Abstract We have developed an analytic approach to investigate the effect of group delayripple of the dispersive devices on the performance of two major building blocks of micro-wave-photonic filters. Firstly, performance of PM-based block in the presence of an arbitrarygroup delay ripple (GDR) is analyzed and compared with the ripple-free case to reveal thedestructive effects of added group delay ripple. In the next step, we repeat the proposedapproach for the AM-based one; again, the performance is compared with the ripple-freecase. Two distortion metrics are also introduced to quantify this distortion. Comparison ofthe performance of two building blocks in the presence of group delay ripple unveils someinteresting characteristics of microwave-photonic filters which have not been mentioned sofar. We also add a general survey of two analyzed building blocks to present their respectivemost significant advantages and shortcomings. The simulated Optisystem results conform toour proposed analytical approach and verify the theoretical model.
High speed microwave signals processing is always a challenge. An interesting solutionto overcome the common problem of low bandwidth of microwave circuits (referred as“electronic bottleneck”) comprises the employment of photonics technology to perform therequired signal processing functionalities of RF signals directly in the optical domain. Imple-mentation of unique properties of photonic devices such as high time-bandwidth product,low loss, electromagnetic interference immunity and light weight leads to a novel researchdirection called “Microwave-Photonics”(Capmany and Novak 2007).
A. Mokhtari (B) · M. AkbariDepartment of Electrical Engineering, Sharif University of Technology, Azadi Avenue,P.O. Box 11155-9363, Tehran, Irane-mail: [email protected]
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404 A. Mokhtari, M. Akbari
In the last decades, there is an increasing effort in researching new microwave photonicstechniques for different applications e.g. photonic generation and processing of microwaveand millimeter signals, optical beam-formers or phased array antennas and radio-over-fibersystems (Yao 2009).
One of the applications that most attracted the interest of the researchers was the possi-bility of using photonic devices to implement flexible filters (Jackson et al. 1985), (Moslehiet al. 2005). Conventionally, the RF signal is modulated by an optical carrier and the com-bined signal is fed to a photonic circuit that samples in the time domain, weighs the samples,and combines them using optical delay lines. At the output, the resulted signals are con-verted optically to RF via various optical receivers. Extensive efforts have been directed tothe design and implementation of photonic microwave filters with different architectures tofulfill a broad range of functionalities recently. Furthermore, the availability of novel com-ponents, such as the arrayed waveguide grating (AWG) (Pastor et al. 2003) and the fiberBragg grating (FBG) (Mora et al. 2002) (Bolea et al. 2011) has opened a new perspectivetoward fully reconfigurable and tunable microwave-Photonic filters. Limitations imposed onthe implementing multi-tap transversal filters via this approach since each tap needs separatededicated laser’s wavelength and bulky delay lines (Pastor et al. 2002; Zeng and Yao 2004;Capmany et al. 2005; Minasian 2006).
Chirped fiber Bragg gratings (CFBG) have been widely investigated due to their tun-able group-delay characteristic and advantages such as compact size, low insertion loss andpotential flexibility (Eggleton et al. 2002; Erdogan 2002; Inui et al. 2002; Komukai et al.2002; Sumetsky et al. 2002; Brennan 2004). They are widely used to compensate the dis-persion of long-haul fiber links. Practically, CFBGs show some characteristics like GDRdistorts angle-modulated links (Ulmer 2004) and optical beam-formers (Thai et al. 2009)since this unfavorable effect causes phase shifts between frequency components of signalthat generates a penalty to be measured (Kashyap 1999).
The object of this paper is to investigate the group delay effect of dispersive devices onthe performance of two microwave-photonic filters. We propose an analytical approach tomodel the microwave-photonic setup assuming quadratic phase response for typical disper-sive devices and small value coefficients in the sinusoid expansion. OptiSystem package isutilized to verify the results of analytical approach.
This paper is structured as follows: Sect. 2 includes the theoretical derivation for the phasemodulator microwave-photonic system both without and with GDR. In Sect. 3, the analysisis repeated for the amplitude modulator microwave-photonic system. Two distortion metricsare presented to help the designers for better selection in Sect. 4. Finally, we compare twopreceding sections and conclude in Sect. 5.
2 The phase modulator microwave-photonic system
We propose to analyze the performance of microwave-photonic filter based on the phasemodulator depicted in Fig. 1. We try to find an equivalent frequency response to describe itsoverall performance. A laser source is modulated by a phase modulator and then the outputwill pass the dispersive device that changes the phase relationship of the output electrical field.A photodiode converts the optical signal to electrical one and eliminates the high frequencycontent. The electrical field after the phase modulator can be modeled as:
E1(t) ∝+∞∑
n=−∞Jn(β)e j (ω0t−n�t) (1)
123
Two building blocks of microwave photonics filters 405
Fig. 1 The PM-based microwave-photonic filter scheme
where ω0 and � are optical and microwave frequencies respectively. βis the phase modula-tor’s index in the form of β = πVpp/(2Vπ ). VppandVπ are peak-to-peak modulator’s voltageand the required voltage to induce π phase shift respectively.
2.1 Ripple-free performance
We assume the induced delay (either by a length of low loss single-mode fiber or LinearlyChirped FBG) is linear (Yao et al. 2007), the quadratic phase function is given by:
The electrical field after the dispersive device can be derived as:
E2(t) =+∞∑
n=−∞Jn(β)e
− j
(θ0+(τ ′−t)(ω0−n�)+τ ′′ (ω0−n�)2
2
)
(3)
We should apply an assemble averaging considering photodiode’s responsivity factor � tofind the photodiode’s current output. Then, we search for a filter around the initial frequency�:
H(�) ∝ �+∞∑
n=−∞Jn+1(β)Jn(β)e
− j{
τ ′′2 (2n+1)�2−(τ ′+τ ′′ω)�
}
(4)
Because of its damping coefficients (Fig. 2), two principal right n = 0 and left n = −1sidebands are sufficient for approximation:
H(�) ∝ � sin
(τ ′′
2�2
)(5)
The filter has a finite-impulse response that its taps are shown in Fig. 2. The transferfunction is also illustrated in Fig. 3 with two poles below 20 GHz.
2.2 Performance in the presence of an arbitrary group delay ripple
We express the group delay (shown in Fig. 4) in terms of cosine terms with amplitude ofτk(in second), frequency of υk (in Hertz) and initial phase of θk (in radian). The expansionof phase leads to:
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406 A. Mokhtari, M. Akbari
Fig. 2 Equivalent filter taps for PM-based microwave-photonic filter
Fig. 3 Equivalent transfer function of PM-based microwave-photonic filter
�GDR(ω) = −[τ ′′ ω2
2+ τ ′ω + θ0 +
∑
k
τkυk cos
(ω
υk+ θk
)](6)
If the amplitudes of cosine terms are small enough to ignore (which is valid for smallbandwidth (Ulmer 2004; Chan and Minasian 2001), (Thai et al. 2009), the expression isapproximated as follows:
exp
(− j
∑
k
τkυk cos
(ω
υk+ θk
))∼= 1 +
∑
k
− jτkυk cos
(ω
υk+ θk
)(7)
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Two building blocks of microwave photonics filters 407
Fig. 4 Group delay characteristic of the dispersive device with 50 ps ripple [extracted form (Ulmer 2004)]
with the insertion of bk = − jτkυk and applying the approach in previous section, the transferfunction will be:
H(�) ∝ �+∞∑
n=−∞Jn+1(β)Jn(β)e
− j{
τ ′′2 (2n+1)�2−(τ ′+τ ′′ω)�
}
(8)
×⎧⎨
⎩1 +∑
k
bk cos
(ω0 − (n + 1)�
υk+ θk
)⎫⎬
⎭ ×⎧⎨
⎩1 +∑
k′b∗
k′ cos
(ω0 − n�
υk′+ θk′
)⎫⎬
⎭
Due to small amplitude of coefficients in summation (|bk | << 1) , the cross-terms couldbe eliminated, so we have b∗
k′ = −bk (Primed coefficients are only necessary for cross-termmultiplications to be ignored). The final transfer function is yielded by:
|H(�)| ∝ � sin
(τ ′′2
�2)
+ �∑
k
2bk sin
(�
2υk
)
⎧⎨
⎩sin
(2ω0 − �
2υk+ θk
)e j τ ′′
2 �2 − sin
(2ω0 + �
2υk+ θk
)e− j τ ′′
2 �2
⎫⎬
⎭ (9)
Comparison of expressions of (5) and (9) shows the distortion effect of GDR on the fil-ter transfer function. Figure 5a depicts the OptiSystem setup used for simulation. Effect ofGDR’s amplitude variation is shown in Fig. 5b; Again, filter transfer function is distorted.As the amplitude of GDR increases, the filter transfer function will be distorted and evennumber of poles of transfer function and their respective frequencies will be changed espe-cially in low frequencies. In order to find an upper limit for the ripple-induced distortion,
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408 A. Mokhtari, M. Akbari
Fig. 5 a The employed OptiSystem setup for simulation of PM-based Microwave-photonic filter. b Effect ofadded amplitude of group delay ripple on the performance of PM-based microwave-Photonic filter (symbolson the line are corresponding OptiSystem results)
the sinusoid terms are considered to be coherent. Practically, the phases of sinusoids arerandomly distributed for a typical FBG that alleviate the overall induced distortion.
3 The amplitude modulator microwave-photonic system
Another block-diagram that is widely used for microwave filter is illustrated in Fig. 6. Again,an equivalent frequency response must be proposed to describe its overall performance.A multi-wavelength laser (MWL) source (e.g. a sliced broadband source or comb laser) ismodulated by an amplitude modulator and then the output will pass the dispersive device andwill specialize each wavelength a group delay. The relative group delays among differentpass-bands make the time delay required for transversal filtering. A photodiode convertsthe optical signal to electrical one and eliminates the high frequency content. If the disper-sive device has a linear shape such as linearly-chirped fiber Bragg grating (LCFBG), theperformance can be analytically formulated.
123
Two building blocks of microwave photonics filters 409
Fig. 6 The AM-based microwave-photonic filter scheme
3.1 Ripple-free performance
The electrical field of N mode-locked laser arrays has the form:∑N
i=1 Ai e jωi tωi (ωi are thelaser’s applied wavelengths). After the AM modulation by frequency of microwave signal�, the electrical field will be changed to:
E1(t) ∝N∑
i=1
Ai ejωi t (1 + β sin(�t)) =
+∞∑
n=−∞Ai
(e jωi t + β
2 j
(e j (ωi +�)t − e j (ωi −�)t
))
(10)
The dispersive device can be modeled as an all-pass filter. Output electrical field of theamplitude modulator is expressed as:
E2(t) = F−1 {E2(ω)}
=N∑
i=1
Ai
⎧⎨
⎩e− j
(θ0+(τ ′−t)ωi +τ ′′ ω2
i2
)
+ β
2 je− j
(θ0+(τ ′−t)(ωi +�)+τ ′′ (ωi +�)2
2
)
− β
2 je− j
(θ0+(τ ′−t)(ωi −�)+τ ′′ (ωi −�)2
2
)⎫⎬
⎭ (11)
=N∑
i=1
Ai e− j
(θ0−ωi
(t−τ ′− τ ′′
2 ωi
))⎧⎨
⎩1 + βe− j τ ′′2 �2
sin(�(t − τ ′ − τ ′′ωi ))
⎫⎬
⎭
The filter’s transfer function around � yields as:
H(�) ∝ − jβ� cos
(τ ′′
2�2
) N∑
n=1
|An |2e− j�(τ ′+τ ′′ωn) (12)
Expecting to behave like a finite impulse response (FIR) periodic filter H̄(�) =∑Nn=1 an exp( jn�T ) (Fig. 7), ωnshould be written in terms of n. If the lasers have equally-
spaced wavelengths, then we can assume that central wavelength is much more than thedistance between two successive optical wavelengths:
ωn = 2πC
λn= 2πC
λ0 + n�λ= 2πC/λ0
1 + n�λ/λ0
∼= ω0(1 − n�λ/λ0) (13)
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410 A. Mokhtari, M. Akbari
Fig. 7 Equivalent transfer function of AM-based microwave-photonic filter and comparison with expectedperiodic filter
Using this result, it is straightforward to simplify the equation (12) With approxima-tion�ω ∼= ω0�λ/λ0:
H(�) ∝ − jβ� cos
(τ ′′
2�2
)e− j�(τ ′+τ ′′ω0)
N∑
n=1
|An |2e− j�nτ ′′�ω (14)
where an = |An |2 ≥ 0 and delay unit isT = −τ ′ω0�λ/λ0 = τ ′�ω. The resulted positivetap weights confirm the general shortcoming of coherent microwave-photonic filters that isthe inability to realize negative taps or complex ones (Yao 2009; Yao et al. 2007).
The resulted filter transfer function H(�) is compared with periodic transfer functionH̄(�) in Fig. 7. The group delay characteristic of investigated dispersive device and thefive applied laser wavelengths are illustrated in Fig. 8. As all lasers have equal powers, theinduced delay is derived as:
T = −τ1ω0/λ0 × �λ = −654 ps/nm × 0.4 nm = −261.6 ps. (15)
3.2 Performance in the presence of arbitrary group delay ripple
Following the same approach developed in Sect. 2.2, the output electrical field of dispersivedevice in the presence of GDR can be expressed as:
E2(t) =N∑
i=1
Ai e− j
(θ0+(τ ′−t)ωi +τ ′′ ω2
i2
) {1 +
∑
k
b′k cos
(ωi
υk+ θk
)}
+Ai
⎡
⎣ β
2 je− j
(θ0+(τ ′−t)(ωi +�)+τ ′′ (ωi +�)2
2
) {1 +
∑
k
b′k cos
(ωi + �
υk+ θk
)}⎤
⎦
123
Two building blocks of microwave photonics filters 411
Fig. 8 Group delay characteristic of the dispersive device and five applies laser applied wavelengths
−Ai
⎡
⎣ β
2 je− j
(θ0+(τ ′−t)(ωi −�)+τ ′′ (ωi −�)2
2
) {1 +
∑
k
b′k cos
(ωi − �
υk+ θk
)}⎤
⎦
(16)
Applying ensemble averaging, the transfer function is calculated as:
H(�) = − j�βe− j�(τ ′+τ ′′ωm )N∑
n=1
|An |2{cos
(τ ′′2
�2)
+∑
k
2b′k cos
(�
2pk
)
×{
e− j τ ′′2 �2
cos
(2ωn + �
2υk+ θk
)+ e j τ ′′
2 �2cos
(2ωn − �
2υk+ θk
) }(17)
Comparing the (14) and (17), GDR induces a distortion in AM-based microwave filtertransfer function just like the PM-based one. Moreover, there is a kind of duality eitherbetween (5) and (14) or (9) and (17) in the PM-based and AM-based microwave-photonicfilter implementations. These two blocks somehow complement each other in generatingarbitrary filter’s transfer functions.
Figure 9a depicts the OptiSystem setup used for simulation; Effect of ripple’s amplitudeis depicted in Fig. 9b. As seen, increasing the amplitude of GDR has no significant effect onthe performance of AM-modulated filter’s transfer function; which is in contrast to Fig. 5b.This is the main instructive point extracted from this survey. Again, the sinusoid terms aresupposed to be coherent to find the worst case (the highest distortion).
4 Distortion metric
Employing the analytical model, the roles of GDR can be discriminated and compared fortwo setups. For comparison, two metrics can be defined to measure the ripple-induced
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412 A. Mokhtari, M. Akbari
Fig. 9 a The employed OptiSystem setup for simulation of AM-based Microwave-photonic filter. b Effect ofadded amplitude of group delay ripple on the performance of AM-based microwave-Photonic filter (symbolson the line are corresponding OptiSystem results)
distortion; First, Mean Square Error (MSE) between the calculated transfer function andthe ripple-free case as:
MSE =fmax∫
0
∣∣Hcalculated − Hripple−free∣∣2 (18)
Second, the correlation between the measured transfer function and the calculated ripple-free case can also defined as a correlation metric:
Two building blocks of microwave photonics filters 413
Fig. 10 Distortion metrics as a function of group delay ripples calculated from analytical model for twosetups, a MSE metric, b Correlation metric
where Corr (x, y) = ∫ fmax0 |x ( f ) y( f )| d f . Two metrics calculation is performed for dif-
ferent GDR values. The normalized results are shown in Fig. 10a for the MSE metric andFig. 10b for the correlation one. The results shows the dominating effect of the GDR on thePM-based filter compared to the AM-based one in deviating the measured filter’s transferfunction from the ripple-free case.
In the PM-based filter, as GDR increases, the filter transfer function will be distorted andeven number of poles of transfer function and their respective frequencies will be changedespecially in low frequencies. Meanwhile, in the AM-based filter, the effect of GDR can bealso imagined as a minor perturbation term added to the overall ripple-free transfer function.
5 Conclusions
Two filtering mechanisms are discussed in this article: the PM-based filtering that induces aphase shift on each pass-band. The phase shift that each pass-band undergoes depends on thefrequency of modulation and index of pass-bands. The relative phase differences of the carrierand other pass-bands create the conversion effect of phase modulation (PM) to amplitudemodulation (AM). During this conversion, amplitude of each sideband is altered; conse-quently, the filtering behavior will be realized. Conversely, AM-based microwave-photonicfilters alter each tap’s coefficient directly and realize an FIR multi-tap filter.
The dispersive devices tend to have an oscillatory group delay nature modeled with sinu-soid over small bandwidth. Two studied blocks behave differently in the presence of GDRs.This contrast is originated from different filtering mechanism in two building blocks; PM-based filtering modulates input signal on phase and utilize PM-AM conversion whilst AM-based one applies it directly on amplitude. Consequently, AM-based one is much morerobust to GDR that distorts the phase. This advantage is earned in exchange for more com-plex implementation of the AM-based one. It is predictable that PM-based filter surpassesthe AM-based one in the presence of the amplitude reflection ripple (another unwanted effectexists in dispersive devices to be modeled separately).
As a rule of thumb, the noise and nonlinearities of the AM-based filter exceeds its counter-part due to more complexity; but, it has more degrees of freedoms that makes the AM-basedone more tunable and flexible.
The GDR effect seems unwanted at the first glance; but if ripple characteristic could beengineered (Yi and Minasian 2008), it may bring about tunable microwave-photonic filter
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Table 1 Comparison of two setups
Sensitivity togroup delay ripple
Sensitivity toamplitude ripple
Flexibility Complexity (cost)
PM-based filter High Low Moderate LowAM-based filter Low Moderate High High
that will have various applications in arbitrary waveform generation (AWG), optical synthe-sizers and equalizers. Moreover, this article shows that we are unable to realize this idea totune the filter in the AM-based one instead; we could employ the amplitude ripple effect toaccomplish this wish. The properties of two setups are summarized and compared in Table 1.
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