# Channel Measurement and Noise Estimation in FBMC/OQAM-Based IoT Networks

Authors: Jun Sun; Xiaomin Mu; Dejin Kong
Journal: Wireless Communications and Mobile Computing
 (2022)

---

## Abstract

Channel measurement plays an important role in the emerging 5G-enabled Internet of Things (IoT) networks, which reflects the channel quality and link reliability. In this paper, we address the channel measurement for link reliability evaluation in filter-bank multicarrier with offset quadrature amplitude modulation- (FBMC/OQAM-) based IoT network, which is considered as a promising technique for future wireless communications. However, resulting from the imaginary interference and the noise correlation among subcarriers in FBMC/OQAM, the existing frequency correlation method cannot be directly applied in the FBMC/OQAM-based IoT network. In this study, the concept of the block repetition is applied in FBMC/OQAM. It is demonstrated that the noises among subcarriers are independent by the block repetition and linear combination, instead of correlated. On this basis, the classical frequency correlation method can be applied to achieve the channel measurement. Then, we also propose an advanced frequency correlation method to improve the accuracy of the channel measurement, by assuming channel frequency responses to be quasi-invariant for several successive subcarriers. Simulations are conducted to validate the proposed schemes.


---

## Body

## 1. Introduction

The evaluation of the link reliability is required in the emerging 5G-enabled Internet of Things (IoT) networks, and based on the evaluation, the network performance can be enhanced by the power control or cell selection [1]. Therefore, the channel measurement plays a very important role in 5G-enabled IoT networks since it reflects the channel quality and link reliability. Generally speaking, the evaluation of the link reliability can be performed by measuring the received power of predefined reference signals [2]. In case of poor link, the base station can conduct the link switch, reducing the probability of network outage. In this study, we focus on the channel measurement for link reliability evaluation in filter-bank multicarrier with offset quadrature amplitude modulation- (FBMC/OQAM-) based IoT networks.As one promising physical-layer technique for IoT networks, FBMC/OQAM exhibits the advantages of asynchronous transmissions owing to the low sidelobe of the spectrum [3, 4] and high spectral efficiency due to the avoidance of the cyclic prefix [5], compared with the classical orthogonal frequency division multiplexing (OFDM). However, due to the real-valued orthogonality, there is imaginary-valued interference among symbols in FBMC/OQAM, called as intrinsic imaginary interference. Resulting from intrinsic imaginary interference [6, 7], many existing algorithms for OFDM cannot be directly used in FBMC/OQAM, such as channel measurement. Since the long-term evolution (LTE) era, reference signal received power (RSRP) has been the key one of channel measurements by utilizing reference signals [8]. Currently, there have existed several algorithms for channel measurement and noise estimation. In [9], a discrete Fourier transform- (DFT-) based time-domain algorithm was presented by utilizing the property that energy of the time-domain channel is only concentrated on the taps corresponding to propagation paths. In [10, 11], the authors proposed a frequency correlation algorithm by computing the correlation of the estimated frequency channel responses at adjacent subcarriers. Due to the low complexity and satisfactory performance, the classical frequency correlation method has been widely adopted for the channel measurement and noise estimation in various 3GPP proposals [10, 11]. However, these existing algorithms were proposed for OFDM-based networks. Due to the imaginary interference, channel measurement and noise estimation are more complicated in FBMC/OQAM systems. To the authors’ knowledge, channel measurement has not been addressed for FBMC/OQAM-based IoT networks.In this paper, we address channel measurement and estimation based on the FBMC/OQAM structure with block repetition. Firstly, we derive the noise distribution mathematically, which indicates that the noises at the receiver subcarriers are independent. On this basis, the classical frequency correlation method can be applied to achieve the channel measurement. Then, we also propose an advanced frequency correlation method to improve the accuracy of the channel measurement in FBMC/OQAM-based wireless networks, by assuming channel frequency responses to be quasi-invariant for several successive subcarriers. Numerical simulations have been conducted to validate the proposed schemes. The innovations are summarized as follows. The corresponding text is shown in the following:(i)
By using the block repetition, the noises at the receiver subcarriers are demonstrated to not be correlated, which is a key difference from the existing concept on noise distribution in FBMC/OQAM systems(ii)
Based on the mentioned-above features, the classical frequency correlation method can be employed in FBMC/OQAM to achieve the channel measurement(iii)
We propose an improved frequency correlation method to improve the accuracy of channel measurement.”


## 2. Classical Frequency Correlation Method

The classical frequency correlation method has been proposed for the channel measurement in OFDM-based networks, which has been widely adopted in various 3GPP proposals [10, 11]. In this section, we firstly present the classical frequency correlation method and then specify the drawbacks when it is directly used for the channel measurement in FBMC/OQAM-based networks.


### 2.1. Channel Measurement in OFDM

For classical OFDM systems, the received signal at the receiver can be obtained [12]
(1)ym=Hmxm+ηm,wherexm is the transmitted symbol at the m-th subcarrier and Hm is the channel frequency response. The noise ηm is usually supposed to satisfy the Gaussian distribution with zero mean and variance σ2 and be independent for different subcarriers in the classical OFDM systems.Based on Equation (1), the frequency-domain channel estimation is obtained as
(2)H^m=ymxm=Hm+ηmxm.Suppose that the transmitted symbolxm has a unit power. The noise term ηm/xm still satisfies the Gaussian distribution with variance σ2 and is independent for different subcarriers, which can be denoted as ξm for simplicity.SupposingM subcarriers used for the estimation and the channel frequency responses are quasi-invariant for adjacent subcarriers, then the channel measurement, i.e., RSRP, can be obtained by the following frequency correlation method [10, 11],
(3)RSRPo=1M−1∑m=0M−2H^mH^m+1∗,whereH^m+1∗ is the conjugate of H^m+1. When Hm=Hm+1 and M goes to infinity, Equation (3) can be rewritten as
(4)RSRPo=limM⟶∞1M−1∑m=0M−2Hm+ξmHm+1+ξm+1∗=limM⟶∞1M−1∑m=0M−2HmHm+1∗+Hmξm+1∗+ξmHm+1∗+ξmξm+1∗.Since the noiseηm is independent and identically distributed in the classical OFDM systems, the following equations hold:
(5)limM⟶∞1M−1∑m=0M−2Hmξm+1∗=0,(6)limM⟶∞1M−1∑m=0M−2ξmHm+1∗=0,(7)limM⟶∞1M−1∑m=0M−2ξmξm+1∗=0.Then, Equation (4) can be rewritten as
(8)RSRPo=limM⟶∞1M−1∑m=0M−2Hm+ξmHm+1+ξm+1∗=limM⟶∞1M−1∑m=0M−2HmHm∗,which represents the quality of the channel.Subsequently, the estimation of noise variance in OFDM systems can be obtained by(9)ε^o=1M∑m=0M−1ymym∗−RSRPo.Note thatε^o will be equal to σ2 when M goes to infinity in the classical OFDM systems.


### 2.2. Channel Measurement in FBMC/OQAM

Different from the classical OFDM system, the transmitted signal of each FBMC/OQAM subcarrier gets through a pulse-shaping filter with low spectrum sidelobe, leading to the better spectral utilization and ability of asynchronous transmission. Figure1 shows the system model of FBMC/OQAM with M subcarriers, in which gk is the pulse-shaping filter, and am,n is the transmitted real-valued symbol at the time-frequency position m,n. Firstly, the real-valued symbols am,n are multiplied by ejπm+n/2, and get through the M/2-point upsampling (corresponding to half symbol duration). Then, the obtained signal is filtered by gk, and the equivalent baseband transmitting signal of FBMC/OQAM can be written as [13, 14]
(10)sk=∑m=0M−1∑n∈ℤam,ngk−nM2ej2πmk/Mejπm+n/2.Figure 1
System model of FBMC/OQAM.After passing a multipath channel,rk is the received signal at the receiver. Then, the demodulation of the received signal can be approximately written as
(11)a^m,n=∑m=0M−1∑n∈ℤrkgk−nM2e−j2πmk/Me−jπm+n/2.Note that the maximum channel delay is usually much smaller than the symbol duration. And it is demonstrated in [13] that Equation (11) can be approximately written as [13]
(12)a^m,n≈Hm,nam,n+jam,no+ηm,n,where Hm,n is the channel frequency response at position m,n. For the slowly time-varying channels, Hm,n≈Hm,n+1. The term jam,no is the imaginary interference to the symbol am,n, i.e.,
(13)jam,no=∑m0∈ℤ∑n0∈ℤam0,n0ζm,nm0,n0,m0,n0≠m,n,where ζm,nm0,n0 is the imaginary interference factor in FBMC/OQAM and is defined as [13]
(14)ζm,nm0,n0=∑k=−∞∞gk−nM2gk−n0M2ej2πkm−m0/M×ejπm+n−m0−n0/2.Specifically whenm,n=m0,n0, ζm,nm0,n0=1. When m,n≠m0,n0, ζm,nm0,n0 is imaginary-valued. It has been revealed that the value of ζm,nm0,n0 is close to zero when ∣m−m0∣≥2 or n−n0≥2 [13], indicating that the imaginary interference to one symbol is mainly determined by its adjacent symbols. Besides, ηm,n is the noise term in FBMC/OQAM systems and is written as
(15)ηm,n=∑k=−∞+∞ηkgk−nM2e−j2πmk/Me−jπm+n/2,where ηk is the additional white Gaussian noise (AWGN) in wireless channels and is assumed to have a variance σ2. It has been proven in [14] that, due to the real orthogonality condition of FBMC/OQAM, ηm,n is of zero mean and has variance σ2, while is related to ηm0,n0 for m−m0≤1 and n−n0≤1, which is different from classical OFDM systems.Motivated by the noise correlation mentioned above, pilots as shown in Figure2 are required if the classical frequency correlation method in (3) is directly used for channel measurement and noise estimation in FBMC/OQAM systems. For the block of Figure 2, the symbols in the first column are set to zero to avoid interference from the previous block, and the symbols in the third column are set to zero to avoid interference from data. The nonzero pilots are in the second column, and it is noted that the pilots are designed to be scattered to ensure that the noise components for nonzero pilots are independent of each other. Assume that a2p,1=±1, and a2p+1,1=0 with p=0,1,⋯,M/2−1. Then, Equation (12) can be rewritten as
(16)a^2p,1=H2p,1a2p,1+η2p,1.Figure 2
Scattered pilot for channel measurement and noise estimation in FBMC/OQAM.By the frequency correlation method, the channel measurement of FBMC/OQAM can be obtained as(17)RSRPf=1M/2−1∑m=0M/2−2a^2p,1a2p,1a∧2p+1,1a2p+1,1∗.Subsequently, the estimation of noise variance in FBMC/OQAM systems can be obtained by(18)ε^f=2M∑m=0M/2−1a^2p,1a^2p,1∗−RSRPf.As presented above, the classical frequency correlation method can be directly used for channel measurement and noise estimation in FBMC/OQAM systems. However, it is noted that the accuracy of channel measurement depends on the number of samples. For a fixed frequency band, all subcarriers in OFDM can be used for the channel measurement, while only half of the subcarriers in FBMC/OQAM can be used for the channel measurement. Therefore, there exists a loss of channel measurement accuracy when the classical frequency correlation method is directly used in FBMC/OQAM systems.


## 2.1. Channel Measurement in OFDM

For classical OFDM systems, the received signal at the receiver can be obtained [12]
(1)ym=Hmxm+ηm,wherexm is the transmitted symbol at the m-th subcarrier and Hm is the channel frequency response. The noise ηm is usually supposed to satisfy the Gaussian distribution with zero mean and variance σ2 and be independent for different subcarriers in the classical OFDM systems.Based on Equation (1), the frequency-domain channel estimation is obtained as
(2)H^m=ymxm=Hm+ηmxm.Suppose that the transmitted symbolxm has a unit power. The noise term ηm/xm still satisfies the Gaussian distribution with variance σ2 and is independent for different subcarriers, which can be denoted as ξm for simplicity.SupposingM subcarriers used for the estimation and the channel frequency responses are quasi-invariant for adjacent subcarriers, then the channel measurement, i.e., RSRP, can be obtained by the following frequency correlation method [10, 11],
(3)RSRPo=1M−1∑m=0M−2H^mH^m+1∗,whereH^m+1∗ is the conjugate of H^m+1. When Hm=Hm+1 and M goes to infinity, Equation (3) can be rewritten as
(4)RSRPo=limM⟶∞1M−1∑m=0M−2Hm+ξmHm+1+ξm+1∗=limM⟶∞1M−1∑m=0M−2HmHm+1∗+Hmξm+1∗+ξmHm+1∗+ξmξm+1∗.Since the noiseηm is independent and identically distributed in the classical OFDM systems, the following equations hold:
(5)limM⟶∞1M−1∑m=0M−2Hmξm+1∗=0,(6)limM⟶∞1M−1∑m=0M−2ξmHm+1∗=0,(7)limM⟶∞1M−1∑m=0M−2ξmξm+1∗=0.Then, Equation (4) can be rewritten as
(8)RSRPo=limM⟶∞1M−1∑m=0M−2Hm+ξmHm+1+ξm+1∗=limM⟶∞1M−1∑m=0M−2HmHm∗,which represents the quality of the channel.Subsequently, the estimation of noise variance in OFDM systems can be obtained by(9)ε^o=1M∑m=0M−1ymym∗−RSRPo.Note thatε^o will be equal to σ2 when M goes to infinity in the classical OFDM systems.


## 2.2. Channel Measurement in FBMC/OQAM

Different from the classical OFDM system, the transmitted signal of each FBMC/OQAM subcarrier gets through a pulse-shaping filter with low spectrum sidelobe, leading to the better spectral utilization and ability of asynchronous transmission. Figure1 shows the system model of FBMC/OQAM with M subcarriers, in which gk is the pulse-shaping filter, and am,n is the transmitted real-valued symbol at the time-frequency position m,n. Firstly, the real-valued symbols am,n are multiplied by ejπm+n/2, and get through the M/2-point upsampling (corresponding to half symbol duration). Then, the obtained signal is filtered by gk, and the equivalent baseband transmitting signal of FBMC/OQAM can be written as [13, 14]
(10)sk=∑m=0M−1∑n∈ℤam,ngk−nM2ej2πmk/Mejπm+n/2.Figure 1
System model of FBMC/OQAM.After passing a multipath channel,rk is the received signal at the receiver. Then, the demodulation of the received signal can be approximately written as
(11)a^m,n=∑m=0M−1∑n∈ℤrkgk−nM2e−j2πmk/Me−jπm+n/2.Note that the maximum channel delay is usually much smaller than the symbol duration. And it is demonstrated in [13] that Equation (11) can be approximately written as [13]
(12)a^m,n≈Hm,nam,n+jam,no+ηm,n,where Hm,n is the channel frequency response at position m,n. For the slowly time-varying channels, Hm,n≈Hm,n+1. The term jam,no is the imaginary interference to the symbol am,n, i.e.,
(13)jam,no=∑m0∈ℤ∑n0∈ℤam0,n0ζm,nm0,n0,m0,n0≠m,n,where ζm,nm0,n0 is the imaginary interference factor in FBMC/OQAM and is defined as [13]
(14)ζm,nm0,n0=∑k=−∞∞gk−nM2gk−n0M2ej2πkm−m0/M×ejπm+n−m0−n0/2.Specifically whenm,n=m0,n0, ζm,nm0,n0=1. When m,n≠m0,n0, ζm,nm0,n0 is imaginary-valued. It has been revealed that the value of ζm,nm0,n0 is close to zero when ∣m−m0∣≥2 or n−n0≥2 [13], indicating that the imaginary interference to one symbol is mainly determined by its adjacent symbols. Besides, ηm,n is the noise term in FBMC/OQAM systems and is written as
(15)ηm,n=∑k=−∞+∞ηkgk−nM2e−j2πmk/Me−jπm+n/2,where ηk is the additional white Gaussian noise (AWGN) in wireless channels and is assumed to have a variance σ2. It has been proven in [14] that, due to the real orthogonality condition of FBMC/OQAM, ηm,n is of zero mean and has variance σ2, while is related to ηm0,n0 for m−m0≤1 and n−n0≤1, which is different from classical OFDM systems.Motivated by the noise correlation mentioned above, pilots as shown in Figure2 are required if the classical frequency correlation method in (3) is directly used for channel measurement and noise estimation in FBMC/OQAM systems. For the block of Figure 2, the symbols in the first column are set to zero to avoid interference from the previous block, and the symbols in the third column are set to zero to avoid interference from data. The nonzero pilots are in the second column, and it is noted that the pilots are designed to be scattered to ensure that the noise components for nonzero pilots are independent of each other. Assume that a2p,1=±1, and a2p+1,1=0 with p=0,1,⋯,M/2−1. Then, Equation (12) can be rewritten as
(16)a^2p,1=H2p,1a2p,1+η2p,1.Figure 2
Scattered pilot for channel measurement and noise estimation in FBMC/OQAM.By the frequency correlation method, the channel measurement of FBMC/OQAM can be obtained as(17)RSRPf=1M/2−1∑m=0M/2−2a^2p,1a2p,1a∧2p+1,1a2p+1,1∗.Subsequently, the estimation of noise variance in FBMC/OQAM systems can be obtained by(18)ε^f=2M∑m=0M/2−1a^2p,1a^2p,1∗−RSRPf.As presented above, the classical frequency correlation method can be directly used for channel measurement and noise estimation in FBMC/OQAM systems. However, it is noted that the accuracy of channel measurement depends on the number of samples. For a fixed frequency band, all subcarriers in OFDM can be used for the channel measurement, while only half of the subcarriers in FBMC/OQAM can be used for the channel measurement. Therefore, there exists a loss of channel measurement accuracy when the classical frequency correlation method is directly used in FBMC/OQAM systems.


## 3. Proposed Scheme for Channel Measurement in FBMC/OQAM

In this section, we propose a novel scheme for channel measurement in FBMC/OQAM systems. Firstly, the imaginary interference factor is analyzed and it is noted that its amplitude is symmetric on the time-frequency position. Then, on the basis, a reversed-order repeated block is designed to remove the correlation among symbols so that all subcarriers can be used for the channel measurement in FBMC/OQAM systems, by the frequency correlation method, improving the RSRP accuracy.


### 3.1. Analysis of Imaginary Interference Factor

According to (14), the imaginary interference factor is defined as
(19)ζm,nm0,n0=∑k=−∞∞gk−nM2gk−n0M2ej2πkm−m0/M×ejπm+n−m0−n0/2,which is the interference coefficient to the symbol am0,n0 from the symbol am,n. It is easily seen that the value of ζm,nm0,n0 only depends on the filter gk and the time-frequency position m,n. Note that the pulse-shaping filter gk in FBMC/OQAM is real-valued and symmetric [15, 16]. Due to the fact that gk is real-valued and symmetric, it can be easily demonstrated that the following equations hold
(20)ζm0,n0m0,n0−1=−ζm0,n0m0,n0+1,(21)ζm0,n0m0−1,n0=−ζm0,n0m0+1,n0,(22)ζm0,n0m0−1,n0−1=ζm0,n0m0−1,n0+1=ζm0,n0m0+1,n0−1=ζm0,n0m0+1,n0+1.Furthermore, letk¯=k−nM/2; then, Equation (19) can be rewritten as
(23)ζm,nm0,n0=∑k¯=−∞∞gk¯gk¯+nM2−n0M2×ej2πk¯+nM/2m−m0/Mejπm+n−m0−n0/2=∑k¯=−∞∞gk¯gk¯+nM2−n0M2×ej2πk¯m−m0/Mejπm−m0nejπm+n−m0−n0/2.According to Equation (23), for adjacent subcarriers, i.e., m−m0=±1, sign of ζm,nm0,n0 is opposite for the odd index n and the even index n. When the isotropic orthogonal transform algorithm (IOTA) is used for pulse shaping in FBMC/OQAM systems [14], the values of ζm,nm0,n0 are depicted in Tables 1 and 2 for the odd index and the even index, respectively. Note that, as mentioned above, the value of ζm,nm0,n0 is close to zero when m−m0≥2 and n−n0≥2 [13]; therefore, Tables 1 and 2 only show the values of ζm,nm0,n0 for m−m0≤1 and n−n0≤1.Table 1
ζm,nm0,n0 with odd time index n.
n0−1n0n0+1m0−10.2280j0.4411j0.2280jm00.4411j1-0.4411jm0+10.2280j-0.4411j0.2280jTable 2
Simulation parameters.
Modulation4QAMSampling frequency (MHz)9.14Number of paths6Power profile (in dB)-6, 0, -7, -22, -16, -20Delay profile (μs)-3, 0, 2, 4, 7, 11


### 3.2. Reversed-Order Repeated Block

In this subsection, a reversed-order repeated block is presented to eliminate the correlation among adjacent subcarrier, utilizing the properties in (20)–(22), Tables 1 and 3. Assume that am,n with m=0,1,⋯,M−1,n=0,1,⋯,N−1 are the transmitted symbols in FBMC/OQAM. Then, these symbols are transmitted repeatedly in the next block, satisfying am,N+n=am,N−1−n with n=0,1,⋯,N−1, as shown in Figure 3 in which M=6 and N=5 are taken as an example. Note that the symbol interval between am,n and am,n+1 is still half of a quadrature amplitude modulation (QAM) symbol, exactly as the symbol interval of normal FBMC/OQAM systems.Table 3
ζm,nm0,n0 with even time index n.
n0−1n0n0+1m0−1-0.2280j-0.4411j-0.2280jm00.4411j1-0.4411jm0+1-0.2280j0.4411j-0.2280jFigure 3
The reversed-order block for FBMC/OQAM.After the modulation at the FBMC/OQAM transmitter and demodulation at the FBMC/OQAM receiver, demodulations of original symbols can be obtained as(24)a^m,n=Hm,nam,n+jam,no+ηm,n,n=0,1,⋯,N−1,and demodulations of repeated symbols can be obtained as
(25)a^m,2N−1−n=Hm,2N−1−nam,2N−1−n+jam,2N−1−no=Hm,nam,n+jam,2N−1−no+ηm,2N−1−n,n=0,1,⋯,N−1.According to (13), the terms jam,no and jam,2N−1−no can be, respectively, written as
(26)jam,no=∑n0=n−1n+1∑m0=m−1m+1am0,n0ζm,nm0,n0,m0,n0≠m,n,(27)jam,2N−1−no=∑n0=2N−2−n2N−n∑m0=m−1m+1am0,n0ζm,2N−1−nm0,n0,m0,n0≠m,2N−1−n,where n=0,1,⋯,N−1.Then, utilizing the properties ofζm,nm0,n0 in (20)–(22), Tables 1 and 3, and considering am,N+n=am,N−1−n with n=0,1,⋯,N−1 as shown in Figure 3, it can be easily demonstrated that the following equation holds:
(28)jam,no+jam,2N−1−no=0.On this basis, it can be obtained as(29)a^m,n+a^m,2N−1−n2=Hm,nam,n+ηm,n+ηm,2N−1−n2.From Equation (29), it is easily realized that the imaginary interference in FBMC/OQAM is removed by the linear combination. Figure 4 depicts the diagram of the imaginary interference cancelation in FBMC/OQAM, and compared with the classical scheme, the key differences are the employment of repeated symbols at the transmitter and the operation of the linear combination at the receiver. At the transmitter, the employment of repeated symbols does not increase additional add operations and multiplication operations. At the receiver, the linear combination only requires add operations, instead of multiplication operations. As a result, the complexity of the proposed scheme is almost the same as the classical scheme.Figure 4
The scheme of the imaginary interference cancelation in FBMC/OQAM.


### 3.3. Analysis of Noise Distribution

It is noteworthy that, although the intersymbol interference is removed in Equation (29), the classical frequency correlation method is still not applicable for the channel measurement in FBMC/OQAM if the noise in (29) is not independent. Note that the noises at the receive subcarriers are correlated in the classical FBMC/OQAM structure without symbol repetition [14], due to its only real-filed orthogonality condition.In this subsection, the distribution of noise termηm,n+ηm,2N−1−n/2 in (29) is analyzed, and it will be demonstrated theoretically that the noises ηm,n+ηm,2N−1−n/2 are independent for different m, which is a key difference from the existing concept of noise distribution in FBMC/OQAM systems.According to Equation (15), ηm,n and ηm,2N−1−n can be, respectively, written as
(30)ηm,n=∑k=−∞+∞ηkgk−nM2e−j2πmk/Me−jπm+n/2,(31)ηm,2N−1−n=∑k=−∞+∞ηkgk−2N−1−nM2e−j2πmk/M×e−jπm+2N−1−n/2,where n=0,1,⋯,N−1. Note that ηk is the AWGN noise in wireless channels and satisfies
(32)Eηk=0,(33)Varηk=σ2,(34)Covηk,ηk0=0,k≠k0,where E·, Var·, and Cov· are the operators of the expectation, variance, and covariance, respectively. As a result, the mean of ηm,n is obtained as
(35)Eηm,n=E∑k=−∞+∞ηkgk−nM2e−j2πmk/Me−jπm+n/2=∑k=−∞+∞Eηkgk−nM2e−j2πmk/Me−jπm+n/2=0.Note that Equation (35) holds for any m and n; hence, the mean of ηm,2N−1−n is also zero.Then, for the sake of simplicity, define(36)χm,n=ηm,n+ηm,2N−1−n2.As mentioned above,ηm,n and ηm,2N−1−n satisfy the Gaussian distribution. Since χm,n is the linear combination of ηm,n and ηm,2N−1−n, it can be realized that χm,n also satisfies the Gaussian distribution. The mean of χm,n can be obtained as
(37)Eχm,n=Eηm,n2+Eηm,2N−1−n2=0.The variance ofχm,n can be obtained as
(38)Varχm,n=Eχm,nχm,n∗−Eχm,nEχm,n∗=Eχm,nχm,n∗=14Eηm,nηm,n∗+Eηm,nηm,2N−1−n∗+Eηm,2N−1−nηm,n∗+Eηm,2N−1−nηm,2N−1−n∗.Note thatEηm,nηm0,n0∗ can be written as
(39)Eηm,nηm0,n0∗=E∑k=−∞+∞ηkgk−nM2e−j2πmk/Me−jπm+n/2∑k=−∞+∞η∗kgk−n0M2ej2πm0k/Mejπm0+n0/2=E∑k=−∞+∞ηkη∗kgk−nM2gk−n0M2×e−j2πm−m0k/Me−jπm+n−m0−n0/2.According to the definition of the imaginary interference factor in Equation (14), we can rewrite (39) as
(40)Eηm,nηm0,n0∗=σ2ζm0,n0m,n.Note that it has been realized in [13] that ζm0,n0m,n is a pure imaginary value when m,n≠m0,n0. And when m,n=m0,n0, the value of ζm0,n0m,n is equal to 1, i.e.,
(41)ζm,nm,n=1.Furthermore, it has been revealed that the value ofζm0,n0m,n is close to zero when ∣m−m0∣≥2 and ∣n−n0∣≥2 [13]. In addition, the term ηm,nηm,2N−1−n∗ is the conjugate of ηm,2N−1−nηm,n∗; hence, it can be obtained as Eηm,nηm,2N−1−n∗+Eηm,2N−1−nηm,n∗=0. Then, Varχm,n can be obtained as
(42)Varχm,n=Eηm,nηm,n∗+Eηm,2N−1−nηm,2N−1−n∗4=σ2+σ24=σ22.Then, the covariance ofχm,n and χm0,n0 with m,n≠m0,n0 is obtained.
(43)Covχm,n,χm0,n0=Eχm,nχm0,n0∗−Eχm,nEχm0,n0∗=Eχm,nχm0,n0∗=14Eηm,nηm0,n0∗+Eηm,nηm0,2N−1−n0∗+Eηm,2N−1−nηm0,n0∗+Eηm,2N−1−nηm0,2N−1−n0∗.Substituting Equations (39) and (40) into (43), it can be obtained as
(44)Covχm,n,χm0,n0=14σ2ζm0,n0m,n+σ2ζm0,2N−1−n0m,n+σ2ζm0,n0m,2N−1−n+σ2ζm0,2N−1−n0m,2N−1−n.Note that, according to the definition ofζm0,n0m,n in (14), the value of ζm0,n0m,n only depends on the values of m−m0 and n−n0, instead of individual m, m0, n, or n0. Based on the properties in (20)–(22), Tables 1 and 3, it can be obtained as
(45)Covχm,n,χm0,n0=0.Therefore, it can be concluded that by the linear combination on the reversed-order block in Figure3, the noise for the demodulated symbol is independent to each other. Note that it has been pointed out in [14] that the noise is correlated to that of its adjacent symbols in normal FBMC/OQAM without the reversed-order block.Above all, Equation (29) in FBMC/OQAM will be equivalent to that of OFDM systems as Equation (2). Then, the classical frequency correlation method can be directly employed for channel measurement and noise estimation in FBMC/OQAM systems.


## 3.1. Analysis of Imaginary Interference Factor

According to (14), the imaginary interference factor is defined as
(19)ζm,nm0,n0=∑k=−∞∞gk−nM2gk−n0M2ej2πkm−m0/M×ejπm+n−m0−n0/2,which is the interference coefficient to the symbol am0,n0 from the symbol am,n. It is easily seen that the value of ζm,nm0,n0 only depends on the filter gk and the time-frequency position m,n. Note that the pulse-shaping filter gk in FBMC/OQAM is real-valued and symmetric [15, 16]. Due to the fact that gk is real-valued and symmetric, it can be easily demonstrated that the following equations hold
(20)ζm0,n0m0,n0−1=−ζm0,n0m0,n0+1,(21)ζm0,n0m0−1,n0=−ζm0,n0m0+1,n0,(22)ζm0,n0m0−1,n0−1=ζm0,n0m0−1,n0+1=ζm0,n0m0+1,n0−1=ζm0,n0m0+1,n0+1.Furthermore, letk¯=k−nM/2; then, Equation (19) can be rewritten as
(23)ζm,nm0,n0=∑k¯=−∞∞gk¯gk¯+nM2−n0M2×ej2πk¯+nM/2m−m0/Mejπm+n−m0−n0/2=∑k¯=−∞∞gk¯gk¯+nM2−n0M2×ej2πk¯m−m0/Mejπm−m0nejπm+n−m0−n0/2.According to Equation (23), for adjacent subcarriers, i.e., m−m0=±1, sign of ζm,nm0,n0 is opposite for the odd index n and the even index n. When the isotropic orthogonal transform algorithm (IOTA) is used for pulse shaping in FBMC/OQAM systems [14], the values of ζm,nm0,n0 are depicted in Tables 1 and 2 for the odd index and the even index, respectively. Note that, as mentioned above, the value of ζm,nm0,n0 is close to zero when m−m0≥2 and n−n0≥2 [13]; therefore, Tables 1 and 2 only show the values of ζm,nm0,n0 for m−m0≤1 and n−n0≤1.Table 1
ζm,nm0,n0 with odd time index n.
n0−1n0n0+1m0−10.2280j0.4411j0.2280jm00.4411j1-0.4411jm0+10.2280j-0.4411j0.2280jTable 2
Simulation parameters.
Modulation4QAMSampling frequency (MHz)9.14Number of paths6Power profile (in dB)-6, 0, -7, -22, -16, -20Delay profile (μs)-3, 0, 2, 4, 7, 11


## 3.2. Reversed-Order Repeated Block

In this subsection, a reversed-order repeated block is presented to eliminate the correlation among adjacent subcarrier, utilizing the properties in (20)–(22), Tables 1 and 3. Assume that am,n with m=0,1,⋯,M−1,n=0,1,⋯,N−1 are the transmitted symbols in FBMC/OQAM. Then, these symbols are transmitted repeatedly in the next block, satisfying am,N+n=am,N−1−n with n=0,1,⋯,N−1, as shown in Figure 3 in which M=6 and N=5 are taken as an example. Note that the symbol interval between am,n and am,n+1 is still half of a quadrature amplitude modulation (QAM) symbol, exactly as the symbol interval of normal FBMC/OQAM systems.Table 3
ζm,nm0,n0 with even time index n.
n0−1n0n0+1m0−1-0.2280j-0.4411j-0.2280jm00.4411j1-0.4411jm0+1-0.2280j0.4411j-0.2280jFigure 3
The reversed-order block for FBMC/OQAM.After the modulation at the FBMC/OQAM transmitter and demodulation at the FBMC/OQAM receiver, demodulations of original symbols can be obtained as(24)a^m,n=Hm,nam,n+jam,no+ηm,n,n=0,1,⋯,N−1,and demodulations of repeated symbols can be obtained as
(25)a^m,2N−1−n=Hm,2N−1−nam,2N−1−n+jam,2N−1−no=Hm,nam,n+jam,2N−1−no+ηm,2N−1−n,n=0,1,⋯,N−1.According to (13), the terms jam,no and jam,2N−1−no can be, respectively, written as
(26)jam,no=∑n0=n−1n+1∑m0=m−1m+1am0,n0ζm,nm0,n0,m0,n0≠m,n,(27)jam,2N−1−no=∑n0=2N−2−n2N−n∑m0=m−1m+1am0,n0ζm,2N−1−nm0,n0,m0,n0≠m,2N−1−n,where n=0,1,⋯,N−1.Then, utilizing the properties ofζm,nm0,n0 in (20)–(22), Tables 1 and 3, and considering am,N+n=am,N−1−n with n=0,1,⋯,N−1 as shown in Figure 3, it can be easily demonstrated that the following equation holds:
(28)jam,no+jam,2N−1−no=0.On this basis, it can be obtained as(29)a^m,n+a^m,2N−1−n2=Hm,nam,n+ηm,n+ηm,2N−1−n2.From Equation (29), it is easily realized that the imaginary interference in FBMC/OQAM is removed by the linear combination. Figure 4 depicts the diagram of the imaginary interference cancelation in FBMC/OQAM, and compared with the classical scheme, the key differences are the employment of repeated symbols at the transmitter and the operation of the linear combination at the receiver. At the transmitter, the employment of repeated symbols does not increase additional add operations and multiplication operations. At the receiver, the linear combination only requires add operations, instead of multiplication operations. As a result, the complexity of the proposed scheme is almost the same as the classical scheme.Figure 4
The scheme of the imaginary interference cancelation in FBMC/OQAM.


## 3.3. Analysis of Noise Distribution

It is noteworthy that, although the intersymbol interference is removed in Equation (29), the classical frequency correlation method is still not applicable for the channel measurement in FBMC/OQAM if the noise in (29) is not independent. Note that the noises at the receive subcarriers are correlated in the classical FBMC/OQAM structure without symbol repetition [14], due to its only real-filed orthogonality condition.In this subsection, the distribution of noise termηm,n+ηm,2N−1−n/2 in (29) is analyzed, and it will be demonstrated theoretically that the noises ηm,n+ηm,2N−1−n/2 are independent for different m, which is a key difference from the existing concept of noise distribution in FBMC/OQAM systems.According to Equation (15), ηm,n and ηm,2N−1−n can be, respectively, written as
(30)ηm,n=∑k=−∞+∞ηkgk−nM2e−j2πmk/Me−jπm+n/2,(31)ηm,2N−1−n=∑k=−∞+∞ηkgk−2N−1−nM2e−j2πmk/M×e−jπm+2N−1−n/2,where n=0,1,⋯,N−1. Note that ηk is the AWGN noise in wireless channels and satisfies
(32)Eηk=0,(33)Varηk=σ2,(34)Covηk,ηk0=0,k≠k0,where E·, Var·, and Cov· are the operators of the expectation, variance, and covariance, respectively. As a result, the mean of ηm,n is obtained as
(35)Eηm,n=E∑k=−∞+∞ηkgk−nM2e−j2πmk/Me−jπm+n/2=∑k=−∞+∞Eηkgk−nM2e−j2πmk/Me−jπm+n/2=0.Note that Equation (35) holds for any m and n; hence, the mean of ηm,2N−1−n is also zero.Then, for the sake of simplicity, define(36)χm,n=ηm,n+ηm,2N−1−n2.As mentioned above,ηm,n and ηm,2N−1−n satisfy the Gaussian distribution. Since χm,n is the linear combination of ηm,n and ηm,2N−1−n, it can be realized that χm,n also satisfies the Gaussian distribution. The mean of χm,n can be obtained as
(37)Eχm,n=Eηm,n2+Eηm,2N−1−n2=0.The variance ofχm,n can be obtained as
(38)Varχm,n=Eχm,nχm,n∗−Eχm,nEχm,n∗=Eχm,nχm,n∗=14Eηm,nηm,n∗+Eηm,nηm,2N−1−n∗+Eηm,2N−1−nηm,n∗+Eηm,2N−1−nηm,2N−1−n∗.Note thatEηm,nηm0,n0∗ can be written as
(39)Eηm,nηm0,n0∗=E∑k=−∞+∞ηkgk−nM2e−j2πmk/Me−jπm+n/2∑k=−∞+∞η∗kgk−n0M2ej2πm0k/Mejπm0+n0/2=E∑k=−∞+∞ηkη∗kgk−nM2gk−n0M2×e−j2πm−m0k/Me−jπm+n−m0−n0/2.According to the definition of the imaginary interference factor in Equation (14), we can rewrite (39) as
(40)Eηm,nηm0,n0∗=σ2ζm0,n0m,n.Note that it has been realized in [13] that ζm0,n0m,n is a pure imaginary value when m,n≠m0,n0. And when m,n=m0,n0, the value of ζm0,n0m,n is equal to 1, i.e.,
(41)ζm,nm,n=1.Furthermore, it has been revealed that the value ofζm0,n0m,n is close to zero when ∣m−m0∣≥2 and ∣n−n0∣≥2 [13]. In addition, the term ηm,nηm,2N−1−n∗ is the conjugate of ηm,2N−1−nηm,n∗; hence, it can be obtained as Eηm,nηm,2N−1−n∗+Eηm,2N−1−nηm,n∗=0. Then, Varχm,n can be obtained as
(42)Varχm,n=Eηm,nηm,n∗+Eηm,2N−1−nηm,2N−1−n∗4=σ2+σ24=σ22.Then, the covariance ofχm,n and χm0,n0 with m,n≠m0,n0 is obtained.
(43)Covχm,n,χm0,n0=Eχm,nχm0,n0∗−Eχm,nEχm0,n0∗=Eχm,nχm0,n0∗=14Eηm,nηm0,n0∗+Eηm,nηm0,2N−1−n0∗+Eηm,2N−1−nηm0,n0∗+Eηm,2N−1−nηm0,2N−1−n0∗.Substituting Equations (39) and (40) into (43), it can be obtained as
(44)Covχm,n,χm0,n0=14σ2ζm0,n0m,n+σ2ζm0,2N−1−n0m,n+σ2ζm0,n0m,2N−1−n+σ2ζm0,2N−1−n0m,2N−1−n.Note that, according to the definition ofζm0,n0m,n in (14), the value of ζm0,n0m,n only depends on the values of m−m0 and n−n0, instead of individual m, m0, n, or n0. Based on the properties in (20)–(22), Tables 1 and 3, it can be obtained as
(45)Covχm,n,χm0,n0=0.Therefore, it can be concluded that by the linear combination on the reversed-order block in Figure3, the noise for the demodulated symbol is independent to each other. Note that it has been pointed out in [14] that the noise is correlated to that of its adjacent symbols in normal FBMC/OQAM without the reversed-order block.Above all, Equation (29) in FBMC/OQAM will be equivalent to that of OFDM systems as Equation (2). Then, the classical frequency correlation method can be directly employed for channel measurement and noise estimation in FBMC/OQAM systems.


## 4. Proposed Channel Measurement in FBMC/OQAM

In this section, the channel measurement in FBMC/OQAM is presented based on the signal model as Equation (29), by using the classical frequency correlation method. Then, an improved frequency correlation method is proposed to improve the channel measurement accuracy.


### 4.1. Channel Measurement Using Classical Frequency Correlation

Figure5 shows the proposed pilot design for channel measurement in FBMC/OQAM, in which am,0 with m=0,1,⋯,M−1 are the pilots and am,n with n=1,2,⋯,N−1 are the data symbols. Note that all symbols in Figure 5 are complex-valued because the imaginary interferences among them can be eliminated completely by the linear combination in (29). As a consequence, compared with Figure 2, a lower pilot overhead is achieved in Figure 5.Figure 5
Proposed pilot design for channel measurement in FBMC/OQAM.According to (29), it can be obtained as
(46)a^m,0+a^m,2N−12am,0=Hm,0+ηm,0+ηm,2N−12am,0.For simplicity, denotea^m,0+a^m,2N−1/2am,0 as ϕm,0, and denote ηm,0+ηm,2N−1/2am,0 as ξm,0. Then, (46) can be rewritten as
(47)ϕm,0=Hm,0+ξm,0,m=0,1,⋯,M−1.Note that, as analyzed in Section3.2, the noise term ξm,0 satisfies independent identical distribution and its variance is σ2/2 when pilots am,0 have the unit transmitting power.Afterwards, the classical frequency correlation method can be directly used to obtain RSRP in FBMC/OQAM systems,(48)RSRPp=1M−1∑m=0M−2ϕm,0ϕm+1,0∗.Subsequently, the estimation of noise variance can be obtained by(49)ε^p=21M∑m=0M−1ϕm,0ϕm,0∗−RSRPp.Different from Figure2 in which only half of subcarriers are valid, all subcarriers are used for the channel measurement in Figure 5, indicating the improvement of accuracy in theory.


### 4.2. Improved Frequency Correlation Method

In this subsection, we propose an improved frequency correlation method to further improve the accuracy of channel measurement in FBMC/OQAM systems, by assuming the channel frequency responses quasi-invariant for several successive subcarriers. Subsequently, we give the theoretical analysis to show its superiority for the accuracy of channel measurement.The frequency correlation method utilizes the properties of quasi-invariant channels for adjacent subcarrier and independent distribution of noises. It is realized that, by the linear combination on the reversed-order block, the noises in (47) are independent for different subcarrier in FBMC/OQAM systems. By assuming the channels to be quasi-invariant for several successive subcarriers, more samples can be used to achieve the channel measurement by the statistical methods. Different from the conventional frequency correlation method, the correlation operations are performed on the several successive subcarriers in the improved frequency correlation method, instead of only two subcarriers, as shown in Figure 6. Hence, the improved frequency correlation method can be expressed as
(50)RSRPi=1Nc∑l=1Nr−1∑m=0M−l−1ϕm,0ϕm+l,0∗,where Nr is the number of subcarriers whose channel frequency responses are quasi-invariant. Nc is the number of sum terms in (50), i.e.,
(51)Nc=M−1+M−2+⋯+M−Nr+1=2M−NrNr−12.Figure 6
Proposed frequency correlation method.
(a)
Classical frequency correlation method(b)
Proposed frequency correlation methodSubsequently, the estimation of noise variance can be obtained by(52)ε^i=21M∑m=0M−1ϕm,0ϕm,0∗−RSRPi.In theory, when the subcarrier numberM goes to infinity, the channel measurement is accurate by the statistical method. However, M is limited in practice, leading to an interference. To evaluate the accuracy of the improved frequency correlation method, the variance of the residual interference is analyzed, in which the AWGN channel is assumed for simplicity. According to (50), the residual interference term can be easily written as
(53)Θi=1Nc∑l=1Nr−1∑m=0M−l−1ξm,0+ξm+l,0∗+ξm,0ξm+l,0∗.Note that, for the classical frequency correlation method in (48), the residual interference term is
(54)Θp=1M−1∑m=0M−2ξm,0+ξm+1,0∗+ξm,0ξm+1,0∗.Since the noise termξm,0 satisfies independent identical distribution, it is easily demonstrated that the expectations of Θp and Θi are equal to zero, i.e., EΘi=EΘp=0. Then, the variances of Θp and Θi can be obtained as
(55)VarΘp=EΘpΘp∗−EΘpEΘp∗=2σ2+σ42M−1,(56)VarΘi=EΘiΘi∗−EΘiEΘi∗=12M−Nr+1+4Nr−22Nr−332M−Nr2Nr−1σ2+12M−NrNr−1σ4.From Equations (55) and (56), when Nr=2, it can be obtained as VarΘi=VarΘp and the proposed frequency correlation method is equivalent to the classical frequency correlation method. When Nr>2, VarΘi will be smaller than VarΘp, and the proposed frequency correlation method can achieve better RSRP measurement performance than the conventional frequency correlation method. Especially when M≫Nr>2, the variance of Θi can be rewritten as
(57)VarΘi≈limM/Nr⟶∞12M−Nr+132M−Nr2Nr−1σ2+12M−NrNr−1σ4=limM/Nr⟶∞4M−Nr+1σ2+2M−Nrσ42M−Nr2Nr−1=limM/Nr⟶∞4σ2+2σ44M−4NrNr−1=limM/Nr⟶∞2σ2+σ42M−1Nr−1=1Nr−1limM/Nr⟶∞2σ2+σ42M−1.According to (55), we have
(58)VarΘi=1Nr−1VarΘp,M≫Nr>2.From Equation (58), the variance of Θi is approximately 1/Nr−1 of that of Θp when M/Nr goes to infinity, which indicates that the proposed frequency correlation method with M pilots can achieve similar accuracy of the classical frequency correlation method with Nr−1M pilots.


## 4.1. Channel Measurement Using Classical Frequency Correlation

Figure5 shows the proposed pilot design for channel measurement in FBMC/OQAM, in which am,0 with m=0,1,⋯,M−1 are the pilots and am,n with n=1,2,⋯,N−1 are the data symbols. Note that all symbols in Figure 5 are complex-valued because the imaginary interferences among them can be eliminated completely by the linear combination in (29). As a consequence, compared with Figure 2, a lower pilot overhead is achieved in Figure 5.Figure 5
Proposed pilot design for channel measurement in FBMC/OQAM.According to (29), it can be obtained as
(46)a^m,0+a^m,2N−12am,0=Hm,0+ηm,0+ηm,2N−12am,0.For simplicity, denotea^m,0+a^m,2N−1/2am,0 as ϕm,0, and denote ηm,0+ηm,2N−1/2am,0 as ξm,0. Then, (46) can be rewritten as
(47)ϕm,0=Hm,0+ξm,0,m=0,1,⋯,M−1.Note that, as analyzed in Section3.2, the noise term ξm,0 satisfies independent identical distribution and its variance is σ2/2 when pilots am,0 have the unit transmitting power.Afterwards, the classical frequency correlation method can be directly used to obtain RSRP in FBMC/OQAM systems,(48)RSRPp=1M−1∑m=0M−2ϕm,0ϕm+1,0∗.Subsequently, the estimation of noise variance can be obtained by(49)ε^p=21M∑m=0M−1ϕm,0ϕm,0∗−RSRPp.Different from Figure2 in which only half of subcarriers are valid, all subcarriers are used for the channel measurement in Figure 5, indicating the improvement of accuracy in theory.


## 4.2. Improved Frequency Correlation Method

In this subsection, we propose an improved frequency correlation method to further improve the accuracy of channel measurement in FBMC/OQAM systems, by assuming the channel frequency responses quasi-invariant for several successive subcarriers. Subsequently, we give the theoretical analysis to show its superiority for the accuracy of channel measurement.The frequency correlation method utilizes the properties of quasi-invariant channels for adjacent subcarrier and independent distribution of noises. It is realized that, by the linear combination on the reversed-order block, the noises in (47) are independent for different subcarrier in FBMC/OQAM systems. By assuming the channels to be quasi-invariant for several successive subcarriers, more samples can be used to achieve the channel measurement by the statistical methods. Different from the conventional frequency correlation method, the correlation operations are performed on the several successive subcarriers in the improved frequency correlation method, instead of only two subcarriers, as shown in Figure 6. Hence, the improved frequency correlation method can be expressed as
(50)RSRPi=1Nc∑l=1Nr−1∑m=0M−l−1ϕm,0ϕm+l,0∗,where Nr is the number of subcarriers whose channel frequency responses are quasi-invariant. Nc is the number of sum terms in (50), i.e.,
(51)Nc=M−1+M−2+⋯+M−Nr+1=2M−NrNr−12.Figure 6
Proposed frequency correlation method.
(a)
Classical frequency correlation method(b)
Proposed frequency correlation methodSubsequently, the estimation of noise variance can be obtained by(52)ε^i=21M∑m=0M−1ϕm,0ϕm,0∗−RSRPi.In theory, when the subcarrier numberM goes to infinity, the channel measurement is accurate by the statistical method. However, M is limited in practice, leading to an interference. To evaluate the accuracy of the improved frequency correlation method, the variance of the residual interference is analyzed, in which the AWGN channel is assumed for simplicity. According to (50), the residual interference term can be easily written as
(53)Θi=1Nc∑l=1Nr−1∑m=0M−l−1ξm,0+ξm+l,0∗+ξm,0ξm+l,0∗.Note that, for the classical frequency correlation method in (48), the residual interference term is
(54)Θp=1M−1∑m=0M−2ξm,0+ξm+1,0∗+ξm,0ξm+1,0∗.Since the noise termξm,0 satisfies independent identical distribution, it is easily demonstrated that the expectations of Θp and Θi are equal to zero, i.e., EΘi=EΘp=0. Then, the variances of Θp and Θi can be obtained as
(55)VarΘp=EΘpΘp∗−EΘpEΘp∗=2σ2+σ42M−1,(56)VarΘi=EΘiΘi∗−EΘiEΘi∗=12M−Nr+1+4Nr−22Nr−332M−Nr2Nr−1σ2+12M−NrNr−1σ4.From Equations (55) and (56), when Nr=2, it can be obtained as VarΘi=VarΘp and the proposed frequency correlation method is equivalent to the classical frequency correlation method. When Nr>2, VarΘi will be smaller than VarΘp, and the proposed frequency correlation method can achieve better RSRP measurement performance than the conventional frequency correlation method. Especially when M≫Nr>2, the variance of Θi can be rewritten as
(57)VarΘi≈limM/Nr⟶∞12M−Nr+132M−Nr2Nr−1σ2+12M−NrNr−1σ4=limM/Nr⟶∞4M−Nr+1σ2+2M−Nrσ42M−Nr2Nr−1=limM/Nr⟶∞4σ2+2σ44M−4NrNr−1=limM/Nr⟶∞2σ2+σ42M−1Nr−1=1Nr−1limM/Nr⟶∞2σ2+σ42M−1.According to (55), we have
(58)VarΘi=1Nr−1VarΘp,M≫Nr>2.From Equation (58), the variance of Θi is approximately 1/Nr−1 of that of Θp when M/Nr goes to infinity, which indicates that the proposed frequency correlation method with M pilots can achieve similar accuracy of the classical frequency correlation method with Nr−1M pilots.


## 5. Simulation Results

In this section, numerical simulations are given to validate the proposed schemes. In our simulations, we consider an FBMC/OQAM system with2048 subcarriers, in which the IOTA function is taken as the pulse-shaping filter [15]. Note that only a partial frequency band, i.e., 180 subcarriers is assigned for one user to achieve channel measurement and noise estimation. In addition, the detailed parameters for the multipath channel are listed as Table 3 [13].Figure7 shows the BER of FBMC/OQAM with the proposed repeated block under the AWGN channel, and for comparison, the performance of FBMC/OQAM without the repeated block is also given. From the results, FBMC/OQAM with the proposed repeated block achieves the same BER performance as the FBMC/OQAM without repeated block, validating the effectiveness of the proposed scheme. As mentioned above, the imaginary interferences among symbols can be eliminated completely by the linear combination in (46); hence, complex-valued symbols can be directly transmitted if the reversed-order repeated block (Figure 5) is adopted. Note that the symbol interval is always T/2 in FBMC/OQAM. Therefore, FBMC/OQAM with the proposed repeated block can transmit M×N complex-valued symbols during the time 2N×T/2=NT, while FBMC/OQAM without the repeated block can transmit M×2N real-valued symbols during the time 2N×T/2=NT. Therefore, the data rate is maintained in the proposed scheme. In addition, compared with FBMC/OQAM without repeated block, only the operation of the linear combination is required in FBMC/OQAM with repeated block; hence, the increase of complexity is negligible.Figure 7
BER comparison in FBMC/OQAM, with or without repeated block.Figures8 and 9 depict the cumulative distribution function (CDF) to evaluate the accuracy of channel measurement in FBMC/OQAM systems, in which the value ΔRSRP represents the offset between the estimated RSRP and the ideal RSRP [17], i.e.,
(59)ΔRSRP=10logRSRPp−10logRSRPideal.Figure 8
CDF of the proposed schemes under the AWGN channel,SNR=−6 dB.Figure 9
CDF of the proposed schemes under the multipath channel,SNR=−6 dB.WhenΔRSRP is closer to 0, the RSRP measurement is more accurate. Otherwise, the accuracy of channel measurement is poor. In simulations, the AWGN channel is adopted in Figure 8 and the multipath channel is adopted in Figure 9, where SNR is set to be -6 dB [18]. From the simulation results, by the classical frequency correlation method, FBMC/OQAM with the proposed repeated block can achieve better RSRP accuracy than FBMC/OQAM without the repeated block. The reason is that, for the repeated block in Figure 5, all subcarriers can be utilized for the channel measurement since the imaginary interference can be completely removed by the linear combination in (46), while only half of subcarriers in Figure 2 can be utilized for the channel measurement. In addition, by the improved frequency correlation method, the RSRP accuracy can be further improved due to the increase of samples as mentioned in Section 4.2.To evaluate the estimation accuracy of noise variance, Figures10 and 11 show the mean square error (MSE) under the AWGN and multipath channels, respectively. From the simulation results, FBMC/OQAM with the proposed repeated block can achieve better estimation accuracy than FBMC/OQAM without the repeated block, due to the fact that all subcarriers are utilized for estimation in the proposed scheme. Especially, compared with Figure 10, there exists the performance floor under the multipath channel in Figure 11. The reason is that the essence of the frequency correlation method is the correlation of adjacent subchannels that are assumed identical. However, multipath channels are frequency-selective. Therefore, there exists a bit of interference in the frequency correlation method. Note that the performance floor is above 23 dB for the conventional scheme while it is below 30 dB for the proposed scheme, indicating the superiority of the proposed scheme.Figure 10
MSE of noise estimation in the proposed schemes under the AWGN channel.Figure 11
MSE of noise estimation in the proposed schemes under the multipath channel.For more adequate evaluation, extended pedestrian A (EPA) model is also used in simulations, and Figure12 depicts the CDF of the proposed scheme under the EPA channel, where SNR is set to be -6 dB. From the simulation results, by the classical frequency correlation method, FBMC/OQAM with the proposed repeated block can achieve better RSRP accuracy than FBMC/OQAM without the repeated block. In addition, by the improved frequency correlation method, the RSRP accuracy can be further improved. The conclusion is consistent to Figures 8 and 9.Figure 12
CDF of the proposed schemes under the LTE EPA channel,SNR=−6 dB.


## 6. Conclusions

In this paper, we addressed the channel measurement for link reliability in the FBMC/OQAM-based IoT networks, in which the block repetition was adopted. Firstly, The noise distribution was derived mathematically, which indicated that the noises at the receiver subcarriers are independent. Then, it was shown that the classical frequency correlation method could be applied to achieve the channel measurement, based on the noncorrelation of noises at the receiver subcarriers. Subsequently, an advanced frequency correlation method was proposed to improve the accuracy of the channel measurement in FBMC/OQAM-based wireless networks. Numerical simulations showed that the accuracy of channel measurement could be significantly improved by the proposed schemes.



---

*Source: 6518066-2022-01-10.xml*