Research Article On Channel Estimation for OFDM/TDM Using MMSE-FDE in a Fast Fading Channel

MMSE-FDE can improve the transmission performance of OFDM combined with time division multiplexing (OFDM/TDM), but knowledge of the channel state information and the noise variance is required to compute the MMSE weight. In this paper, a performance evaluation of OFDM/TDM using MMSE-FDE with pilot-assisted channel estimation over a fast fading channel is presented. To improve the tracking ability against fast fading a robust pilot-assisted channel estimation is presented that uses time-domain filtering on a slot-by-slot basis and frequency-domain interpolation. We derive the mean square error (MSE) of the channel estimator and then discuss a tradeoff between improving the tracking ability against fading and the noise reduction. The achievable bit error rate (BER) performance is evaluated by computer simulation and compared with conventional OFDM. It is shown that the OFDM/TDM using MMSE-FDE achieves a lower BER and a better tracking ability against fast fading in comparison with conventional OFDM.


Introduction
Orthogonal frequency division multiplexing (OFDM) is receiving considerable interest because of its high capacity and robustness against the channel frequency-selectivity [1]. OFDM signals, however, have a problem with a high peakto-average power ratio (PAPR). OFDM combined with time division multiplexing (OFDM/TDM) [2] can reduce the PAPR, and furthermore, with the application of minimum mean square error frequency domain equalization (MMSE-FDE) it can bridge the performance of conventional OFDM and single carrier (SC)-FDE [3]. In OFDM/TDM, the inverse fast Fourier transform (IFFT) time window (i.e., OFDM/TDM frame) of conventional OFDM is divided into K slots; within each slot an OFDM signal with reduced number of subcarriers is transmitted. At the receiver, MMSE-FDE is applied over the entire frame to exploit the channel frequency-selectivity. MMSE-FDE, however, requires knowledge of the channel state information and the noise variance.
Various channel estimation techniques for conventional OFDM and SC-FDE have been presented in [4][5][6][7][8][9][10][11]. There are two approaches to estimate the channel state information. In one approach, the channel estimation is done by time-domain multiplexed pilot (TDM-pilot), but the tracking ability against fast fading degrades. In another approach, the channel estimation is done by frequencydomain multiplexed pilot (FDM-pilot) to improve the tracking ability against fast fading, but the bit error rate (BER) increases due to increased noise after frequency-domain interpolation. To date, the effect of channel estimation on the performance of OFDM/TDM using MMSE-FDE has not been fully discussed. This paper deals with the channel estimation for OFDM/TDM using MMSE-FDE over a fast fading channel. We adopt a pilot-structure for SC transmission [8] and apply recursive least-square (RLS) algorithm [12] in time domain (in this paper called time-domain first-order filtering (TDFF)) on a slot-by-slot basis within the OFDM/TDM frame to improve the tracking ability against fast fading. We develop an expression for the mean square error (MSE) of the channel estimator and then discuss a tradeoff between improvement of the tracking ability against fast fading and 2 EURASIP Journal on Wireless Communications and Networking the noise reduction. The BER performance improvement is achieved through the frequency diversity gain and improved tracking ability against fast fading without sacrificing transmission data-rate.
The rest of the paper is organized as follows. In Section 2, we describe the system model. Pilot-assisted channel estimation techniques are presented in Section 3, while the estimator performance analysis is presented in Section 4. We evaluate the performance by computer simulation in Section 5. Section 6 concludes the paper.

System Model
The OFDM/TDM system model is illustrated in Figure 1. Throughout this paper, T c -spaced discrete-time signal representation is used, where T c represents FFT sampling period.
The gth signaling interval of the conventional OFDM with N c subcarriers is divided into K slots (i.e., gth OFDM/TDM frame) as shown in Figure 2 is the ensemble average operation). The gth frame OFDM/TDM signal can be expressed using the equivalent lowpass representation as for t = 0 ∼ N c − 1, where x and E s , respectively, denote the largest integer smaller than or equal to x and the datamodulated symbol energy. We note here that OFDM/TDM signal with K = 1 (i.e., N m = N c ) reduces to the conventional OFDM system with N c subcarriers. After insertion of an N m -sample guard interval (GI), the OFDM/TDM signal is transmitted over a fading channel. The gth frame OFDM/TDM signal propagates through the channel having a discrete-time channel impulse response h g (τ) given as where h g,l and τ l , respectively, denotes the gth frame's path gain and the time delay of the lth path having the samplespaced exponential power delay profile with channel decay factor α (i.e., E[|h g,l | 2 ] = ((1−α)/(1−α L ))α l ). The maximum time delay of the channel is assumed to be less than the GI length.
The received gth frame OFDM/TDM signal is represented by {r g (t); t = −N m ∼ N c − 1}. After removal of the GI, {r g (t); t = 0 ∼ N c − 1} is fed to N c -point FFT to decompose the received signal into N c frequency components {R g (n); n = 0 ∼ N c − 1} given by    where S g (n), H g (n), I g (n), and N g (n), respectively, denote the Fourier transforms of the gth frame transmit OFDM/TDM signal, the channel impulse response, the inter-slot interference (ISI) and the additive white Gaussian noise (AWGN) having single sided power spectral density N 0 . Note that in case of conventional OFDM (K = 1) the ISI term vanishes. One-tap MMSE-FDE similar to SC-FDE [8] is applied over the entire frame with K concatenated OFDM signals as [3] R g (n) = R g (n)W g (n) = S g (n) H g (n) + I g (n) + N g (n) (4) with H g (n) = H g (n)W g (n) and I g (n) = I g (n)W g (n), where W g (n) denotes the MMSE equalization weight given by EURASIP Journal on Wireless Communications and Networking (g 2)th frame gth frame (g 1)th frame  where σ 2 g and (·) * denote the noise variance and the complex conjugate operation, respectively. We need to estimate H g (n) and σ 2 g . Channel estimation techniques will be described in the following section.
The time-domain OFDM/TDM signal is recovered by applying N c -point IFFT to { R(n); n = 0 ∼ N c −1}, and then, the demodulation of OFDM signals with N m subcarriers is done by N m -point FFT [3].

Channel Estimation
The channel and the noise variance in OFDM/TDM system can be estimated with TDM-pilot, but the tracking ability significantly degrades in a fast fading channel [13]. To improve tracking against fast fading the channel estimation with FDM-pilot [5] can be directly applied to OFDM/TDM, but the GI must be inserted between the slots within the OFDM/TDM frame that will reduce the data-rate and the frequency diversity gain cannot be obtained. Former is due to the fact that the GI length must be kept the same as conventional OFDM, while latter is because the insertion of GI will destroy the property of OFDM/TDM. Moreover, if FDM-pilot is directly applied to OFDM/TDM, the received pilot subcarriers within the slots will be corrupted by ISI (see (4)) leading to the estimator performance degradation.
We focus on three estimation schemes: (i) channel estimation with TDM-pilot and time-domain first-order filtering (TDFF), (ii) channel estimation with TDM-pilot [13], and (iii) channel estimation with FDM-pilot [5]. These methods are described next (note that our aim is to keep the transmission data-rate the same as conventional OFDM). Then, pilot signal selection and noise variance estimation are discussed.

Channel Estimation with TDM-Pilot and TDFF. The
and into the GI as a cyclic prefix as illustrated in Figure 3. Since the H e,g (n) Figure 5: Pilot-assisted channel estimation using time-delay domain windowing.
same pilot is used for all frames, the (g − 1)th frame's pilot slot acts as a cyclic prefix for the gth frame's GI. Thus, the channel estimation can be performed using the gth frame's N m -sample GI. Similar frame structure was presented for SC transmission in [8]. The channel gain estimate and noise variance estimate to be used for FDE are denoted by H e,g (n) and 2σ 2 e,g , respectively. H g (n) and σ 2 g in (5) are replaced by H e,g (n) and σ 2 e,g , respectively. The received pilot {r g (t); t = −N m ∼ −1} in the GI is filtered on a slot-by-slot basis by the time-domain first-order filtering as illustrated in Figure 3 to increase the signal-tonoise power ratio (SNR) of the pilot signal. The filtered pilot signal is obtained as where γ is the forgetting factor with the initial condition r 0 (t) = r 0 (t). Then, N m -point FFT is applied with q = n/K for n = 0 ∼ N c − 1 and the initial condition R 0 (q) = R 0 (q). The instantaneous channel gain estimate at the qth subcarrier is obtained by removing the pilot modulation as where N g (q) = N g (q)/P(q) and P(q) denotes the qth Since q = n/K , the channel estimates are obtained only at the frequencies n = 0, N m , 2N m , . . . , (N c − 1) as shown in Figure 6. Hence, an interpolation is necessary to obtain the channel gains for all frequencies (i.e., n = 0 ∼ N c − 1). We  apply the following frequency-domain interpolation. First, Then, N c -point FFT is applied to obtain the interpolated channel gain estimates {H e,g (n);

Channel Estimation with TDM-Pilot.
For OFDM/TDM with pilot-assisted channel estimation using TDM-pilot [13], a pilot signal is transmitted followed by N d OFDM/TDM data frames is illustrated in Figure 4. The block diagram of channel estimation is illustrated in Figure 5. Notice that in this case all N c subcarriers are used as pilots. First, by reverse modulation, the instantaneous channel gain estimate H g (n) at the nth subcarrier is obtained by (8). Then, N c -point IFFT is applied to {H g (n); n = 0 ∼ N c −1} to obtain the instantaneous channel impulse response {h(τ); τ = 0 ∼ N c − 1}. Assuming that the actual channel impulse response is present only within the GI, the estimated channel impulse response beyond the GI is replaced with zeros to reduce the noise [11]. Finally, N c -point FFT is applied to obtain the improved channel gain estimates {H e,g (n); n = 0 ∼ N c − 1}.

Channel Estimation with FDM-Pilot.
For pilot-aided channel estimation with FDM-pilot [5] using frequencydomain interpolation an N m equally-spaced pilot subcarriers among N c subcarriers are used.
First, by reverse modulation, the instantaneous channel gain estimate {H g (q); q = n/N m for n = 0 ∼ N c − 1} at the pilot subcarriers is obtained by (8), where N m is the number of pilot subcarriers. Since q = n/N m , the channel estimates are obtained only at the frequencies n = 0, N m , 2N m , . . . , (N c − 1) as shown in Figure 6. Hence, the frequency-domain interpolation described in Section 3.1 is used to obtain the channel gains for all frequencies (i.e., n = 0 ∼ N c − 1). N m -point IFFT is performed on {H g (q); q = 0 ∼ N m − 1} to obtain the instantaneous channel impulse response {h(τ); τ = 0 ∼ N m − 1} as in (9) and then, N cpoint FFT is applied to obtain the interpolated channel gain estimates {H e,g (n); n = 0 ∼ N c − 1}.

Pilot Sequence Selection.
A selection of pilot sequence is an important design issue. If the amplitude of P(n) drops at some frequencies, the noise component in the channel estimate will be enhanced and thereby the estimation accuracy will degrade leading to poor performance. To avoid the noise enhancement, it is desirable that P(n) has constant amplitude irrespective of n. On the contrary, if P(n) is constant for all n, a large amplitude variation may appear in p(t) and consequently, the pilot signal may be distorted due to nonlinear power amplification. In this paper we use Chu sequence [14] as the pilot since the PAPR problem is not present as shown in Figure 7.

Noise Power Estimation.
The noise component at the qth pilot subcarrier can be estimated by subtracting the received pilot component H e,g (q)P(q) from R g (q) as N e,g q = R g q − H e,g q P q (11) for q = 0 ∼ N m − 1. The noise variance estimate can be obtained as Note that in the case of channel estimation scheme presented in Section 3.2, all subcarriers are used as pilots (i.e., N m = N c ).
In what follows, we present MSE analysis of the channel estimator with use of both time-domain first-order filtering and frequency-domain interpolation.

Performance Analysis
We have presented pilot-assisted channel estimation schemes in Section 3. The performance of channel estimators presented in Sections 3.2 and 3.3 has already been theoretically investigated in literature; hence, we analyze only the performance of the estimator presented in Section 3.1.
The gth frame's filtered channel gain estimate at the qth subcarrier can be expressed as with the initial condition H 0 (q) = H 0 (q) and 0 < γ ≤ 1. We define the MSE at nth subcarrier component of the gth frame as MSE g (n) = E[|e g (n)| 2 ] = E[|H e,g (n) − H g (n)| 2 ]. Substituting (9) and (10) into (13), the channel gain estimate is given as H e,g (n) = γH g (n) + 1 − γ H g−1 (n) EURASIP Journal on Wireless Communications and Networking Real Imaginary P (n) (Chu) p (t) (Chu) (c) is the frequency-domain interpolation function with q = n/K for n = 0 ∼ N c − 1. Using (14), the MSE g (n) of the gth frame's filtered and interpolated channel gain estimate at the nth subcarrier is obtained as where R H (m − n) = E[H g (n)H * g (m)] is the channel frequency-correlation function, J 0 (x) is the zeroth order Bessel function of the first kind, and f D T f is the normalized Doppler frequency with 1/T f = 1/T c N m . We assume Jake's isotropic scattering model [15]. It can be seen from (16) that the gth frame's MSE g (n) of the channel estimator is a function of the previous (g − 1)th frame's MSE. To obtain average MSE in a steady-state condition, we let g → ∞ and thus, we obtain of AWGN, while the second term represents the influence of the channel time-selectivity. Thus, a tradeoff is present; as the filter coefficient γ increases (decreases), the channel estimator becomes more (less) robust against the channel time-selectivity while on the other hand, the estimator ability to reduce the noise decreases (improves). This tradeoff property computed using (17) is illustrated in Figure 8.

Simulation Results and Discussions
Simulation parameters are shown in Table 1. We assume QPSK data-modulation with N c = 256 and N m = 16. Chu sequence is used as the pilot given by p(t) = exp( jπt 2 /N m ) for t = 0 ∼ N m − 1 [14]. Propagation channel is an L = 16-path block Rayleigh fading channel having exponential power delay profile with decay factor α as shown in Figure 9. The zero-mean independent complex path gains {h l ; l = 0 ∼ L − 1} remain constant over one OFDM/TDM frame length and vary frame-by-frame. Without loss of generality, we assume τ 0 = 0 < τ 1 < · · · < τ L−1 and that the lth path time delay is τ l = lΔ, where Δ(≥ 1) denotes the time delay separation between adjacent paths. The maximum time delay of the channel is equal to the GI length (i.e., L = N g ). Figure 10   N g )T c ) is OFDM/TDM frame length. The figure shows that the proposed channel estimation provides a good MSE performance. As the value of γ decreases, the noise is reduced, but the tracking ability against fading tends to be lost. As f D T s increases, the channel varies faster and consequently, the higher γ is required to better track the fading variations. Thus, the optimum value of γ is not only a function of E b /N 0 but also the channel time-selectivity. As shown in Figure 10, the optimum γ opt , for minimizing the MSE for E b /N 0 = 5 (15) dB is about γ opt = 0.03 (0.04), 0.1 (0.15), and 0.3 (0.5) for f D T s = 0.0001, 0.001, and 0.01, respectively. Figure 11 illustrates the average BER performance using the proposed channel estimation as a function of E b /N 0 for f D T s = 0.0001 and α = 0 dB. The optimum γ is used for each E b /N 0 value. It can be seen from Figure 10 that the OFDM/TDM with the proposed channel estimation achieves a much better BER performance than OFDM; the required E b /N 0 for BER = 10 −3 reduces by about 6.5 dB in comparison with OFDM using TDM-pilot when The normalized Doppler frequency f D T s is an important parameter that affects the BER performance with pilotassisted channel estimation. To show the advantage of OFDM/TDM over the conventional OFDM, Figure 12 shows the BER performance with proposed channel estimation as a function of f D T s for E b /N 0 = 15 dB and α = 0 dB. The optimum γ opt is used for each f D T s value. For comparison, we also plot curves for OFDM with channel estimation using both FDM-pilot and TDM-pilot. The achievable BER of OFDM/TDM is lower than OFDM, but it slightly increases as f D T s increases.

BER Performance.
The above performance improvement is obtained at the cost of a slight increase in computational complexity in comparison with conventional OFDM [13].

Impact of Channel Frequency-Selectivity.
The channel frequency-selectivity is a function of channel decay factor α; as α increases the channel becomes less frequency-selective and when α → ∞ it approaches a frequency-nonselective channel (single-path channel). Figure 13 illustrates the average BER performance using the proposed channel estimation as a function of the channel decay factor α.
First we examine the impact of α on the channel estimation accuracy by evaluating the MSE. Figure 13(a) illustrates the dependency of MSE of channel estimation on α when E b /N 0 = 15 dB. The MSE is almost insensitive to f D T s . Figure 13(b) illustrates the BER dependency on α when E b /N 0 = 15 dB. The BER curve of ideal channel estimation case is also plotted as a reference. It can be seen that, as α increases, the BER slightly increases due to the reduced frequency diversity gain resulting from a weaker channel frequency-selectivity.

Fixed γ.
Since γ is one of the key parameters in the estimator, in this section, we discuss the robustness of the algorithm when γ is fixed. Figure 14 illustrates the average bit error rate (BER) performance with: (i) ideal CE, (ii) optimum γ (i.e., γ opt ) and (iii) fixed γ. The performance with ideal CE is illustrated as a reference.   The figure shows that for a lower E b /N 0 (i.e., E b /N 0 < 15 dB), the BER with fixed γ = 0.5 is almost the same as with γ opt . As expected, γ opt and fixed γ = 0.5 give the same BER at E b /N 0 = 15 dB because γ = 0.5 is optimum value at E b /N 0 = 15 dB and f D T s = 10 −2 . However, as E b /N 0 increases (i.e., E b /N 0 > 15 dB), the BER with fixed γ = 0.5 approaches a floor value of about BER = 10 −3 , while the performance with γ opt consistently improves. Figure 14(b) illustrates the BER performance as a function of f D T s at E b /N 0 = 15 dB. The figure shows that in a slow fading environment (i.e., f D T s < 10 −2 ) the BER with fixed γ = 0.5 slightly increases in comparison with γ opt ; for f D T s = 10 −3 the BER performance degradation with fixed γ = 0.5 is as small as 2 × 10 −3 in comparison with γ opt . As expected, γ opt and fixed γ = 0.5 give the same BER at f D T s = 10 −2 because γ = 0.5 is optimum value at E b /N 0 = 15 dB and f D T s = 10 −2 . However, as the value of f D T s further increases (i.e., the channel becomes more time selective) the BER rapidly increases as well.
The above results indicate that if a fixed γ is to be used, γ opt at the nominal value of E b /N 0 and f D T s in a considered scenario should be selected as the fixed γ.

Conclusions
In this paper, the performance evaluation of OFDM/TDM using MMSE-FDE with practical channel estimation in a fast fading channel was presented. A tracking against fast fading is improved by robust pilot-assisted channel estimation that uses time-domain first-order filtering on a slot-by-slot basis and frequency-domain interpolation. The MSE of the channel estimator using time-domain first-order filtering and frequency-domain interpolation was derived and then, a tradeoff between improving the tracking ability against fading and the noise reduction was discussed. It was shown that the OFDM/TDM using MMSE-FDE provides a lower BER and a very good tracking ability against fading in comparison with conventional OFDM while keeping the same data-rate transmission.