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Blind Signal Separation-Based Frequency-dependent I/Q Imbalance Compensation for Direct-conversion Receivers
Research Paper / Jan 2009
Blind Signal Separation-Based Frequency-dependent I/Q Imbalance Compensation for Direct-conversion Receivers �
Chia-Pang Yen, Yingming Tsai, and Robert Olesen
InterDigital Communications LLC
2 Huntington Quadrangle

Melville, NY 11747, USA

Chia-Pang.Yen@, Yingming.Tsai@,

Abstract— Frequency-dependent I/Q imbalance is one of the concerns in the design of a high data rate wireless wideband direct-conversion receiver (DCR). To address this challenge, a blind signal separation method, based on a conjugate signal model is proposed. Necessary and sufficient conditions for signal separation are derived, and a gradient-based method is used to solve this problem. The performance of our approach is evaluated for orthogonal frequency division multiplexing (OFDM) systems.
Keywords: Radio frequency (RF), direct-conversion receiver (DCR), I/Q imbalance, blind signal separation, orthogonal frequency division multiplexing (OFDM) signals.


The evolution of wireless communication systems has driven the design and implementation of the radio transceivers. Next generation of wireless communication networks (for example, 3GPP LTE system) will support a high data rate, which requires efficient wideband radio design in the terminals. Driven by those demands, I/Q down-conversion have become a key design issue in addressing the need for cost- and size-efficient transceiver implementation.

In commonly used direct-conversion front-end receivers, the received signal is I/Q down-converted from RF directly to a baseband signal. Due to an imperfect oscillator, and low pass filters in the RF front-end receiver, the In-phase and Q-phase signal paths will inevitably have mismatches of amplitude and phase. This causes interference upon down conversion to baseband. Direct-conversion receivers are vulnerable to I/Q imbalance since I/Q separation is performed early in the RF/analog portion. The impact of I/Q imbalance is more severe to a system which uses high order modulations, and a high coding rate. Therefore, I/Q imbalance correction is essential for the design of higher data rate systems.
Many I/Q imbalance estimation and compensation methods that require specific training signals patterns have been proposed, see for example [5], [6]. However, the requirement for specific training patterns may not be a practical assumption. Thus, there is a need for blind signal processing methods. To the best of our knowledge, there are not many blind frequency-dependent I/Q imbalance compensation methods for DCR published. Authors in [3] and [4] propose methods using the statistical properties of the source signal. The contributions of this paper are two-fold. First, a necessary and sufficient condition for separation of the convolutive mixtures of the desired signal and its image using blind estimation is proved. Second, based on this condition, a compensation method is proposed and demonstrated.
The rest of the paper is organized as follows. In Section II, the frequency-dependent I/Q imbalance model are described. In Section III, The conditions for signal separation are derived and a gradient based method is used to solve for a separation filter. In Section IV., the simulation results are obtained for an OFDM system. Finally, conclusions are drawn in Section V.
Notation: Bold face upper case symbols, e.g.,
denotes matrices, and bold face lower case symbols, e.g.,
, denotes column vectors. The
the element of matrix
is denoted as
is a composite vector obtained by stacking column vectors together. The asterisk * denotes complex conjugate.
is the expected value.
is the Frobenius norm of
denotes conjugate transpose of a matrix.
denotes the trace of a matrix.
is used to denote the linear convolution.
I/Q imbalance Signal model and second-order statistic property
System Model

A typical block diagram of the RF front-end for DCR is presented in Fig. 1(a), and its equivalent mathematical model is given in Fig. 1(b). The received RF signal
is expressed as,

is the received baseband signal
, and
, and
is the transmitted signal. The received signals
are direct downconverted by a local oscillator signal
are the the mismatched gain and phase.
The downconverted signal
is expressed as

denote low pass filters for the
branch shown in Fig. 1(b) and
. After carrying out the low pass filtering and transforming equation (2) to the frequency domain, the I and Q branch signals are



. (4)

Then the received frequency domain baseband signals
can be expressed as



. (6)

or equivalantly in the time-domain,

. (7)
Hence, we can see that due to the I/Q imbalance, the received down-conversion signal
is distorted by its image signal
To evaluate the I/Q imbalance distortion, the analog front-end image-reject ratio (IRR) [1], in decibels (dB), is defined as

After sampling the continuous time signal
with sampling interval
, the discrete time signal is

. (9)

Second-order statistics
The discrete-time autocorrelation function (ACF) of a complex random signal
is defined as
. Another second-order statistics, the complementary autocorrelation function (CACF), is defined as
. If a complex random signal
is proper, then its CACF is equal to zero for all lag
[7], i.e.,
. In this paper, we assume the transmitted complex signal
is a zero-mean white process, i.e.,
and is proper, i.e.,

denote the Dirac delta function. The above assumption is a reasonable assumption because most communication systems, specifically those which have encoders or interleavers to achieve this condition.
In this paper, we further assume that the channels are wide-sense stationary uncorrelated scattering (WSSUS). It can be shown that a white and proper process, after a WSSUS channel, remains white and proper. The signal after a time-varying channel can be expressed as,
. The autocorrelation of a WSSUS channel impulse response is given by [14],

�� EMBED Equation.3

is a function that depends on
is the zero-order Bessel function of first kind, and
represents the maximum Doppler shift. Using this autocorrelation, the ACF of received signal
can be computed as,

, thus it is still a white process. Similarly, its CACF can be evaluated as,

The properness of
follows from the fact that
is proper.
Blind signal separation for I/Q imbalance using conjugate signal model
If the filters
are modeled by
tap length FIR filters [8], the time-domain frequency-selective I/Q imbalance in (9) can be rewritten in the following conjugate model:


The objective of signal separation is to make the filter output a scaled and delayed version of the input [13]. Mathematically, the goal is to find a separation filter
with length
such that the signals and its image,
, are separated at the filter output, i.e.,


is a
permutation matrix, either
is a
diagonal scaling matrix and
is a delay.
In order to achieve the separation, prior information about the source signals, i.e., conjugation between these two signals needs to be imposed. This motivates us to come up with a cost function that incorporates this conjugate constraint. With this constraint, and the goal to restore the property of signal
, a necessary and sufficient condition for signal separation can be proved as in Lemma 1.
Remark 1: After the separation filter, the output is a delayed and filtered version of
.Therefore, the delay and the effective channel should be estimated then equalized prior to decoding. This is not a big issue since although a known training signal does not exist but most systems do have pilot symbols. For example, in LTE, the pilot symbols are inserted every six subcarriers in the frequency domain. These pilot symbols can be used for estimating the delay and equalization. Since these are not the scope of this paper, they are assumed to be perfectly known in the simulations. The permutation matrix
can also be identified in this manner.
Remark 2: Since
is white and proper, it can be shown that the autocorrelation matrix of
equal to a diagonal matrix,

Conditions for signal separation
In order to satisfy the conjugate condition for the elements of the filter output, i.e.
, we need to impose the following constraints on the elements of the filter matrices, i.e.

Define the composite vector
. By equation (12), this composite vector
can be written as


is a Toeplitz matrix composed of
We are now ready to derive the conditions which lead to signal separation.
Lemma 1: Under the constraints
are separator matrices, that is
, if and only if

Proof: The sufficient condition can be shown by plugging equation (12) into (14) then using equation (13). That is

, when
Now we proceed to prove the necessary condition. By looking at the matrix equation

element by element, i.e.


we can obtain the following conditions for




From equation (12), the filter output
can be written as

are filters depend on
. Then from (15)


is signal power and
. One can conclude that equation (18) can be satisfied when
is an anti-symmetric thus must be a non-causal filter. The latter case can be excluded and we can conclude that
. Without loss of generality, let us assume
. Then
in (17) becomes


Plug equation (18) into equation (15), and we have

, (20)

which implies
is an all-pass filter. It can be shown that for FIR all-pass filters, all the zeros are at the origin of the
-plane, thus, must have a linear phase. Therefore,
for delay
. One can see that the output of the filter is just a scaling and delayed version of
, i.e.
. Similarly, if
, then
which leads to
. Recall that, by construction,
, therefore
In Lemma
, we prove that the autocorrelation matrix of filter output signal
is the necessary and sufficient condition for signal separation. Therefore, we define the following cost function

. (21)
Then the separation matrices
can be obtained by minimizing the cost function

. (22)

Here we proposed a gradient-based method [11] to iteratively solve the problem. At each step, the gradient of the cost function, i.e.
, is computed and the argument
is updated by stepping along the gradient direction. Note that in complex domain, the gradient direction is the derivative with respect to
instead of
Based on the fact that
equation (21) can be expressed as


where matrix
is defined as the
submatrix of
, where
is composed of the

The term
can be expanded as

��EMBED Equation.3

Similarly, the term
is expressed as


Observing equations (24) and (25), we find out that the derivatives needed are of the form
, where
submatrices from
. Based on [12], the results of these derivatives are given as follows, without lengthy step by step details. Note that the gradient is the derivative w.r.t.
instead of



The cost function is minimized iteratively by the learning rule


is the matrices at
the step,
is the learning rate and

At each step,
is also updated by


Performance Analysis and Numerical Results
In this section, simulation results are presented to demonstrate the performance of the proposed blind compensation scheme for OFDM systems. The simulated OFDM system is with symbol length
subcarriers, guard symbol
, and subcarrier spacing
kHz [9]. Hence - the total symbol length is equal to
samples per OFDM symbol. We consider two different cases. Case 1_3-tap LPF: The I/Q imbalance parameters are; gain mismatch
, phase mismatch
and I and Q branch LPF
respectively [8] where the analog IRR is in the range of 18-25 dB [Fig. 2(a)]. Case 2_ 2-tap LPF: The frequency-dependent I/Q imbalance parameters are
[8] where the analog front end IRR is around 26-35 dB [Fig. 3(a)]. Both simulations use the learning rate
and with initial values
. This is justified by the fact that, in practical, the LPFs on both I and Q branch are close to an ideal, thus its impulse response is close to a delta function. In the simulations, 50,000 samples are used for estimating the autocorrelation function, 64-QAM modulation is used and the received SNR is 20dB. The convergence rate is shown in Fig. 2(b) and 3(b). The proposed method converges in approximately
iterations for case 1 and 2 respectively. In case 2 the LPFs are more ideal than in case 1, therefore it is not surprising that the simulation converges faster in case 2. In Fig.2 and 3, the performance of the proposed method is shown in terms of IRR. Figure 2(a) and 3(a) show that the IRR is improved by around 20 dB after I/Q imbalance compensation. Fig. 4 and 5 show the Symbol Error Rate (SER) performance before and after compensation. The results show that there is a need to compensate for the I/Q imbalance in both cases, since without compensation, the performance is unacceptable. Fig. 4 and 5 also show that the proposed compensation scheme provides an effective removal of the imbalance, and also improves the BER performance.

In this paper, a blind signal separation approach is proposed to compensate for a frequency-dependent I/Q imbalance in a wideband direct-conversion receiver. Necessary and sufficient conditions for signal separation are derived. Based on the conditions a gradient-based method is used to solve for the separation filter. Simulation results show the proposed approach effectively removes distortion caused by the I/Q imbalance.
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Figure 1: (a) Frequency-dependent I/Q imbalance of direct-conversion receiver (b) Mathematical model of a DRR with I/Q imbalance (c) Equivalent mathematical model of (b).

Figure 2: 3-tap LPF case
(a) IRR in before and after compensation (b) Convergence of the cost function.

Figure 3: 2-tap LPF case
a) IRR before and after compensation (b) Convergence of the cost function.

Figure 4: Symbol Error Rate for 3-tap LPF case

Figure 5: Symbol Error Rate for 2-tap LPF case


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I/Q LO (gd, Φ)

I/Q Imbalance compensator

Analog Front End

Digital Baseband