Abstract

Arrayed waveguide gratings (AWG) are widely used as wavelength division multiplexers (MUX) and demultiplexers (DEMUX) in optical networks. Here we propose and demonstrate that conventional AWGs can also be used as integrated spectral filters to realize a Fast Fourier transform (FFT) and its inverse form (IFFT). More specifically, we point out that the wavelength selection conditions of AWGs when used as wavelength MUX/DEMUX also enable them to perform FFT/IFFT functions. Therefore, previous research on AWGs can now be applied to optical FFT/IFFT circuit design. Compared with other FFT/IFFT optical circuits, AWGs have less structural complexity, especially for a large number of inputs and outputs. As an important application, AWGs can be used in optical OFDM systems. We propose an all-optical OFDM system with AWGs and demonstrate the simulation results. Overall, the AWG provides a feasible solution for all-optical OFDM systems, especially with a large number of optical subcarriers.

©2011 Optical Society of America

1. Introduction

The Fast Fourier transform (FFT) is one of the most important tools in digital signal processing, which has applications almost everywhere in the digital world. In optical communications, it also has been used as the core process for optical orthogonal frequency division multiplexing (OFDM) transmission. Optical OFDM provides a promising solution for future high-speed and long-haul data transmission [1,2], because of its demonstrably favorable performance, including its tolerance to chromatic and polarization mode dispersion [3,4], as well as its high spectral efficiency. The main principle of optical OFDM is the generation of analog symbol signals, whose spectral components include multiple subcarriers modulated in parallel by independent data streams with relatively slow rates. All the subcarriers are orthogonal to each other, without introducing crosstalk. At the receiver the data in each subcarrier can be retrieved without interference from others. Overall, optical OFDM provides a flexible and efficient transmission platform for high rate optical communications.

Conventional OFDM signal generation and reception processes are essentially an inverse Fast Fourier transform (IFFT) at the transmitter, followed by a FFT at the receiver. Currently many systems implement optical OFDM with an electronic digital signal processing (DSP) based FFT/IFFT. Figure 1 displays a schematic diagram of such an OFDM system. An IFFT is performed to generate OFDM symbols. Guard intervals (GIs), or cyclic prefixes (CPs), are typically inserted between OFDM symbols to maintain the orthogonality of the subcarriers in the presence of dispersion and other signal distortions. The generated digital OFDM samples are converted into analog signals through a digital-to-analog converter (DAC), which are afterwards used to modulate an optical carrier. At the receiver side, the optical signal is detected and digitally sampled by an analog-to-digital converter (ADC). An FFT is performed to demultiplex the subcarriers, and other DSP techniques are applied to correct dispersions and other distortions.

 

Fig. 1 Schematic diagram of an OFDM transmission system with electronic IFFT and FFT. S/P: serial-to-parallel conversion. GI: Guard interval; DAC: Digital-to-analog converter; ADC: analog-to-digital converter; P/S: Parallel-to-serial conversion

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While optical OFDM presents a lot of advantages, real-time optical OFDM systems are severely limited by the speed of the electronic processing in the ADC, DAC, and FFT/IFFT modules. Currently, real-time electronically implemented optical OFDM systems have their symbol rates limited to the range of megasymbols/s [57]. Most recently a real-time optical OFDM system with a total rate of 110Gb/s was demonstrated utilizing optical band-multiplexing [8]. In [8], the receiving system only processed the transmitted signal with a bandwidth of 2GHz.

To overcome the speed limitation of the electronics, all-optical OFDM was previously proposed, to implement the optical OFDM transmitter and receiver with primarily optical devices [911]. One method to implement all-optical OFDM transmitters is generating multiple optical subcarriers, separating them with optical devices and modulating each subcarrier at the baud rate close to the subcarrier spectral spacing [9,1113]. The spectra of the modulated optical subcarriers will overlap with each other, forming one optical OFDM channel band (sometimes called superchannel [12]). The net data rate of whole optical OFDM channel strongly depends on the number of subcarriers. An OFDM superchannel of 1.2Tb/s with 24 subcarriers [12] and an optical OFDM transmitter at rate of 10.8Tb/s with 75 subcarriers [9] have been reported.

As for the all-optical OFDM receiver, optical FFT circuits have been proposed [14] and experimentally demonstrated [1517], which imitate the FFT (and also IFFT) algorithms, to achieve FFT function all-optically without O-E or E-O conversion. Such all-optical FFT circuits can be applied to demultiplex the optical OFDM superchannel signal. The optical IFFT/FFT circuits are usually comprised of an integrated passive optical network with several stages, each of which contains appropriate temporal delays and phase shifts. Many of these optical FFT circuits have special designs to simplify the circuit complexity [15,17]. For example, in [17], cascaded delay interferometers are proposed to achieve FFT/IFFT functions, which largely reduces the number of phase shifters and delay. However, to realize a full FFT /IFFT function, a large number of modules are still required inside the optical circuit, especially when the transform order N becomes large. So far the implemented optical FFT circuits with N up to 8 are reported [15,17].

In this paper, we propose that both the IFFT and FFT functions can be realized all-optically with arrayed waveguide gratings (AWGs), which are conventionally used as wavelength filters. This approach provides a fully integrated and scalable optical FFT/IFFT solution on a single chip that utilizes existing technologies which have been developed over decades. The feasibility of this concept was previously established as an OFDM demultiplexer through simulations [18]. Different from Ref. 18, here we focus on the specific AWGs parameter designs required to implement FFT/IFFT functions, and illustrate that the design of AWGs in FFT/IFFT circuits has an important correspondence to existing AWG designs for wavelength filters. Also we present the critical design rules for AWGs as FFT/IFFT filters.

Compared with other optical FFT/IFFT circuits, our proposed AWG has less structural complexity. Moreover, current AWG fabrication technologies already enable the implementation of AWGs with a large N. Therefore our work provides a feasible way of implementing all-optical OFDM systems. The AWG as FFT/IFFT circuits is especially suitable for optical OFDM superchannel system consisting of a large number of optical subcarriers.

In the following discussion, we propose two types of all-optical OFDM systems using AWGs and display the simulation results, to demonstrate the principles of IFFT and FFT operation and the applications in optical OFDM systems. Moreover, we investigate the effects of practical devices on the OFDM system’s performance.

2. Operational principles of the AWG as FFT/IFFT filters and AWG parameter design

AWGs are usually intended for wavelength division multiplexing (WDM) and demultiplexing, as described in [19,20]. The use of AWGs as an FFT circuit and an OFDM demultiplexer was theoretically proposed in [18]. In this work we show that by imposing a few additional conditions on the design parameters of an AWG as a WDM filter, we can satisfy the conditions required for FFT/IFFT operation. Whereas only the FFT function was addressed in [18], we also address the realization of the IFFT function.

A typical structure of an AWG is shown in Fig. 2 . An AWG consists of input/output waveguides, two focusing slab regions and one arrayed multi-channel waveguide between the two slab regions, with constant path increment ΔL between the channels. Usually the two slab regions are identical with the details shown in Fig. 2(b). Denote the AWG’s input/output waveguide separation as D. Also denote the arrayed waveguide separation as d (for input) and d1 (for output), and the radius of the curvatures as f (for input) and f 1(for output). Here d = d1, and f = f 1 . Denote the number of waveguide channels as N for the input/out waveguides and the arrayed waveguides.

 

Fig. 2 Waveguide structure of an AWG (a) the whole structure (b) the enlarged picture of the output slab region [17].

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The wavelength selection at one output waveguide (meaning at the focal point x) is based on the constructive interference of the beams from the arrayed waveguides [19], which is,

2πns(λ0)λ0(f1x1d12f1)+2πns(λ0)λ0(f+xd2f)=2πns(λ0)λ0(f1+x1d12f1)+2πns(λ0)λ0(fxd2f)+2πnc(λ0)λ0ΔL+2mπ,

where m is an integer, and ns and nc are the refractive index in the slab and arrayed waveguide region, respectively. Based on Eq. (1), a series of formulas are derived as following [19]:

λ0=ncΔLm,        Δλ=nsdDλ0NcfΔL,      Nch=λ0fnsdD,

where λ0 is the center wavelength, Δλ is the channel spacing of the AWG transmission spectrum, and Nch is the number of channels in a free spectral range (FSR). Nc = nc-λdnc/dλ represents the group index of the arrayed waveguide.

Now we rewrite the channel spacing Δλ in frequency:

Δf=cλ02Δλ=nsdDfλ0cΔLNc=1Nch1τ,     τ=ΔLNcc.

It is apparent that τ is the time difference of light traveling between adjacent channels in the arrayed waveguide. Also 1/τ defines the FSR in frequency domain.

We now show how the AWG can perform both the FFT and IFFT operations. Consider the AWG as a spectral filter. Based on the parameters in Fig. 2, and taking the vertical symmetric axis as the phase reference in Fig. 2(b), we can write the impulse response function between the ith input and the kth output as [21]:

hik(t)=m=0N1exp[πjnsdλ0(2mN+1)(sinθi+sinθo)]δ(tnsfcNcL+mΔLc)                                                                                                  (m=0,1,...,N1).

In the expression of Eq. (4), the phase terms originate from the slab regions and the delay terms come from both the slab region and arrayed waveguide. To simplify Eq. (4), we can take the following approximations:

sinθi(2iN+1)D2f,    sinθo(2kN+1)D2f.

Now let Nch = N, and neglect the constant time delay terms, since they do not change the final transfer function form. A simplified formula of Eq. (4) can be obtained as:

hik(t)=m=0N1ejπnsdDλ0f(2mN+1)(i+k+1)δ(tmτ)=m=0N1ejπNch(2mN+1)(i+k+1)δ(tmτ)        =m=0N1ejπN(2mN+1)(i+k+1)δ(tmτ)                                (m=0,1,, N1) 

Here we emphasize that Eq. (6) is obtained by introducing Eq. (3), which is the wavelength selection condition. Nch = N means that the AWG transmission spectrum repeats after every NΔλ periods. In other words, the FSR of the AWG 1/τ exactly matches the spacing between different frequency bands, each band containing N channels. Such AWGs are called cyclic AWGs. Equation (6) is interpreted as follows: the N arrayed waveguides provide temporal delays and the input/output slab regions produce phase shifts.

If we just consider the phase shifts at the output slab region (from port j in the arrayed waveguide to the output port k), we have

θj,k=2πnsdDλ0f(mN+12)k.

Equation (7) is a similar expression to Eq. (5) in [18], with a different phase reference point. However, we claim that to realize FFT/IFFT functions, the parameter Nch is critical and Eq. (2) and (3) are required.

To get a FFT formula from Eq. (6), set input i = N-1. For an input signal si(t), take the observation period as . The signal at the each output k is

Sk(t)=si(τ)h(tτ)dτ=m=0N1ej2πN(mN12)ksi(tmτ).

At the sampling point t = (N-1) τ,

Sk[(N1)τ]=m=0N1ej2πN(N12m)ksi(mτ)                    n=N/2m¯¯ (n=0N1si'[(N2n)τ]ej2πnkN)ejπNk=FFT(si'[(N2n)τ])ejπNk.

It is clearly seen that Sk[(N-1)τ] is the FFT output of the signal series si[(N/2-n)τ],which is a cyclic shift of si() by N/2 within period Nτ, in a reverse order. The extra term ejπk/N just provides an additional cyclic shift of the sequence si[(n + N/2)τ]. Both the cyclic shift and the extra phase term can be eliminated by a slight design modification in the AWG input/output waveguides and the proper selection of the phase reference.

Note that Eq. (9) is only valid when all of the N copies of si(t) overlap with different time delay. In other words, since we only observe si(t) at 0≤ t<T = , there is only a time window with width τ, during which the FFT is realized, as shown in Fig. 3(a) . Therefore a time gating device is usually required to sample the signal at that window.

 

Fig. 3 Configuration of the AWG to realize the operations of (a) FFT and (b) IFFT

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To realize the IFFT function, set k = N-1. For different inputs, the impulse response becomes

hi(t)=m=0N1expj2πN(mN12)iδ(tmτ).

Suppose multiple input signals Si (t) (i = 0,1,…N-1) are present, and Si (t) is the time sample.

Si(t)={Si(mT),        t=mT0,                 otherwise

Still consider the time period 0≤ t<T = . Then only Si(0) is significant, and the output signal is derived as:

sout(t)=i=0N1Si(τ)hi(tτ)dτ.

Taking samples of sout(t) at t = mτ (m = 0, 1,…, N-1), we have

                      sout(mτ)=i=0N1ej2πN(mN12)iSi(0)=i=0N1Si(0)ej2πN(N12)iej2πmiNn=N/2+m  s'out[(n+N2)τ] = i=0N1Si(0)ejπNiej2πniN=IFFT(Si(0)ejπNi).

s’out[(n + N/2)τ] is a cyclic shift of sequence sout(mτ) by N/2. Again, the term ejπi/N just adds an additional cyclic shift to sout().

In the IFFT configuration, the input signal must be discrete signals at each subcarrier frequency, or at least having time interval less than τ. Otherwise there will be interference from other symbols at some of the N samples sout(mτ), because of the time delay in the arrayed waveguide. In practice, we can use N data sequences Si(m) to individually modulate N optical short-pulse sequences with repetition period T = . The IFFT result will be obtained at one AWG output at sampling points t = mτ (m = 0, 1,…, N-1).

From the above discussion, we recognize several important AWG design parameters for FFT/IFFT circuits. For example, if we want the optical OFDM signal to have 16 subcarriers with frequency spacing of 10GHz, then the AWG should be a cyclic AWG with 16 channels in the arrayed waveguides, with channel spacing of 10GHz. The FSR of the AWG should be 16 × 10GHz = 160GHz, implying τ = 1/160GHz = 6.25ps. The path increment ΔL can be calculated accordingly from Eq. (3). Additionally, a 1 × N AWG is enough to realize the FFT/IFFT function, instead of an N × N AWG. Here We discuss N × N AWGs to provide configuration flexibility and also to enable it to be used in other applications [21]. Meanwhile an N × N configuration does not induce significant design or implementation complexity compared to 1 × N AWGs, because of its structural symmetry.

Now we investigate the AWG transfer function in the spectral domain. The filter transfer function can be derived by taking the Fourier transform of Eq. (6) and is written as [21]:

Hik(f)=ejπ(N1)fτm=0N1ejπ(2m+N1)(i+k+1N+fτ)=ejπ(N1)fτsin[π(i+k1+Nfτ)]sin[π(i+k1)/N+fτ].

Figure 4 (a) and (b) show the theoretical transmission spectrum of an AWG and the spectral measurement of a commercially fabricated AWG that we have, respectively. Figure 4(a) is obtained from Eq. (14) and shows that each channel has a free spectral range of 1/τ, and is orthogonal to all the other channels, because it vanishes at the peaks of all the other channels. For practical AWGs, the transmission spectral shape for each channel is usually a Gaussian function [19]. In Fig. 4(b) the spectral shape at each channel has some distortion and is somewhat asymmetric with respect to the center frequency. The adjacent channel crosstalk is approximately −18dB. This distortion and channel crosstalk are mainly due to the phase errors induced during AWG manufacturing. When the AWG is used for optical OFDM demultiplexer, the channel crosstalk will induce coherent interference by other subcarriers. To address this problem, phase error compensation approaches have already been investigated to rectify the spectral shape and reduce the channel crosstalk [19,20]. In research field, the implemented AWGs have better performances than the ones that we use, as shown in Fig. 4 (c) [19]. This AWG is designed for wavelength filters, containing 32 channels with 10GHz channel spacing. Note that the spectral shape is close to the theoretical results in Fig. 4 (a), which proves the validity of Eq. (6). The adjacent channel crosstalk is about −30dB, even without post-processing for phase error correction. Thus current AWG fabrication technologies can achieve very low channel crosstalk.

 

Fig. 4 Theoretical and experimental spectrum of different channels of a cyclic AWG (a) theoretical calculation (b) experimental measurements. Both the spectra have a FSR of 160GHz, and channel spacing of 10GHz. (c) The spectral transmittance of a 32-channel AWG experimentally demonstrated in [19], with 10GHz channel spacing

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Figure 4(b) and (c) also displays a 3dB power difference within the AWG FSR. As explained in [19], there is usually 2.5-3dB power difference between the peripheral ports and central ports for a cyclic AWG. The impact of the response non-uniformity on the FFT function is analyzed in [18]. It can also be mitigated through pre-channel equalization.

In summary, the AWG along with time samplers can be used to realize both the FFT and IFFT functions. The key design rules to have the AWG work as both FFT and IFFT filters are consistent with designing a cyclic AWG as a wavelength MUX and DEMUX. Therefore previous research results on general AWG design can be applied here for FFT/IFFT operations, such as phase error compensation techniques to reduce the adjacent channel cross-talk.

The AWG has less structural complexity compared with other approaches, especially for a large N. Ref. 15 designed an optical circuit based on an FFT butterfly algorithm, achieving higher order FFT functions by arranging multiple FFT circuits with a smaller order. But the complexity of that circuit increases quickly as N increases. Ref. 17 simplifies the FFT circuit design and uses N-1 cascaded delay interferometers to achieve a full functional N-point FFT/IFFT circuit. However, it stills consists of N-1 interferometers and N-1 phase stabilizers. By contrast, the AWG-based FFT/IFFT circuit here only requires the design of the arrayed waveguide and the two slab regions, for any size of N. Even for a transform order N = 8, the fabrication of a AWG as an FFT/IFFT circuit can be easier than other approaches. Therefore our approach provides a feasible solution for an integrated optical FFT/IFFT circuit with a large N. For example, current AWGs having 128 and even 512 inputs/outputs have already been reported, with 10GHz channel spacing [22,23], and can potentially be applied to implement large-order optical FFT/IFFT circuits.

3. All-optical OFDM systems with AWGs and simulation results

Since AWGs can be applied to all-optical IFFT and FFT operations, we propose an all-optical OFDM transmitter and receiver with AWGs, as shown in Fig. 5 . In this setup, the two AWGs are both cyclic AWGs, with 10GHz channel spacing as an example. The AWG at the transmitter is used as an IFFT filter, and the one at the receiver is utilized as an FFT filter. The mode-locked laser (MLL) servers as a pulse train source and has a repetition rate R of 10GHz. After the pulse train passes through the 1 by N coupler, multiple synchronous pulse trains are generated and are modulated individually by the OFDM data at each subcarrier, at rate of 10Gbaud. When these modulated data streams pass through the AWG, an IFFT operation is performed. At the receiver end, the transmitted signal is injected to another AWG. An FFT process is conducted through the AWG. The demultiplexed signals at different subcarriers are presented at different AWG outputs and can be collected after optical sampling.

 

Fig. 5 The proposed AWG-based all-optical OFDM system setup. MLL: mode locked laser; EDFA: erbium-doped fiber amplifier; S: optical sampler. The AWG at the transmitter performs IFFT and the one at the receiver performs FFT.

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Figure 6 shows the simulation results based on the setup in Fig. 5, using on-off-keying (OOK) modulation. The results display the eye diagrams at the receiver, with N = 4 and 16, respectively. As shown in Fig. 6, an open eye can be obtained for both N = 4 and 16, which validates the operations of AWGs functioning as IFFT and FFT circuits, as discussed in Section 2. Therefore these AWGs can be utilized in the applications where optical FFT or IFFT process is needed. These applications are not restricted in optical OFDM systems but can be other areas, such as fast digital signal processing in which large order FFT/IFFT are required.

 

Fig. 6 Eye diagrams of the demultiplexed OFDM signal at the receiver end in Fig. 5 for (a) N = 4 (b) N = 16

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On the other hand, it can be seen from Fig. 6 that the duration of the eye is very short, which requires short and precise optical sampling techniques to extract the transmitted data. Such a narrow sampling window can be explained by the discussions in Section 2. In Section 2, we have already shown that the IFFT results are only correct at the sampling points t = mτ (m = 0, 1,…, N-1). As for the FFT operation, the FFT results are only valid during the interval T/N. Figure 7 provides a better picture of explanation. After IFFT and FFT operations the transmitted data can only be retrieved at the right sampling point t0, as shown in Fig. 7. Therefore, this all-optical OFDM system works well only under rigorous experimental conditions and is sensitive to the signal distortions and AWG fabrication imperfections.

 

Fig. 7 (a) The generated optical OFDM signal waveform after the transmitter in Fig. 5; (b) the schematic of OFDM signal demultiplexing process in the system shown in Fig. 5

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A modified all-optical OFDM system setup with AWGs is shown in Fig. 8 . The AWG at the transmitter end in Fig. 8 now functions as a regular WDM-DEMUX and is used to separate all the subcarriers, generated by the MLL. Each subcarrier afterwards is modulated by the OFDM data. The modulated subcarriers are combined together through a coupler to form an OFDM signal. This all-optical OFDM transmitter configuration has been proposed in [9,11,12]. Here we focus on using the AWG at the receiver end to perform the FFT operation and demultiplex the OFDM signal. Figure 9 shows the simulation results of the demultiplexed OFDM signal, with the same parameter configuration as in Fig. 5. As shown in Fig. 9, clear eyes are obtained with duration of T/N, because only FFT operation is conducted in the system. The demultiplexed OFDM signal can be obtained after optical samplers.

 

Fig. 8 The modified AWG-based all-optical OFDM system setup. MLL: mode locked laser; EDFA: erbium-doped fiber amplifier; S: optical sampler. The AWG at the transmitter performs WDM demultiplexing and the one at the receiver performs FFT.

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Fig. 9 Eye diagrams of the demultiplexed OFDM signal at the receiver end in Fig. 7 for (a) N = 4 (b) N = 16

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The sampling window will shrink as N increases. If we assume that an ideal sampler is placed, with sampling window exactly matching T/N, the sampled signal’s amplitude will decrease by 3dB when N doubles. The signal-to-noise ratio (SNR) will also drop by 3dB, if the noise level keeps the same. In practical experiments, the sampler’s gate window does not necessarily match the sampling window width, and the imperfections of the FFT filter and the optical sampler will also degrade the SNR [15]. All of these practical factors will induce additional power penalty.

In real all-optical OFDM systems, the transmission performance also depends on the devices in use and their specific parameters, such as the modulator bandwidth. In order to successfully demultiplex an OFDM with AWG, the modulated signal at each subcarrier should have constant value during the modulation interval. But limited modulator bandwidth affects the modulated signal, such as the rise and fall times. Therefore modulation bandwidth affects the all-optical OFDM system’s performance. Figure 10 (a) shows the simulation results of the demultiplexed OFDM signal, based on the setup of Fig. 8, with an modulator bandwidth of 50GHz (N = 16). Compared with Fig. 9 (b), the duration of the eye opening becomes much shorter, due to the limited modulator bandwidth. Figure 10(b) shows the demultiplexed eye diagram when the 16 modulated subcarriers are not symbol-aligned. The eye opening becomes much smaller compared to Fig. 9 (a), which indicates that the OFDM system requires symbol alignment for all subcarriers. The effects of symbol-alignment on optical OFDM system’s performance are discussed elaborately with more details in [13,24].

 

Fig. 10 Eye diagrams of the demultiplexed OFDM signal with 50GHz modulator bandwidth in Fig. 7 for N = 16 (a) the modulated subcarrers are symbol-aligned (b) the modulated subcarriers are not symbol-aligned

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4. Conclusion

We illustrated that AWGs can perform both FFT and IFFT functions passively and all-optically, and assert that the FFT/IFFT filter conditions are satisfied when the cyclic AWG’s wavelength selection conditions are met. Therefore AWGs can be used for realization of all-optical OFDM systems providing high-throughput, modulation-transparent operation. Previous work on the AWG design and fabrication can be substantially applied here to implement the optical FFT/IFFT circuits. The simple waveguide structure of AWG makes it a feasible solution for FFT/IFFT with a large N. We propose two types of all-optical OFDM systems with AWGs. Our simulation results prove the AWG’s operation as both FFT and IFFT circuits.

Acknowledgment

The authors would like to thank Shahab Etemad, Zhensheng Jia, and Anjali Agarwal of Telcordia Technologies for their help of providing part of the equipment.

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17. D. Hillerkuss, M. Winter, M. Teschke, A. Marculescu, J. Li, G. Sigurdsson, K. Worms, S. Ben Ezra, N. Narkiss, W. Freude, and J. Leuthold, “Simple all-optical FFT scheme enabling Tbit/s real-time signal processing,” Opt. Express 18(9), 9324–9340 (2010). [CrossRef]   [PubMed]  

18. A. J. Lowery, “Design of arrayed-waveguide grating routers for use as optical OFDM demultiplexers,” Opt. Express 18(13), 14129–14143 (2010). [CrossRef]   [PubMed]  

19. C. R. Doerr and K. Okamoto, “Advances in silica planar lightwave circuits,” J. Lightwave Technol. 24(12), 4763–4789 (2006). [CrossRef]  

20. M. K. Smit and C. Van Dam, “PHASAR-based WDM-devices: Principles, design and applications,” IEEE J. Sel. Top. Quantum Electron. 2(2), 236–250 (1996). [CrossRef]  

21. G. Cincotti, N. Wada, and K. Kitayama, “Characterization of a full encoder/decoder in the AWG configuration for code-based photonic routers—part I: modeling and design,” J. Lightwave Technol. 24(1), 103–112 (2006). [CrossRef]  

22. K. Takada, M. Abe, and K. Okamoto, “Low-cross-talk polarization-insensitive 10-GHz-spaced 128-channel arrayed-waveguide grating multiplexer-demultiplexer achieved with photosensitive phase adjustment,” Opt. Lett. 26(2), 64–65 (2001). [CrossRef]  

23. K. Takada, M. Abe, M. Shibata, M. Ishii, and K. Okamoto, “Low-Crosstalk 10-GHz-spaced 512-channel arrayed-waveguide grating multi/demultiplexer fabricated on a 4-in wafer,” IEEE Photon. Technol. Lett. 13(11), 1182–1184 (2001). [CrossRef]  

24. G. Goldfarb, G. Li, and M. G. Taylor, “Orthogonal wavelength-division multiplexing using coherent detection,” IEEE Photon. Technol. Lett. 19(24), 2015–2017 (2007). [CrossRef]  

References

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  1. H. Sanjoh, E. Yamada, and Y. Yoshikuni, “Optical orthogonal frequency division multiplexing using frequency/time domain filtering for high spectral efficiency up to 1 bit/s/Hz,” in Conference on Optical Fiber Communication, OFC (Anaheim, CA, 2002), paper ThD1, 401–402 (2002).
  2. W. Shieh and I. Djordjevic, OFDM for Optical Communications, Academic Press, 2009.
  3. A. J. Lowery and J. Armstrong, “Orthogonal-frequency-division multiplexing for dispersion compensation of long-haul optical systems,” Opt. Express 14(6), 2079–2084 (2006).
    [Crossref] [PubMed]
  4. W. Shieh, W. Chen, and R. S. Tucker, “Polarization mode dispersion mitigation in coherent optical orthogonal frequency division multiplexed systems,” Electron. Lett. 42(17), 996–997 (2006).
    [Crossref]
  5. Y. Benlachtar, P. M. Watts, R. Bouziane, P. Milder, D. Rangaraj, A. Cartolano, R. Koutsoyannis, J. C. Hoe, M. Püschel, M. Glick, and R. I. Killey, “Generation of optical OFDM signals using 21.4 GS/s real time digital signal processing,” Opt. Express 17(20), 17658–17668 (2009).
    [Crossref] [PubMed]
  6. Q. Yang, S. Chen, Y. Ma, and W. Shieh, “Real-time reception of multi-gigabit coherent optical OFDM signals,” Opt. Express 17(10), 7985–7992 (2009).
    [Crossref] [PubMed]
  7. F. Buchali, R. Dischler, A. Klekamp, M. Bernhard, and Y. Ma, “Statistical Transmission Experiments Using a Real-Time 12.1 Gb/s OFDM Transmitter”, in Conference on Optical Fiber Communication, OFC (San Diego, CA, 2010), paper OMS3 (2010).
  8. S. Chen, Y. Ma, and W. Shieh, 110-Gb/s Multi-Band Real-Time Coherent Optical OFDM Reception after 600-km Transmission over SSMF Fiber”, in Conference on Optical Fiber Communication, OFC (San Diego, CA, 2010), paper OMS2 (2010).
  9. D. Hillerkuss, T. Schellinger, R. Schmogrow, M. Winter, T. Vallaitis, R. Bonk, A. Marculescu, J. Li, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, K. Weingarten, T. Ellermeyer, J. Lutz, M. Möller, M. Hübner, J. Becker, C. Koos, W. Freude, and J. Leuthold, “Single source optical OFDM transmitter and optical FFT receiver demonstrated at line rates of 5.4 and 10.8Tb/s,” in Conference on Optical Fiber Communication, OFC (San Diego, CA, 2010), paper PDPC1 (2010).
  10. H. Chen, M. Chen, and S. Xie, “All-optical sampling orthogonal frequency-division multiplexing scheme for high-speed transmission system,” J. Lightwave Technol. 27(21), 4848–4854 (2009).
    [Crossref]
  11. F. C. Garcia Gunning, S. K. Ibrahim, P. Frascella, P. Gunning, and A. D. Ellis, “High symbol rate OFDM transmission technologies”,in Conference on Optical Fiber Communication, OFC (San Diego, CA, 2010), paper OThD1 (2010).
  12. S. Chandrasekhar, X. Liu, B. Zhu, and D. W. Peckham, “Transmission of a 1.2 Tb/s 24-carrier no-guad-interval coherent OFDM superchannel over 7200-km of Ultra-Large-Area Fiber,” in Proc. ECOC 2009 (Vienna, Austria), paper PD2.6 (2009).
  13. S. Chandrasekhar and X. Liu, “Experimental investigation on the performance of closely spaced multi-carrier PDM-QPSK with digital coherent detection,” Opt. Express 17(24), 21350–21361 (2009).
    [Crossref] [PubMed]
  14. K. Lee, C. T. D. Thai, and J.-K. K. Rhee, “All optical discrete Fourier transform processor for 100 Gbps OFDM transmission,” Opt. Express 16(6), 4023–4028 (2008).
    [Crossref] [PubMed]
  15. K. Takiguchi, M. Oguma, H. Takahashi, and A. Mori, “PLC-based eight-channel OFDM demultiplexer and its demonstration with 160 Gbit/s signal reception,” in Conference on Optical Fiber Communication, OFC (San Diego, CA, 2010), paper OThB4 (2010).
  16. Y. Huang, D. Qian, R. E. Saperstein, P. N. Ji, N. Cvijetic, L. Xu, and T. Wang, “Dual-polarization 2x2 IFFT/FFT optical signal processing for 100-Gb/s QPSK-PDM all-optical OFDM,” in Conference on Optical Fiber Communication, OFC (San Diego, CA, 2009), paper OTuM4 (2009).
  17. D. Hillerkuss, M. Winter, M. Teschke, A. Marculescu, J. Li, G. Sigurdsson, K. Worms, S. Ben Ezra, N. Narkiss, W. Freude, and J. Leuthold, “Simple all-optical FFT scheme enabling Tbit/s real-time signal processing,” Opt. Express 18(9), 9324–9340 (2010).
    [Crossref] [PubMed]
  18. A. J. Lowery, “Design of arrayed-waveguide grating routers for use as optical OFDM demultiplexers,” Opt. Express 18(13), 14129–14143 (2010).
    [Crossref] [PubMed]
  19. C. R. Doerr and K. Okamoto, “Advances in silica planar lightwave circuits,” J. Lightwave Technol. 24(12), 4763–4789 (2006).
    [Crossref]
  20. M. K. Smit and C. Van Dam, “PHASAR-based WDM-devices: Principles, design and applications,” IEEE J. Sel. Top. Quantum Electron. 2(2), 236–250 (1996).
    [Crossref]
  21. G. Cincotti, N. Wada, and K. Kitayama, “Characterization of a full encoder/decoder in the AWG configuration for code-based photonic routers—part I: modeling and design,” J. Lightwave Technol. 24(1), 103–112 (2006).
    [Crossref]
  22. K. Takada, M. Abe, and K. Okamoto, “Low-cross-talk polarization-insensitive 10-GHz-spaced 128-channel arrayed-waveguide grating multiplexer-demultiplexer achieved with photosensitive phase adjustment,” Opt. Lett. 26(2), 64–65 (2001).
    [Crossref]
  23. K. Takada, M. Abe, M. Shibata, M. Ishii, and K. Okamoto, “Low-Crosstalk 10-GHz-spaced 512-channel arrayed-waveguide grating multi/demultiplexer fabricated on a 4-in wafer,” IEEE Photon. Technol. Lett. 13(11), 1182–1184 (2001).
    [Crossref]
  24. G. Goldfarb, G. Li, and M. G. Taylor, “Orthogonal wavelength-division multiplexing using coherent detection,” IEEE Photon. Technol. Lett. 19(24), 2015–2017 (2007).
    [Crossref]

2010 (2)

2009 (4)

2008 (1)

2007 (1)

G. Goldfarb, G. Li, and M. G. Taylor, “Orthogonal wavelength-division multiplexing using coherent detection,” IEEE Photon. Technol. Lett. 19(24), 2015–2017 (2007).
[Crossref]

2006 (4)

2001 (2)

K. Takada, M. Abe, and K. Okamoto, “Low-cross-talk polarization-insensitive 10-GHz-spaced 128-channel arrayed-waveguide grating multiplexer-demultiplexer achieved with photosensitive phase adjustment,” Opt. Lett. 26(2), 64–65 (2001).
[Crossref]

K. Takada, M. Abe, M. Shibata, M. Ishii, and K. Okamoto, “Low-Crosstalk 10-GHz-spaced 512-channel arrayed-waveguide grating multi/demultiplexer fabricated on a 4-in wafer,” IEEE Photon. Technol. Lett. 13(11), 1182–1184 (2001).
[Crossref]

1996 (1)

M. K. Smit and C. Van Dam, “PHASAR-based WDM-devices: Principles, design and applications,” IEEE J. Sel. Top. Quantum Electron. 2(2), 236–250 (1996).
[Crossref]

Abe, M.

K. Takada, M. Abe, and K. Okamoto, “Low-cross-talk polarization-insensitive 10-GHz-spaced 128-channel arrayed-waveguide grating multiplexer-demultiplexer achieved with photosensitive phase adjustment,” Opt. Lett. 26(2), 64–65 (2001).
[Crossref]

K. Takada, M. Abe, M. Shibata, M. Ishii, and K. Okamoto, “Low-Crosstalk 10-GHz-spaced 512-channel arrayed-waveguide grating multi/demultiplexer fabricated on a 4-in wafer,” IEEE Photon. Technol. Lett. 13(11), 1182–1184 (2001).
[Crossref]

Armstrong, J.

Ben Ezra, S.

Benlachtar, Y.

Bouziane, R.

Cartolano, A.

Chandrasekhar, S.

Chen, H.

Chen, M.

Chen, S.

Chen, W.

W. Shieh, W. Chen, and R. S. Tucker, “Polarization mode dispersion mitigation in coherent optical orthogonal frequency division multiplexed systems,” Electron. Lett. 42(17), 996–997 (2006).
[Crossref]

Cincotti, G.

Doerr, C. R.

Freude, W.

Glick, M.

Goldfarb, G.

G. Goldfarb, G. Li, and M. G. Taylor, “Orthogonal wavelength-division multiplexing using coherent detection,” IEEE Photon. Technol. Lett. 19(24), 2015–2017 (2007).
[Crossref]

Hillerkuss, D.

Hoe, J. C.

Ishii, M.

K. Takada, M. Abe, M. Shibata, M. Ishii, and K. Okamoto, “Low-Crosstalk 10-GHz-spaced 512-channel arrayed-waveguide grating multi/demultiplexer fabricated on a 4-in wafer,” IEEE Photon. Technol. Lett. 13(11), 1182–1184 (2001).
[Crossref]

Killey, R. I.

Kitayama, K.

Koutsoyannis, R.

Lee, K.

Leuthold, J.

Li, G.

G. Goldfarb, G. Li, and M. G. Taylor, “Orthogonal wavelength-division multiplexing using coherent detection,” IEEE Photon. Technol. Lett. 19(24), 2015–2017 (2007).
[Crossref]

Li, J.

Liu, X.

Lowery, A. J.

Ma, Y.

Marculescu, A.

Milder, P.

Narkiss, N.

Okamoto, K.

Püschel, M.

Rangaraj, D.

Rhee, J.-K. K.

Shibata, M.

K. Takada, M. Abe, M. Shibata, M. Ishii, and K. Okamoto, “Low-Crosstalk 10-GHz-spaced 512-channel arrayed-waveguide grating multi/demultiplexer fabricated on a 4-in wafer,” IEEE Photon. Technol. Lett. 13(11), 1182–1184 (2001).
[Crossref]

Shieh, W.

Q. Yang, S. Chen, Y. Ma, and W. Shieh, “Real-time reception of multi-gigabit coherent optical OFDM signals,” Opt. Express 17(10), 7985–7992 (2009).
[Crossref] [PubMed]

W. Shieh, W. Chen, and R. S. Tucker, “Polarization mode dispersion mitigation in coherent optical orthogonal frequency division multiplexed systems,” Electron. Lett. 42(17), 996–997 (2006).
[Crossref]

Sigurdsson, G.

Smit, M. K.

M. K. Smit and C. Van Dam, “PHASAR-based WDM-devices: Principles, design and applications,” IEEE J. Sel. Top. Quantum Electron. 2(2), 236–250 (1996).
[Crossref]

Takada, K.

K. Takada, M. Abe, M. Shibata, M. Ishii, and K. Okamoto, “Low-Crosstalk 10-GHz-spaced 512-channel arrayed-waveguide grating multi/demultiplexer fabricated on a 4-in wafer,” IEEE Photon. Technol. Lett. 13(11), 1182–1184 (2001).
[Crossref]

K. Takada, M. Abe, and K. Okamoto, “Low-cross-talk polarization-insensitive 10-GHz-spaced 128-channel arrayed-waveguide grating multiplexer-demultiplexer achieved with photosensitive phase adjustment,” Opt. Lett. 26(2), 64–65 (2001).
[Crossref]

Taylor, M. G.

G. Goldfarb, G. Li, and M. G. Taylor, “Orthogonal wavelength-division multiplexing using coherent detection,” IEEE Photon. Technol. Lett. 19(24), 2015–2017 (2007).
[Crossref]

Teschke, M.

Thai, C. T. D.

Tucker, R. S.

W. Shieh, W. Chen, and R. S. Tucker, “Polarization mode dispersion mitigation in coherent optical orthogonal frequency division multiplexed systems,” Electron. Lett. 42(17), 996–997 (2006).
[Crossref]

Van Dam, C.

M. K. Smit and C. Van Dam, “PHASAR-based WDM-devices: Principles, design and applications,” IEEE J. Sel. Top. Quantum Electron. 2(2), 236–250 (1996).
[Crossref]

Wada, N.

Watts, P. M.

Winter, M.

Worms, K.

Xie, S.

Yang, Q.

Electron. Lett. (1)

W. Shieh, W. Chen, and R. S. Tucker, “Polarization mode dispersion mitigation in coherent optical orthogonal frequency division multiplexed systems,” Electron. Lett. 42(17), 996–997 (2006).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

M. K. Smit and C. Van Dam, “PHASAR-based WDM-devices: Principles, design and applications,” IEEE J. Sel. Top. Quantum Electron. 2(2), 236–250 (1996).
[Crossref]

IEEE Photon. Technol. Lett. (2)

K. Takada, M. Abe, M. Shibata, M. Ishii, and K. Okamoto, “Low-Crosstalk 10-GHz-spaced 512-channel arrayed-waveguide grating multi/demultiplexer fabricated on a 4-in wafer,” IEEE Photon. Technol. Lett. 13(11), 1182–1184 (2001).
[Crossref]

G. Goldfarb, G. Li, and M. G. Taylor, “Orthogonal wavelength-division multiplexing using coherent detection,” IEEE Photon. Technol. Lett. 19(24), 2015–2017 (2007).
[Crossref]

J. Lightwave Technol. (3)

Opt. Express (7)

A. J. Lowery and J. Armstrong, “Orthogonal-frequency-division multiplexing for dispersion compensation of long-haul optical systems,” Opt. Express 14(6), 2079–2084 (2006).
[Crossref] [PubMed]

D. Hillerkuss, M. Winter, M. Teschke, A. Marculescu, J. Li, G. Sigurdsson, K. Worms, S. Ben Ezra, N. Narkiss, W. Freude, and J. Leuthold, “Simple all-optical FFT scheme enabling Tbit/s real-time signal processing,” Opt. Express 18(9), 9324–9340 (2010).
[Crossref] [PubMed]

A. J. Lowery, “Design of arrayed-waveguide grating routers for use as optical OFDM demultiplexers,” Opt. Express 18(13), 14129–14143 (2010).
[Crossref] [PubMed]

S. Chandrasekhar and X. Liu, “Experimental investigation on the performance of closely spaced multi-carrier PDM-QPSK with digital coherent detection,” Opt. Express 17(24), 21350–21361 (2009).
[Crossref] [PubMed]

K. Lee, C. T. D. Thai, and J.-K. K. Rhee, “All optical discrete Fourier transform processor for 100 Gbps OFDM transmission,” Opt. Express 16(6), 4023–4028 (2008).
[Crossref] [PubMed]

Y. Benlachtar, P. M. Watts, R. Bouziane, P. Milder, D. Rangaraj, A. Cartolano, R. Koutsoyannis, J. C. Hoe, M. Püschel, M. Glick, and R. I. Killey, “Generation of optical OFDM signals using 21.4 GS/s real time digital signal processing,” Opt. Express 17(20), 17658–17668 (2009).
[Crossref] [PubMed]

Q. Yang, S. Chen, Y. Ma, and W. Shieh, “Real-time reception of multi-gigabit coherent optical OFDM signals,” Opt. Express 17(10), 7985–7992 (2009).
[Crossref] [PubMed]

Opt. Lett. (1)

Other (9)

F. Buchali, R. Dischler, A. Klekamp, M. Bernhard, and Y. Ma, “Statistical Transmission Experiments Using a Real-Time 12.1 Gb/s OFDM Transmitter”, in Conference on Optical Fiber Communication, OFC (San Diego, CA, 2010), paper OMS3 (2010).

S. Chen, Y. Ma, and W. Shieh, 110-Gb/s Multi-Band Real-Time Coherent Optical OFDM Reception after 600-km Transmission over SSMF Fiber”, in Conference on Optical Fiber Communication, OFC (San Diego, CA, 2010), paper OMS2 (2010).

D. Hillerkuss, T. Schellinger, R. Schmogrow, M. Winter, T. Vallaitis, R. Bonk, A. Marculescu, J. Li, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, K. Weingarten, T. Ellermeyer, J. Lutz, M. Möller, M. Hübner, J. Becker, C. Koos, W. Freude, and J. Leuthold, “Single source optical OFDM transmitter and optical FFT receiver demonstrated at line rates of 5.4 and 10.8Tb/s,” in Conference on Optical Fiber Communication, OFC (San Diego, CA, 2010), paper PDPC1 (2010).

F. C. Garcia Gunning, S. K. Ibrahim, P. Frascella, P. Gunning, and A. D. Ellis, “High symbol rate OFDM transmission technologies”,in Conference on Optical Fiber Communication, OFC (San Diego, CA, 2010), paper OThD1 (2010).

S. Chandrasekhar, X. Liu, B. Zhu, and D. W. Peckham, “Transmission of a 1.2 Tb/s 24-carrier no-guad-interval coherent OFDM superchannel over 7200-km of Ultra-Large-Area Fiber,” in Proc. ECOC 2009 (Vienna, Austria), paper PD2.6 (2009).

H. Sanjoh, E. Yamada, and Y. Yoshikuni, “Optical orthogonal frequency division multiplexing using frequency/time domain filtering for high spectral efficiency up to 1 bit/s/Hz,” in Conference on Optical Fiber Communication, OFC (Anaheim, CA, 2002), paper ThD1, 401–402 (2002).

W. Shieh and I. Djordjevic, OFDM for Optical Communications, Academic Press, 2009.

K. Takiguchi, M. Oguma, H. Takahashi, and A. Mori, “PLC-based eight-channel OFDM demultiplexer and its demonstration with 160 Gbit/s signal reception,” in Conference on Optical Fiber Communication, OFC (San Diego, CA, 2010), paper OThB4 (2010).

Y. Huang, D. Qian, R. E. Saperstein, P. N. Ji, N. Cvijetic, L. Xu, and T. Wang, “Dual-polarization 2x2 IFFT/FFT optical signal processing for 100-Gb/s QPSK-PDM all-optical OFDM,” in Conference on Optical Fiber Communication, OFC (San Diego, CA, 2009), paper OTuM4 (2009).

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Figures (10)

Fig. 1
Fig. 1 Schematic diagram of an OFDM transmission system with electronic IFFT and FFT. S/P: serial-to-parallel conversion. GI: Guard interval; DAC: Digital-to-analog converter; ADC: analog-to-digital converter; P/S: Parallel-to-serial conversion
Fig. 2
Fig. 2 Waveguide structure of an AWG (a) the whole structure (b) the enlarged picture of the output slab region [17].
Fig. 3
Fig. 3 Configuration of the AWG to realize the operations of (a) FFT and (b) IFFT
Fig. 4
Fig. 4 Theoretical and experimental spectrum of different channels of a cyclic AWG (a) theoretical calculation (b) experimental measurements. Both the spectra have a FSR of 160GHz, and channel spacing of 10GHz. (c) The spectral transmittance of a 32-channel AWG experimentally demonstrated in [19], with 10GHz channel spacing
Fig. 5
Fig. 5 The proposed AWG-based all-optical OFDM system setup. MLL: mode locked laser; EDFA: erbium-doped fiber amplifier; S: optical sampler. The AWG at the transmitter performs IFFT and the one at the receiver performs FFT.
Fig. 6
Fig. 6 Eye diagrams of the demultiplexed OFDM signal at the receiver end in Fig. 5 for (a) N = 4 (b) N = 16
Fig. 7
Fig. 7 (a) The generated optical OFDM signal waveform after the transmitter in Fig. 5; (b) the schematic of OFDM signal demultiplexing process in the system shown in Fig. 5
Fig. 8
Fig. 8 The modified AWG-based all-optical OFDM system setup. MLL: mode locked laser; EDFA: erbium-doped fiber amplifier; S: optical sampler. The AWG at the transmitter performs WDM demultiplexing and the one at the receiver performs FFT.
Fig. 9
Fig. 9 Eye diagrams of the demultiplexed OFDM signal at the receiver end in Fig. 7 for (a) N = 4 (b) N = 16
Fig. 10
Fig. 10 Eye diagrams of the demultiplexed OFDM signal with 50GHz modulator bandwidth in Fig. 7 for N = 16 (a) the modulated subcarrers are symbol-aligned (b) the modulated subcarriers are not symbol-aligned

Equations (14)

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2 π n s ( λ 0 ) λ 0 ( f 1 x 1 d 1 2 f 1 ) + 2 π n s ( λ 0 ) λ 0 ( f + x d 2 f ) = 2 π n s ( λ 0 ) λ 0 ( f 1 + x 1 d 1 2 f 1 ) + 2 π n s ( λ 0 ) λ 0 ( f x d 2 f ) + 2 π n c ( λ 0 ) λ 0 Δ L + 2 m π ,
λ 0 = n c Δ L m ,          Δ λ = n s d D λ 0 N c f Δ L ,        N c h = λ 0 f n s d D ,
Δ f = c λ 0 2 Δ λ = n s d D f λ 0 c Δ L N c = 1 N c h 1 τ ,       τ = Δ L N c c .
h i k ( t ) = m = 0 N 1 exp [ π j n s d λ 0 ( 2 m N + 1 ) ( sin θ i + sin θ o ) ] δ ( t n s f c N c L + m Δ L c )                                                                                                    ( m = 0 , 1 , ... , N 1 ) .
sin θ i ( 2 i N + 1 ) D 2 f ,      sin θ o ( 2 k N + 1 ) D 2 f .
h i k ( t ) = m = 0 N 1 e j π n s d D λ 0 f ( 2 m N + 1 ) ( i + k + 1 ) δ ( t m τ ) = m = 0 N 1 e j π N c h ( 2 m N + 1 ) ( i + k + 1 ) δ ( t m τ )          = m = 0 N 1 e j π N ( 2 m N + 1 ) ( i + k + 1 ) δ ( t m τ )                                  ( m = 0 , 1 , ,   N 1 )  
θ j , k = 2 π n s d D λ 0 f ( m N + 1 2 ) k .
S k ( t ) = s i ( τ ) h ( t τ ) d τ = m = 0 N 1 e j 2 π N ( m N 1 2 ) k s i ( t m τ ) .
S k [ ( N 1 ) τ ] = m = 0 N 1 e j 2 π N ( N 1 2 m ) k s i ( m τ )                      n = N / 2 m ¯ ¯   ( n = 0 N 1 s i ' [ ( N 2 n ) τ ] e j 2 π n k N ) e j π N k = F F T ( s i ' [ ( N 2 n ) τ ] ) e j π N k .
h i ( t ) = m = 0 N 1 exp j 2 π N ( m N 1 2 ) i δ ( t m τ ) .
S i ( t ) = { S i ( m T ) ,          t = m T 0 ,                   o t h e r w i s e
s o u t ( t ) = i = 0 N 1 S i ( τ ) h i ( t τ ) d τ .
                       s o u t ( m τ ) = i = 0 N 1 e j 2 π N ( m N 1 2 ) i S i ( 0 ) = i = 0 N 1 S i ( 0 ) e j 2 π N ( N 1 2 ) i e j 2 π m i N n = N / 2 + m    s ' o u t [ ( n + N 2 ) τ ]   =   i = 0 N 1 S i ( 0 ) e j π N i e j 2 π n i N = I F F T ( S i ( 0 ) e j π N i ) .
H i k ( f ) = e j π ( N 1 ) f τ m = 0 N 1 e j π ( 2 m + N 1 ) ( i + k + 1 N + f τ ) = e j π ( N 1 ) f τ sin [ π ( i + k 1 + N f τ ) ] sin [ π ( i + k 1 ) / N + f τ ] .

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