## Abstract

A new technique to implement a photonic microwave delay-line filter based on nonuniform tap spacing with arbitrary bandpass response is proposed and experimentally demonstrated. Being different from a regular photonic microwave delay-line filter where the taps are uniformly spaced, the proposed filter in this paper has nonuniformly-spaced taps. The key feature of this technique is that a photonics microwave delay-line filter with arbitrary bandpass response can be realized with only positive taps via nonuniform tap spacing. The use of the proposed technique to implement a flat-top bandpass filter is experimentally demonstrated.

© 2008 Optical Society of America

## 1. Introduction

Optical microwave signal processing has been a topic of interest for many years [1, 2]. The key advantage of processing microwave signals in the optical domain is that the signal processors can operate at very high speed, which eliminates the bottleneck due to the limited sampling rate of electronics. Most of the photonic microwave filters proposed in the literature have a structure with a finite impulse response (FIR). To avoid optical interference, photonic microwave filters are usually designed to operate in the incoherent regime. A photonic microwave delay-line filter that is operating in the incoherent regime would usually have all positive coefficients. It is known that a photonic microwave delay-line filter with all-positive coefficients can only operate as a lowpass filter, or a special design must be incorporated to achieve bandpass functionality. In addition, to design a photonic microwave delay-line filter with arbitrary bandpass characteristics, such as a flat top and sharp transition bands, the filter coefficients should not be positive only; but negative or even complex coefficients are required. Although the implementation of a photonic microwave delay-line filter with negative [3–12] or complex coefficients [13–14] have been demonstrated recently, these filters usually have a complicated structure, which may limit their potential for practical applications.

In this paper, we propose a new and simple technique to design and implement a photonic microwave delay-line filter with arbitrary bandpass response using a microwave delay-line filter with all positive taps that are nonuniformly spaced. A photonic microwave filter with nonuniformly spaced taps was proposed in [15] to increase the free spectral range (FSR). In this paper, however, the nonuniformly spaced time delays are introduced to generate equivalent negative or complex tap coefficients, to implement a photonic microwave filter with an arbitrary bandpass response. We show that negative or complex coefficients could be equivalently generated by simply adjusting the tap spacing. We demonstrate that the passband response concerned would have the same spectral characteristics as a photonic microwave delay-line filter with true negative or complex coefficients. The key advantage of this technique is that a photonic microwave delay-line filter with arbitrary bandpass response can be realized through nonuniform spacing with only true positive taps. The use of the proposed technique to implement a seven-tap bandpass photonic microwave delay-line filter with a flat-top bandpass response is experimentally demonstrated.

## 2. Principle

It is known that a regular, uniformly-spaced FIR filter has an impulse response given by

where *N* is the number of taps, *a _{k}* is the filter coefficient of the

*k*th tap,

*T*=2

*π*/Ω is the time delay difference between two adjacent taps, and Ω is the FSR of the filter.

By applying the Fourier transform to the two sides of Eq. (1), we have the frequency response of the filter, which is given by

It is known that *H*(*j*ω) has a multi-channel frequency response with adjacent channels separated by an FSR, with the *m*th channel located at *ω*=*m*Ω. Note that except for the different central frequencies, the frequency responses of all the channels are exactly identical.

In a regular photonic microwave delay-line filter based on incoherent detection, the coefficients are usually all positive. However, a phase term can be introduced to a specific coefficient by adding a proper time delay at the specific tap, which is termed time-delay-based phase shift, as illustrated in Fig. 1.

For example, at *ω*=*m*Ω a time delay shift of Δ*τ* will generate a phase shift given by

Obviously such a phase shift is frequency-dependent, which is accurate only for the frequency at *m*Ω, but approximately accurate for a narrow frequency band at around *m*Ω. As a result, if the *m*th bandpass response, where *m*≠0, is considered, one can then achieve the desired phase shift at the *k*th tap by adjusting the time delay shift by Δ*τ _{k}*. Following Eq. (2), one can get the frequency response of the nonuniformly-spaced FIR filter at around

*ω*=

*m*Ω,

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}=\sum _{k=0}^{N-1}{\alpha}_{k}\mathrm{exp}\left(-j\omega \Delta {\tau}_{k}\right)\times \mathrm{exp}\left(-jk\frac{2\pi}{\Omega}\omega \right)$$

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\approx \sum _{k=0}^{N-1}{\alpha}_{k}\mathrm{exp}\left(-jm\Omega \Delta {\tau}_{k}\right)\times \mathrm{exp}\left(-jk\frac{2\pi}{\Omega}\omega \right)$$

As can be seen from Eq. (4), one can get an equivalent phase shift for each tap coefficient. Specifically, if the desired phase shift for the *k*th tap is φ_{k}, based on Eq. (4) the total time delay τ_{k} for the *k*th tap is

As a result, if the time delay of each tap is adjusted based on Eq. (5), the filter coefficients would have the required phase shifts to generate the required passband with the desired bandpass characteristics.

The same concept using nonuniform tap spacing to introduce phase shifts to an RF pulse has been recently demonstrated by the authors to implement RF pulse phase encoding [16]. We pointed out in [16] that the time-delay-based phase shift is frequency dependent, which is shown here in Eq. (3). As a result, the phase shifts due to the additional time delays would be different for different passbands. Therefore, we can conclude that in a nonuniformly-spaced photonic microwave delay-line filter, the frequency responses for different channels are different. In addition, within the *m*th passband, the time-delay-based phase shift is accurate only for the frequency at *m*Ω, and approximately accurate for a narrow band at around *m*Ω. The maximum error of the phase shift is determined by the maximum bandwidth of the bandpass concerned and the central frequency. For many applications, it is usually required that the bandwidth of the passband is narrow, then the error due to the frequency-dependent phase shift is small and negligible, which ensures the effectiveness of the proposed technique.

## 3. PM-IM conversion for the elimination the baseband

Based on Eq. (4), for *m*=0 no phase modulation is introduced to *H*′(ω). Therefore, *H*′(ω) at ω≈0 is still an all-positive-coefficient filter with a baseband resonance around dc. For many applications, we expect the filter is operating as a bandpass filter. Therefore, the baseband resonance at *m*=0 should be eliminated. To do so, in the proposed filter we use a phase modulator instead of an intensity modulator. It has been demonstrated that the phase modulation to intensity modulation (PM-IM) conversion in a dispersive fiber would generate a notch at dc, which could be used to eliminate the baseband resonance, to achieve bandpass functionality [17]. The PM-IM conversion in a dispersive fiber can be expressed

where *c* is the light velocity in free space, *χ* is the total dispersion of the dispersive device, and λ is the central wavelength of the optical carrier. By properly selecting the total dispersion, the peak of *H _{PM-IM}*(

*ω*) can be designed to locate at the same location as the

*m*th channel. If

*m*=1 is selected and the first peak of

*H*(

_{PM-IM}*ω*) is located at the 1

^{st}-order channel of the filter, then we can get the required dispersion, which is given by

## 4. Experiment

A seven-tap photonic microwave delay-line filter with nonuniformly-spaced taps to produce a flat-top bandpass frequency response is investigated experimentally. Assume the passband concerned is at *m*=1 and the frequency response of the bandpass has a shape of a rectangle, then the corresponding impulse response should be a *sinc* function, which has both positive and negative values extending to infinity along the horizontal axis. For practical implementation, the calculated impulse response should be cut off to enable a physically realizable filter. If a regular true-negative-coefficient photonics microwave filter is employed to produce the frequency response, the filter coefficients can be selected to be [-0.12, 0, 0.64, 1, 0.64, 0, -0.12]. The frequency response is shown as dotted line in Fig. 2, where *T*=82.6 ps is used in our numerical simulation, which corresponds to an FSR of 12.1 GHz. The 3-dB bandwidth of the filter is 5.0 GHz with a central frequency of 12.1 GHz.

With the proposed technique, the same bandpass characteristics at *m*=1 can be obtained by using nonuniform spacing. The filter is designed based on Eq. (5) with all-positive coefficients of [0.12, 0, 0.64, 1, 0.64, 0, 0.12]. The time delays for the seven taps are then [-2.5*T*, -2*T*, -*T*, 0, *T*, 2*T*, 2.5*T*], with the taps nonuniformly spaced. The frequency response of the PM-IM conversion is designed to make its first peak be located at *f*=12.1 GHz, which is shown as the dash-dot line in Fig. 2. The overall frequency response of the nonuniformly-spaced filter is calculated and shown as solid line in Fig. 2. It is clearly seen a flap-top frequency response is achieved in a photonic microwave delay-line filter with all-positive coefficients. The 3-dB bandwidth is 4.9 GHz and the central frequency is 12.1 GHz. The frequency response of the passband is close to that generated by a regular photonic microwave filter with true negative taps.

The seven-tap filter is then experimentally demonstrated. Since two of the seven coefficients are zero, a laser array with five wavelengths is employed. The five wavelengths are multiplexed and sent to a phase modulator. The phase modulated signal is then applied to a photodetector (PD) via a length of standard single-mode fiber. The experimental setup is shown in Fig. 3.

The frequency response of the filter is measured using a vector network analyzer (VNA, Agilent E8364A). In our design, the center frequency of the passband is located at 12.1 GHz. Based on Eq. (7) the total dispersion of the single mode fiber is calculated to be 425 ps/nm (corresponding to a length of 25 km standard single-mode fiber). The frequency response of the PM-IM conversion is shown as dash-dot line in Fig. 4(a). Since the time delay difference is proportional to the wavelength spacing between adjacent wavelengths, the optical wavelength corresponding to the *k*th tap is given by

where *D* is the chromatic dispersion parameter of the standard single-mode fiber (*D*=17 ps/nm/km at 1550 nm), *L* is the length of the standard single-mode fiber, and λ_{0} and λ_{k} are the wavelengths for the 0th and the *k*th taps. In the experiment, we tune the wavelengths to achieve the required nonuniformly-spaced time delay differences based on Eq. (8). The power of each laser is also adjusted to achieve the desired tap coefficients. The output of the laser array is measured using an optical spectrum analyzer, which is shown Fig. 4(b). The frequency response of the photonic microwave delay-line filter is shown as solid line in Fig. 4(a). A bandpass filter with a flat-top frequency response is then experimentally demonstrated.

## 5. Conclusion

We have proposed and demonstrated a new technique to implement a photonic microwave delay-line filter with all-positive coefficients which can provide an arbitrary bandpass characteristics using nonuniformly-spaced time delays. Being different from a regular uniformly-spaced photonic microwave delay-line filter, the proposed filter here achieved the same bandpass characteristics with equivalent negative or complex coefficients obtained using nonuniformly-spaced taps. A seven-tap microwave bandpass filter with a frequency response having a flat-top passband was experimentally demonstrated. The key advantage of this technique is that a microwave delay-line filter with an improved bandpass characteristics can be easily achieve by nonuniform spacing with all-positive coefficients.

## Acknowledgement

The work was supported by The Natural Sciences and Engineering Research Council of Canada (NSERC).

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