45dB

Fig. 7. Analytical and simulated pdfs of the calibrated𝜀𝑇.

for all the results that follow.

In Figures 4 and 5, the calibrated E [𝐼𝑅𝑅𝑇(𝑓)] and

E [𝐼𝑅𝑅𝑅(𝑓)] are investigated with four periodic training

designs. Training-1 uses the sequence in (53) but with𝑆 (𝑘) = 𝑒𝑗𝜙𝑘∀𝑘, and 𝜇 = 23/(3 ⋅ 64), and Training-2, Training-3 and Training-4 use the sequence in (53) with 𝜇 = 24/(3 ⋅ 64),

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-10 -5 0 5 10 -10 -8 -6 -4 -2 0 2 4 6 8 10 64QAM, ● : Calibrated, × : No Calibration I-Part Q-Part

Fig. 8. Sample signal constellation with and without calibrations

(SNR𝑙𝑜𝑜𝑝𝑏𝑎𝑐𝑘= 35dB, SNR𝑐ℎ𝑎𝑛𝑛𝑒𝑙= ∞). 5 10 15 20 25 10−5 10−4 10−3 10−2 10−1 100 64QAM SNRchannel (dB)

BER No CalibrationCalibrated Ideal System

Fig. 9. Bit error rate performance with and without calibration

(SNR𝑙𝑜𝑜𝑝𝑏𝑎𝑐𝑘= 35dB ).

𝜇 = 11.4/(3 ⋅ 64) and 𝜇 = 23/(3 ⋅ 64) , respectively. For com-parison purpose, Training-1 is selected to violate Conditions-A.2 and Condition-A.3, Training-2 and Training-3 are selected to violate Condition-B while Training-4 is the optimal train-ing under the simplified criterion of (40) that satisfies both Condition-A and Condition-B. As can be seen in the figures, violation of Condition-A or Condition-B may incur a large performance loss. In addition, the proposed method provides around 20-35 dB performance improvement over the whole frequency band, as compared to the case of no calibration. The figures also show a nearly perfect match between simulation and analytical results. In the rest of this section, Training-4 is used for the calibration training.

Figure 6 shows the analytical and simulated pdfs of the calibrated𝐼𝑅𝑅𝑇(𝑓) at 𝑓= 4MHz, where it shows very good

match between simulation and analysis. The smallest simu-lated𝐼𝑅𝑅𝑇 is at 36 dB which are around 16 dB better than

the cases of no calibration. Figure 7 shows the analytical and simulated pdfs of the calibrated 𝜀𝑇, where the largest 𝜀𝑇 is

at -24 dB. Very significant improvements are observed with

the proposed method. Table 2 gives example simulated and analytical means and standard deviations of 𝐼𝑅𝑅𝑇, 𝐼𝑅𝑅𝑅, 𝜀𝑇 and𝜀𝑅 under different SNRs. Again, it shows very good

match between simulation and analysis.

Figures 8 and 9 show a sample received signal constellation and bit error rate performance respectively for an un-coded 64-QAM OFDM (orthogonal frequency-division multiplexing) system with and without calibration. The simulations are obtained under normal communication where transmitter and receiver are located at different devices with 𝜇 = 0 and 𝑐 (𝑡) = 𝛿 (𝑡). A one-tap equalizer is employed at the receiver for the simulated OFDM system that uses 64-point FFT (fast Fourier transform) with 52 subcarriers carrying data. In Fig. 8, SNR𝑐ℎ𝑎𝑛𝑛𝑒𝑙 = ∞ is adopted because, in doing so, the

sole effect of residual radio impairments on the constellation points can be investigated. As are shown in the figures, the adverse effects due to radio impairments are removed almost completely by the proposed calibration.

VI. CONCLUSION

A digital calibration method is proposed for the direct-conversion radio transceiver to calibrate its own transmit-ter and receiver radio impairments, including frequency-independent I-Q imbalance, frequency-dependent I-Q imbal-ance, and dc offset. By introducing a shift between transmit and receive frequencies, the radio impairments appearing at the transmitter and receiver can be calibrated simultaneously without a dedicated analog circuitry in the feedback loop. The calibration parameters are estimated based on the non-linear least-squares principle, and the calibration performance is analyzed that agrees very well with the simulations. The issue of training design is also investigated; sufficient con-ditions for optimal training are provided under a simplified criterion, and an example of optimal training is given for the periodic training structure. Analytical and simulation results show significant improvement is obtained with the proposed method, as compared to the non-calibrated systems.

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Chen-Jui Hsu was born in HsinChu, Taiwan, in

1982. He received the B.S. degree in electronics engineering in 2004, the M.S. degree in communi-cation engineering in 2006, and the Ph.D degree in electrical engineering in 2010, all from the National Chiao Tung University, Taiwan.

Since 2010, he has been a principal engineer with digital communication division, MStar Semiconduc-tor, Inc., Hsinchu, Taiwan. His research interest is in the inner receiver design of wireless communi-cations, particularly in RF front-end impairments estimation and compensation algorithms.

Wern-Ho Sheen (M’91) Prof. Wern-Ho Sheen

re-ceived his Ph.D. degree from the Georgia Institute of Technology, Atlanta, USA in 1991. From 1991 to 1993, he was with Chunghwa Telecom Labs as an associate researcher. From 1993 to 2001, he was with the National Chung Cheng University, where he held positions as Professor in the Department of Electrical Engineering and the Managing Director of the Center for Telecommunication Research. From 2001 to 2009, he was a Professor in the Department of Communications Engineering, National Chiao Tung University. Currently he is with the Department of Information and Communication Engineering, Chaoyang University of Technology. Prof. Sheen has been an active researcher in the areas of communication theory, mobile cellular systems, signal processing for wireless communications, and chip implementation of wireless communications systems.

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