Estimation and Compensation of Impairments caused by Quadrature Modulators and Demodulators

Transcription

1 Estimation and Compensation of Impairments caused by Quadrature Modulators and Demodulators Gregory T. Nash School of Electrical and Computer Engineering University of Illinois, Chicago Chicago, Illinois Abstract Several techniques have been proposed for estimating and compensating for the effects of quadrature modulators and demodulators. Several of them will be discussed in this survey including those that observe the envelope of signals, the pilot tones of OFDM signals, and others. I. INTRODUCTION A. Basic introduction and motivation 1) The Quadrature Modulator and Demodulator: The quadrature modulator is an arrangement radio frequency (RF) circuit comprised of two mixers which are summed together. Mathematically, it is represented as: s(t) =I t (t) cos(f c t) Q t (t) sin(f c t) (1) Here s(t) is the output signal that is generally output and transmitted to the channel. I t (t) and Q t (t) are analog signals to be transmitted, and f c is the carrier frequency. These signals are analog, but they could, or course, be modulated and shaped digital signals. In practice, I t (t) and Q t (t) will be bounded in frequency by an ω B << f c. Similarly, a quadrature demodulator performs the opposite functionality on the signal r(t) which would be s(t) after received through the channel, specifically: I r (t) =h(t) r(t) cos(f c t) Q r (t) =h(t) r(t) sin(f c t) (2) Here, the h(t) is an ideal low pass filter with a cutoff frequency greater than ω B that removes the upper mixing image. It is easy to see that in this configuration we can have I t (t) = I r (t) and Q t (t) = Q r (t) to complete the communication system. When I t (t) and Q t (t) are baseband signals, the transmitter is said to be a zero intermediate frequency (ZIF) transmitter and the receiver is said to be a direct downconverstion (DDC) receiver. If I t (t) and Q t (t) are passband signals, the transmitter is said to be a complex IF architecture. 2) Cost and Size: The cost and size of hardware are two of the primary motivations to use this transmitter and receiver architecture. The most alternative would be a heterodyne where the intermediate frequency (IF) could be generated in the digital or analog domain. This leads to extra local oscillators (LO s), filters, amplifiers, mixers. This alternative inevitably has additional cost, and size. 3) Flexibility: Another motivation for the use of quadrature modulators is flexibility in frequency. Mixers, whether in a quadrature modulator or not, produce additional undesired products according to: f RF = m (mf LO + nf IF ) (3) n Most often m =1and n =1for low side injection or n = 1 for high side injection. For all other integers of m and n, additional, undesired products. If these additional products cannot be filtered or are non inherently at a low enough level, they are referred to as spurious emissions. They will interfere with other users of the spectrum, and thus transmitters are regulated from emitting them. The end result is that transmitters using a non-zero f IF will have a operational range that is set by the location and level of these other terms. Since, ZIF transmitters only produce images at mf RF they have a very high tuning range. B. The problem of IQ imbalance and DC feedthrough 1) Mathematical Model: There are non-idealities associated with quadrature modulators and demodulators that affect ZIF and DDC architectures. Namely, the problems of IQ imbalance and DC feedthrough or offset. There are several models of this problem, but the are generally equivalent. One model will be used for this paper. Starting with the ideal model in 1, the non-ideal model is: s(t) =((1 + α)i t (t)+βq t (t)) cos(f c t)+... ((1 α)q t (t)+βi t (t)) sin(f c t) χ I cos(f c t)+χ Q sin(f c t) (4) This is shown in figure 1. To account for non-ideal scaling in the digital to analog converters (DAC s) and gains in the mixers, α is introduced and referred to as scaling error. For a non-ideal phase between the cosine and sine, β is introduced and referred to as quadrature error. Lastly, to compensate for bad bias on the I or Q branch, χ I and χ Q are introduced. These factors are unknowns and will vary from radio to radio on a manufacturing process. In modern communication systems, these non-idealities must be accounted for if a low bit error rate

2 I t(t) cos(f ct) 1+α χ I shift by 90 + β s(t) The envelope of the impaired signal might look as in figure 3. We will show that the estimation of these impairments amounts to estimating aspects of this waveform. 2) DC estimation: The most intuitive example of estimating the impairments is found in [1]. As is true for many techniques, two different signals are used to determine the estimates. One tone will be used to estimate χ I and χ Q. The signal is no signal at all, I(t) =Qt =0. Then: 1 α Q t(t) I (t) =(1 + α)i(t) βq(t)+χ I =χ I (6) Fig. 1. χ Q Quadrature Modulator Block Diagram with impairments (BER) is to be maintained. Accounting for this error eventually boils down to estimating these parameters. As is common, the passband signal is represented by a complex representation at baseband. Using this representation, we have: s(t) =(1 + α)i t (t)+βq t (t)+j((1 α)q t (t)+βi t (t))+ + χ I + jχ Q (5) 2) What are the impacts: The impact of quadrature impairments will have several effects. The first and most direct is that is skews signal constellations. When not compensated, this adds to the noise and reduces the system performance. This error also interferes with most other algorithms running in a communication system. Some examples include power control algorithms, carrier frequency estimation, and predistortion algorithms. C. Brief overview of proposed techniques Several techniques have been published for estimating and compensating for quadrature error. The two main categories of techniques. Some use known signals and estimate the constants. The others are blind techniques which work using only knowledge of the original signal statistics. Different known signals are used. Some use tones, and others use aspects of the signals which are defined by the communication standards. II. ESTIMATION FOR MODULATORS A. Using a Tone and Ideal Envelope Detector 1) System Definition: This algorithm works in a system as shown in 2. This work proposes a methods by which signals are presented to the quadrature modulator, QMC, in this figure, and a envelope detector, shown as ED in this figure, is used to observe the output. It also puts this in framework of a system with Digital Predistorion, DPD. DPD account for the compensation for the non-linear power amplifier. Q (t) =(1 α)q(t) βi(t)+χ Q =χ Q (7) Here, I(t) and Q(t) are the ideal tones, and I (t) and Q (t) are the impaired signals. And the envelope will be: s(t) 2 =s(t)s (t) =(χ I + jχ Q )(χ I jχ Q ) =χ 2 I + jχ I χ Q jχ I χ Q + jχ 2 Q =χ 2 I + jχ 2 Q (8) As always, these measurements are made in the presence of noise although this is not discussed in [1]. If this noise were added prior to the quadrature modulator and the distributions were different on the I and Q channels, one might be able to estimate χ I and χ Q with enough samples. However, this is not usually true, and an iterative approach is taken. This approach is outlined: 1) I(t) and Q(t) are initialized to 0. 2) I(t) is adjusted by δ and power is measured at output. 3) Q(t) is adjusted by δ and power is measured at output. 4) If power increased at previous change of I(t), adjust I(t) by δ, otherwise adjust by δ. 5) If power increased at previous change of Q(t), adjust Q(t) by δ, otherwise adjust by δ. 6) Repeat from step 4 until power is sufficiently low 7) Then χ I will be I(t) and Then χ Q will be Q(t), Though this step is better studied in control theory than estimation, it is important to simplify the mathematics when estimating α and β. All parameters are dealt with simultaneously in [2], but it requires known phase delay in the loop. 3) Scaling estimation: The next set of estimation will be to estimate the scaling error. Again, the procedure in [1] can be shown and an estimation problem which will account for noisy measurements. For this test the signals are I 1 (t) =A and Q 1 (t) = 0 and then I 2 (t) = 0 and Q 2 (t) =A. With impairments this comes to: I 1(t) =(1 + α)a β0 =(1 + α)a Q 1(t) =(1 α)0 βa = βa

3 and I 2(t) =(1 + α)0 βa = βa Q 2(t) =(1 α)a β0 =(1 α)a (9) Calculating the envelope and adding the noise in observations: and s 1 (t) 2 =s 1 (t)s 1(t) =((1 + α)a + jβa)((1 + α)a jβa) =(1 + α) 2 A 2 + β 2 A 2 =A 2 ((1 + α) 2 + β 2 )+n(t) =A 2 (1 + 2α + α 2 + β 2 )+n 1 (t) (10) s 2 (t) 2 =s 2 (t)s 2(t) =((1 α)a + jβa)((1 α)a jβa) =(1 α) 2 A 2 + β 2 A 2 =A 2 ((1 α) 2 + β 2 )+n(t) =A 2 (1 2α + α 2 + β 2 )+n 2 (t) (11) To make this estimation, take the difference of these equations and apply them to a matrix. This yields: If we extend this to multiple samples, we have the matrix case: s 1 [0] 2 s 2 [0] 2 s 1 [1] 2 s 2 [1] 2. s 1 [N 1] 2 s 2 [N 1] 2 =21 {N 1}α+ +[n 1 [n] n 2 [n]] (12) Using this linear model and assuming AWGN, the MVUE is: ˆα = 1 2N N 1 n=0 [ ] n 2 [ ] s 1 n 2 s 2 4) Quadrature: This tone is represented as: f s f s x(t) =I(t)+jQ(t) = exp(jf c t) I(t) = cos(f t t) I (t) = cos(f t t) β sin(f t t) Q(t) = sin(f t t) (13) Q (t) = sin(f t t) β cos(f t t) (14) To see how this works, first the envelope of the ideal signal is examined s(t) 2 =s(t)s (t) = cos 2 (f t t) + sin 2 (f t t)=1 (15) Now, the technique in question is used step by step so χ and α can be neglected. Again, the envelope with noise is: s(t) 2 =s(t)s (t) =I 2 (t)+q 2 (t) =(cos(f t t) β sin(f t t)) 2 + (sin(f t t) β cos(f t t)) 2 = cos 2 (f t t) 2β cos(f t t) sin(f t t)+ + β 2 sin(f t t) + sin 2 (f t t)+ 2β cos(f t t) sin(f t t)+β 2 cos(f t t) =1 + β 2 2β sin(2f t t)+n(t) (16) Using this technique, the time varying part of the signal is 2β sin(2f t t)+n(t). Again, [1] does not take noise into account, which would mean the variance of the process n(t) would be 0 and all that is needed is to subtract the max from the minimum and find β. It has been shown that the MVUE of β under AWGN will be: ˆβ = N 1 n=0 [ ] n 2 s f s ( ) n sin 2f t f s (17) where f s denotes the sampling frequency. 5) Using Tones with Envelope Detector Error: The previous work assumed that we had perfect knowledge of the envelope and the only impairment was knows. In [2], many impairments are to the envelope detector are introduced. Among these are envelope detector non-linearities, bias, and quantization error. With the non-linearities, the previous methods of estimating α, the difference in I and Q gain, is impaired. The same is true for trying to estimate β, which boiled down to estimating the amplitude of the residual sinusoid, this is also complicated by non-linearities. The assumption of zero mean noise with then be invalid, and the estimator will be biased. Likewise, quantization error which can often be represented by scaling the signal and assigning the value of the integer below the value of the signal. This error will also be biased and lead to a biased estimation. The solution stated in [3] is to use two tones and in [2] this is extended to use a set of tones. When this is the case, a similar construction as referenced in the single tone summary but with an least squares estimate using all the tones as its elements can provide an accurate estimate in the presence of impairments. Essentially, this is a repetition of the signal tone that allows for the detector non-idealities to be explored and adjusted for.

4 Fig. 2. From [2], a modern transmitter with DPD 6) Tone Summary: Though this method has closed form expressions for the estimators. It is also attractive because envelope detectors exist in nearly all transmitters for power or gain control, and would not add cost for it. In the signal tone case, it is easily seen that the DC offset as the estimate of how far off the DC of the signal is. On the scaling error, the estimate is clearly the difference in gain on I and Q, and on quadrature, the estimate is the correlation between I and Q. These are important concepts in understanding the other schemes. In practice, this requires some down time in the transmitter which is frequently inconvenient. Also, it imposes some constraints on how an envelope detector might be sampled which may add cost to the system. Lastly, it is typical in regulations that calibration tones may not be sent over the channel, so some switching would have to be incorporated. B. Using an envelope detector for a modulator One of the major drawback of sending tones is the question of whether or not regulations and standards will allow for it. Training sequences may overcome this by using signals that are standards compliant. A method which works on an arbitrary signals is shown in [4]. Using the knowledge of the transmitted signal s sign and amplitude, estimates of the of the impairments are made. Each uses a LMS like convergence on the value. In [4], a joint estimation is not evaluated, but if all impairments are alternately estimated and compensated for, full convergence can be achieved. Another method shown to work is in [2]. It includes the envelope non-linearities, bias, and quantization, and yields a complicated non-linear estimation. It too is shown to work over converging iterations. The impaired envelope is shown in 3 with non-linearities and quantization. C. Using Complex Feedback In earlier radio architectures, the idea of a dedicated receiver for observing the transmitter was deemed to be an inefficient solution. As complex digital modulation has become dominant over FM, PSK, and FSK due to spectral efficiency considerations, large amplifiers have had to be linearized. At first, this was through analog schemes such as Cartesian feedback and feed forward. Currently, the predominant architecture for high power radios is called digital predistortion. In this architecture, Fig. 3. From [2], this shows the envelope with quadrature impairments such a dedicated receiver is required and thus could be used to enhance quadrature modulator estimation. Further, up to this point, the quadrature impairments have been considered to be linear and frequency invariant. For small bandwidth signals, frequency invariance is a mostly valid assumption. As signal bandwidths increase, this assumptions is less valid. We can then replace α and β from our model with α(ω) and β(ω). A method using an observation receiver which makes such an estimation is in [5]. By using the observation receiver, the problem becomes: [e I e Q ]= [ α T, β T ] [x I [0], x I [1],..., x I [N 1], x Q [0], x Q [1],..., x Q [N 1]] (18) In this equation, e I and e Q represent the error between what was transmitted and what was received. The scalars α and β have been replaced with the vectors α and β to represent the frequency dependency. Then an LSE solution is employed via the Least Means Squared (LMS) iterative method. III. ESTIMATION FOR DEMODULATORS A. Using a training sequence 1) Using training sequence from OFDM: The series of tones described in the previous works have an obvious extension to orthogonal frequency domain modulation, or OFDM. In this modulation scheme, data is mapped into symbols. The inverse discrete Fourier transform is calculated of these symbols and this is transmitted as a composite symbol. The result is that output signal looks like he sum of closely spaced tones. In many OFDM based schemes, especially those in IEEE s and 16, preamble precedes data. The preamble is a composite symbol which contains the information required to decode the subsequent composite symbols. In, [6], a scheme is proposed which a special preamble is designed to easily estimate quadrature modulator impairments.

5 In the preamble the symbols, d 1 and d 2 are given by: d 1,k {1, 1} for all k, (19) { d1,k for k {1, 2,..., K} d 2,k = (20) d 1,k for k { K, K +1,..., 1} With this structure, an matrix equation of symbols and impairments can be made in the frequency domain which can be solved to make an estimator of the impairments. Do to the structure of alternately inverting the tones, this can be seen as similar to the first examples in [1] and [2]. When the data from the same tone in d 1 and d 2 are added together, they should add, then cancel. In [7], an alternate sequence is proposed where in the preamble the symbols, d 1 and d 2 are given by: { 0 for k {1, 2,..., K} d 1,k = (21) d 1,k for k { K, K +1,..., 1} where in this preamble d 1,k can be anything, and possibly fulfilling traditional preamble information. This again has a similar effect where what appears in the null carrier is quite clearly, the result of α and β make estimating them easy. Other LS formulations are in [8]. 2) Using training sequence from OFDM with CFO: Another potential impairment is carrier frequency offset, or CFO. This impairment is quite common because when to communication systems are separated, a common reference is not possible, and the carrier frequency must be estimated. In [9], a preamble is designed such that CFO and quadrature impairments might be simultaneously estimated. B. Using Signal Statistics There is a powerful assumption that can be made regarding the received signals. It can be assumed that the data on I and Q can be independent. This is true of most communication standards. This leads to a structure in [10]. In [11], this assumption leads to an investigation using the theory of Blind Source Separation, and extends this problem to the frequency varying analogous to the quadrature modulator. Specifically, using a class of algorithms know as equivalent adaptive separation via independence, the impairments can be iteratively estimated. This is analogous to assuming a covariance structure and using that in and LS sense, then using sequential algorithms to find the inverse directly. The algorithms are shown to approach the Cramer-Rao Lower Bound in [12]. IV. USING MODULATORS AND DEMODULATORS SIMULTANEOUSLY IS ADDITIONALLY DIFFICULT A. Compensating with the LO Where there are both quadrature modulators and demodulators, a total estimation and compensation may be employed. This system is shown in 4. The signal r(t) is sent through a quadrature modulators, then to a quadrature demodulator which is handled by a joint compensation. In [13], a novel solution is proposed by which changing the amplitude and quadrature of the LO such that the resulting RF signal will be compensated for already. A ML scheme is proposed to determine estimates of the impairments. It goes on to discuss more classical methods of compensating the receiver. This scheme would require some non-typical implementation of the quadrature modulator and demodulator. In general, the LO s are buffered so that ripple does not introduce impairments to the transmitted signal. B. Using Expectation Maximization Much work assumes that all compensation will be done at the receiver. What makes the complicated is that after the modulator adds its impairments, the channel masks these and then the receiver can add it s own impairments. This complicated problem is discussed, in [14]. The proposed solution is to use the Expectation Maximization algorithm. First and estimate of the channel and quadrature modulator effects is made. Then the estimate is made of the of the demodulator effects. This is iterated until good estimates exist of both. Simulated results show that while signal to noise with classical demodulator estimation reaches a floor governed by the the quadrature modulator and channel effects, this algorithm does not. V. MIMO BASED SCHEMES In the previous works, the impairments due to the quadrature modulators were assumed to be from one modulators. Further, they have assumed a single quadrature demodulator. This system in shown in 5. Transmitters represent by s nt are sent to a receivers, ỹ nt. In a multiple input / multiple output system, or MIMO, we allow for several transmitters each with it s own impairments. Also, we have a plurality of receivers to make this measurements. This increases the difficulty of estimation. In [15], a method is used to simultaneously estimate all of these parameters. These MIMO OFDM system incorporate codes specifically designed to allow the receiver to discriminate transmitters which are orthogonal in only space, which manifests itself as a unique collection of channels. In particular, the Alamouti Scheme bares a striking resemblance to the training sequence in [6]. Using these known sequences, a large matrix calculation is capable of removing all impairments. Similarly, in [16], space frequency block codes, SFBC s, are used as the know signal for the estimation. These codes are incorporated into the Long Term Evolution standard. VI. CONCLUSION As the need for transceiver flexibility and low cost increase, ZIF and DDC architectures are becoming more prevalent High order modulation schemes require the the impairments be estimated and compensated for to an increasing degree. Where once, a calibration could be used, now selfcalibrating and real-time estimation is needed. This has been

6 Fig. 4. From [11], a full TX and RX system Fig. 5. From [15], a full MIMO system shown to be possible with envelope detectors, but modern architectures can used complex feedback to make the estimation. Further, advanced techniques at the receivers may allow the constraints to be loosened on the transmitters. These techniques included blind ones that use only the principle of independence to estimate the values. The state-of-the-art is the joint estimation of many sets of errors from many transmitters and receivers simultaneously. While this make make for a complicated receiver, only the central communication node make be required to do this make much more cost effective user equipment possible. REFERENCES [1] M. Faulkner, T. Mattsson, and W. Yates, Automatic adjustment of quadrature modulators, Electronics Letters, vol. 27, no. 3, pp , jan [2] J. Cavers, New methods for adaptation of quadrature modulators and demodulators in amplifier linearization circuits, Vehicular Technology, IEEE Transactions on, vol. 46, no. 3, pp , aug [3] S. Burglechner, G. Hueber, and A. Springer, On the estimation and compensation of iq impairments in direct conversion transmitters, oct. 2008, pp [4] R. Marchesani, Digital precompensation of imperfections in quadrature modulators, Communications, IEEE Transactions on, vol. 48, no. 4, pp , apr [5] A. Lim, V. Sreeram, and G.-Q. Wang, Digital compensation in iq modulators using adaptive fir filters, Vehicular Technology, IEEE Transactions on, vol. 53, no. 6, pp , nov [6] T. Schenk, P. Smulders, and E. Fledderus, Estimation and compensation of tx and rx iq imbalance in ofdm-based mimo systems, Radio and Wireless Symposium, 2006 IEEE, pp , Jan [7] S. Fouladifard and H. Shafiee, A new technique for estimation and compensation of iq imbalance in ofdm receivers, vol. 1, nov. 2002, pp vol.1. [8] A. Tarighat and A. Sayed, Ofdm systems with both transmitter and receiver iq imbalances, june 2005, pp [9] S. De Rore, E. Lopez-Estraviz, F. Horlin, and L. Van der Perre, Joint estimation of carrier frequency offset and iq imbalance for 4g mobile wireless systems, vol. 5, june 2006, pp [10] A. Pascht, T. Bitzer, T. Bohn, and F. Endress, Estimation and compensation of iq-imbalances in direct down, nov. 2003, pp [11] M. Valkama, M. Renfors, and V. Koivunen, Advanced methods for i/q

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