Internal note 9: Blind carrier frequency offset estimation for OFDM/OQAM over multipath channel

Transcription

1 Internal note 9: Blind carrier frequency offset estimation for OFDM/OQAM over multipath channel Gang Lin, Lars Lundheim, Nils Holte Department of Electronics and Telecommunications Norwegian University of Science and Technology NTNU 79 Trondheim, Norway {lingang; lundheim; March 8, 006 Introduction It is well known that multicarrier systems are much more sensitive to carrier frequency offset CFO then single carrier systems. The effect caused by CFO for OFDM/QAM system was analyzed in [. In [, the author indicated that CFO should be less than % of the band width of subchannel to guarantee the signal to interference ratio be higher than 30-dB. A critically sampled OFDM/OQAM system is also not robust to CFO [, even when optimal pulses are used as shaping filters [3. Bolcskei presented a blind CFO and STO estimation algorithm for OFDM system in [, which is a natural extension of the estimator for single carrier QAM transmitting system designed by Gini [5. For OFDM/OQAM system, Bolcskei s estimator can estimate only CFO and has relatively large mse thus can be used only for coarse CFO estimation. A fine CFO estimator for OFDM/OQAM is presented by Ciblat and Serpedin in [6. This estimator is an extension of the estimator introduced by Ciblat and Vandendorpe [7. Both Bolcskei and Ciblat estimators are based the second order statistics of the received sequence before demodulator. For Bolcskei estimator, cyclic correlation functions are used to estimate CFO, while channel impulse response needs to be known over multipath fading channel. Ciblat estimator utilizes the cyclic conjugate correlation functions. While for non-weighting OFDM/OQAM system over AWGN channel, it can be strictly proved that the received sequence before demodulator is circular, which leads the conjugate cyclic correlation function to be zero. Thus subchannel weighting is also needed for Ciblat estimator, which is as same as Bolcskei estimator. For OFDM/OQAM systems, time-frequency well-localized shaping pulses can be used [8[9. Thus large enough subchannel number, the equivalent channel can be approximated as flat-fading. It is obvious that flat-fading channel is easier to deal than frequency selective channel since flat-fading channel is just a special case of frequency selective channel. This motives us to estimate CFO based on the sequence out of receiver filters. In addition, since the sampling rate of the signal out of receiver filter is N/ times lower than that of immediately received signal, lower implementation complexity can be achieved. In this note, we just use cyclic conjugate correlation functions to estimate CFO the method based on correlation functions has been presented separately. It is unfortunately shown that our method also doesn t work for non-weighting OFDM/OQAM systems over AWGN channel. While for practical OFDM systems, null-subchannels are always in presence, which can be used to estimate CFO blindly. Simulation shows that Ciblat estimator doesn t work fine for OFDM/OQAM systems with sparsely distributed nullsubchannels, then we also consider interleave weighted OFDM/OQAM systems to make a comparison with Ciblat estimator. This work is supported by BEATS.

2 System model. Overview and definitions The discrete model of OFDM/OQAM is shown in Figure. This model has N subchannels, and subchannels are weighted by factors {w k } N k0. The weighting factor w k is real-valued to maintain the orthogonality. Figure : The time discrete model of base band OFDM/OQAM with frequency offset Each subchannel transmits one QAM symbol a k [n a R k [n + j ai k [n per T seconds. The OQAM symbols are formed by shifting the imaginary part of QAM symbols by T/. By summing up all the subchannels, the modulator generates a T/N sampled output sequence x[l N w m a Rm[n g[l nn + j a Im[n g[l nn N e j π N l+ π m that is also the input to the channel.

3 Here we consider stationary channel. If the number of subchannels N is large enough, the equivalent channel of the each subchannel can be approximated as flat-fading. The fading factors are denoted by {µ k } N k0. The channel model also includes an independent zero-mean white Gaussian additive noise ν[l with correlation function c ν [τ σν δ[τ. The carrier frequency offset f e is normalized with respect to /T. Then we can write the received sequence out of channel as N r[l e j π N f el w m µ m a Rm[n g[l nn + j a Im[n g[l nn N e j π N l+ π m + ν[l. In the subchannel k of receiver, the received sequence is first down-converted by modulator e j π N l+ π k, then filtered by the receiver filter f[l and N/ times down-sampled to generate a T/ spaced sequence b k [s r[l e j π N l+ π k f[l l N e j π N fel l ls N w m µ m e j π N l l + π m k f[l + ν[l e j π N l+ π k f[l N e j π N fel w m µ m l a R m [n g[l l nn + j a I m[n g[l l nn N ls N a R m [n g[l l nn + j a I m[n g[l l nn N e j π N l l + π m k f[l e j π N f el + ν[l e j π N l+ π k f[l ls N. 3 Note that although the sequence immediately before decimator or immediately out of receiver filter contains more information than the N/ down-sampled sequence b k [s, it is not possible in practice since for FFT based OFDM/OQAM systems [0[[[3, such decimator is merged into the FFT modular and the only output signal is b k [s. We define and p o m,k [l g[l ej π N l+ π m k f[l e j π N f el, ν o k [l ν[l e j π N l+ π k f[l. 5 We can see that p o m,k [l is actually the equivalent overall impulse response from subchannel m in the transmitter side to the subchannel k in the receiver side over ideal transmitting channel, and ν o k [l is the filtered noise. If g[l and f[l are bandlimited to [/T, /T as usually assumed for OFDM/OQAM systems and f e <, p o m,k [l 0 for m k >. Then we can simplify 3 as N b k [s e j π N fel N e jπfes w m µ m w m µ m N e jπf es w m µ m a R m[n p o m,k [l nn + j m k a I m[n p o m,k [l nn N + ν o k [l ls N a Rm[n p m,k [s N nn + j m k a Im[n p m,k [s N nn N + ν o k [sn a R m[n p m,k [s n + j m k a I m[n p m,k [s n + ν k [s, 6 where p m,k [s p o m,k [s N and ν k[s ν o k [s N are N/ times down-sampled versions of po m,k [l and νo k [l respectively. We assume that the data symbols are statistical independent between different subchannels, different symbols, and the real and imaginary parts are i.i.d., i.e. E [ a R m[n a R k [n E [ a I m[n a I k[n { σ a /, if m k and n n 0, otherwise, 3

4 and E [ a R m[n a I k[n 0, m, k, n, n, where the operator E[ represents statistical expectation and σ a is the average power of the QAM symbol.. The conjugate correlation function It has been reported that very precise estimated CFO can be attained based on the cyclic conjugate correlation function of the received sequence r[l [6. Now we will check the conjugate correlation function of b k [s, which is the output sequence of receiver filter in subchannel k. The conjugated correlation function is defined as c k [s, τ E [b k [s + τ b k [s. Based on the expression of b k [s shown in 6 and the assumptions above, we have c k [s, τ σ N a ejπfes+τ wm µ m + E [ν k [s + τ ν k [s. p m,k [s + τ n p m,k [s n p m,k [s + τ n p m,k [s n 7 By further assuming that the real and imaginary parts of ν[l are i.i.d., then based on 5, we find after a few intermediate steps that E [ν k [s + τ ν k [s 0. It s also possible to simplify the summation in 7. For s q, i.e. s is even, the summation can be written as p m,k [q + τ n p m,k [q n p m,k [q + τ n p m,k [q n p m,k [n + τ p m,k [n p m,k [n + τ + p m,k [n +. Similarly, for s q +, we have p m,k [q + + τ n p m,k [q + n p m,k [q + τ n p m,k [q n p m,k [n + τ p m,k [n p m,k [n + τ + p m,k [n +. Thus we have that c k [s, τ c k [s + q, τ and c k [s + q +, τ c k [s + q, τ. Then by defining A m,k τ, f e we can simplify 7 as where p m,k [n + τ p m,k [n p m,k [n + τ + p m,k [n +, 8 c k [s, τ σ N a ejπfes+τ/ wm µ m s A m,k τ, f e σ N a ejπfeτ e jπfe+/s wm µ m A m,k τ, f e r k τ, f e e jπf e+/s, 9 r k τ, f e σ N a ejπf eτ wm µ m A m,k τ, f e. 0 We can see clearly that c k [s, τ is cyclic stationary with a period f e +/. Thus for r k τ, f e 0, the spectrum with respect to s of c k [s, τ has a sharp peak at f e + /, which can be used to estimate

5 f e. By assuming that the transmitter f[l and receiver g[l are identical real-valued and symmetric pulses, and band-limited to [/T, /T as usually assumed in OFDM/OQAM systems, we find unfortunately that see appendix A that N A m,kτ, f e 0 for any value of f e. Then going back to 9, in the case of non-weighting system and ideal transmitting channel, i.e. w m, µ m, we have that c[s, τ 0. This implies that the information of f e will be totally lost. 3 Estimation algorithm In the previous section, we have proved that for the case of w k and µ k, the conjugate correlation function c k [s, τ will be zero. One method to remain frequency offset information is subchannel weighting, i.e., distributing individual subchannel different power. Here we consider two kinds of weighting methods: Method : Set w k 0 for L selected subchannels k, k,, k L, which can be referred as null-subchannel, while the other factors chosen as. If the null-subchannels are sparsely distributed and the kth subchannel is one null subchannel, it can be easily verified that c b,k [s, τ, c k [s, τ and c b,k+ [s, τ contain the information of f e. This implies that we can estimate f e from subchannels k, k and k +. While the useful signal power of subchannels k and k + is much lower than k, here we use only the output sequence of subchannel k to estimate CFO. Another practical problem is to choose τ. Recall that N A m,kτ, f e 0. If subchannel k is null-subchannel, we have c k [s, τ σ a µ k A k,kτ, f e they are actually equal for flat-fading channel immediately based on 9. If the shaping pulses are known, we can calculate A k,k τ, f e directly based on 8. Here we show an example of g[l is the square root raised cosine pulse with a roll-off factor α. The 3-D curves of A k,k τ, f e are shown in Figure. We can see that smaller roll off factor α leads to wider spread over τ. For the case of α.0, A k,k τ, f e is close to zero if τ >, thus we choose τ up to. Note that negative τ is just a shifted version of positive τ and thus contains no extra information. For the case of α 0., we can choose τ up to Amplitude Amplitude τ fe τ fe 0.5 a α 0. b α.0 Figure : The amplitude of A k,k τ, f e In practice only one set of finite-length data records {b k [s} M s0 is available, since c k [s, τ is a periodic with respect to s, it can be estimated by only one sample. We may denote this one sample estimation of c k [s, τ by y k [s, τ b k [s + τ b k [s. Then by defining y k [s [ y k [s, 0, y k [s,,, y k [s, L τ T, we can write the estimation algorithm as ˆf e,m arg max J M f f 0,, where J M f def 5 L M l M s0 y kl [s e jπfs.

6 We can see that J M f contains the contribution from different τs and different null-subchannels. Method : w {w, w, w, w,...}, where w w, and they should be real-valued to guarantee the orthogonality between subchannels. It is shown in appendix A that for f e < 0.5 and m k >, A m,k τ, f e 0. Thus we have c k [s, τ σ a µ k w w A k,k τ, f e they are actually equal for flat-fading channel. Thus CFO can be estimated over all subchannels, and we can write the estimation algorithm as ˆf e,m arg max J M f f 0,, where J M f def N M k M s0 y k [s e jπfs. 3 Weighting method is more practical than method since null-subchannel is always in presence for OFDM systems. Here we introduce weighting method is just to make a comparison with Ciblat estimator. Thus in the next section, we will only give theoretical analysis for weighting method. Asymptotic analysis In previous section, we have suggested two weighting methods and the corresponding estimation algorithms and 3. In this section, we will present the asymptotic analysis for weighting method, i.e. inserting of null-subchannels. Recall that y k [s, τ defined in is just the estimation of c k [s, τ by one sample. It is obvious that E [y k [s, τ c k [s, τ. Thus we can write y k [s, τ as y k [s, τ c k [s, τ + e k [s, τ. In fact e k [s, τ is just the estimation error and caused by two independent parts: one part comes from channel noise, another part comes from input symbol. It will be verified later that e k [s, τ is actually stationary with respect to time instant s. In practice, the duration of g[l should be finite. We define e k [s [ e k [s, 0 e k [s, e k [s, L τ T, then e k [s satisfies mixing condition: L N, s Z, ν i {0, }, cum L e ν k [s,, e ν L k [s L ML <, s,,s L where e 0 k [s i e k [s i, e k [s i e k [s i. We may explain more about mixing condition. If e k [s and e k [s become statistically independent as s s > P, then cum L e ν k [s,, e ν L k [s L 0, for max{ s i s j : i, j {,, L}} > P ref. [: Theorem.3. iii, p.9. This means a finite memory system can always satisfy mixing condition. Then based on this mixing condition and using a reasoning similar to Ciblat et al. [5, we find that see appendix that ˆf e,m is an unbiased estimation for f e, and the variance is asymptotically given by { L 3 l Re Ψ kl f e Ψ } kl f e γ π M 3 L, 5 l Φ k l f e where Ψ k f e r H k f e P k r k f e Ψ k f e r H k f e P k r kf e Φ k f e r H k f e r k f e. 6 6

7 Here r k f e is a vector given by r k f e [ r k 0, f e, r k, f e,, r k L τ, f e T and Pk, P k are matrices with entries given by [ Pk τ,τ S ek f e + /, τ, τ [ Pk τ,τ S ek f e + /, τ, τ 7 respectively, where S ek f e + /, τ, τ and S ek f e + /, τ, τ are given by 70 and 7 in appendix B respectively. 5 Simulation results First we present some preliminary simulation conditions which are used through out all simulations: 6OQAM modulation with power σa 0; g[l and p[l are square root raised cosine pulses with a roll off factor α.0; 3 Each result is obtained by averaging over 000 Monte Carlo trials; SNR def σa/σ ν 5 The number of subchannels: N 6. We further assume that the input QAM symbols are uniformly distributed, then E [ a R k [n 0. σ a. To find the maximum point of J M f, 6 times over sampling FFT is used to find the coarse peak, then simplex method is used to find the precise maximum point. In Figure 3, examples of J M f for M 56, fe 0., SNR 0 and 30 db are shown. We can see that for high SNR, there is a sharp peak around f e + 0.5, as shown in 3a. While for low SNR shown in 3b, the desired peak around f e is lower than peaks caused by noise, then false detection occurs. False detection can be avoided by appropriate initialization f /T f /T a SNR30dB b SNR0dB Figure 3: The curves of J M f Another practical problem is to choose L τ. Here we just show the theoretical mse versus L τ of estimator for a special case of M 56 and SNR 30 db. The results are shown in Figure. We can see that for α.0, the mse becomes saturated after L τ. For smaller roll-off factor α, we can gain more from increasing L τ. This is accordant to the results shown in Figure. Since the roll-off factor α.0, we will set L τ for both estimator and 3 in the following simulations. Simulation : false detection ratio versus SNR In this simulation, we set f e 0., L, L τ 3 and the false detection ratio is gotten over Monte Carlo trials. The simulated results are shown in Figure 5. We can see that the false detection ratio falls quickly with increasing SNR after a specific threshold, which can be used to explain the threshold effects of mse. 7

8 mse α.0, fe 0 α.0, fe 0. α.0, fe 0. α 0.5, fe 0 α 0.5, fe 0. α 0.5, fe 0. α 0., fe 0 α 0., fe 0. α 0., fe Lτ Figure : Theoretical mse versus L τ for estimator M8 M56 M5 false detection ratio SNR db Figure 5: False detection ratio versus SNR for different M Simulation : performance of estimator versus SNR Here we choose M 56, L. The bias and mse versus SNR for different f e are shown in Figure 6. We can see that after a specific threshold, the theoretical curves matches well with the simulated results. The threshold effect is caused by the false peak detection of J M f for high lever noise. Such false peak detection can be avoided by appropriate coarse CFO estimation. We also note that for f e 0. and 0., the simulated values deviate from theoretical prediction for high SNR. It will be shown soon that such a deviation disappears asymptotically with increasing M. Simulation 3: performance of estimator versus f e We still keep M 56, L. The bias and mse versus f e for different SNR are shown in Figure 7. We can see that in the acquisition range f e < 0.5, the estimator performance is almost independent to f e. It also shows that the estimator can work only after a specific SNR, which can be also observed in simulation. For high SNR, there exists gap between theoretical and simulated results. Simulation : performance of estimator versus M In this simulation, we set L. The bias and mse versus M for different SNR are shown in Figure 8. We can see that there exists obvious gap between theoretical and simulated results, while this gap decreases with increasing M. Simulation 5: performance of estimator versus L From 9, we can see that the theoretical mse should be inversely proportional to the number of nullsubchannels if the null-subchannels are sparsely distributed. We simulate one case f e 0., M 56. In our simulation, subchannels,6,0, are null-subchannels for L. The simulation results are shown 8

9 0 0 fe 0: theoretical fe 0.: theoretical fe 0.: theoretical fe 0: simulated fe 0.: simulated fe 0.: simulated 0 6 mse SNR db Figure 6: Mse versus SNR for different f e SNR 0 db: theoretical SNR 0 db: theoretical SNR 0 db: theoretical SNR 60 db: theoretical SNR 0 db: simulated SNR 0 db: simulated SNR 0 db: simulated SNR 60 db: simulated mse fe /T Figure 7: Mse versus f e for different SNR in Figure 9. We can see that more null-subchannels will lead to lower MSE and lower SNR threshold. It is interesting that although there exists obvious gap between theoretical and simulated results for high SNR, the relationship of simulated mse versus the number of null-subchannels is as same as theoretical prediction. Simulation 6: Comparison of estimators 9 and 0 In this simulation, we set M 56, f e 0. for both estimator 9 and 0. For estimator 0, we simulate two cases: w /, w 6/; w /, w 7/. The total power is kept to the same. For estimator 9, we simulate the case of L. The results are shown in Figure 0. We can see that estimator 0 is less sensitive to SNR if the SNR has exceeded the threshold. While the floor mse of estimator 9 is much lower than estimator 0. Simulation 7: performance of estimator over multipath channel In this simulation, we set M 56, f e 0.. First we simulate the estimator performance over time-invariant multipath channel. Here we consider a five paths channel, which has the impulse response h[l λ d δ[l d, 8 d0 where λ d is the complex-valued amplitude. First we consider a specific stationary multipath channel. Just for the purpose of simulation, we set [ [ λ 0 λ λ λ 3 λ Its frequency response is shown in Figure. Here we just use the frequency response at f k to approximate the flat-fading attenuation 9

10 0 5 SNR 0dB: theoretical SNR 0dB: theoretical SNR 0dB: theoretical SNR 60dB: theoretical SNR 0dB: simulated SNR 0dB: simulated SNR 0dB: simulated SNR 60dB: simulated 0 5 SNR 0dB: theoretical SNR 0dB: theoretical SNR 0dB: theoretical SNR 60dB: theoretical SNR 0dB: simulated SNR 0dB: simulated SNR 0dB: simulated SNR 60dB: simulated mse mse M M a f e 0. b f e 0. Figure 8: Mse versus M for different SNR and f e 0 L : theoretical L : theoretical L : simulated L : simulated mse SNR db Figure 9: Mse versus SNR for different L factor µ k of subchannel k, i.e. µ k d0 λ d e jπkd/6. Subchannel is set as the only null-subchannel. The simulation results is shown in Figure. We can see that the simulated curve matches well with theoretical prediction. This verifies that the flat-fading approximation of each subchannel is reasonable. Comparing the mse for AWGN and multipath channels, we find they are close. For low SNR section, the mse for multipath channel is even better than AWGN. This can be explained by that the amplitude of frequency response of multipath channel around subchannel is largely higher than as shown in Figure, which implies that the equivalent SNR is higher than AWGN channel. Anyway, we can conclude that estimator is robust to multipath effects. Now we change the coefficients λ d over each trial. Just for the purpose of simulation, we set the covariance of λ d to be /5 to keep the average power of received signal as same as AWGN channel. The coefficients λ d is independently randomly selected over each trial. The simulation results of estimator and 3 are shown in Figure 3a and 3b respectively. We can see that the unknown multipath channel will degrade estimator performance which lies in two aspects: higher SNR threshold; slightly larger mse in high SNR section. For estimator, the degradation is less obvious for larger L. For large L, the probability of that all null-subchannels for CFO estimation are in low power is much smaller, then the performance should be less sensitive to frequency-selective fading. Then we simulate estimator over multipath Rayleigh-fading channel. Now the amplitude λ d in 8 will be relative to time instant l. In our simulation, λ d [l are identical distributed for all paths and modelled as a lowpass autoregressive precess with fifth order multiple poles at ρ, which can be expressed as λ d [l 5ρλ d [l 0ρ λ d [l + 0ρ 3 λ d [l 3 5ρ λ d [l + ρ 5 λ d [l 5 + n s [l, 9 0

11 0 Estimator 9: L Estimator 0: w /, w 6/ Estimator 0: w /, w 7/ 0 mse SNR db Figure 0: Mse versus SNR for different methods Hf f /T Figure : The frequency response of transmitting channel where n s [l is the complex-valued zero-mean Gaussian white noise with variance σn. The frequency response of such a filter is Hf and the 3-dB bandwidth B ρe jπf 5 λ T is related to ρ by the equation 5 ρ [ 5 cosπb λ T ρ Since σλ d c λd [0 σn Hf df, we set σn 0./ Hf df to let s0 σ λ d. We simulate two cases: B λ T 0.00 slow fading and B λ T 0.0 fast fading. The simulation results are shown in Figure. We can see that estimator is not robust to multipath Rayleigh-fading channel. However, for slow Rayleigh-fading channel, estimator can still get reliable CFO estimation. Simulation 8: Comparison with Bolcskei estimator Here we compare estimator to Bolcskei estimator [. The formula 5 in [ is used for estimation of carrier frequency offset θ e and timing offset n e. Note that for OFDM/OQAM system, M N, which was clearly indicated in 3 of [. Replacing M by N, we copy 5 in [ as where C r [k, τ N ejπθ eτ e j π N kn e Γ N [τ A g,g [ τ, k N Γ N [τ [ A g,g τ, k N N k0 l [σ c,r + k σ c,i + cρ [τ δ[k, 0 w k e j π N kτ g[l g[l τe j π N kl,

12 0 AWGN: theoretical Multipath: theoretical AWGN: simulated Multipath: simulated 0 mse SNR db Figure : Mse versus SNR over multipath channel 0 0 AWGN: L AWGN: L Multipath: L Multipath: L AWGN: w /, w 6/ AWGN: w /, w 7/ Multipath: w /, w 6/ Multipath: w /, w 7/ 0 mse 0 6 mse SNR db SNR db a Estimator b Estimator 3 Figure 3: The performance of estimator and 3 over AWGN and time-invariant multipath channel and σc,r and σ c,i are respectively the average power of the real and imaginary parts of sent QAM symbols. The frequency offset θ e in [ is normalized with respect to N/T, while our expression f e is with respect to /T, thus f e Nθ e. Now we assume σc,r σ c,i and the transmitter pulse gt is band-limited to [/T, /T. By using Parseval s relation, we can rewrite A [ g,g τ, k N shown in as [ A g,g τ, k g[l g[l τ e j π N kl N l Gf Gf + k N e jπfτ df, where Gf l g[l e jπfl. Since g[l is the discrete form of gt with a sampling interval T/N, Gf should be band-limited to [/N, /N. This leads to A [ g,g τ, k N is nonzero only if k 0, ±. While for k ±, σc,r + k σc,i 0 because of the assumption of σ c,r σ c,i. Thus we conclude that C r [k, τ can be used for estimation only for k 0. In this case, only frequency offset can be recovered and the accuracy will be affected by the term c ρ [τ caused by noise. But the weighting factor w k is still important to keep Γ N [τ nonzero for τ [0, N. At last, the estimator can be expressed as ˆθ e π L τ τ τ arg {Ĉr [0, τ Γ N [τ }. 3 Note that since A g,g [ τ, k N is real-valued and positive, we don t need to consider it s effect.

13 0 0 mse AWGN: L AWGN: L Multipath channel B λ T0.00: L Multipath channel B λ T0.00: L Multipath channel B λ T0.0: L Multipath channel B λ T0.0: L SNR db Figure : Mse versus SNR of estimator over AWGN and multipath Rayleigh-fading channel In simulation, g[l is square root raised cosine pulse with a roll off factor and L τ 5. The weighting factors are chosen as: w k except w 9 0. One can verify that Γ N [τ 0 for all τ [0, N. The comparison of performance versus SNR is shown in Figure 5. The simulation conditions are the same: M 56, f e 0.. From the simulation results, we can see that estimator is far more accurate than Bolcskei estimator except the low SNR section of SNR< db. 0 0 Estimator 9 with L Bolcskei estimator mse SNR db Figure 5: The comparison of performance versus SNR between estimator 9 and Bolcskei estimator Then we compare the performance versus M for different SNR. Here we simulate two cases: SNR 0 db low SNR and 0 db high SNR. The frequency offset f e 0.. The simulation results are shown in Figure 6. We can see that for SNR 0 db, estimator will exceed Bolcskei estimator when M > 850; while for SNR 0 db, estimator works above the SNR threshold and is always superior to Bolcskei estimator. This implies that estimator is better than Bolcskei estimator for large enough M. In fact from Figure6b, we can see clearly that mse level of estimator decreases much faster than that of Bolcskei estimator with increasing M. Simulation 9: Comparison of estimator 3 to Ciblat estimator In [6, Ciblat and Serpedin designed a blind CFO estimation algorithm based on conjugate cyclostationarity of the received sequence. However, the authors made a little small mistake on the model of OFDM/OQAM, thus their model is equal to adding a weighting factor j m to the mth subchannel. If there is no weighting, it can be proved that Ciblat and Serpedin s estimator doesn t work either, which is same to Bolcskei estimator [ and our estimator. Simulation shows that Ciblat and Serpedin s estimator works poorly for weighting method, i.e. adding some null-subchannels. Thus we compare only estimator 3 to Ciblat and Serpedin s. We set f e 0. and M 56. The simulation results are shown in Figure 7. We can see that for the same weighting factors, estimator 3 exceeds Ciblat and Serpedin s 3

14 Estimator 9 with L: 0dB Estimator 9 with L: 0dB Bolcskei estimator: 0dB Bolcskei estimator: 0dB bias Estimator 9 with L: 0dB Estimator 9 with L: 0dB Bolcskei estimator: 0dB Bolcskei estimator: 0dB mse M M a Bias versus M b Mse versus M Figure 6: The comparison of performance versus M between estimator 6 and Bolcskei estimator estimator. From Figure 5, we can see that estimator 3 has lower SNR threshold and mse. Especially for w /, w 6/, estimator 3 s SNR threshold is about 6dB while Ciblat and Serpedin s doesn t work well even for every high SNR 60dB. For this case, we should increase M to make Ciblat estimator work Estimator 0: w /, w 6/ Estimator 0: w /, w 7/ Ciblat estimator: w /, w 6/ Ciblat estimator: w /, w 7/ 0 mse SNR db Figure 7: The comparison of mse versus SNR of estimator 3 to Ciblat estimator A Proof of N A m,kτ, f e 0 Proof. First by defining P m,k f s p m,k[s e jπsf note here P m,k f is normalized with respect to /T instead of the sampling rate /T, thus P m,k f should be periodic to f with a period instead of, we can use the decimator formula see formular..3 in [6 to get P m,k f def p m,k [n e jπnf Pm,k f + P m,k f. It is more difficult to get the frequency response of the decimated filter p m,k [n +. We can view p m,k [n+ as the times decimated version of p m,k [s+, while s p m,k[s+ e jπsf e jπf P m,k f, then we have P m,k f def p m,k [n + e jπnf ejπf P m,k f P m,k f. 5

15 We can use the discrete form of Parseval s relation, i.e. g [n g [n G f G f df, where G f g [n e jπfn and G f g [n e jπfn. For the case of τ q, i.e. τ is even, by using and 5, we can rewrite 8 as A m,k τ, f e Similarly, for τ q +, we have A m,k τ, f e By combining 6 and 7, we have A m,k τ, f e Then we define p m,k [n p m,k [n + q p m,k [n + p m,k [n + q + P m,k f P m,k f P m,k f P m,k f e jπfq df P m,k f P m,k f + P m,k f P m,k f e jπfτ df. 6 p m,k [n p m,k [n + q + p m,k [n + p m,k [n + q + P m,k f P m,k f e jπfq P m,k f P m,k f e jπfq+ df P m,k f P m,k f + P m,k f P m,k f e jπfτ df. 7 P m,k f P m,k f e jπfτ + P m,k f P m,k f e jπf+τ df 0.5 P m,k f P m,k f e jπfτ df P m,k f P m,k f e jπfτ df P m,k f P m,k f e jπfτ df. 8 T m,k f P m,k f P m,k f 9 that is the integral kernel of 8. From the definition of p m,k [l shown in and the relationship of p m,k [s p m,k [ N s, we can write the spectrum of p m,k [s as P m,k f e j π m k Gf m k n Gf + f e n, 30 where Gf is the spectrum of square root raised cosine pulse with a roll off factor α. Note that Gf is band limited to [ +α, +α normalized with respect to /T, then P m,kf is just a periodic extension of Gf m k Gf + f e. Substituting 30 into 9, we have [ T m,k f m k Gf m k n Gf + f e n [ n n m k G f m k n G f + f e n n n [ Gf m k n Gf + f e n Gf + m k + n + Gf + n + f e. 3 5

16 Without the loss of generality, we may assume 0 f e < 0.5. Then P m,k f is nonzero only if m {k, k, k, k + }. First we check the case of m k. Substituting m k into 3, we have T k,k f n n Gf n + Gf + f e n Gf + n Gf + n + f e. 3 Since Gf is nonzero only if f [,, it can be easily verified that the products of Gf n + Gf + n and Gf + f e n Gf + n + f e are nonzero only if < n + n < 5 and 3 + f e < n + n < + f e respectively. Since 0 f e < 0.5, and n and n is integer, the two conditions can t be satisfied in the same time, we can conclude that T k,k f 0. This implies that A k,k τ, f e 0. Then we try to analyze the case of m k: T k,k f a n n n n n n Gf n Gf + f e n Gf + n + Gf + n + f e Gf n Gf + f e n Gf + n + Gf + n + f e [ Gf n Gf + fe n Gf n Gf n f e + Gf n Gf + f e n Gf n + Gf n + f e, 33 where a follows from the fact that the product of Gf n Gf + f e n Gf + n + Gf + n + f e is nonzero only if n { n, n }. For the case of m k : T k,k f a n n Gf n + Gf + f e n Gf + n Gf + n + f e Gf n + Gf + f e n Gf n Gf n + f e Gf n Gf + f e n Gf n + Gf n + f e, 3 where a follows from the fact that the product of Gf n + Gf + f e n Gf + n Gf + n + f e is nonzero only if n n. At last for the case of m k + : T k+,k f n n Gf n Gf + f e n Gf + n + Gf + n + f e Gf n Gf + f e n Gf n Gf n f e Gf n Gf + f e n Gf n Gf n f e. 35 Thus from 33, 3 and 35, we can get that N T m,k f k+ mk T m,k f 0 36 which implies that N A m,kτ, f e 0. 6

17 B Derivation of asymptotical variance Since e k [s satisfies the mixing condition, we can use the Lemma introduced by Ciblat [5, which can be written as: Lemma. Let then s K def M,k f K N, sup M M K+ s K f [0, s0 s K e k [s e jπfs, M,k f a.s. 0, as M. 37 Here a.s. stands for almost sure. Then we have the below theorems: Theorem. For e k [s, τ y k [s, τ ĉ k [s, τ satisfies mixing condition, we have ˆf e,m f e and M ˆf e,m f e a.s. 0 as M. a.s. 0 Proof. Recalling that y k [s, τ ĉ k [s, τ + e k [s, τ, we immediately have y k [s r k f e e jπf e+/s + e k [s and can write the objective function in as J M f M L M M s0 L r k l f e l l r kl f e e jπfe+/s + e kl [s e jπfs M M s0 e jπf e+/ fs + s 0 M,k l f As M, M s0 ejπfe+/ fs δ[f e + / f; and by Lemma, s 0 0. Then arg max f 0, J M f a.s. a.s. f e + /, ˆfe,M f e 0. Also based on Lemma 3 shown in [5, M ˆf e,m f e a.s. 0 as M ; otherwise M M s0 ejπfe+/ fs will not converge to δ[f e + / f. Theorem. M 3/ ˆf e,m f e is asymptotically Gaussian.. M,k l f a.s. Proof. Since ˆf e,m arg max f 0, J M f, we have dj M f df f ˆfe,M + 0. Then using the first order Taylor expansion of dj M f df dj M f df f ˆfe,M + dj M f df ffe + around f e +, we have + ˆf e,m f e d J M f df ffξ +, where f ξ lies between f e and ˆf e,m. Then we have where M 3/ ˆf e,m f e A M B M, 38 A M M d J M f df B M dj M f M df ffξ + ffe

18 Now we try to calculate A M for M. First we define Y kl f M J M f L l Y k l f and d J M f L d Yk H df l f Y kl f L Y kl f df f l It can be easily calculated that K Y kl f f K M l M s0 y k l [s e jπfs, then { } + Re Yk H l f Y kl f f. M s0 jπsk y kl [s e jπfs. Then we have A M L { } Y kl f M f + Re Yk H l f Y kl f f l ffξ + L M M jπs y kl [s e jπfs M l s0 ff ξ + L M H + M Re M y kl [s e jπfs jπs y kl [s e jπfs M M l s0 s0 ffξ +. Recalling that y kl [s r kl f e e jπfe+ s + e kl [s, then as M, f ξ f e, and based on Lemma, the contribution of e kl [s disappear, we have A M a.s. π M π 3 M s0 L s r kl f e l π M 3 M s0 s L l r kl f e L r kl f e. 0 l Now we will show that B M is asymptotically Gaussian distributed as M. First we have B M dj M f L { M df Re Yk H ffe+ l M l f Y } k l f f ff e+ L M H Re j M y kl [s e jπfs π M M s y kl [s e jπfs M l s0 s0 ffe + L Im r kl f e + M H M e kl [s e jπfe+ s π r kl f e M M s + π M M M s e kl [s e jπfe+ s M l s0 s0 s0 L π r H M k Im l f e M M H s e kl [s e jπf e+ s π M + M M e kl [s e jπf e+ s r kl f e M l + l M M s0 s0 e kl [s e jπf e+ s H π M M M s0 s0 s e kl [s e jπf e+ s L { Im π r H k l f e E M,k l f e + π M + E 0H M,k M l f e + r k l f e + π s 0H M,k l f e + E M,k l f e + }, where M E K M,k f M K s K e k [s e jπfs. M s0 8

19 It can be proved that E K M,k f e + is asymptotically zero-mean Gaussian distributed confer [5 for more details. As M, we have s 0H M,k f e + a.s. 0 according Lemma, and M a.s. M. Therefore we can rewrite as B M a.s. j π L R H k l E M,kl, 3 l where R k E M,k [ r H k fe r H k f e rt k fe [ E 0T M,k f e + ET M,k f e + H r T k f e E0H M,k f e + EH M,k f e + T. a.s. Then B M N 0, σb, as M, where σ B is asymptotic covariance of B M. Thus we can conclude that M 3/ ˆf e,m f e a.s. N 0, σ with covariance σ 9 σ B π. L l r kl f e Base on Theorem, the mse of ˆfe,M can be immediately written as γ π M. 3 L l r kl f e Then the only work left is to calculate σb. For sparsely distributed null-subchannel indices {k l} L l, e k l [s are independent since the shaping pulse f[l and g[l are bandlimited to [/T, /T. Then based on 3, we have σ B E [B M B M π [ The matrix E E M,k E H M,k can be expressed as L l 9 σ B R H k l E [ E M,kl E H M,k l Rkl. 5 P M,k 0, 0 P M,k 0, PM,k 0, 0 PM,k 0, E [ E M,k E H P M,k, 0 P M,k, PM,k, 0 PM,k, M,k P M,k 0, 0 P M,k 0, P M,k 0, 0 P M,k 0,, 6 P M,k, 0 P M,k, P M,k, 0 P M,k, where [ P M,k K, K E E K M,k f e + EK H M,k f e + P M,k K, K E M M M K +K + s 0 s 0 [ E K M,k f e + EKT M,k f e + M K +K + M M s 0 s 0 s K s K E [ e k [s e H k [s e jπfe+ s s s K s K E [ e k [s e T k [s e jπf e+ s +s. We further define R ek [λ, τ, τ E [e k [s + λ, τ e k [s, τ and R ek [λ, τ, τ E [e k [s + λ, τ e k [s, τ e jπf es. It will be shown later that R ek [λ, τ, τ, Rek [λ, τ, τ are not a function of s, i.e. wide sense stationary, so we omit the time instant s. As M, by using the mixing condition, we can write the entries of 9

20 P M,k K, K and P M,k K, K as [ lim PM,k K, K M τ,τ lim M M K+K+ M M s 0 s 0 M lim M M K +K + s 0 λ lim [ PM,k K, K M τ,τ lim M where s K s K R ek [s s, τ, τ e jπfe+ s s s λs M+ R ek [λ, τ, τ e jπf e+ λ K + K + S e k f e +, τ, τ M K +K + M M s 0 s 0 M lim M M K +K + s 0 λ s K s K s λs M+ R ek [λ, τ, τ e jπfe+ λ s K s λ K R ek [λ, τ, τ e jπf e+ λ lim M M M K+K+ s 0 s K+K R ek [s s, τ, τ e jπfe+s e jπfe+ s+s s K s λ K Rek [λ, τ, τ e jπf e+ λ lim M M M K+K+ s 0 s K +K K + K + S ek f e +, τ, τ, 7 S ek f, τ, τ def S ek f, τ, τ def λ λ R ek [λ, τ, τ e jπfλ R ek [λ, τ, τ e jπfλ. 8 Using 7, we immediately have that P M,kl 0, P M,kl, 0 P M,k l 0, 0, P M,kl, 3 P M,k l 0, 0 and P M,kl 0, P M,kl, 0 P M,kl 0, 0, PM,kl, P 3 M,kl 0, 0 as M. Then by substituting 6 into 5, we have σ B π 3 L i { Re r H k l f e P M,kl 0, 0 r kl f e r H k l f e P } M,kl 0, 0 r k l f e. 9 By shortening the denotements P M,k 0, 0, P M,k 0, 0 to P k, P k respectively, and substituting the expression of σb into γ, we then have 9. While the work is still not finished since S ek f, τ, τ, S ek f, τ, τ are not known. First we need to calculate R ek [λ, τ, τ and R ek [λ, τ, τ. A. calculation of cross correlation function of e k [s, τ Recalling that e k [s, τ y k [s, τ c k [s, τ b k [s + τ b k [s r k τ, f e e jπfe+/s, we have R ek [λ, τ, τ E [e k [s + λ, τ e k[s, τ E [b k [s + λ + τ b k [s + λ r k τ, f e e jπf e+/s+λ b k [s + τ b k [s r k τ, f e e jπf e+/s E [ b k [s + λ + τ b k [s + λ b k[s + τ b k[s r kτ, f e E [b k [s + λ + τ b k [s + λ e jπf e+/s r k τ, f e E [b k[s + τ b k[s e jπf e+/s+λ + r k τ, f e r kτ, f e e jπf e+/λ E [ b k [s + λ + τ b k [s + λ b k[s + τ b k[s r kτ, f e c k [s + λ, τ e jπf e+/s r k τ, f e c k[s, τ e jπf e+/s+λ + r k τ, f e r kτ, f e e jπf e+/λ E [ b k [s + λ + τ b k [s + λ b k[s + τ b k[s r k τ, f e r kτ, f e e jπf e+/λ. 50 0

21 Then based on 6, we can continue the first term in the right-hand side of 50 as E [ b k [s + λ + τ b k [s + λ b k[s + τ b k[s [ E e jπf es+λ+τ N w m µ m a R m[n p m,k [s + λ + τ n + j m k a I m[n p m,k [s + λ + τ n e jπf es+λ N e jπfes+τ N w m µ m w m µ m + νk[s + τ N e jπfes w m µ m e jπf eλ+τ τ n,n,n 3,n N m,m,m 3,m 0 E + ν k [s + λ + τ a R m[n p m,k [s + λ n + j m k a I m[n p m,k [s + λ n + ν k [s + λ a R m[n p m,k[s + τ n j m k a I m[n p m,k[s + τ n a R m[n p m,k[s n j m k a I m[n p m,k[s n + νk[s w m w m w m3 w m µ m µ m µ m 3 µ m [ a R m [n p m,k[s + λ + τ n + j m k a I m [n p m,k[s + λ + τ n a R m [n p m,k[s + λ n + j m k a I m [n p m,k[s + λ n a R m 3 [n 3 p m 3,k[s + τ n 3 j m3 k a I m 3 [n 3 p m 3,k[s + τ n 3 a R m [n p m,k[s n j m k a I m [n p m,k[s n + σ N ejπf eλ E [ν k [s + λ + τ νk[s + τ w m µ m p m,k [n + λ p m,k[n + σ N a ejπf eλ τ E [ν k [s + λ + τ νk[s w m µ m + σ N ejπfeλ+τ E [ν k [s + λ νk[s + τ w m µ m + σ N ejπf eλ+τ τ E [ν k [s + λ νk[s w m µ m p m,k [n + λ τ p m,k[n p m,k [n + λ + τ p m,k[n p m,k [n + λ + τ τ p m,k[n + E [ν k [s + λ + τ ν k [s + λ ν k[s + τ ν k[s. 5

22 Then we can continue the summation in the first term in the right-hand side of 5 as N m,m,m 3,m 0 n,n,n 3,n w m w m w m3 w m µ m µ m µ m 3 µ m [ E a R m [n p m,k[s + λ + τ n + j m k a I m [n p m,k[s + λ + τ n a R m [n p m,k[s + λ n + j m k a I m [n p m,k[s + λ n a R m 3 [n 3 p m 3,k[s + τ n 3 j m3 k a I m 3 [n 3 p m 3,k[s + τ n 3 a R m [n p m,k[s n j m k a I m [n p m,k[s n [ a R E k [n 3 N σ w m µ m p m,k [n + λ + τ p m,k [n + λ p m,k[n + τ p m,k[n [ N + σ w m µ m p m,k [n + λ + τ p m,k [n + λ p m,k [n + λ + τ + p m,k [n + λ + [ N w m µ m p m,k [n + τ p m,k[n p m,k[n + τ + p m,k[n + + σ a + σ a [ N [ N w m µ m w m µ m pm,k [n + λ + τ p m,k[n [ N pm,k [n + λ + τ τ p m,k[n [ N w m µ m w m µ m pm,k [n + λ τ p m,k[n pm,k [n + λ p m,k[n. 5 Since ν k [s is the filtered and N/ times down-sampled noise, and f[l is bandlimited to [/T, /T, we have ν k [s N l ν[l e j π N l+ π k f[ N s l s Then we can rewrite the last term in the right-hand side of 5 as ν[ N s e jπs+ π k f[ N s s. 53 E [ν k [s + λ + τ ν k [s + λ νk[s + τ νk[s [ N E ν[ N s e jπs+ π k f[ N s + λ + τ s ν[ N s e jπs+ π k f[ N s + λ s s s ν[ N s 3 e jπs3+ π k f[ N s + τ s 3 ν[ N s e jπs+ π k f[ N s s E s 3 [ ν[l σ ν s N f[ N s f[n s + τ f[ N s + λ f[n s + λ + τ s f[ N s f[n s + λ [ N + σν f[ N s f[n s + λ + τ τ s s + f[ N s f[n s + λ + τ f[ N s f[n s + λ τ. 5 s s

23 We assume that f[l and g[l are identical real-valued symmetric pulses. Reminding that ν[l is zeromean white Gaussian noise with correlation function σν δ[τ, and the real and imaginary parts of ν[l are i.i.d., we can write the correlation function of filtered noise ν d k [s as [ c νk [τ E ν d k [s + τ ν k[s [ E ν[l e j π N l + π k f[ N s + τ l ν[l e j π N l + π k f[ N s l σ ν l l l f[l + N τ f[l σ ν p d t [τ, 55 where p d t f[l f[ l l N s f[l g[l l N s is N/ times down-sampled filter of the overall response of the cascade of transmitter and[ receiver filters. Similarly, the fourth order moment E ν[l can be written as [s def [ E ν[l [ E Re {ν[l} + Im {ν[l} + Re {ν[l} Im {ν[l} σν. 56 3

24 Then by substituting the results of 5, 5, 5, 55 and 56 into 50, we have [ a R ek [λ, τ, τ e {E jπfeλ+τ τ R k [n 3 σ a N w m µ m p m,k [n + λ + τ p m,k [n + λ p m,k[n + τ p m,k[n [ N + σ w m µ m p m,k [n + λ + τ p m,k [n + λ p m,k [n + λ + τ + p m,k [n + λ + [ N w m µ m p m,k [n + τ p m,k[n p m,k[n + τ + p m,k[n + + σ a + σ a + σ a σ ν + σ a σ ν + σ a σ ν [ N [ N w m µ m w m µ m pm,k [n + λ + τ p m,k[n [ N pm,k [n + λ + τ τ p m,k[n [ N N e jπfeλ p d t [λ + τ τ w m µ m N e jπfeλ τ p d t [λ + τ w m µ m N e jπf eλ+τ p d t [λ τ w m µ m w m µ m p m,k [n + λ p m,k[n w m µ m p m,k [n + λ τ p m,k[n p m,k [n + λ + τ p m,k[n pm,k [n + λ τ p m,k[n + σ a σν N e jπf eλ+τ τ p d t [λ w m µ m p m,k [n + λ + τ τ p m,k[n [ N + σν f[ N s f[n s + λ + τ τ f[ N s f[n s + λ s s + f[ N s f[n s + λ + τ f[ N s f[n s + λ τ s r k τ, f e r kτ, f e e jπfe+/λ s pm,k [n + λ p m,k[n } 57 We note that R ek [λ, τ, τ is not a function of s. This confirms that e k [s, τ is stationary. We can further simplify the expression. Based on 8, we have p m,k [n + λ + τ p m,k [n + λ p m,k [n + λ + τ + p m,k [n + λ + λ A m,k τ, f e p m,k [n + τ p m,k[n p m,k[n + τ + p m,k[n + A m,kτ, f e. 58

25 Then based on the definition of r k τ, f e shown in 0, we have σa ejπf eλ+τ τ [ N [ N w m µ m σ a λ e jπf eλ w m µ m p m,k [n + λ + τ p m,k [n + λ p m,k [n + λ + τ + p m,k [n + λ + p m,k [n + τ p m,k[n p m,k[n + τ + p m,k[n + e jπf eτ N w m µ m A m,k τ, f e r k τ, f e r kτ, f e e jπf e+/λ. e jπf eτ N w m µ m A m,k τ, f e Thus we can simplify R ek [λ, τ, τ as { [ R ek [λ, τ, τ e jπf eλ+τ τ a R E k [n 3 σ a N + σ a + σ a w m µ m [ N [ N w m µ m p m,k [n + λ + τ p m,k [n + λ p m,k[n + τ p m,k[n w m µ m pm,k [n + λ + τ p m,k[n [ N + σ N ejπfeλ p d t [λ + τ τ w m µ m pm,k [n + λ + τ τ p m,k[n [ N + σ N ejπf eλ τ p d t [λ + τ w m µ m + σ N ejπf eλ+τ p d t [λ τ w m µ m w m µ m p m,k [n + λ p m,k[n p m,k [n + λ τ p m,k[n p m,k [n + λ + τ p m,k[n w m µ m pm,k [n + λ τ p m,k[n pm,k [n + λ p m,k[n } + σ N ejπfeλ+τ τ p d t [λ w m µ m p m,k [n + λ + τ τ p m,k[n [ N + σν f[ N s f[n s + λ + τ τ f[ N s f[n s + λ s s + f[ N s f[n s + λ + τ f[ N s f[n s + λ τ. 59 s s B. calculation of conjugate cross correlation function of e k [s, τ 5

26 First based on the definition of R ek λ, τ, τ e jπfes E [e k [s + λ, τ e k [s, τ, we have R ek [λ, τ, τ e jπfes E [e k [s + λ, τ e k [s, τ e jπfes E [b k [s + λ + τ b k [s + λ r k τ, f e e jπf e+/s+λ b k [s + τ b k [s r k τ, f e e jπf e+/s e jπfes E [ b k [s + λ + τ b k [s + λ b k [s + τ b k [s r k τ, f e E [b k [s + λ + τ b k [s + λ e jπf e+/s [ r k τ, f e E b k [s + τ b k [s e jπfe+/λ s + r k τ, f e r k τ, f e e jπfe+/λ e jπfes E [ b k [s + λ + τ b k [s + λ b k [s + τ b k [s r k τ, f e c k [s + λ, τ e jπfe+/s r k τ, f e c k [s, τ e jπfe+/λ s + r k τ, f e r k τ, f e e jπfe+/λ e jπfes E [ b k [s + λ + τ b k [s + λ b k [s + τ b k [s r k τ, f e r k τ, f e e jπfe+/λ. 60 Then based on 6, we can continue the first term in the right-hand side of 60 as e jπfes E [ b k [s + λ + τ b k [s + λ b k [s + τ b k [s [ e jπf es E e jπf es+λ+τ N w m µ m a R m[n p m,k [s + λ + τ n + j m k a I m[n p m,k [s + λ + τ n e jπfes+λ N e jπf es+τ e jπf es N w m µ m N w m µ m w m µ m N e jπfeλ+τ+τ n,n,n 3,n m,m,m 3,m 0 E + ν k [s + λ + τ a R m[n p m,k [s + λ n + j m k a I m[n p m,k [s + λ n + ν k [s + λ a R m[n p m,k [s + τ n + j m k a I m[n p m,k [s + τ n + ν k [s + τ a R m[n p m,k [s n + j m k a I m[n p m,k [s n w m w m w m3 w m µ m µ m µ m3 µ m + ν k [s [ a R m [n p m,k[s + λ + τ n + j m k a I m [n p m,k[s + λ + τ n a R m [n p m,k[s + λ n + j m k a I m [n p m,k[s + λ n a R m 3 [n 3 p m3,k[s + τ n 3 + j m3 k a I m 3 [n 3 p m3,k[s + τ n 3 a R m [n p m,k[s n + j m k a I m [n p m,k[s n + e jπf es E [ν k [s + λ + τ ν k [s + λ ν k [s + τ ν k [s. 6 6

27 The summation in the first term of 6 can be continued as N m,m,m 3,m 0 n,n,n 3,n w m w m w m3 w m µ m µ m µ m3 µ m [ E a R m [n p m,k[s + λ + τ n + j m k a I m [n p m,k[s + λ + τ n a R m [n p m,k[s + λ n + j m k a I m [n p m,k[s + λ n a R m 3 [n 3 p m3,k[s + τ n 3 + j m3 k a I m 3 [n 3 p m3,k[s + τ n 3 a R m [n p m,k[s n + j m k a I m [n p m,k[s n [ a R E k [n 3 N σ w m µ m p m,k [n + λ + τ p m,k [n + λ p m,k [n + τ p m,k [n [ N + σ w m µ m p m,k [n + λ + τ p m,k [n + λ p m,k [n + λ + τ + p m,k [n + λ + [ N w m µ m p m,k [n + τ p m,k [n p m,k [n + τ + p m,k [n + [ N + σ a w m µ m p m,k [n + λ + τ p m,k [n + τ p m,k [n + λ + τ + p m,k [n + τ + [ N w m µ m p m,k [n + λ p m,k [n p m,k [n + λ + p m,k [n + [ N + σ w m µ m p m,k [n + λ + τ p m,k [n p m,k [n + λ + τ + p m,k [n + [ N w m µ m p m,k [n + λ p m,k [n + τ p m,k [n + λ + p m,k [n + τ +. 6 Now we try to calculate the second term in the right-hand side of 6 as E [ν k [s + λ + τ ν k [s + λ ν k [s + τ ν k [s E [ν k [ N s + λ + τ ν k [ N s + λ ν k[ N s + τ ν k [ N s [ E ν[l e j π N l+ π k f[ N s + λ + τ l l l 3 ν[l 3 e j π N l 3+ π k f[ N s + τ l 3 l l ν[l e j π N l+ π k f[ N s + λ l ν[l e j π N l + π k f[ N s l E [ν[l f[l f[l + N τ f[l + N λ f[l + N λ + τ e j π N l+ π k 0. l [ E Re {ν[l} 6 E [Re {ν[l} l f[l f[l + N τ f[l + N λ f[l + N λ + τ e j π N l+ π k 63 7

28 Then substituting 6 and 63 into 6, then into 60, we have R ek [λ, τ, τ { [ a e jπfeλ+τ+τ R E k [n 3 N σ w m µ m p m,k [n + λ + τ p m,k [n + λ p m,k [n + τ p m,k [n [ N + σ w m µ m p m,k [n + λ + τ p m,k [n + λ p m,k [n + λ + τ + p m,k [n + λ + [ N w m µ m p m,k [n + τ p m,k [n p m,k [n + τ + p m,k [n + [ N + σ w m µ m p m,k [n + λ + τ p m,k [n + τ p m,k [n + λ + τ + p m,k [n + τ + [ N w m µ m p m,k [n + λ p m,k [n p m,k [n + λ + p m,k [n + [ N + σ w m µ m p m,k [n + λ + τ p m,k [n p m,k [n + λ + τ + p m,k [n + [ N } w m µ m p m,k [n + λ p m,k [n + τ p m,k [n + λ + p m,k [n + τ + r k τ, f e r k τ, f e e jπfe+/λ. 6 We can see that R ek [λ, τ, τ is independent to time instant s. In addition, different from R ek [λ, τ, τ, it is also unrelated to noise ν[l. Similarly, based on 58 and the definition of r k τ, f e, we can simplify the expression of R ek [λ, τ, τ as { [ R ek [λ, τ, τ e jπf eλ+τ +τ a R E k [n 3 σ a N w m µ m p m,k [n + λ + τ p m,k [n + λ p m,k [n + τ p m,k [n [ N + σ w m µ m p m,k [n + λ + τ p m,k [n + τ p m,k [n + λ + τ + p m,k [n + τ + [ N w m µ m p m,k [n + λ p m,k [n p m,k [n + λ + p m,k [n + [ N + σ w m µ m p m,k [n + λ + τ p m,k [n p m,k [n + λ + τ + p m,k [n + [ N } w m µ m p m,k [n + λ p m,k [n + τ p m,k [n + λ + p m,k [n + τ C. Calculation of S ek f, τ, τ and S ek f, τ, τ Having got the expressions for R ek [λ, τ, τ shown in 57 and R ek [λ, τ, τ shown in 6, we are now ready to calculate S ek f, τ, τ and S ek f, τ, τ based on 8. First we consider S ek f, τ, τ. It is more convenient to rewrite the summations in 57 in frequency domain. By recalling that P m,k f 8

29 s p m,k[s e jπfs, and using Parseval s relation, we have 8 8 p m,k [n + λ + τ p m,k [n + λ p m,k[n + τ p m,k[n e jπfλ P m,k f P m,k f f e jπfτ df P m,k f P m,k f f e jπfτ df df P m,k f P m,k f f Pm,kf Pm,kf f e jπf τ f τ df df e jπfλ df, 66 and p m,k [n + λ p m,k[n p m,k [n + λ + τ p m,k[n p m,k [n + λ τ p m,k[n p m,k [n + λ + τ τ p m,k[n P m,k f e jπfλ df P m,k f e jπfτ e jπfλ df P m,k f e jπfτ e jπfλ df P m,k f e jπfτ τ e jπfλ df. 67 Similarly, by noting that Gf N s f[ N s e jπfs, we have N N f[ N s f[n s + λ s N f[ N s f[n s + λ + τ s N f[ N s f[n s + λ τ s f[ N s f[n s + λ + τ τ s G f e jπfλ df G f e jπfτ e jπfλ df G f e jπfτ e jπfλ df G f e jπfτ τ e jπfλ df. 68 9

30 Then we have S ek f, τ, τ ejπfeτ τ E [ a R k [n 3 N σ a w m µ m P m,k f P m,k f f e f P m,kf P m,kf f e f e jπfτ fτ df df + σ a 8 ejπf eτ +τ fτ + σ a 8 ejπfeτ τ + σ a σ ν + σ a σ ν + σ a σ ν + σ a σν + σ ν N e jπfτ e jπfτ N N N w m µ m P m,k f N w m µ m P m,k f w m µ m P m,k f f e f w m µ m P m,k f f e f w m µ m P m,k f f f e G f e jπfτ τ df N N e jπfτ τ w m µ m P m,k f f f e G f e jπfτ+τ df w m µ m P m,k f f f e G f e jπfτ+τ df N w m µ m P m,k f f f e G f e jπf τ τ df e jπf τ +τ df e jπfτ τ df e jπfτ τ + e jπfτ e jπfτ+τ G f G f f df. 69 Then we immediately have S ek f e +, τ, τ ejπfeτ τ E N [ a R k [n 3 σ a w m µ m P m,k f P m,k f e jπfτ df + σ a 8 τ e jπf eτ τ + σ a 8 ejπfeτ τ + σ a σ ν + σ a σ ν + σ a σ ν + σ a σν + σ ν N N e jπfe+τ e jπfe+τ N N w m µ m P m,k f N w m µ m P m,k f P m,k f P m,k f e jπfτ df w m µ m P m,k f w m µ m P m,k f w m µ m P m,k + f e f G f e jπfτ τ df N N e jπf e+τ τ G f G f e + f w m µ m P m,k + f e f G f e jπfτ+τ df w m µ m P m,k + f e f G f e jπfτ+τ df N w m µ m P m,k + f e f G f e jπfτ τ df e jπfτ τ df e jπfτ +τ df e jπfτ τ + e jπfe+τ e jπfτ+τ df

31 Now we will calculate S ek f, τ, τ. First we have p m,k [n + λ + τ p m,k [n + λ p m,k [n + τ p m,k [n 8 P m,k f P m,k f f P m,k f P m,k f f e jπfτ+fτ df df e jπfλ df.7 Also based on the definition of A m,k τ, f e in 8, we have p m,k [n + λ + τ p m,k [n + τ p m,k [n + λ + τ + p m,k [n + τ + τ A m,k λ + τ τ, f e p m,k [n + λ p m,k [n p m,k [n + λ + p m,k [n + A m,k λ, f e p m,k [n + λ + τ p m,k [n p m,k [n + λ + τ + p m,k [n + A m,k λ + τ, f e p m,k [n + λ p m,k [n + τ p m,k [n + λ + p m,k [n + τ + τ A m,k λ τ, f e. 7 Thus by using 8, we have S ek f, τ, τ ejπfeτ+τ Then we have [ a R E k [n 3 N σ a w m µ m P m,k f P m,k f f e f P m,k f P m,k f f e f e jπf τ +f τ df df N + τ e jπf eτ +τ σ a w m µ m P m,k f P m,k f 8 N w m µ m P m,k f f e f P m,k f f e f e jπf τ τ df N + τ σ a 8 e jπf feτ e jπf eτ +τ w m µ m P m,k f P m,k f N w m µ m P m,k f f e f P m,k f f e f e jπfτ+τ df. 73 S ek f e +, τ, τ [ a E ejπfeτ+τ R k [n 3 σ a N w m µ m P m,k f P m,k f e jπfτ df + τ σ a 8 ejπf eτ +τ N + σ a 8 ejπfeτ+τ N P m,k f P m,k f e jπfτ df w m µ m P m,k f P m,k f e jπfτ τ df w m µ m P m,k f P m,k f e jπfτ+τ df. 7 3