Chapter 2 Overview of MIMO-OFDM Systems 7
3.4 Implementation Hints
3.4.3 Verification Platform
A verification platform was constructed to measure the performance of both the proposed method and the conventional method (two-repeat preamble-based method), as shown in Figure 3-14. The design was directly mapped onto the FPGA chips (Xilinx XtremeDSP, Virtex-II) with on-board 14-bit digital to analog converters (DACs) to transform the digital data into analog signals. Then the signals were transmitted by in-house RF front end. After down-converting RF signals to baseband at RX part, the analog signals are fed to 14-bit analog to digital converters (ADCs). The received data from ADCs was processed to calculate the CFO value using both the proposed method and the conventional method. In the meantime, the received data from ADCs was also applied to the chip testing, which was performed to verify the full functionality of the chip using Agilent 93000 SOC Test System. After the CFO estimation and chip testing processes, the results were collected to evaluate the performance of both the proposed method and the conventional method.
Figure 3-14. The photo of platform.
Figure 3-15. Estimated CFO of measurement vs. simulation.
-1.5 -1 -0.5 0 0.5 1 1.5 -1.5
-1 -0.5 0 0.5 1 1.5
I
Q
(a)
-1.5 -1 -0.5 0 0.5 1 1.5
-1.5 -1 -0.5 0 0.5 1 1.5
I
Q
(b)
Figure 3-16. Measurements of QPSK constellation: (a) P-CFO method. (b) Two-repeat preamble-based method.
The observed gain error and phase error are ~ 0.9 dB and ~10D, respectively.
Figure 3-15 shows both the measured and simulated CFO value. It indicates that the measured performance is a little worse than simulations since there are other spurious effects, noise, and nonlinearities in the front end. Although there is degradation in measured performance, the proposed method is still more accuracy. Figure 3-16 shows the baseband constellation for the QPSK modulation scheme. The error vector magnitude (EVM) for the proposed method is 7.4% (Figure 3-16(a)), while the EVM for the two-repeat preamble-based is 24.4% (Figure 3-16(b)). So the proposed solution is more robust than the conventional method under IQ-M conditions.
3.5 Summary
In this chapter, a novel algorithm is proposed to estimate the carrier frequency offset under the conditions of IQ-M in direct-conversion OFDM receivers. The proposed algorithm can adopt three training symbols to estimate the frequency offset from 50− ppm to 50+ ppm under a 2.4 GHz carrier frequency with 2 dB gain error and 20 degree phase error in multipath environments. Simulation results indicate that the average estimation error of the proposed P-CFO algorithm can fulfill many system requirements, preventing obvious performance loss under different IQ-M conditions.
The proposed design is implemented in an ASIC with 3.3 0.4 mm× 2 core area and 10 mW power dissipation at 54 Mbits/s data rate. Hence, the proposed algorithm can enhance the performance of wireless OFDM systems, enabling small low-cost systems to be achieved.
Chapter 4
Preamble-Assisted Estimation for I/Q Mismatch
In direct-conversion orthogonal frequency division multiplexing (OFDM) receivers, the impact of I/Q mismatch (IQ-M) with carrier frequency offset (CFO) must be considered. A preamble-assisted estimation is developed to circumvent IQ-M with CFO.
Both simulation and experiment results show that the proposed method could provide good estimation efficiency and enhance the system performance. Moreover, the proposed scheme is compatible with current wireless local area network (WLAN) standards.
OFDM is a spectrally efficient technique for high-speed wireless communications.
OFDM systems are, however, susceptible to imperfect synchronization and non-ideal front-end effects, which cause serious performance degradation. Generally, OFDM systems are highly sensitive to CFO, which is the frequency mismatch of local oscillators between the transmitter and the receiver. CFO introduces inter-carrier interference in OFDM systems due to the loss of orthogonality between sub-carriers.
Recent research has also focused on developing monolithic OFDM receivers, particularly for low-cost technology. Direct-conversion architecture is one potential candidate for simple integration among different architectures. However, direct-conversion receivers suffer from mismatch between the I and Q channels, such as IQ-M of non-ideal radio frequency (RF) circuits [24], [25], [52]. Specifically, IQ-M arises when the phase and gain differences between I and Q branches are not exactly 90 degrees and 0 dB, respectively. IQ-M introduces the image interference into the desired signal and then degrades the system performance.
Due to the impairment in the analog components, the low-pass filters (LPFs) of I and Q channels are not identical. The mismatched LPFs result in frequency-dependent IQ-M. Frequency-dependent IQ-M means that the imbalances can vary with frequency.
In an OFDM system with frequency-dependent IQ-M, the IQ-M parameters for every sub-carrier are different. Several schemes have been proposed to estimate constant IQ-M [63]-[66]. Although these methods can work well under constant IQ-M, they do not consider non-ideal LPFs and CFO phenomenon. Therefore, these methods are not robust to frequency-dependent IQ-M and CFO. Estimation for frequency-dependent IQ-M has also been studied in the open literature [48], [57]-[60], [62]. The effect and analysis of frequency-dependent IQ-M are given in [57]-[58]. Xing et al. [48] presented a method for frequency-dependent IQ-M. An optimal training sequence for frequency-dependent IQ-M has been proposed in [59]. However, these two methods have specific packet formats, making them incompatible with current WLAN standards, such as IEEE 802.11a/g [4]-[5]. Cetin et al. proposed an adaptive self-calibrating image rejection receiver [60]. This method needs a long converged time. Therefore it is not suitable for packet-based WLAN systems because the packet length is always limited.
In practice, frequency-dependent IQ-M and CFO arise simultaneously. However, only some references consider the frequency-dependent IQ-M with CFO. This study
mainly concentrates on the estimation of frequency-dependent IQ-M with CFO. To maintain and realize systems with imperfect front-end modules, a preamble-assisted estimation scheme is developed. Simulation results show that the performance loss is in the range of 1.6 to 1.8 dB at 10−4 bit error rate. Experiment results demonstrate that the proposed method could overcome joint impairments of frequency-dependent IQ-M and CFO, enabling a high performance receiver.
The remainder of the this chapter is organized as follows. Section 4.1 introduces the system model. Section 4.2 develops the constant IQ-M estimation. Section 4.3 then presents the frequency-dependent IQ-M estimation. Section 4.4 discusses the transmitter IQ-M. Conclusions are finally drawn in Section 4.5.
4.1 System Model
Figure 4-1 depicts the block diagram of a direct-conversion architecture. The received RF signal is down-converted to baseband, and then filtered out by the LPFs. Let r n ( ) and w n be the received signal and additive white Gaussian noise (AWGN), ( ) respectively. The baseband signal1 with CFO fΔ and IQ-M is given by [48], [57]
1 Equation (4.1) is based on discrete-time representation. Continuous time representation of (4.1) is expressed as
represent the frequency-dependent IQ-M parameters. Note that the phase rotation is inversed in the direction between original signal and the conjugate signal if the CFO is present. The signal r n can be further expressed as ( )( ) r n =h n( )⊗x n( ), where x n ( ) and ( )h n are the transmitted signal and channel impulse response, respectively. The symbol “⊗ ” represents the convolution operator. From (4.1), the received signal can be regarded as the original signal added to the conjugate signal. The constant IQ-M parameters are expressed as [54]-[56]
( )
where ε and ϕ denote the constant amplitude and phase mismatch, respectively. If neither gain nor phase error exists, α remains at unity, and β decreases to zero.
Figure 4-1. Direct-conversion receiver with I/Q mismatch and CFO.
(a)
(b)
Figure 4-2. Received signal with I/Q mismatch: (a) Amplitude. (b) Angle.
Moreover, the average filter response and response mismatch are defined as [48], [57]
( )
( )
( ) 0.5 ( ) ( ) ( ) 0.5 ( ) ( )
I Q
I Q
n l n l n
n l n l n
ψ ξ
= +
= − (4.3)
where l n and ( )I( ) l n denote the LPFs of I and Q channels. If the LPFs of I and Q Q channels are identical, the response mismatch can be ignored. In Figure 4-2, it is clear that the signal with IQ-M has amplitude variations and phase jumps compared with the case without IQ-M. This effect causes the demodulation error and further degrades the system performance.
4.2 Constant IQ-M Estimation
4.2.1 Constant IQ-M Estimation without CFO
Here we analysis the effect of constant IQ-M in OFDM systems and propose a scheme to estimate IQ-M. If there is no CFO, (4.1) can be simplified to
( ) ( ) ( ) ( )
y n =αr n +βr n∗ +w n (4.4)
After the fast Fourier transform (FFT), the frequency-domain data can be expressed as
{ }
where the subscript N denotes the FFT size. Equation (4.5) shows that IQ-M can cause the symbol at the sub-carrier k to be scaled by the complex factor α . Moreover, the complex conjugate of the symbol at sub-carrier –k multiplied by another complex factor β will be present. The desired sub-carrier k will include the unwanted interference related to the sub-carrier –k, implying that IQ-M can distort the accuracy of received signal. For instance, the channel estimation can be inaccuracy due to the interference caused by IQ-M. The effect of IQ-M on channel estimation can then be expressed as
ˆ k
where P denotes the preamble patternk 2. In order to simplify the notation, the noise term is ignored in the following derivation. If X is the training symbol, (4.6) is k rewritten as
From (4.7), it is obvious that IQ-M has large impact on the channel frequency response (CFR). If the influenced CFR is used to equalize the received data, the system
2 The preamble pattern adopted in this study is shown as follows
26 1,1 26 {1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1 -1 -1 1 1 -1 1 -1 1 1 1 1,
P− ∼− ∼ =
performance could be terrible especially for high order QAM constellation, e.g., 64-QAM. Thus, the challenge is to estimate precision IQ-M parameters and CFR at the same time.
The IQ-M is estimated by taking advantage of the relationship between desired sub-carriers and image sub-carriers. Figure 4-3 shows the estimated CFR. Note that only sub-carriers 26 to 1 and –1 to –26 of FFT output are shown in Figure 4-3. From Figure 4-3, there is a distinct difference between sub-carriers H and 25 H . The 24 difference between sub-carriers H and 24 H can be expressed as 25
Channel Frequency Response (RM S:50)
Index
Magnitude (|H|)
W ith IQ imbalance W ithout IQ imbalance
26 -26
Index
Figure 4-3. The estimated channel frequency response.
Because the coherency bandwidth of the channel is much larger than the inter-carrier spacing (channel bandwidth is 20 MHz, and the bandwidth of the inter-carrier spacing is 312.5 kHz), the variation of two consecutive channel carriers can be assumed to be small. It implies that H24 ≈H25. Because P−∗25 P25 = −P−∗24 P24 = , 1 α≈ and 1
Using the approximation in (4.9) leads to
( )
24 25 24 25
H −H = −β H−∗ +H−∗ (4.10)
The result for IQ-M parameters is
(
24 25) (
24 25)
After calculating IQ-M parameters and the CFR from the 1st long training symbol, we can use the information from the 2nd long training symbol to update IQ-M parameters.
The final IQ-M parameters are expressed as (see Appendix C for details)
1,24 24 1,24 25 2nd long training symbol, respectively. Equation (4.12) shows that IQ-M parameters depend on not only the current CFR but also the previous one. From Figure 4-3, there are 20 transitions in the long preamble if the IQ-M exists. The proposed method can get 20 estimates and then average them to get accuracy value.
The proposed method can estimate exact IQ-M parameters and CFR at the same time. Since IQ-M parameters are static over many symbols, the influenced signal can be corrected by the IQ-M compensation. In this method, the IQ-M is estimated in the frequency domain, but the compensation scheme is applied in the time domain. It means the influenced signal can be compensated in the time domain, i.e. before the FFT operation. Naturally, time-domain compensation is inherently better than frequency-domain compensation because time-domain compensation can avoid some decision errors, which are inevitable in the decision directed approach. With the mathematical derivation, the received data can be corrected by
2 2
The data flow of the proposed method is depicted in Figure 4-4.
Auto Frequency
Figure 4-4. Data flow of the constant IQ-M estimation and compensation.
4.2.2 Constant IQ-M Estimation with CFO
In this section, an IQ-M estimation with CFO is developed. Some WLAN standards have specified training symbols for channel parameter estimation [4]-[5]. The CFR can thus be estimated by training symbols. However, the CFR estimation could be inaccuracy if IQ-M is involved. Equation (4.5) shows that the desired sub-carrier is distorted by the image part under IQ-M conditions. In addition, if there is CFO and constant IQ-M, (4.1) can be simplified to
( )
2 2
( ) ( ) j fnTs ( ) j fnTs ( )
y n =αr n e πΔ +β r n e πΔ ∗+w n (4.14)
After the FFT operation, the frequency-domain data is expressed as
{ }
( )
where ε is the normalized frequency offset. From (4.16), Dk is the complex conjugate of D−k. The estimated CFR is thus given by transition on the CFR. IQ-M parameters can then be estimated by taking advantage of this property illustrated in Figure 4-3. As can be seen from Figure 4-3, there are several sharp transitions, and we take the average of IQ-M parameters, which are estimated from these transitions. In order to simplify the notation, the noise term and ICI are ignored in the following derivation. To explain the proposed method, two sub-carriers, H and 24 H , are utilize to complete the mathematical derivation. Thus, the following 25 equation holds
24 24 24 24 24 24
Subtracting H from 24 H , the following equation holds 25
( )
Substituting equation (4.21) into equation (4.19) leads into
( )
( )Before solving the equation for IQ-M parameters, the estimated CFO value based on P-CFO scheme is used to multiply the received signal with an inverse direction, i.e., CFO compensation. The compensated signal is written as
( )
2 4
( ) j fnTs ( ) ( ) j fnTs ( )
y n e− πΔ =αr n +β r n e πΔ ∗ +w n (4.23)
After the FFT operation, the frequency-domain data is expressed as
{
2} ( )
By the same technique described above, the following equation holds
24 24 24 24 24
The IQ-M parameters are thus given by
( )
Substituting equation (4.29) into equation (4.27) leads into
( )
( )The approximation in (4.30) is based on the assumption that the variation of two consecutive channel carriers can be assumed to be small. Equation (4.30) can thus be expressed in the general form
( )
To simplify the notation, let
( )
Equation (4.31) is rewritten as
2 0
In (4.33), all parameters are complex numbers, and the solution can be found by setting the real part and imaginary part to be equal to zero. Each complex number is denoted separately as x = (xR, xI) . Therefore, the following equation holds
(
βR2 +βI2)
nR −(
βRmR−βImI)
+pR = 0 (4.34)w = m n +m n . Substituting equation (4.36) into equation (4.34) leads into
2 2 of the quadratic equation can be obtained by the quadratic formula
2 4
The parameter βR can thus be calculated by the proposed method. After calculating βR, β can then be calculated by the same way. However, we cannot carry out the I calculation in the beginning since the IQ-M parameter α is included in (4.32).
According to the simulation, we can set the parameter α to be equal to unit at first.
After the first calculation, we use the result to calculate IQ-M parameters again, and then accurate IQ-M parameters could be found in enough iterations (five iterations).
With the information of estimated CFR and CFO value, we can calculate accuracy IQ-M parameters for data compensation. The received data can then be corrected by using (4.13). Figure 4-5 illustrates the overall estimation and compensation procedure.
1 ( 1)
Figure 4-5. (a) The block diagram for the IQ-M estimation. (b) The compensation blocks for IQ-M and CFO.
4.3 Frequency-Dependent IQ-M Estimation 4.3.1 The Proposed Method
The goal here is to estimate the frequency-dependent IQ-M with CFO. The basic strategy for extracting IQ-M parameters is to employ long training symbols. The packet format used in this method is based on IEEE 802.11a/g standards. Let ( )x n denote a long training symbol distorted by IQ-M and CFO, as defined in (4.1). The long training symbol after fast Fourier transform (FFT) is given by
{ }
where D is the distortion, which is accompanied with inter-carrier interference (ICI) k due to CFO [28]. X is the data sub-carrier and k H stands for the channel frequency k
Equation (4.40) clearly indicates that IQ-M results in a mutual interference between symmetric sub-carriers in OFDM systems, as shown in Figure 4-6. Let x n and ( )
( l)
x n +N represent two consecutive long training symbols, where N is the samples l of a long training symbol. In order to simplify the notation, the noise term is ignored in the following derivation.
Index
...
...
Channel Frequency Response
X-k
Xk
Figure 4-6. Mutual interference due to I/Q mismatch.
In practice, IQ-M not only introduces unwanted image interference into the desired signal, but also restricts the accuracy of CFO estimation. A basic strategy for computing the CFO is to employ two repeated training symbols [28]. However, the estimation error increases when the gain or phase error is not zero. In [61], a pseudo-CFO (P-CFO) algorithm, which rotates three training symbols by adding extra frequency offset into the received sequence, is developed to improve CFO estimation.
Since the P-CFO method can estimate the frequency offset more accurately than the two-repeat preamble-based scheme, it is adopted to estimate the CFO value in this
work. The training symbols are then rotated by the estimated CFO. Because the long training symbols are periodic, the received signals, ( )r n and (r n +Nl), can be replaced by r n . Therefore, the following equations holds: ( )
(2 ) (2 ( ) )
The IQ-M parameters can be obtained as follows.
1
Equations (4.41) and (4.42) indicate that the two training symbols are multiplied by the estimated CFO and then are subtracted to eliminate the image interference. Once the IQ-M parameters are estimated, the effect of IQ-M could be corrected by the frequency-domain compensation. With the estimated IQ-M parameters, the frequency domain compensation is given by
{
(2 )} {
(2 )}
The compensation gain could be balanced by the channel equalizer. Moreover, the
proposed method could be directly applied to IEEE 802.11a/g standards because it does not require any special packet format.
In the open literature [58], [59], [62], most of them consider the IQ-M only.
Therefore, these methods may not be suitable for joint CFO and IQ-M. In other words, these methods can tolerate slight CFO value. By contrast, the proposed method is capable of tolerating large CFO value. In the condition of slight CFO value, the inter-carrier interference is smaller than large CFO value. For instance, if the estimated CFO value is within ± 2 ppm, the existing methods [58], [59], [62] for frequency-dependent IQ-M could be adopted. In the frequency domain, phase tracking is still performed. Since some rotation operations (CFO estimation and compensation) are eliminated, this approach can maintain the system performance with reason-able complexity.
Figure 4-7. The proposed frequency-dependent IQ-M estimation architecture.
Figure 4-7 shows the frequency-dependent IQ-M estimation architecture. The estimator begins to work when long training symbols (preambles) arrive. Firstly, the long preambles are multiplied by the estimated CFO. In order to reduce the image interference due to the IQ-M effect, the 1st training symbol should subtract the 2nd training symbol. At the same time, the conjugate preamble follows the same procedure
described above. After calculating the difference between the first and the second preambles, a complex divider is used to eliminate the phase rotation. Finally, these estimated parameters are applied to compensate for the frequency-dependent IQ-M in the frequency domain.
4.3.2 Simulation and Experiment Results
A typical OFDM system for wireless LAN was adapted as the referred specification to evaluate the performance [4]-[5]. The simulation parameters were OFDM symbol length 64, and cyclic prefix 16. There were ten short training symbols and two long training symbols. The number of taps of frequency-selective fading was of order 8, simulating as an independent and identically distributed (i.i.d.) complex Gaussian random variable.
All data are modulated by 16-QAM and 64-QAM. The CFO was set at 50 ppm, and the constant IQ-M parameters were 1 dB gain error and 10 degree phase error. Two-tap FIR filters were adopted to model the frequency-dependent IQ-M [48]. These two LPFs were modeled as
( ) 0.1 ( ) ( 1) and packet error rate (PER) performance. In these two figures, the following scenarios are considered. “Multipath only” legend refers to the system with perfect RF front-end and “without comp” means that the estimation scheme is not applied to the system
with frequency-dependent IQ-M. “FD IQ-M” legend means that the estimation scheme is applied to the system with frequency-dependent IQ-M and “constant IQ-M” means that the estimation scheme is applied to the system with constant IQ-M. Figures 4-8 and 4-9 show that the degradation in BER and PER owing to IQ-M was significant, particularly for 64-QAM modulation. After compensation, the performance under the condition of constant IQ-M was close to the case with multipath only. The SNR loss of
with frequency-dependent IQ-M. “FD IQ-M” legend means that the estimation scheme is applied to the system with frequency-dependent IQ-M and “constant IQ-M” means that the estimation scheme is applied to the system with constant IQ-M. Figures 4-8 and 4-9 show that the degradation in BER and PER owing to IQ-M was significant, particularly for 64-QAM modulation. After compensation, the performance under the condition of constant IQ-M was close to the case with multipath only. The SNR loss of