LMS Algorithm Based Error Correction Technique in In-Phase and Quadrature Channel Signal Processing

Amritakar Mandal #1, B. K. Kaushik*2, Brijesh Kumar*3,K. C. Tyagi@4, , R. P. Agarwal$5

#Radar Laboratory, PMG, New Delhi, New Delhi, INDIA

*Department of Electronics and Computer Engineering, Indian Institute of Technology Roorkee, INDIA

1amritkar2k@gmail.com

3bk228dpt@iitr.ernet.in @Electronics and Communication Engineering,

Dev Bhoomi Institute of Technology, Dehradun, INDIA

$Department of Computer Science and Engineering, Shobhit University,Meerut, INDIA

2bkk23fec@iitr.ernet.in 4prajanag@gmail.com

Abstract—-Every communication receiver that uses in-phase and quadrature channel signal processing technique encounters problems related to matching of gain and phase in both the channels. The gain and phase imbalances occur between Low Pass Filter and Local Oscillator used in both the channels and as a result the performance of the receivers and the quality of the received signals are degraded. The imbalances produced may cause insufficient attenuation in image frequency band leading to interference. The problem needs to be compensated. This paper represents the design and implementation of Coordinate Rotation Digital Computer (CORDIC) algorithm based Adaptive FIR filter using Least Mean Square (LMS) algorithm to give a solution for correcting I/Q imbalances. The design of CORDIC in the vector rotation mode results in high system throughput due to its pipelined architecture where latency is reduced in each of the pipelined stage and finds its appropriate application in real time signal processing. The good convergence of CORDIC in the given LMS algorithm based adaptive filter facilitates in easy calculation of new filter weights.

Keywords— I/Q imbalance, Low Pass Filter, Local Oscillator, CORDIC, FIR filter, LMS algorithm.

I. INTRODUCTION A quadrate receiver consists of two distinct channels to

form in-phase and quadrate components of the received signal. Each channel consists of a mixer, low pass filter, A/D converter and digital signal processor unit. Coherent oscillator (COHO) is common to both the channel with 90o phase shift. The mismatch between low pass filters can severely limit the performance of the adaptive canceller and the matched filters and hence reduces the quality of the signal [1]. If noisy oscillators are used for both channels, then I/Q error may be generated because of gain and phase imbalances in mixer output [2]. But the chances of generation of error due to

oscillator are negligible as both the channel uses same coherent oscillator. This paper presents a feasible design of a LMS algorithm based adaptive filter using CORDIC for the correction of I/Q channel mismatch to increase Signal to Noise ratio (SNR) significantly. Reformulated trigonometric LMS algorithm has been used for the realization of adaptive FIR filter [3]. The sine and cosine terms generated are used in filtering and weight updating purpose. The sine and cosine terms can be calculated using polynomial approximation, e.g. Taylor series. But it requires a considerable amount of hardware space on the silicon substrate. Interpolation method using table look-up may be the other solution. But it also requires large number of gates and ROM memory. The CORDIC offers the opportunity to calculate the desired errors in a simple and efficient way due to its good convergence property as shown in Fig. 2. This paper first describes the I/Q channel demodulation, errors in section II; I/Q compensation technique in section III; and finally algorithms and adaptive filter architecture design in section IV and V respectively.

II. I/Q CHANNEL DEMODULATION AND ERRORS As per the Euler’s theorem, vector sum of cosine’s

component is completely real whereas the spectrum of sine component is totally imaginary. If the cosine and sine components are combined, the resultant spectrum becomes one sided with direction of rotation (positive or negative frequency) and with known real (cosine) and imaginary (sine) components [4].

tjcc

cetjt ωωω =+ sincos The process of recovering both real and imaginary signal

component is known as I/Q demodulation. I stand for in-phase channel which processes cosine (real) components. Q stands for quadrate channel which processes sine (imaginary)

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component. The input of I/Q channel is Intermediate Frequency (IF). If the carrier frequency of IF is fc with a time varying amplitude )(ta and time varying phase )(tφ , then input signal )(ts will be:

)](2cos[()()( ttftatS c φπ += . In I channel, the IF signal is multiplied by reference carrier

frequency produced by crystal oscillator at zero phase reference. The output of the I channel mixer is, I(t) , given by :

)2cos()].(2cos[()()( tfttftatI cc πφπ += )](4cos[)()](cos[)( ttftatta c φπφ ++=

The first term is the average value (DC) of the product and represents cosine of the signal phase and amplitude. The second term with high frequency component is suppressed by Low Pass Filter (LPF). So the output of the I channel is

)](cos[)()( ttatI φ= . Similarly, the Q channel output can be derived. The LPF

output at Q channel is: )](sin[)()( ttatQ φ= .

Thus, I and Q channels are necessary for any angle modulated signals because the two side bands of the RF spectrum contain different information and there is a strong possibility of corruption of signals if they overlap each other to create Inter Symbol Interference (ISI).

The receiver possesses a Coherent Oscillator which is at the same phase with the transmitter frequency. The low pass filters in both the paths of the receiver must have identical characteristics as any mismatch in their characteristics will lead to I/Q errors. In practice, the errors are unavoidable and the outcome is insufficient attenuation of the image frequency [1,4]. Due to this phenomenon, an error compensation technique is required.

III. ERROR COMPENSATION TECHNIQUE IN I/Q CHANNELS In the receiver model same coherent oscillator is used for

both in-phase and quadrate channel. So error due to mismatch in oscillator phase or gain is negligible. The total error may be produced by the LPFs in both channels. The basic principle is to make the input signal through the filters identical and then compensate for the error between the two outputs. The compensation filter is an adaptive filter which uses LMS algorithm to reduce the error.The calibration of the signal can be performed with random input signal as shown in Fig. 3. To perform the calibration in the circuit, both the mixers are isolated. This means both switches 1SW and 2SW are open. A known calibrated signal source is then incorporated to the input of LPFs . Let the quadrate channel low pass filter is mismatched. So it is obvious that I/Q error will persist. To compensate the mismatch between the channels, adaptive algorithm based filter is incorporated. Now the circuit is needed to be trained so that the two signals at the filter output would become identical.

After training, the weights are calculated and the transfer function of new filter is obtained and added to the network to neutralize the error [1]. It is observed that the errors have been decreased significantly as the LMS filter output showed

improvement in Mean Squared Error (MSE) with respect to iterations, as shown in Fig. 5, and thus mismatch is neutralized.

IV. LMS ALGORITHM AND CORDIC The LMS algorithm [3] is one of the simplest well known

adaptive algorithms and with the help of this algorithm a robust adaptive filter can be devised. Let the FIR filter is linear discrete time filter. x(n) And d(n) is input and training sequence of the FIR filter. e(n) is the error between desired output and the obtained output. Let w be an N-tap filter

coefficient vector, then, T]w,........,,w[ww N 110 −=

and T)]x(n),.......,[x(n),x(nx(n) N 11 +−−= . The mean square

error is (n)]E[e(n)e(n)]E[eε T== 2 . The minimum mean square error will give the optimum filter weight. Using the steepest descent search algorithm, an optimum filter weight in the form of 1w R popt

−= is obtained. The i -th iteration using

steepest descent search algorithm is given by the expression: Rw]μ[pw(i))w(i −+=+1 . Replacing R and p by

(n)x(n)xT and x(n)d(n) respectively, LMS algorithm is obtained in the form: n) x (n) e (+ ) = w (n) w (n+ μ1 . Here μ is called the step size.

The Delayed LMS (DLMS) is equivalent to the LMS algorithm but the gradient is evaluated at some previous iteration, say L iterations before present iteration. Then the updated equation of DLMS algorithm can be expressed as, (n-L)x (n-L) e ) =w (n) +w (n+ μ1 .

Let the tap weight kw in the equation kkk θAw sin= be such that each kw , kkk AwA +≤≤− , maps uniquely to a

kθ in the interval ]π,π[22

+− .Then using LMS algorithm,

recursively updated Trigonometric LMS (TLMS) algorithm can be formulated as (n)μΔ(n)x(n)eθ(n))θ(n +=+1 where

k)(n) x(nθAd(n)e(n) kk

N

k−∑−=

−

=sin

1

0. The Trigonometric

LMS algorithm is suitable for CORDIC based realization [3], since the two quantities, such as k)x(nθk −sin and

k)x(nθk −cos in, k = 0, 1, …., N-1, required for filtering and updatation 10 −≤≤ Mi , denote a sequence of ±1s that determine the direction of each elementary rotation.

When M is the total number of elementary rotation angles, i-th angle αi is given by:

⎪⎩

⎪⎨

⎧

−−−

−−

−

−− ==),1(1

),1(1

),0(

),(1,

2tanh

2tan

2

]2[tan1

is

is

is

imsim m

mα

Where m=0, 1 and 1− correspond to the rotation operation in linear, circular, and hyperbolic coordinate system respectively. For a given value of θ, the CORDIC iteration is given

227

by:⎥⎦⎤

⎢⎣⎡

⎥⎦

⎤⎢⎣

⎡⎥⎦⎤

⎢⎣⎡

−−−=

+

+

i

ii

i

i

i

i

yx

δ

iδyx

1221

1

1 and iiii αδzz −=+1 ,

where iαi−−= 2tan 1 .

In case of counter clockwise rotation of a vector, the recursively updated equations are generated as: iiiiiii αδyαδxx sincos1 −=+

and iiiiiii αδxαδyy sincos1 +=+ . The above equations can be simplified in the following

form: )y(x iiiiiii αδαδx tancos1 −=+ )αδx(yαδy iiiiiii tancos1 +=+ .

Here, iαtan is restricted to .2 i−± Thus, multiplication is transformed to an arithmetic right shift. Since cosine is an even function, therefore ( ) ( )αα −= coscos The iterative equation can be reduced to-

)y(x iδKx iiiii−= −+ 21 and )iδx(yKy iiiii

−+=+ 21 .

where )221(2arctancos iiK i−+=⎟

⎠⎞⎜

⎝⎛ −= is known as

gain factor for each iteration. If M iterations are performed, then scale factor, K, is defined as the multiplication of every Ki. So,

∏ −+=∏=−

=

−

=

1

0

1

0)221(

M

i

M

ii

iKK .

The elementary functions sine and cosine can be computed using the rotation mode of the CORDIC algorithm if the initial vector starts at )0,( K with unit length [7].

V. PIPELINED ARCHITECTURE OF CORDIC In Pipelined CORDIC architecture, a number of rotational

modules are incorporated and each module is responsible for one elementary rotation [8-10]. The modules are cascaded through intermediate latches (Fig. 1). Every stage within the pipelined CORDIC architecture, only adder/subtraction is used. The shift operations are hardwired using permanent oblique bus connections to perform multiplications by 2-i. The pre-computed values of i-th iteration angle αi require at each module are stored at a ROM memory location as shown in Table-I. The delay is adjusted by using proper bit-length in the shift register. Since no sign detection is needed to force zf=0, the carry save adders are well suited in this architecture. The use of these adders reduces the stage delay significantly. With the pipelining architecture, the propagation delay of the multiplier is the total delay of a single adder. So ultimately the throughput of the architecture is increased by many folds as the throughput is given by: “1/(delay due to a single adder)”. If an iterative implementation of the CORDIC is used, the processor would take several clock cycles to give output for a given input. But in the pipelined architecture, each pipeline stage takes exactly one clock cycle to pass one output [3].

The most recurrent problem for a CORDIC implementation is overflow. Since the first tangent value is 20=1 then rotation

range will be [ ]2,2 ππ− . The difference in binary representation between these two angles is one bit.

Overflow arises when a rotational angle crosses a positive right angle to a negative one. To avoid overflow, an overflow control is added. It checks for the sign of the operands involved in addition or subtraction and the result of the operation. If an overflow is produced, the result keeps its last sign without affecting the final result. In the overflow control, the sign of zi determines whether addition or subtraction is to be performed.

of the k-th coefficient can be computed by using single CORDIC processor in adaptive filter as shown in Fig. 4.

The theory of CORDIC computation is to decompose the desired rotation angle into the weighted sum of a set of predefined elementary rotation angles [5, 6]. Each of them can be accomplished with simple shift-add operation for a desired rotational angle θ. It can be represented for M iterations of an input vector (x,y)T setting initial conditions: x0=x, y0=y, and

z0=θ as ∑−=−

=

1

0

M

iiif αδθz . If zf=0 holds, then ∑=

−

=

1

0

M

iiiαδθ ,

i.e. the total accumulated rotation angle is equal to θ. δi

Figure 1. Pipelined CORDIC Architecture

Figure 2. Convergence of CORDIC w.r.t number of iterations

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TABLE I. PRE-COMPUTED ANGLES

Figure 3. Calibration techniques for filter mismatch correction

Figure 4. CORDIC based adaptive filter

VI. CONCLUSION This paper presented the error correction technique in in-

phase and quadrature channel with a CORDIC based adaptive FIR filter using LMS algorithm. The use of pipelined CORDIC computational architecture makes implementation of this kind of adaptive filter easier. Numbers of micro-rotations have been adjusted so as to achieve better convergence and speed of operation while minimizing delay in error computation. CORDIC algorithm is used to achieve high degree of computational accuracy facilitating better application in I/Q channel calibration. The inherent issue of CORDIC i.e. overflow is quite appropriately resolved using

proposed design. The property of good convergence of CORDIC is efficiently used in adaptation and filtering with LMS algorithm.

Figure 5. Convergence of error at the output of adaptive filter with step size

REFERENCES [1] Isis Mikhael, Wasfy. B, Pravinkumar, “Adaptive IQ Mismatch

Cancellation For Quadrature IF Receiver”, Proceedings of 4th WSEAS International Conference on Signal Processing, Robotics and Automation, 2005.

[2] Fred Harris, “Digital filter equalization of analog gain and phase mismatch in I-Q receivers”, 5th IEEE International Conference on Universal Personal Communications, vol. 2, pp. 793-796, 1996.

[3] M. Chakraborty, A. S. Dhar and Moon Ho Lee, “A Trigonometric Formulation of the LMS Algorithm for Realisation of Pipelined CORDIC,” IEEE Trans. Circuits and Systems, vol. 52, no. 9, pp. 530-534, Sep.2005.

[4] Mikko Valkama, M. Renfors, V. Koivunen, “Advanced Methods For I/Q Imbalance Compensation in Communication Receivers”. IEEE Transactions on Signal Processing, vol. 49, No. 10, pp. 2335-2344, Oct. 2001.

[5] J.E. Volder. "The CORDIC Trigonometric Computing Technique". IRE Transactions on Electronic Computing, vol EC-8, pp 330-334, Sept. 1959.

[6] Y.H. Hu. "CORDIC-Based VLSI Architectures for Digital Signal Processing" IEEE Signal Processing Magazine, Vol. 9, No. 3, pp. 16-35, 1992.

[7] Andraka R.A., "Survey of CORDIC Algorithms for FPGA Based Computers”, Proceedings of the 1998 ACM/SIGDA 6th International Symposium on FPGAs, pp 191-200, Monterey, California, Feb.22-24, 1998.

[8] A. Mandal, K.C. Tyagi, B.K. Kaushik, “VLSI Architecture Design and Implementation for Application Specific CORDIC Processor”, 2nd IEEE International Conference on Advances in Recent Technologies in Communicaion and Computing (ARTCom), pp. 191-193, Oct 16-17, 2010.

[9] S. Wang, V. Piuri, E. E. Swartzlander. Jr., "Granularly-pipelined CORDIC processors for sine and cosine generators”, IEEE International Conference on Acoustics, Speech, and Signal Processing ICASSP, vol. 6, pp. 3298-3301, 1996.

[10] Y.H. Hu. "The Quantization Effects of the CORDIC Algorithm" IEEE Trans. Signal Processing, vol. 40, No. 4, pp. 834-844, Apr. 1992.

I ii αtan2 =− )2arctan( i

i−=α αi in radian

0 1 45o 0.7854 1 0.5 26.565o 0.4636 2 0.25 14.063o 0.2450 3 0.125 7.125o 0.1244 4 0.0625 3.576o 0.0624 5 0.03125 1.7876o 0.0312 .. ….. …….. …….

229