Chapter 2 Doubly Selective Fading Channels
2.3 OFDM Systems over Doubly-Selective Fading Channels . 21
OFDM is a promising transmission technique to achieve high data rates over wireless mobile channels. Besides bandwidth efficiency, OFDM offers many advantages over conventional single carrier systems, for example, robustness against multipath delay
spread, all-digital-FFT implementation, the possibility of adaptive channel allocation and adaptive modulation of the subcarriers for maximizing the capacity [41]. Thanks to cyclic prefix (CP), the ISI can be avoided, and the channel effect can easily be com-pensated by applying one-tap frequency domain equalizers since the frequency selective fading channel is transformed into parallel flat fading channels. However, if there ex-ists channel variations within one OFDM symbol will introduce interferences among subcarriers, called ICI. OFDM is vulnerable to ICI due to tightly spacing between subcarriers.
Consider the baseband equivalent OFDM system model with N subcarriers. By doing N -point IDFT of the data sequence of the OFDM symbol, {S(m)}, the time-domain transmitted signal x(k) at the k-th time instant can be written as
x(k) = 1 N
N −1
X
m=0
S(m)ej2πmkN , k = 0, 1, · · · , N − 1. (2.27)
Assume a CP with length of NCP longer than the maximum delay spread of the channel, denoted as L; the total length of a transmitted OFDM symbol is NS = N + NCP. Let h(k; l) represent the l-th delay path of the time-varying dispersive channel impulse response at the k-th sampling instant. The channel path {h(k; l)} is assumed to follow the WSSUS model, i.e., E[h(k; l)h∗(k; l+l0)] = σl2ξl0 where ξl0denotes the 1-D Kronecker delta and σ2l is the average power of the l-th path. Assuming CP removal and perfect synchronization, the sampled baseband equivalent received signal y(k) of the OFDM symbol can be expressed as [9,10]
y(k) =
L−1
X
l=0
h(k; l)x(k − l) + n(k), 0 ≤ k ≤ N − 1 (2.28)
where n(k) is the additive white Gaussian noise.
Taking discrete Fourier transform of Equation (2.28), the frequency-domain
re-ceived signal on the i-th subcarrier is Note that H(m; k) can be interpreted as the CFR on the m-th subcarrier at the k-th time instant. The ICI on the i-th subcarrier which is caused by the signal on the m-th subcarrier comes through the ICI channel Hi,m.
Assume that the time variation of CIR is linear within one OFDM symbol [13,15].
It is a reasonable assumption in most practical cases when the normalized Doppler frequency (normalized to subcarrier spacing) is smaller than 0.1, which corresponds to 500 km/h in WiMAX standard. The CIR of the p-th OFDM symbol can be written as
h(pNS− N + k; l) = h(pNS −N − 1
2 ; l) +k − N −12
NS δ(pNS; l) (2.32) where NS denotes the length of CP plus N . h(pNS−N −12 ; l) represents the center point of the p-th OFDM symbol and δ(pNS; l) is the slope of the l-th channel path at the p-th OFDM symbol. Note that with the linear variation assumption the channel at the center point is also the averaged channel in one OFDM symbol.
From (2.27) and (2.28), the received signal can be compactly expressed in vector
form as
Using (2.32), the CIR matrix in (2.33) can be decomposed as
Ht = Mt+ ψDt (2.34)
Substituting (2.34) into (2.33) and applying DFT matrix to y, the received signal r = [R(0), R(1), · · · , R(N − 1)]T in the frequency domain is given by
where Havg+G∆ is the CFR matrix and z is the AWGN in the frequency domain. The term G∆s represents ICI which is determined by ∆ and a fixed matrix G where ∆ is DFT of the slope of the CIR, Dt. Note that since Mt and Dt are circulant matrix, Havg and ∆ will be diagonal. On the contrary, G is circulant since ψ is diagonal.
Consequently, the CFR matrix can be decomposed as
Now, a tiny yet crucial rearrangement of this well-known ICI signal model is in order. By multiplying both sides of (2.35) by the diagonal matrix (∆H−1avg), we can rearrange (2.35) into:
where IN denotes the N ×N identity matrix, and ˜s acts as the transmitted signal, ˜r the received signal and (∆H−1avg)G as the ICI channel. Note that in (2.35), ∆ multiplies G from the right; it follows that the column vectors of the CFR matrix are weighted by different ∆k. This model registers how the signal on each subcarrier is spread over and comes to interfere other subcarriers. Different weights represent different spreading effects for each subcarrier. In (2.37), however, G is multiplied by the diagonal matrix (∆H−1avg) from the left; this time, row vectors of the CFR matrix are weighted with a different (∆k/Hk). This model registers how a subcarrier is interfered by its neighboring subcarriers. Different weights indicates different ICI levels experienced by each subcarrier. These insights motivate us the following analysis.
2.4 ICI Indicator
The ICI model in (2.35) and (2.37) provides more insights when it is examined at the subcarrier-level granularity. There is no uniform band structure in the CFR matrix since each subcarrier faces very different ICI situations. In (2.37), (∆H−1avg)G represents the ICI channel; this motivates us to propose the metric |∆k/Hk| to indicate the ICI level on each subcarrier. From (2.37), define the Signal-to-Interference Channel power Ratio (SICR) on the k-th subcarrier as
SICRk = | ˜Hk,k|2 PN −1
m=0,m6=k| ˜Hk,m|2, (2.38)
which can be further approximated to
SICRk≈ proportional to |∆k/Hk|2. For example, if a subcarrier suffers little ICI, the SICR (≥ 22 dB) will be quite high and |∆k/Hk| (≤ −5 dB) is low. On the other hand, when deep fading occurs at some subcarrier, |∆k/Hk| becomes large (≥ 0 dB) and SICR (≤ 12 dB) is low, and the demodulated symbol tends to be wrong due to severe ICI.
To better utilize |∆k/Hk| as the ICI indicator, its statistical properties are inves-tigated. Assuming Rayleigh fading channels, δ(nN s; l) and h(nNS− N −12 ; l), the CIR slope and the averaged CIR, are zero-mean circular complex Gaussian random variables (RVs) with variances σδ(nN s;l)2 and σ2
h(nNS−N −12 ;l). It follows that ∆kand Hkare also zero-mean circular complex Gaussian RVs with variances P
lσδ(nN s;l)2 and P
lσ2
h(nNS−N −1
2 ;l)
respectively. Note that denoting A as a zero-mean circular complex Gaussian RV with
variance σ2, the PDF of A is given by
pA(a) = 1
πσ2 exp{−|a|2
σ2 }. (2.40)
It is well known that the magnitude of A, B = |A|, follows a Rayleigh distribution with parameter σ2/2 as
pB(b) = b
σ2/2exp{−b2
σ2}, if b ≥ 0, (2.41)
and pB(b) = 0 elsewhere.
To simplify the derivation, we further assume ∆k and Hk are uncorrelated, on the grounds that ∆k is caused by the mobility of scatterers while Hk is mainly determined by the reflection property of scatterers. This property is also shown in [13] in which the zeroth-order and the first-order derivative in the power series expansion of the channel correspond to our Hk and ∆k. Define X , 10log10(|∆k|) and Y , 10log10(|Hk|). Since X and Y follow the Log-Rayleigh distribution with the PDF [42], which can be derived from (2.41) by using the technique of function of RV, is given by
p(ϕ|η2) = (eϕ/c)2
cη2 exp (−(eϕ/c)2
2η2 ) (2.42)
where the constant c = 10 log10(e) and η2 is the so-called localization parameter that equals to 12P
ηX2 and ηY2 being the localization parameters of X and Y . Using the property that the mean of Log-Rayleigh PDF is 10 log10(η2) + c0 where c0 = 2c(ln 2 +R+∞
0 ln v exp(−v)dv) is a constant and the variance is c224π2 [42], it follows that the PDF of 10log10(|∆|Hk|
k|) only
depends on the ratio between P
lσ2δ(nN s;l) and P
lσ2
h(nNS−N −1
2 ;l). Intuitively, channel variation is caused by mobility rather than depends on the channel PDP. We expect the ratio to be insensitive to PDPs. In this case, the PDF of the ICI indicator will be consistent over a wide range of channels with different PDPs.
−30 −25 −20 −15 −10 −5 0 5 10 15
Figure 2.6: Probability density functions (solid lines) and two histograms (dash lines for the ITU Vehicular-A channel and dots for two-path equal gain channel) of 10log10(|∆Hk
k|) for moving speeds at 60, 120 and 350 km/h. Bell-shape Doppler power spectrum and uncorrelated scattering are used.
Simulation is conducted to verify the theoretical result, using the WiMAX stan-dard with 10 MHz bandwidth, 2.5 GHz central frequency, 1024 subcarriers. Two channel models with very different PDPs, the ITU Vehicular-A and a two-path equal gain channel, are used. Fig. 2.6 shows the theoretical PDF and the histogram of 10log10(|∆Hk|) for various moving speeds, and they coincide closely so that our
theo-retical analysis of PDF is valid. Recall that larger |∆k/Hk| means higher ICI level.
As the moving speed gets higher, more subcarriers experience severe ICI; yet even at 350 km/h, there still is a major portion of subcarriers experiencing mild ICI. It is also observed from Fig. 2.6 that the distribution of |∆k/Hk| remains essentially the same when two quite different PDPs are used. The PDF can be used in evaluating the ben-efit of reducing complexity by adapting ICI equalizers according to the ICI indicator.
Details can be seen in Chapter 3.
−15 −10 −5 0 5 10
0 0.02 0.04 0.06 0.08 0.1 0.12
| k/Hk| (dB)
Theoretical Flat Doppler PSD Correlated scattering
Figure 2.7: Probability density functions (solid lines) and two histograms (dash lines for the flat Doppler spectrum and dots for the correlated scattering with correlation 0.7 between paths) of 10log10(|∆Hk
k|) for moving speed at 350 km/h. Two-path equal gain channel is used.
Other factors that may affect the PDF are also investigated. In Fig. 2.7, the effects of Doppler spectrum shape and correlated scattering are shown. As can be
seen, the PDF is also quite insensitive to the Doppler spectrum shape. In contrast, the correlation existing among channel paths can change the PDF. However, the overall trend is still that the PDF’s center becomes larger when the moving speed gets larger, so does the maximum Doppler spread. Therefore, the ICI indicator can still offer useful information on channel variations.
In practice, the ICI indicator is obtained as the ratio between the estimates of
∆k and Hk. An estimation method that uses consecutive OFDM symbols is provided in [15]. Here, with the goal of keeping complexity low, a simpler method is used.
Instead of estimating the complete ICI model as in [15], we use the estimate of the averaged CFR, ˆHk, which is always needed for signal demodulation. The estimate of
∆k is simply obtained as the difference between ˆHk of adjacent OFDM symbols due to the fact that CFR varies linearly across time-domain:
∆ˆk,n= ˆHk,n− ˆHk,n−1
where second index in the subscript denotes the symbol index. Though the estimate is not very accurate, the degradation in simulated BER performance is insignificant.
Channel estimation for time-varying channels may also be used [8,10–13], but we re-gard this as unnecessary in the current setting. To support higher Doppler spreads in the future, better channel estimation will be needed and advanced channel estimation techniques can be found in [43–47] and extended for MIMO channels in [48,49]. Fur-thermore, to jointly estimate the channel and other synchronization parameters has been proposed in [50–52].