### 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.

Direct-Conversion
RF Transmitter
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*r n*
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Direct-Conversion
RF Receiver
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( )*
Joint Estimator
of RF
Impairments
ˆ ( )*n*
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*w n*

Receiver Calibration Circuit

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real signal
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Transmitter Calibration Circuit _{To Power }

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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 *𝑆*1, *𝑆*2, *𝑆*3 and *𝑆*4 are used to

control the signal flow paths; during calibration training,*𝑆*1 ,

*𝑆*3 and*𝑆*4 are at the upper positions, and*𝑆*2 is at the lower

position to form an internal loopback, whereas during normal
communication,*𝑆*1,*𝑆*3 and*𝑆*4are at the lower positions, and

*𝑆*2is 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) ⊗ ℎ𝑄 _{𝑇}(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 by*𝛼𝑇* and*𝜃𝑇*, respectively. And,

the carrier feed-through induces dc offset at the receiving
side [1][2] and is characterized by Re{*𝑏*0*𝑒𝑗2𝜋𝑓𝑇𝑡*}, where

*𝑏*0 *= 𝑏𝐼*0*+ 𝑗𝑏𝑄*0 , *𝑓𝑇* is the transmit center frequency, and

*𝑗 =* *√−1. In the following, 𝑏*0 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, 𝑇𝑠*is the symbol duration,

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

Re{*𝑥 (𝑡) 𝑒𝑗2𝜋𝑓𝑇𝑡*}with its base-band equivalent given by [15]
*𝑥 (𝑡) = ℎ𝑇,+(𝑡) ⊗ 𝑠𝑝(𝑡) + ℎ𝑇,−(𝑡) ⊗ 𝑠∗𝑝(𝑡) + 𝑏*0*, (2)*
where
*ℎ𝑇,±(𝑡) = 1/2 ⋅*
[
*ℎ𝐼*
*𝑇(𝑡) ± 𝛼𝑇𝑒𝑗𝜃𝑇ℎ𝑄𝑇* *(𝑡)*
]
*⊗ ℎ _{𝑇}(𝑡) , (3)*
and

*𝑎∗*

_{denote the complex conjugate of}

_{𝑎. 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
response*ℎ𝑇,+(𝑡), and the other is the mirror-frequency path*

with impulse response*ℎ𝑇,−(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* *= 𝑏*0*= 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 by*𝛼𝑅*and*𝜃𝑅*, frequency-dependent I-Q imbalance,

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

and dc offset, *𝑑*0 *= 𝑑𝐼*0*+ 𝑗𝑑𝑄*0. *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 ˜𝑣*0*(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) + 𝑣*0*(𝑡)*]

*+ ℎ𝑅,−(𝑡) ⊗*[*𝑒𝑗2𝜋Δ𝑓𝑡𝑦 (𝑡) + 𝑣*0*(𝑡)*]*∗+ 𝑑*0*,* (4)

where*𝑦(𝑡) and 𝑣*0*(𝑡) are the low-pass equivalents of ˜𝑦(𝑡) and*

*˜𝑣*0*(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) + 𝑣*0*(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)
1_{In practice, the intentional frequency shift can be implemented precisely}
with two digital frequency synthesizers from a single reference oscillator [20].
2_{The 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)
where*𝑢 (𝑛) = 𝑢 (𝑡)∣ _{𝑡=𝑛𝑇}_{𝑠}*

*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∗+ 𝑏*0

*.*(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= −𝑏*0

*⊗*(

*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 of*𝑤𝑜𝑝𝑡(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) = 10log*10*∣𝐺𝑇,+(𝑓)∣*
2

*∣𝐺𝑇,−(𝑓)∣*2*dB,*

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

*𝜀𝑇* = 10log10*∣Δ𝑏∣*
2

*∣𝑏*0*∣*2dB

(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) + 𝑣*0*(𝑛)*]
*+ 𝑔𝑅,−(𝑛) ⊗*[*𝑒𝑗2𝜋𝜇𝑛𝑦 (𝑛) + 𝑣*0*(𝑛)*]*∗+ Δ𝑑, (16)*
where
*𝑟 (𝑛) = ℎ𝑅,+(𝑛) ⊗*[*𝑒𝑗2𝜋𝜇𝑛𝑦 (𝑛) + 𝑣*0*(𝑛)*]
*+ ℎ𝑅,−(𝑛) ⊗*[*𝑒𝑗2𝜋𝜇𝑛𝑦 (𝑛) + 𝑣*0*(𝑛)*]*∗+ 𝑑*0*, (17)*

*𝜇 = Δ𝑓𝑇𝑠* is the normalized frequency offset,

*Δ𝑑 = (𝑑*0*− 𝑑) − 𝜌 (𝑛) ⊗ (𝑑*0*− 𝑑)∗,* (19)

and *{𝑣*0*(𝑛)} are i.i.d. (independent, and identically *

dis-tributed) zero-mean Gaussian noise with variance of *𝜎*2
0. In

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

*𝜌𝑜𝑝𝑡(𝑛) =*(*ℎ∗𝑅+(𝑛)*

)*† _{⊗ ℎ}*

*𝑅−(𝑛) ,* (20)

and

*𝑑𝑜𝑝𝑡= 𝑑*0*.* (21)

Similar to the transmitter case, the receiver calibration
perfor-mance is evaluated by
*𝐼𝑅𝑅𝑅(𝑓) = 10log*10*∣𝐺𝑅,+(𝑓)∣*
2
*∣𝐺𝑅,−(𝑓)∣*2dB
(22)
and
*𝜀𝑅*= 10log10*∣Δ𝑑∣*
2
*∣𝑑*0*∣*2*dB,*
(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)) + 𝑣*0

*(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 on

*𝑠(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) + 𝑏*1]

*+ 𝑒−𝑗2𝜋𝜇𝑛*

_{[𝑓}*2,+(𝑛) ⊗ 𝑠 (𝑛) + 𝑓2,−(𝑛) ⊗ 𝑠∗(𝑛) + 𝑏*2]

*+ 𝑑*0

*+ 𝑣 (𝑛) ,*(25) where

*𝑓1,±(𝑛) =*(

*ℎ𝑅,+(𝑛) 𝑒−𝑗2𝜋𝜇𝑛*)

*⊗ ℎ𝑇,±(𝑛) ,*(26)

*𝑓2,±(𝑛) =*(

*ℎ𝑅,−(𝑛) 𝑒𝑗2𝜋𝜇𝑛*)

*⊗ ℎ∗𝑇,∓(𝑛) ,*(27)

*𝑏*1=(

*ℎ𝑅,+(𝑛) 𝑒−𝑗2𝜋𝜇𝑛*)

*⊗ 𝑏*0

*,*(28)

*𝑏*2=(

*ℎ𝑅,−(𝑛) 𝑒𝑗2𝜋𝜇𝑛*)

*⊗ 𝑏∗*0

*,*(29) and

*𝑣 (𝑛) = ℎ𝑅,+(𝑛) ⊗ 𝑣*0

*(𝑛) + ℎ𝑅,−(𝑛) ⊗ 𝑣*0

*∗(𝑛) .*(30)

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

from*𝑟(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), 𝑏*0 and*𝑑*0 directly from

(25)-(30) and then apply them to (11)-(13), (20) and (21). Direct
estimation of *ℎ𝑇,±(𝑛), ℎ𝑅,±(𝑛), 𝑏*0 and*𝑑*0, 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)*

*= −*(

*𝑓*)

_{1,+}(𝑛)*†(𝑛) ⊗ 𝑓1,−(𝑛) ,*(31)

*𝑏𝑜𝑝𝑡= −𝑏*0

*⊗*(

*𝑔𝑇,+,𝑜𝑝𝑡(𝑛)*)

*†*

*= −𝑏*1

*⊗ [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 𝑏*1 which along with *𝑑*0 can

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

*𝑏*1,*𝑏*2and*𝑑*0will all be estimated under the principle of

least-squares with *𝑓2,+(𝑛) and 𝑏*2 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,

**f***𝑖,±= [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−1 _{𝑛=−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,*

and**f =**[**f***𝑇*

*1,+ , f1,−𝑇*

*, 𝑏*1

**, f**2,+𝑇

**, f**2,−𝑇*, 𝑏*2

*, 𝑑*0]

*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** =
[

**Γ***𝑁 (𝜇) T Γ𝑁(−𝜇) T 1𝑁* ],

**T =**[

**S S**

*∗*

**1**

*𝑁*],

**Γ**

*𝑁(𝜇) = diag*{

*1,𝑒𝑗2𝜋𝜇, . . . , 𝑒𝑗2𝜋𝜇(𝑁−1)*} is the diagonal 3

_{Generally,}

*perfor-mance and complexity. Given a design of a radio transceiver where the worst values of radio impairments are specified,*

_{𝑁 is selected based on a tradeoff between calibration }*𝑁 can be selected according to the*designer’s own tradeoff on the performance vs. complexity.

matrix with elements {*1,𝑒𝑗2𝜋𝜇 _{, ⋅ ⋅ ⋅ , 𝑒}𝑗2𝜋𝜇(𝑁−1)*}

_{,}

**1***𝑁* 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)

where**Υ =**(**Φ***𝐻*** _{Φ}**)

*−1*

_{Φ}*𝐻*

_{is the pseudo inverse of}

_{Φ. Note}that**Φ has to have full-rank in order to assure identifiability.**

**After obtaining ˆ*** f, 𝑤𝑜𝑝𝑡(𝑛)*,

*𝑏𝑜𝑝𝑡*,and

*𝜌𝑜𝑝𝑡(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**[**vv***𝐻*]**Υ***𝐻*}*= 𝑡𝑟*{**ΥCvΥ***𝐻*}*,*

(38)
where**E [⋅] denotes the operation of taking expectation, C****v** =

E[**vv***𝐻*] _{is the noise correlation matrix, and}_{𝑡𝑟 {X} denotes}

the trace of the square matrix**X. Notice that with no frequency**

shift, i.e.,*𝜇 = 0, (35) becomes*

**r = S**(**f**_{1,+}**+ f*** _{2,+}*)

**+ S**

*∗*(

_{f}*1,−***+ f***2,−*

)

*+ (𝑏*1*+ 𝑏*2*+𝑑*0**) 1***𝑁.*

(39)
In such an undesirable case,**f***1,+*and**f***2,+*are not identifiable,

so are**f***1,−*and**f***2,−*, and*𝑏*1,*𝑏*2, and*𝑑*0, and, therefore, it is not

possible to estimate*𝑤𝑜𝑝𝑡(𝑛), 𝜌𝑜𝑝𝑡(𝑛) and 𝑏𝑜𝑝𝑡*as in (31)-(33),

respectively. This explains the necessity of the introduction of
the frequency shift*𝜇 during calibration training.*

*B. Training Sequence Design*

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

]

*= 𝑡𝑟*{**ΥCvΥ***𝐻*}. As is seen

in (30), however,**Cv** is a function of*ℎ𝑅,+(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*{**ΥΥ***𝐻*}* _{= 𝑡𝑟}*{(

_{Φ}*𝐻*

**)**

_{Φ}*−1*}

_{(40)}

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

Let ∑*𝑁−1 _{𝑛=0}*

*∣𝑠 (𝑛)∣*2/

*𝑁 = 1. It can be shown that*

**𝑡𝑟{Φ**𝐻_{Φ} = (4𝐿}

*𝑓+ 3) ⋅ 𝑁,* (41)

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

**Φ***𝐻_{Φ = 𝑁 ⋅ I}*

*4𝐿𝑓*+3 (42)

which in turns leads to

**T***𝐻_{T = 𝑁 ⋅ I}*

*2𝐿𝑓*+1

*,*(43)

**T**

*𝐻*

_{Γ (−2𝜇) T = 0}*2𝐿𝑓+1×2𝐿𝑓*+1

*,*(44) and

**T**

*𝐻*

_{Γ (±𝜇) 1}*𝑁*

**= 0**

*2𝐿𝑓+1×1,*(45)

where**I***𝑚*is the identity matrix with dimension**𝑚, and 0**𝑚×𝑛

is the all zero matrix of size*𝑚 × 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 of*𝑃 +1*
periods with*𝐾 samples in each period, i.e., 𝑠(𝑛) = 𝑠(𝑛+𝐾),*
*𝑛 = −𝐾, ⋅ ⋅ ⋅ , 0, ⋅ ⋅ ⋅ , 𝑁 − 𝐾 − 1, where 𝑁 = 𝐾𝑃 . Define*

**S**1 be the signal matrix for one period (from the second

period), with its * 𝑖 − 𝑗 element given by [S*1]

*𝑖,𝑗*

*= 𝑠 (𝑖 − 𝑗),*

*0 ≤ 𝑖 ≤ 𝐾 − 1, and 0 ≤ 𝑗 ≤ 𝐿𝑓 − 1, T*1

**= [S**1

*1*

**, S**∗*], and*

**, 1**𝐾**T = [T***𝑇*

1* , . . . , T𝑇*1]

*𝑃*

*𝑇*_{, then the matrix} _{Φ can be decomposed}

as follows.
**Φ =**[ **Γ***𝑁 (𝜇) T Γ𝑁(−𝜇) T 1𝑁* ]
=
⎡
⎢
⎢
⎢
⎣

**Γ**

*𝐾*1

**(𝜇) T****Γ**

*𝑁*1

**(−𝜇) T****1**

*𝐾*

*𝑒𝑗2𝜋𝜇𝐾*

_{Γ}*𝐾*1

**(𝜇) T***𝑒−𝑗2𝜋𝜇𝐾*

**Γ**

*𝐾*1

**(−𝜇) T****1**

*𝐾*.. . ... ...

*𝑒𝑗2𝜋𝜇(𝑃 −1)𝐾*

_{Γ}*𝐾*1

**(𝜇) T***𝑒−𝑗2𝜋𝜇(𝑃 −1)𝐾*

**Γ**

*𝐾*1

**(−𝜇) T****1**

*𝐾*⎤ ⎥ ⎥ ⎥ ⎦

**.= Φ2****Φ**1, (46) where

**Φ**1= ⎡ ⎣

**Γ**

_{0}

_{𝐾×2𝐿}𝐾**(𝜇) T**_{𝑓}_{+1}1

_{Γ}0*+1*

_{𝐾}𝐾×2𝐿**𝑓**_{(−𝜇) T}_{1}

_{0}0*𝐾×1*

_{𝐾×1}**0**

*1×2𝐿𝑓*+1

**0**

*1×2𝐿𝑓*+1

*√*

*𝐾*⎤

*⎦ ,*(47)

**Φ**2= ⎡ ⎢ ⎢ ⎢ ⎢ ⎣

**I**

*𝐾*

**I**

*𝐾*

*√*1

_{𝐾}

**⋅ 1**𝐾*𝑒𝑗2𝜋𝜇𝐾*

_{I}*𝐾*

*𝑒−𝑗2𝜋𝜇𝐾*

**I**

*𝐾*

*√*1

_{𝐾}*.. . ... ...*

**⋅ 1**𝐾*𝑒𝑗2𝜋𝜇(𝑃 −1)𝐾*

_{I}*𝐾*

*𝑒−𝑗2𝜋𝜇(𝑃 −1)𝐾*

**I**

*𝐾*

*√*1

_{𝐾}*⎤ ⎥ ⎥ ⎥ ⎥ ⎦*

**⋅ 1**𝐾*.*(48) Thus,

**Φ**

*𝐻*1

**Φ**1

*+3 and*

**= 𝐾 ⋅ I**4𝐿𝑓**Φ**

*𝐻*2

**Φ**2

*are to constitute a sufficient condition of (42). Furthermore, the condition*

**= 𝑃 ⋅ I**2𝐾+1**Φ**

*𝐻*

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

*𝐶𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛 − 𝐴.1 :* **S***𝐻*
1**S**1* = 𝐾 ⋅ I𝐿𝑓,* (49)

*𝐶𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛 − 𝐴.2 :*

**S**

*𝑇*1

**S**1

**= 0**

*𝐿𝑓×𝐿𝑓,*(50) and

*𝐶𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛 − 𝐴.3 :*

**S**

*𝑇*1

**1**

*𝑁*

**= 0**

*𝐿𝑓×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) =*1 63 ∑

_{𝑁}*𝑘=0*

*𝑆 (𝑘) 𝑒𝑗2𝜋𝑛𝑘*64

*, 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)

The condition **Φ***𝐻*
2**Φ**2 * = 𝑃 ⋅ I2𝐾+1*, called Condition-B,
amounts to
⎡
⎣

_{𝛾}_{1}

**𝑃 ⋅ I****𝐾**_{(𝜇) ⋅ I}_{𝐾}*𝛾*1

**(−𝜇) ⋅ I**_{𝑃 ⋅ I}_{𝐾}*𝐾*

*𝛾*2

_{𝛾}_{2}

**(−𝜇) ⋅ 1**_{(𝜇) ⋅ 1}_{𝐾}𝐾*𝛾*2

**(𝜇) ⋅ 1**𝐻𝐾*𝛾*2

**(−𝜇) ⋅ 1**𝐻𝐾*𝑃*⎤

*(54) That is,*

**⎦ = 𝑃 ⋅I**2𝐾+1.*𝛾*1

*(𝜇) =*

*1 −*(

*𝑒𝑗4𝜋𝜇𝐾*)

*𝑃*

*1 − 𝑒𝑗4𝜋𝜇𝐾*

*= 0,*(55) and

*𝛾*2

*(𝜇) =*

*√*1

*𝐾⋅*

*1 −*(

*𝑒𝑗2𝜋𝜇𝐾*)

*𝑃*

*1 − 𝑒𝑗2𝜋𝜇𝐾*

*= 0.*(56)

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

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

In (49)-(51) and (57), we have successfully separated the
design of*𝑠(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* and*𝜀𝑅*. For brevity, only*𝐼𝑅𝑅𝑇(𝑓)*

and*𝜀𝑇* will be treated explicitly here; similar procedures can

be applied to analyze*𝐼𝑅𝑅𝑅(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** *.=*

[

**Υ***𝑇*

**f***1,+* **Υ***𝑇***f***1,−* **Υ***𝑇𝑏*1 **Υ***𝑇***f***2,+* **Υ***𝑇***f***2,−* **Υ***𝑇𝑏*2 **Υ***𝑇𝑑*0
]*𝑇*

.
**From (37), then we have ˆf***1,+* **= f***1,+* **+ Υf***1,+***v,**

**ˆf***1,−***= f***1,−***+Υf***1,− v, ˆ𝑏*1

*= 𝑏*1

**+Υ**

*𝑏*1

**v, ˆf**

*2,−*

**= f**

*2,−*

**+Υf**

*2,−*

**v,**and ˆ

*𝑑*0

*= 𝑑*0

**+ Υ**

*𝑑*0

**v. During the internal loopback, the**signal-to-noise ratio (SNR

*𝑙𝑜𝑜𝑝𝑏𝑎𝑐𝑘*) is usually very high,

say SNR*𝑙𝑜𝑜𝑝𝑏𝑎𝑐𝑘> 30dB, and, therefore, it is reasonable*

to assume that ˆ**f***1,+* * ≈ f1,+*. Recall that

**f**

*1,+*is the desired

channel response from transmitter to receiver, as is given
in (26). Using this approximation, the estimated calibration
filters*𝑤(𝑛) in (31) is*ˆ
ˆ
*𝑤 (𝑛) = −*(*𝑓*ˆ* _{1,+}(𝑛)*)

*†⊗ ˆ𝑓1,−(𝑛) ≈ −*(

*𝑓1,+(𝑛)*)

*†*

_{⊗ ˆ}_{𝑓}*1,−(𝑛) .*(58) Without loss of generality,

*𝑤 (𝑛) will be modeled as*ˆ an FIR filter in this analysis and denoted by

**w =**ˆ [ ˆ

*𝑤 (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 ≈ −F**_{1,+}**× ˆf**_{1,−}**= −F**_{1,+}×**f**

_{1,−}**+ Υf**

*1,−*

**v**)

**= w + v**

*𝑤,*(59) where

**F**

*1,+*is the

*𝐿 × 𝐿𝑓*sized convolution matrix of the

truncated inverse filter(*𝑓 _{1,+}(𝑛)*)

*†*,

**w .= [𝑤 (0) , 𝑤 (2) , ⋅ ⋅ ⋅, 𝑤 (𝐿 − 1)]**𝑇**= −F**_{1,+}**f*** _{1,−},* (60)

(see (31)), and

**v***𝑤* *.= [𝑣𝑤(0) , 𝑣𝑤(1) , . . . , 𝑣𝑤(𝐿 − 1)]𝑇* **= −F**1,+**Υf***1,− 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)*
*= ℎ*_{}*𝑇,−(𝑛) + 𝑤 (𝑛) ⊗ ℎ*_{} *𝑇,+(𝑛)*_{}
*≈0*
*+𝑣 _{𝑤}(𝑛) ⊗ ℎ𝑇,+(𝑛)*

*≈ 𝑣*(63)

_{𝑤}(𝑛) ⊗ ℎ𝑇,+(𝑛) .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*

calibrated*𝐼𝑅𝑅𝑇(𝑓) in (14) is evaluated by*
*𝐼𝑅𝑅𝑇(𝑓) ≈ 10log*10 *∣𝐻𝑇,+(𝑓)∣*
2
*∣𝐻𝑇,+(𝑓)∣*2*∣𝑉𝑤(𝑓)∣*2
*= −10log*_{10}*∣𝑉 _{𝑤}(𝑓)∣*2

*(dB).*(64) Furthermore,

*𝑣(𝑛) in (30) is approximated by ℎ𝑅,+(𝑛) ⊗*

*𝑣*0

*(𝑛) because ∣ℎ𝑅,+(𝑛)∣ ≫ ∣ℎ𝑅,−(𝑛)∣. Using this*

approxi-mation in (61), we have

*𝑉 _{𝑤}(𝑓) .= FT [v𝑤] ≈ −Ψ𝐻(𝑓) F1,+*

**Υf**

*1,−*

**Gv**0

*0*

**= 𝝌**𝐻𝑤**(𝑓) v***,*

(65)
where * Ψ (𝑓)* = [

*1, 𝑒𝑗2𝜋𝑓*]

_{, . . . , 𝑒}𝑗2𝜋𝑓(𝐿−1)*𝑇*

_{,}

**G is the 𝑁 × 𝑁 sized convolution matrix of***ℎ𝑅,+ (𝑛), v*0 =

*[𝑣*0

*(0) , 𝑣*0

*(1) , . . . , 𝑣*0

*(𝑁 − 1)]𝑇*, and

**𝝌**𝐻*𝑤 (𝑓) = −Ψ𝐻(𝑓) F1,+*

**Υf**

*1,−*0

**G. Since {𝑣***(𝑛)}𝑁−1𝑛=0*

are zero mean i.i.d complex circular symmetric Gaussian
variables with variance *𝜎*2

0, *∣𝑉𝑤(𝑓)∣*2 is an exponentially

distributed random variable with the pdf

*𝑝*(*∣𝑉 _{𝑤}(𝑓)∣*2)=

*1*

_{𝝌}_{𝐻}*𝑤*02

**(𝑓) 𝝌**𝑤(𝑓) 𝜎*𝑒*

*−*1

**𝝌𝐻**_{𝑤 (𝑓)𝝌𝑤(𝑓)𝜎}_{0}2

*∣𝑉𝑤(𝑓)∣*2

_{,}*∣𝑉*2

_{𝑤}(𝑓)∣*≥ 0,*(66) and the pdf of

*𝐼𝑅𝑅𝑇(𝑓) is*

*𝑝 (𝐼𝑅𝑅𝑇(𝑓)) =*log

_{10 ⋅ 𝝌}_{𝐻}*𝑒*10

*𝑤*2010

**(𝑓) 𝝌**𝑤(𝑓) 𝜎*−𝐼𝑅𝑅𝑇 (𝑓)*10

*10*

_{𝑒}−*−𝐼𝑅𝑅𝑇 (𝑓)*10

*2*

**𝝌𝐻**_{𝑤 (𝑓)𝝌𝑤(𝑓)𝜎}_{0}

_{,}*− ∞ < 𝐼𝑅𝑅𝑇(𝑓) < ∞,*(67)

Finally, it can be shown from (66) and [24] that
*E {𝐼𝑅𝑅𝑇(𝑓)} = −10 ⋅ E*
{
log_{10}*∣𝑉 _{𝑤}(𝑓)∣*2}

*= −10log*

_{10}

*02*

**𝝌**𝐻𝑤**(𝑓) 𝝌**𝑤(𝑓) 𝜎*𝑒*

**C**

*,*(68) E{

*(𝐼𝑅𝑅𝑇(𝑓))*2 }

*= 100 ⋅ E*{(log

_{10}

*∣𝑉*2)2 } =(10log

_{𝑤}(𝑓)∣_{10}

*𝑒√𝜋*6 )

_{2}

*+ (E {𝐼𝑅𝑅𝑇(𝑓)})*2

*,*(69) and

*VAR {𝐼𝑅𝑅𝑇(𝑓)} =*( 10log

_{10}

*𝑒√𝜋*6 )2

*,*(70)

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*= 5*𝑜*)
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 offset*𝑏0*and*𝑑0*, with signal *𝑏0= −0.1 × (1 + 𝑗)/√*2,

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

To analyze *𝜀𝑇*, first note that ˆ*𝑏 ≈ −ˆ𝑏*1*⊗*(*𝑓1,+(𝑛)*

)* _{†}*
in
(32) because

*𝑓*1=

_{1,+}(𝑛) ≫ 𝑓_{1,−}(𝑛) in real systems. Using ˆ𝑏*𝑏*1**+ Υ***𝑏*1**v, then ˆ𝑏 is approximated as***ˆ𝑏 ≈ −𝑏*_{∑}1* − Υ𝑏*1

**v**

*𝑛*

*𝑓1,+(𝑛)*

*≈ 𝑏 + 𝑣𝑏,*(71) where

*𝑣𝑏= −*∑

**Υ**

*𝑏*1

**v**

*𝑛*

*𝑓1,+(𝑛)*

*.*(72)

Furthermore, from (10), (71) and (62),*Δ𝑏 is approximated as*
*Δ𝑏 ≈ ˆ𝑔𝑇,+(𝑛) ⊗ ˆ𝑏 + ˆ𝑔𝑇,−(𝑛) ⊗ ˆ𝑏∗+ 𝑏*0
*= ˆ𝑔*_{}*𝑇,+(𝑛) ⊗ 𝑏 + ˆ𝑔*_{}*𝑇,−(𝑛) ⊗ 𝑏∗+ 𝑏*0_{}
*≈0*
*+ˆ𝑔𝑇,+(𝑛) ⊗ 𝑣𝑏*
*+ ˆ𝑔𝑇,−(𝑛) ⊗ 𝑣∗𝑏*
*≈ ˆ𝑔𝑇,+(𝑛) ⊗ 𝑣𝑏*
*≈ ℎ𝑇,+(𝑛) ⊗ 𝑣𝑏*
*= −*
(_{∑}
*𝑛* *ℎ𝑇,+(𝑛)*
)
* ⋅ Υ𝑏*1

**v**∑

*𝑛*

*𝑓1,+(𝑛)*

**≈ 𝝌**𝐻*Δ𝑏*

**v**0

*,*(73) where

**𝝌**𝐻*Δ𝑏= −*(

_{∑}

*𝑛*

*ℎ𝑇,+(𝑛)*)

*1*

**⋅ Υ**𝑏**G**∑

*𝑛*

*𝑓1,+(𝑛)*(74) Similar to (67)-(70), we have

*𝑝 (𝜀𝑇*) =

*∣𝑏*0

*∣*2

_{log}

*𝑒*10

*Δ*

**10 ⋅ 𝝌**𝐻*𝑏*Δ

**𝝌***𝑏𝜎*02 10

*𝜀𝑇*10

*𝑒*

*−∣𝑏0∣210𝜀𝑇*10

*2 0*

**𝝌𝐻**_{Δ𝑏}_{𝝌Δ𝑏𝜎}*,*

*− ∞ < 𝜀𝑇*

*< ∞,*(75)

*E {𝜀𝑇} = E*{ 10log

_{10}

*∣Δ𝑏∣*2

*∣𝑏*0

*∣*2 }

*≈ 10log*

_{10}

*20*

**𝝌**𝐻Δ𝑏**𝝌**Δ𝑏𝜎*∣𝑏*0

*∣*2

*𝑒*

**C**

*,*(76) and

*VAR {𝜀𝑇} =*( 10log

_{10}

*𝑒√𝜋*6 )2

*.*(77) −8 −6 −4 −2 0 2 4 6 8 20 25 30 35 40 45 50 55 Simulation, SNR

*= 35dB Frequency (MHz) E[*

_{loopback}*IRR*

*T*

*] (dB)*

*L*

*f*= 5

*L*= 6

_{f}*L*

*f*= 7

*L*= 8

_{f}*L*

*f*= 12

Fig. 2. Performance of the calibrated*E [𝐼𝑅𝑅𝑇] with different 𝐿𝑓′𝑠.*

−8 −6 −4 −2 0 2 4 6 8
20
25
30
35
40
45
50
55
Simulation, SNR* _{loopback}* = 35dB
Frequency (MHz)
E[

*IRR*

*R*

*] (dB)*

*L*= 5

_{f}*L*= 6

_{f}*L*= 7

_{f}*L*= 8

_{f}*L*= 12

_{f}Fig. 3. Performance of the calibrated*E [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)*∑*𝑁 _{𝑛=0}∣𝑠 (𝑛)∣*2/

*𝜎*2

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: SNR*𝑙𝑜𝑜𝑝𝑏𝑎𝑐𝑘* and

SNR*𝑐ℎ𝑎𝑛𝑛𝑒𝑙*.SNR*𝑙𝑜𝑜𝑝𝑏𝑎𝑐𝑘* is the SNR defined for calibration

training (internal loopback) whereas SNR*𝑐ℎ𝑎𝑛𝑛𝑒𝑙* defined for

normal communication. In real systems, SNR*𝑙𝑜𝑜𝑝𝑏𝑎𝑐𝑘* *≫*

SNR*𝑐ℎ𝑎𝑛𝑛𝑒𝑙* 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), and*𝜇 = 23/(3 ⋅ 64). Recall that 𝐿𝑓* is the

TABLE II

EXAMPLE MEANS AND STANDARD DEVIATIONS OF THE CALIBRATED*𝐼𝑅𝑅 _{𝑇}*,

*𝐼𝑅𝑅𝑅*,

*𝜀𝑇*,AND

*𝜀*. Parameters SNR

_{𝑅}

_{𝑙𝑜𝑜𝑝𝑏𝑎𝑐𝑘}_{Simulation}Mean (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 SNR

*= 35dB Frequency (MHz) E[*

_{loopback}*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
SNR* _{loopback}* = 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. Since*𝐿𝑓* may not be

known exactly in advance, it is advisable to use a sufficiently
large*𝐿𝑓* 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{ IRR _{T} (4MHz) } = 35dB*

*IRR*(4MHz) (dB) Simulation Analysis SNR

_{T}*= 35dB SNR*

_{loopback}*= 55dB SNR*

_{loopback}*= 45dB*

_{loopback}Fig. 6. Analytical and simulated pdfs of the calibrated*𝐼𝑅𝑅𝑇* 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
ε* _{T} (dB)*
max{ε

*Simulation Analysis SNR*

_{T}} = −24dB*= 55dB SNR*

_{loopback}*= 35dB SNR*

_{loopback}*= 45dB*

_{loopback}Fig. 7. Analytical and simulated pdfs of the calibrated*𝜀𝑇*.

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 with*𝑆 (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),*

-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

(SNR*𝑙𝑜𝑜𝑝𝑏𝑎𝑐𝑘*= 35dB, SNR*𝑐ℎ𝑎𝑛𝑛𝑒𝑙= ∞).*
5 10 15 20 25
10−5
10−4
10−3
10−2
10−1
100 64QAM
SNR* _{channel}* (dB)

BER No CalibrationCalibrated Ideal System

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

(SNR*𝑙𝑜𝑜𝑝𝑏𝑎𝑐𝑘*= 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
calibrated*𝐼𝑅𝑅𝑇(𝑓) at 𝑓= 4MHz, where it shows very good*

match between simulation and analysis. The smallest
simu-lated*𝐼𝑅𝑅𝑇* 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* and*𝜀𝑅* 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, SNR*𝑐ℎ𝑎𝑛𝑛𝑒𝑙* *= ∞ 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.