Chapter 2 OFDM System and Channel Estimation
2.3 Conventional Time-Invariant Channel Estimator
2.3.3 Frequency Domain Interpolation Methods
As discussed, channel responses in the frequency domain can be obtained in pilot tone positions. To obtain response in data tone positions, we need to use the interpolation technique. Note that the interpolation is highly system dependent;
different pilot allocation methods result in different interpolation strategies. In this section, we will introduce dome interpolation techniques and explain how to deal with different systems.
The interpolation problem involves two-dimensional operations, i.e., operations in the frequency temporal domains. The dimension in frequency domain is defined by the subcarrier index, and the temporal domain is the symbol time index. Note that the channel delay spread determines how fast the channel response varies in the frequency domain. It is difficult to obtain a good interpolator if the channel response variation is large. The classical approach for channel interpolation is to construct a polynomial interpolator fitting responses in known samples. The polynomial interpolator can be formulated in various ways, such as the power series, Lagrange interpolation and Newton interpolation. These various forms are mathematically equivalent and can be transformed from one to another. Assume that samples are available, denoted as
known samples, can be written in a power series form as
tn N N+1
Figure 2-11 A polynomial interpolator
Note that the polynomial is unique (see in Figure 2-11). Using Equation (2.33) and a set of known samples, we can solve the polynomial coefficients by
linear equations which can be formulated as a matrix form:
1
The matrix in Equation (2.34) is called a Vandermonde matrix. The dimension of the matrix becomes large when N is large; its inverse is then difficult to compute. For this reason, the polynomial order used is usually small (e.g., linear or quadratic interpolation).
For signals in pilot tones, we can rewrite Equation (2.34) as:
x Ta= (2.35)
pilot signals, is the polynomial coefficient, and is the order of polynomial interpolator. Note that in Equation (2.36) is no longer a Vandermonde matrix. Also, it becomes overdetermined since
0 1 1 The simplest polynomial interpolator is the first-order polynomial interpolator.
However, the performance is usually not satisfied. The quadratic interpolator is a second-order polynomial interpolator. It can have better performance in general. Since the number of unknowns is larger, more pilots are also required. Consider the cluster structure in the IEEE 802.16e system in Figure 2-9. As we can see, only 2 pilot signals are available in a single OFDM symbol. In order to have a quadratic estimate, at least we have to use two clusters at a time such that 4 pilot signals can be used.
Using Equation (2.37), we can then obtain the polynomial coefficients with the LS algorithm. Finally, all frequency response across two clusters in a single OFDM symbol can be evaluated with the fitted quadratic function. The estimation scheme is shown in Figure 2-12.
Figure 2-12 Channel estimation with 2-cluster quadratic interpolator
The method mentioned above is a one-dimensional interpolation approach, which only uses one OFDM symbol. Although a higher order may have better results,
reduce the effectiveness of the interpolation. Another way to enhance the interpolation performance is to use a two-dimension interpolator.
In [12], a simple method has been proposed to perform a two-dimension interpolation in the IEEE802.16e system. The two-dimension interpolator is separated into two one-dimension polynomial interpolators; one is linear and the other is quadratic. Since the frequency response varies slowly with time, the interpolation between OFDM symbols is linear. This method is called a Linear and Quadratic (LAQ) method.
The basic idea of the method is to use pilot signals from other clusters to aid the interpolation in a particular cluster. From Figure 2-9, we can see that the pilot signals in consecutive clusters (along with the symbol time index) are only decimated by a factor of two. Thus, we use a linear interpolation, averaging two consecutive pilot signals, to obtain the one in between. By doing so, 2 extra (pseudo) pilot signals are obtained in a cluster. Figure 2-13 shows the operation described above. Now, there are 4 pilot signals in a cluster. The matrix-vector form in Equation (2.36) can be reformulated as
where is channel response of the vector of 4 pilot signals, are location index of the 4 pilot signals (1,5,9,13, respectively), and is the polynomial coefficient vector. Using Equation (2.37), we can then obtain the polynomial coefficients by the LS method.
x { , , , }t t t t0 1 2 3
a
Figure 2-13 Channel estimation with LAQ interpolator
Chapter 3
Proposed Channel Estimators
In Chapter 2, we have reviewed some channel estimation methods for the pilot-aided OFDM systems. Specifically, we have described the LAQ frequency-domain channel estimator developed for the IEEE 802.16e system.
Although the LAQ interpolation method can have good performance, the results in the boundary areas of a cluster may not be satisfactory. This is because channel responses interpolated in those areas only use one-sided data. In this chapter, we will first propose a modified LAQ interpolation to solve the problem. The details are discussed in Section 3.1.
Due to limited pilot signals, channel estimation in frequency domain interpolation will be degraded when channel delay spread is large. In this case, the interpolation method cannot recover the frequency response completely, even with a high order of polynomial. For this reason, we focus the channel estimation in time domain approach. However, the conventional MMSE and LS channel estimator in time domain require high computational complexity. In addition, the MMSE channel estimator needs to know the channel statistical characteristics, which are usually not available. In this Chapter, we will develop a high-performance yet low-complexity LS channel estimator. This method works well with large channel delay spread and high
mobile speed. In typical wireless channels, the delay spread may be large, but the number of non-zero taps is small. The key idea of the proposed method is locate and estimate the responses at non-zero positions. Detailed description can be found in Section 3.2 and Section 3.3. Combining both temporal and frequency domain operations, we finally propose an efficient channel estimator. A recursive procedure is also introduced to further reduce the complexity and improve performance. The details are discussed in Section 3.4.
In high-speed mobile communications, the coherent time of the channel can be less than an OFDM symbol duration. In this case, the channel in the OFDM symbol is not time-invariant anymore, and ICI effect is introduced. Many ICI suppression methods have been proposed recently [13]-[15]. All the methods rely on good channel estimates. For the last section of this chapter, we extend the method developed in Section 3.4 and propose a low-complexity time-variant channel estimator. The details are discussed in Section 3.5.
3.1 Sliding-Windowed LAQ Channel Estimator
As mentioned in Section 2.3.3, the LAQ interpolation method (as shown in Figure 2-13) has been developed for the IEEE 802.16e system. In the determination of the order of the polynomial, two aspects must be considered. First, the number of pilot signals must be larger than the order of the polynomial to make sure the matrix is over-determined (see Equation 2.37). Second, the order of the polynomial must be high enough such that it can properly describe the frequency response variation, which depends on the maximum channel delay spread. The LAQ interpolation method chooses the quadratic function to conduct channel estimation, compromising these two factors.
T
Although the LAQ interpolation method can have good performance, the results in the cluster boundaries may not be satisfactory. This is because channel responses interpolated in the areas only use one-sided data. As a result, the boundary areas will be discontinuous as shown in Figure 3-1. In the figure, the estimation result with the LAQ method is denoted by the solid curve, the discontinuousness in the boundary areas is indicated by the dashed box.
Figure 3-1 Discontinuousness in boundary areas
Thus, the LS fitting in Equation (2.37) may not be optimal for tones in both cluster ends. For this reason, we propose an extended method, called the sliding window method, to remedy this problem. The method is depicted in Figure 3-2, and the corresponding operations are summarized in the following steps:
1) Select a window size (the dashed box in Figure 3-2).
2) Use the channel responses of pilot and pseudo pilot signals in the window, and construct the matrix in Equation (2.36). Then, use the LS algorithm to obtain the polynomial coefficient vector . a
3) Calculate the frequency response in the output region (the colored part in Figure 3-2) by the LAQ interpolation method, and output the estimates in the region.
4) Slide the window along the subcarrier index by tones and then go to 2) until the subcarrier index reach its maximum.
This extended interpolation method can solve the problem mentioned above and the performance can be further enhanced, though its computational complexity is somewhat higher than conventional LAQ interpolation method. The window length, the output region, and the step of sliding are parameters needed to be determined. The Simulation results are shown in Chapter 4.
3.2 Low-Complexity Time Domain Channel Estimator
We have described the conventional time domain LS channel estimator in Section 2.3.1. Figure 3-3 illustrates the equation for OFDM receive signal in frequency domain. With the pilot-aided system, we take the signals on pilot locations (the grey area in Figure 3-3) to fit the LS channel estimate. The estimated time domain channel impulse response is shown in Equation (2.23). Since the solution involves a matrix inversion, we require Ο
( )
L3 arithmetic operations where L is the maximum channel delay spread. In OFDM systems, is usually the CP size which is large in general. As a result, the required computational complexity is large.We now take the advantage of a property in typical wireless channels to develop low-complexity algorithms. In outdoor wireless environments, the number of major multipath reflections is usually limited. In other words, the number of non-zero channel tap is small, though the delay spread may be large. Let the number of these particular taps be
L
L%, which is always much less than L. The number of unknowns to be solved is L% instead of L. (the dotted area in Figure 3-3). Then the channel impulse response on particular position can be estimated by Equation (2.23) and only requires Ο %
( )
L3 arithmetic operations [16].The only problem is we need to locate the channel tap positions which have significant values. Using this approach, we can estimate channel more precisely, and reduce the computational complexity significantly. How to locate the channel tap positions will be introduced in next section.
Figure 3-3 Equation for OFDM receive signal in frequency domain
3.3 Channel Tap Search Algorithm
In this section, we propose two methods to locate significant (nonzero) channel taps. The idea is to construct a preliminary time-domain channel impulse response first, and then locate the significant taps by some searching algorithm. Finally, we can estimate the response at those taps by the LS method. The details are described below.
3.3.1 Preliminary Channel Estimation with Preamble
The preamble is the first symbol of the downlink OFDM transmission. A simple way to construct the channel impulse response is using the preamble, which is the known sequence. Consider the preamble passing through a multipath channel, which is shown in Figure 3-4(a). Because of the multipath effect, the received signal will be a superposition of signals from different delay paths and AWGN. We can then conduct the correlation operation between the received signal and the known preamble (as shown in Figure 3-4(b)). When the preamble matches the original signal, the peaks can then be located. Since the channel responses are complex, we will use their absolute values in the peak detection.
Figure 3-4 Preliminary channel estimation (with preamble)
The correlation operation can be made equivalent to the convolution operation.
So, the procedure can be transformed to frequency domain.
h⊗ ⊗ =f f IFFT H F F{ ⋅ ⋅ } (3.1) where and are the channel response respectively in the time and frequency domains, and
h H
f and F are the preamble sequence respectively to in the time and frequency domains. We can then multiply the received signal and the preamble in the frequency and transfer the result back to the time domain to replace the time-domain correlation operation. However, not all communication systems have the preamble structure. So, we will introduce another method to search channel taps.
3.3.2 Preliminary Channel Estimation with Pilots
The time-domain channel impulse response can be constructed if its frequency-domain response can be estimated. In pilot-tone systems, the channel frequency response can be estimated and interpolated as discussed in Section 3.1.
Since we only need to know the rough channel impulse response shape, we can take the IFFT of the interpolated channel response (see in 2.3.3) to obtain the time-domain response. Based on the result, we can then locate peaks of channel impulse response.
3.3.3 Channel Tap Search Method
Here, we propose two simple methods to locate the positions of significant channel taps. Figure 3-5 shows the result of the preliminary channel estimation using the preamble of the IEEE802.16e. We can clearly see the peaks of the channel response. Furthermore, the channel tap is complex value so we only see the magnitude part.
-5 0 5 10 15 20 25 30 35 40 45
0 0.1 0.2 0.3 0.4 0.5 0.6
Figure 3-5 Result of the preliminary channel estimation
The first method is simply to compare the value of a tap with a threshold. If it is larger than the threshold, the tap is deemed as a peak (significant tap). Otherwise, it will be considered as a zero tap (Figure 3-6). As we can see, there is a low-passed signal embedded in the channel response. So some fake taps will occur near the significant taps (the gray line in Figure 3-6). For this reason, the threshold will be difficult to determine and the smaller taps may not be detected.
Figure 3-6 Channel tap searching method (threshold dependent)
The second method takes a first-order differentiation to the channel response. As a result, the low-passed signal will be removed. We can then compare the result with a threshold. The tap with a value higher than the threshold is deemed as a significant tap.
Note that, a significant tap will result two peaks; one is positive and the other is negative (Figure 3-7). We need only consider the positive one. The definition of the differentiation operation is given by
d k[ ]=h k%[ + −1] h k%[ ] (3.2)
Figure 3-7 Channel tap searching method by first-order differentiation
However, consecutive channel taps may not be detected. For this reason, we further propose a more robust method for the detection. The idea is to locate the taps in an iteratively manner, rather than in one short. Figure 3-8 shows the iterative procedure. First, locate channel taps with Method 1 or 2 with a higher threshold value.
In this case, smaller or consecutive taps may not be detected. Then, remove the located channel taps from the channel response (as shown in Figure 3-8(a)), and relocate taps (the threshold may be changed). Repeat this process until no more taps are detected. With proper thresholds, this iterative method we can almost locate all channel taps correctly.
Figure 3-8 Iterative channel tap searching method
3.4 Joint Time and Freq Domain Channel Estimator
In Section 3.2, we propose a modified time-domain LS channel estimator for channels with known tap positions. In Section 3.3, we proposed two channel tap search algorithms; one uses the preamble and the other uses pilots. In IEEE802.16e system, the channel response estimated with the preamble may not be able to be applied in the following data bursts since the receiver is mobile. For this reason, we only consider the preliminary channel estimation using pilot signals. Since the Method 2 for the tap search performs better, we use that in following development.
Combining the preliminary channel estimation, the channel tap search algorithm, and the modified time-domain LS channel estimator, we are able to obtain a low-complexity yet high-performance channel estimator. We called this a joint time and frequency domain channel estimator. The estimation flowchart is shown in Figure 3-9. The procedure is summarized as follows:
Figure 3-9 Proposed channel estimation flowchart
1) Use the conventional frequency domain interpolation method to obtain the preliminary channel frequency response.
2) Take the IFFT of the response to the time domain, and then locate significant channel taps (the threshold is iteration dependent).
3) Use the LS algorithm to estimate channel impulse response in those taps.
4) Compute the least-squared error (LSE) with a threshold. If the LSE is greater than the threshold, reconstruct the frequency response of those located taps by FFT, and force the response in the guardband region to zero. Then, subtract it from the original channel frequency response, and go to step (2). If LSE is smaller than the threshold, stop the iteration.
Note that the LSE is a good indicator telling us when to stop the searching. In other words, it can avoid the redundant channel taps to be detected and reduce the computational complexity for the LS algorithm. In mobile environments, the channel tap positions may change with time slowly or suddenly. The LSE can also help us to
check channel tap positions have changed or not.
The iteratively operation not only locates the channel taps more precisely, but also requires reduces computational complexity of the LS algorithm. However, a FFT/IFFT operation is required for each iteration and this will increase the computational complexity significantly. This can be remedied with the following approach. The main idea is to transfer the response-subtraction operation in the frequency domain to the time domain. Note that the operation conducted in the frequency domain is windowing and subtraction, which can be transferred into convolution and subtraction in the time domain. The function to be convolved is the sinc filter. In practice, the sinc filter may be difficult to implement. So, we may replace it by some lowpass filter. Since the number of detected taps is expected to be small, the required computational complexity of the convolution operation will not be significant. The modified flowchart is shown in Figure 3-10.
3.5 Time-Variant Channel Estimation
The maximum Doppler spread of a mobile station is shown to be
C light. The coherent time is known to be the inverse of Doppler spread (
c
tC 1
C D
t ≅ f ).
If the mobile speed is high, the coherent time becomes small. When the coherent time is less than an OFDM symbol duration, the channel in the OFDM symbol is no longer time-invariant. As a result, intercarrier-interference (ICI) effect will be induced, degrading the system performance. There is a considerably amount of research in ICI mitigation. It has been known that the performance of ICI mitigation heavily depends on the accuracy of channel estimation. In this section, we focus on the estimation of time-variant channel. We will extend method proposed in previous sections to obtain a low-complexity and high-performance algorithm.
In [15], a linear time-variant channel model is used to develop a simple channel estimation method. The received signal with ICI effect is modeled as
(3.4)
where is the received signal in the frequency domain, is the transmitted signal in the frequency domain, denotes the FFT of the AWGN, and is the FFT size. The middle term in the right hand side of (3.4) represents the ICI term. It has been shown in [15] that can be seen as the frequency response of an equivalent channel, averaging the channel taps over the time duration of
. Let be the averaged time-domain channel tap. Thus,
where L is the length of channel. Note that ˆ ,0 can be estimated with .
Hi Y Xi / i
Using the model, we can treat the channel as a time-invariant channel with time domain taps ’s (Its frequency response is ). Thus, the channel estimation
Using the model, we can treat the channel as a time-invariant channel with time domain taps ’s (Its frequency response is ). Thus, the channel estimation