Chapter 3 Proposed Maximum Doppler Shift Estimation Method
3.5 Summary
In this chapter, a maximum Doppler shift estimation method using Doppler power spectrum is proposed. It is shown that the proposed method works well over wide Doppler shift at low SNR condition (< 10 dB). This method provides a tradeoff between computational complexity and estimation accuracy by changing the size of observation interval and accumulation. Furthermore, two schemes of Fourier transform (FFT scheme and CTA scheme) are proposed to provide a tradeoff between computational complexity and the memory size. The FFT scheme has lower computational complexity than the CTA scheme, but it needs larger memory size than the CTA scheme.
Chapter 4
ICI Cancellation in OFDM Systems over Time-Varying Channels
In this chapter, we propose an adaptive ICI cancellation scheme based on [5]. The proposed method provides a tradeoff between the performance of output signals and processing time. Through the ICI analysis, the relation between ICI and the maximum Doppler shift will be introduced. Then, from Chapter 3, we can utilize the estimated maximum Doppler shift as a parameter of the adaptive ICI cancellation scheme.
First, Section 4.1 presents the ICI analysis. In Section 4.2, the proposed adaptive ICI cancellation scheme is described. The computer simulations and the summary are shown in Section 4.3 and Section 4.4, respectively.
4.1 ICI Analysis
Signals transmitted in a mobile environment are often impaired by the delay spread and Doppler shift. In order to keep a near-constant channel in each subcarrier of OFDM systems, the length of an OFDM symbol should be increase as the delay spread increases. Since the delay spread increased, the length of the guard interval should also be expanded to prevent the ISI. However, because of the symbol duration increased, the
OFDM system became sensitive to time-varying channels. The orthogonality between subcarriers is lost in time-varying channels, resulting in intercarrier interference (ICI), which stains the optimal data detection, and degrades the bit-error-rate (BER) performance in OFDM systems.
From Equation (3.9), the k-th subcarrier signal after the FFT at the receiver side can be rewritten as
, 0 1
k k k k k
Y =H X +I +W ≤ ≤k N − (4.1)
In Equation (4.1), the ICI term, Ik, is caused by the time-varying channel. In a time-invariant channel, the Ik is zero and E H{| k | }2 = . In a slow time-varying 1 channel (i.e., the normalized maximum Doppler shift is small), the ICI power can be assumed as E I{| k | }2 ≈ . Here, the normalized maximum Doppler shift, f0 n, is denoted the maximum Doppler shift normalized to the OFDM symbol duration (N T⋅ s).
If fn is small, the ICI power can be neglected compared to the background noise power [24]. On the other hand, when the normalized maximum Doppler shift is high, the ICI effects are too critical to be ignored. In addition, the power of the desired signal spreads out to neighboring subcarriers, because of E H{| k | }2 < . 1
4.1.1 ICI in OFDM Systems
Here, an OFDM system model over time-varying channels is considered. Form Equation (3.3), the channel is modeled by the time-varying discrete impulse response,
( , )
h n l , which is defined as the response of time n to an impulse response applied at l-th tap. After transmitting through the channel, the received sample sequences which are the convolution results between transmitted symbols and time-varying channels.
Therefore, Equation (3.5) can be expressed in a matrix form:
In Equation (4.2), h represents the channel response of the l-th tap at the time n l, instance n. Then, the above equation can be rewritten as
TD ν
= ⋅ +
yK H xK K
, (4.3) where HTD is the time-varying channel matrix. From Equation (4.3), the received signal
matrix can be derived as
TD ν In the above equation, F is the N-dimensional DFT matrix, and it can be written as
( )
Here, the frequency domain channel matrix, HFD, is defined as
FD = ⋅ TD⋅ H
H F H F , (4.6)
For the ISI free case, the delay spread of the channel is shorter than the length of CP. After removing the ISI corrupted guard interval, the received sequence computes an N-point DFT operation to obtain the frequency domain signal. Therefore, Equation (4.4)
can be rewritten as
( TD ν)
Denote HFD as an “equivalent frequency domain channel matrix.” In time-varying environments, non-diagonal elements of HFD are not zero, and these elements induce the ICI. Let HFD(d k is the d-th row, k-th column of H, ) FD. Defined
4.1.2 Analysis of ICI Power
In time-invariant channels, the matrix of HFD in Equation (4.8) is a diagonal matrix, thus the received signals can be compensated with a simple one-tap equalizer per subcarrier. But in time-varying channels, the matrix of HFD is no longer a diagonal matrix, and if one-tap equalizers are still used, the signals will be probably detected
into wrong decisions. Therefore, the non-diagonal terms in the matrix HFD (ICI terms) should be mitigated to obtain a set of more reliable decisions.
The ICI power based on the theorem in [5], [20], [25] will be analyzed here.
Assuming a typical WSSUS channel model as below
( ) ( )
{
n l, *n q l m,}
hh( s) l2 ( )E h ⋅h − − =φ q T⋅ σ δ m , (4.10)
where φhh(q T⋅ s)=J0(2πf q Td ⋅ ⋅ s) denotes the normalized tap autocorrelation, Ts is sampling time, and σ denotes the variance of the l-th tap. Since the non-diagonal l2 terms of HFD are the ICI terms, we will compute the power of these ICI terms. Denote u n( ) an N-point rectangular window
Further define the (2N-1)-point triangular window
( ) else.
Equation (4.12) can be rewritten as
Figure 4.1 shows the computer simulations according to Equation (3.14) and Equation (4.14). From Figure 4.1, we can find the coefficients of the frequency domain channel matrix have the most power on the central band and the edges of the channel matrix.
When the maximum Doppler shift increases, more signal power leaks out to the neighboring subcarriers. Therefore, by the relations between the ICI power and the maximum Doppler shift, we can know how many neighboring subcarriers should be taken account to the ICI eliminating procedure.
-100 -50 0 50 100
Figure 4.1 (a): ICI variance versus d (d is the index of super- or sub- diagonal)
-15 -10 -5 0 5 10 15
Figure 4.1 (b): Zoomed view of Figure 4.1 (a)
4.2 Adaptive ICI Cancellation Scheme
In this session, we will introduce an adaptive ICI cancellation scheme based on [5], which provides a low-complexity linear minimum mean square error (LMMSE) estimator. According to the analysis in Section 4.1, we can know that the main ICI comes form the several neighboring subcarriers. In order to reduce the computational complexity and memory size requirement, we can only use some elements of matrix HFD. Therefore, to squeezes the significant coefficient of HFD (Equation 4.7) into the 2D +1 central diagonals, a D× lower triangular matrix in the bottom-left D corner, and a D× upper triangular matrix in the top-right corner, illustrated in D Figure 4.2. It shows that the desired structure of HFD which passes the shaded region and zeros the non-shaded region. When the maximum Doppler shift is given, the value of D can be determined. The parameter D can be chosen to tradeoff between the performance and the complexity. The choice of parameter D will be described later in Section 4.3.
D+1
D+1
D+1 D+1
D
D
D
D
0
0
Figure 4.2: Desired structure of frequency domain channel matrix
4.2.1 Low-Complexity MMSE ICI Cancellation
With linear time-invariant channels, HFD is diagonal, and the LMMSE equalization scheme can be implemented in O N operations. This simple frequency ( ) domain equalization is the conventional motivation for the use of OFDM. However, with time-invariant channels, HFD is not diagonal any more, the LMMSE estimator requires matrix inversion. In classical method, the LMMSE estimator [11], [12] is given by
(
FD FD)
FD1
H N0 − H
= ⋅ + ⋅
XJK H H I H YJK
(4.15)
The implementation of the MMSE estimator requires O N
( )
3 operations, resulting in impractical implementation for large N. The major computation in Equation (4.15) is in calculation the inverse. In order to reduce the computational complexity, a low-complexity MMSE equalizer [5] is used here.The low-complexity LMMSE estimator is applied to reduce the ICI and noise.
Using parameter D to determine the desired structure of frequency domain channel matrix, we can obtain
lFD
M FD
= ⋅
H C H (4.16)
where CM denotes a mask matrix that pass the shaded region and zeros the non-shaded region in Figure 4.2. Let U = ⋅2 D+1, and a U × vector with i-th entry is defined 1
. The MMSE estimator for detecting based on Equation (4.17) is bu = R pu−1 u , where Ru can be recursively calculated, and the computational complexity can be greatly reduced. The total computational complexity of this low-complexity method is only
(
2)
O N U operations [5]. On the other hand, using an MMSE receiver based on the whole frequency domain channel matrix, as suggested in [24], requires O N
( )
3operations. Since U N, the computational complexity decreases greatly.
4.3 Computer Simulations
In this section, the simulation results of the adaptive ICI cancellation scheme will be presented. The simulation parameters are listed in Table 4.1. The channel model is the Jakes model. The normalized maximum Doppler shift fn is changed between 0 and 0.15. The configuration we consider here is an OFDM system with a bandwidth of 6 MHz and 2K mode of DVB-T systems. The QPSK modulation scheme is adopted in the below simulations. Through out simulations, the timing synchronization and channel state information (CSI) are assumed to be known. Here, we still used the proposed maximum Doppler shift estimation method to estimated Doppler shift.
Moreover, The CTA method is used and the observation interval K is 128 symbols with Q = 5. The maximum frequency difference Δfmax is 50 Hz.
Table 4.1: Parameters of computer simulations
System DVB-T
Transmission mode 2K
Channel bandwidth 6 MHz
CP length 1/4 OFDM symbol
Modulation QPSK
Carrier frequency 710 MHz
Normalized maximum Doppler shift 0 ~ 0.15
Channel model Jakes model
Figure 4.3 to Figure 4.7 compare the BER performance of the low-complexity MMSE and the conventional one-tap equalizer for different normalized Doppler shift.
The performance of the known CSI is also shown in these figures. If the parameter D of the low-complexity MMSE increases, the BER performance improves correspondingly.
In Figure 4.3, when the normalized maximum Doppler shift is smaller than 0.01, the MMSE and the one-tap equalizer both have the similar BER performance. When the Doppler shift increased, more signal power leaks out to the neighboring subcarriers.
Therefore, the parameter D must be appropriately chosen to maintain the BER performance under a desired level.
0 5 10 15 20 25 30
10-4 10-3 10-2 10-1 100
SNR(dB)
BER
One tap MMSE, D = 1 MMSE, D = 2 MMSE, D = 3 Known CSI
fn = 0.01
Figure 4.3: BER performance versus SNR. The normalized maximum Doppler shift is 0.01.
0 5 10 15 20 25 30
Figure 4.4: BER performance versus SNR. The normalized maximum Doppler shift is 0.05.
Figure 4.5: BER performance versus SNR. The normalized maximum Doppler shift is 0.1.
0 5 10 15 20 25 30
Figure 4.6: BER performance versus SNR. The normalized maximum Doppler shift is 0.15.
An adaptive ICI cancellation scheme can be achieved, because the information of the maximum Doppler shift is known by the estimation. A threshold scheme is proposed to provide a tradeoff between the performance and the computational complexity. For example, we choose that the BER performance is smaller than 10-3 in SNR = 25 dB. Therefore, the threshold of the adaptive ICI cancellation scheme can be determined by using the above simulation results. Table 4.2 lists the thresholds and their corresponding equalization type. When the normalized maximum Doppler shift is equal to or smaller than 0.01, the one-tap equalization can sufficiently satisfy the requirement. But, if the normalized maximum Doppler shift is larger than 0.01, the MMSE equalization with appropriate parameter D is needed such that the BER performance can be hold.
Table 4.2: Adaptive ICI cancellation scheme
Normalized max. Doppler shift Equalizer
n 0.01
f ≤ One-tap equalizer
0.01<fn ≤0.05 MMSE with D = 1
0.05< fn ≤0.1 MMSE with D = 2
0.1<fn MMSE with D = 3
Figure 4.7 shows the BER performance of different equalization type with variation of normalized maximum Doppler shift, which shows in Figure 4.8. As the result in Figure 4.7, the BER performance of the adaptive ICI cancellation scheme is better than the BER performance of the MMSE with D = 2. Although the BER performance of the MMSE with D = 3 is better than other equalization types, the computational complexity of the MMSE with D = 3 is larger than others. The maximum average computational complexity per OFDM symbol of the MMSE with D
= 3 and the MMSE with D = 2 is respectively O N and
(
7 2)
O N(
5 2)
operations; the maximum average computational complexity per OFDM symbol of the adaptive method is about O(
4.71⋅N2)
operations, as shown in Figure 4.9. As can be seen, the MMSE with D = 3 and the MMSE with D = 2 waste the computational complexity and processing time in low Doppler shift (fn < 0.05), rather than improve the BER performance appreciably. The adaptive ICI cancellation scheme can use the appropriate computational complexity, and satisfies the performance requirement.0 5 10 15 20 25 30
Figure 4.7: BER performance of different equalization type with variation of normalized maximum Doppler shift.
0 10 20 30 40 50 60 70 80 90 Normalized maximum Doppler shift (fn) Estimated
Actual K = 128, Q = 5
Figure 4.8: Variation of normalized maximum Doppler shift for Figure 4.7.
0 20 40 60 80 O(3*N^2)
O(5*N^2) O(7*N^2)
Time(Sec.)
Average computational complexity (per OFDM symbol)
MMSE, D = 3 MMSE, D = 2 Adaptive
Figure 4.9: Average computational complexity of MMSE with D = 3, MMSE with D = 2, and the adaptive ICI cancellation scheme
4.4 Summary
In this chapter, the ICI analysis is introduced and shows the relation between ICI and the maximum Doppler shift. By this relation and simulation results, we can know that how many neighboring subcarriers should be taken account to the ICI eliminating procedure. Therefore, an adaptive ICI cancellation scheme be proposed, and this method can use the appropriate computational complexity to mitigate the ICI effects.
Chapter 5 Conclusion
In this thesis, we proposed a maximum Doppler shift estimation method for mobile OFDM systems. This proposed method can work well over a wide Doppler shift range at low SNR. Moreover, it can provide more flexibility between the estimation accuracy and computational complexity by changing the observation interval and accumulation size. In Chapter 2, the technique of OFDM systems and DVB systems were introduced. In Chapter 3, the proposed maximum Doppler shift estimation method was presented. This method employs the characteristics of Doppler power spectrum, which is the frequency domain of the ACF of the estimated CIRs. Moreover, the FFT scheme and the CTA scheme were proposed, and these two schemes provide a tradeoff between the computational complexity and the memory size requirement. Then, the complexity of these schemes is analyzed; the FFT scheme has lower computational complexity than the CTA scheme, but it needs a larger memory size than the CTA scheme. To reduce the variation of noise and interference power spectrum, the estimated Doppler power spectrum is accumulated. When the proposed method utilizes a larger observation interval and accumulation size, the estimation results achieve a better accuracy and work well at low SNR. In Chapter 4, ICI analysis and an adaptive ICI cancellation scheme were presented. When the maximum Doppler shift increases,
more signal power leaks out to the neighboring subcarriers. By the relations between the ICI power and the maximum Doppler shift, we can know how many neighboring subcarriers should be taken account to the ICI eliminating procedure. Therefore, the adaptive method can utilize the appropriate equalizer to mitigate ICI as the maximum Doppler shift is given. Then, we analyze the complexity of the adaptive ICI cancellation scheme and compare this scheme with other schemes.
In new generation wireless communication systems, OFDM systems are considered for both fixed and mobile environment. Those new generation systems should be capable of working efficiently in a wide operating range, such as different carrier frequencies in licensed and licensed-exempt bands, large speeds rang of subscribers. The above reasons motivated the use of adaptive techniques, and one key parameter in adaptive mobile communication systems is the maximum Doppler shift.
Knowing Doppler shift also can aid handoffs and power control. Therefore, our goal is to combine the proposed maximum Doppler shift estimation method with these applications in the future work.
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