chapter 4 synthesis of lattice form iir optical...

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CHAPTER 4

SYNTHESIS OF LATTICE FORM IIR OPTICAL DELAY

LINE FILTERS

4.1 INTRODUCTION

IIR filters are digital filters with infinite impulse response. Unlike FIR

filters, they have the feedback (a recursive part of a filter) and therefore they are

known as recursive digital filters. Infinite Impulse Response (IIR) filters are the

first choice when speed is paramount and phase non-linearity is acceptable. IIR

filters are computationally more efficient than FIR filters as they require fewer

coefficients due to the fact that they use feedback or poles. However feedback

can result in the filter becoming unstable if the coefficients deviate from their

true values. The general difference equation for an IIR digital filter is

y(n) = ak y(n k) + bk x(n k) (4.1)

where ak is the kth feedback tap. The first in the filter function denotes

summation from k = 1 to k = N -1 where N is the number of feedback taps in

the IIR filter. The second denotes summation from k = 0 to k = M -1 where

M is the number of feed forward taps. IIR filters consist of zeros and poles,

and require less memory than FIR filters

In this chapter, the synthesis of two lattice form IIR optical delay

line filters, viz., three port (1 x 3) and five port (1 x 5), for RF filtering is

presented.

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The synthesis of these filters have been carried out using

constrained least square (CLS) method and compared with the existing

method. The synthesis methods are compared in terms of the number of

stages, coupling coefficient values, phase angles, pass band and stop band

attenuation levels.

4.2 CIRCUIT CONFIGURATION FOR 1 x 3 AND 1 X 5

LATTICE FORM IIR OPTICAL DELAY LINE FILTER

In general, 1 x M structure has one input and M outputs that are

composed of (MN+M-1) directional couplers, (MN + M 1) phase shifters

and an external phase shifter ex. The circuit configuration of 1 x 3 IIR optical

delay line filter is shown in Figure 4.1. The delay difference in each path has

a time delay . The five port optical delay line filter is composed of various

canonical filter components as shown in Figure.4.1. The circuit configuration

of 1 x 5 IIR optical delay line filter differ from FIR architecture by number of

components present in the circuit.

Figure 4.1 Circuit configuration of a Nth stage 1 x 5 IIR optical delay line filter

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The multi-port optical delay-line circuit has a number of cascaded

unit elements. Each unit element is composed of one symmetric Mach-

Zehnder interferometer and one delay line.

The transfer function of the first element is given as

(4.2)

The transfer functions of the directional coupler between the

waveguides are given in equation 4.3 and 4.4.

(4.3)

(4.4)

The third and fifth components are the phase shifters having phase

angle A and B and the transfer functions are given as

(4.5)

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(4.6)

The total filter characteristic is expressed as the multiple products

of all these basic components.

4.3 SYNTHESIS METHOD

The vector elements H(z) , F(z) , G(z) and Q(z) of a three-port

optical delay-line circuit with ring waveguides can be expressed using

complex expansion coefficients ak , bk , ck ( k = 0 ~ N ) and dk ( k = 1~ N ) as

follows:

(z)= akZ-kNk=0

F(z)= bkZ-kNk=0

G(z)= ckZ-kNk=0 (4.7)

Q(z)=1+ dkZ-kN

k=1

where ( )( )

, ( )( )

and ( )( )

indicate the transfer function from input port to port

1, port 2 and port 3 respectively.

It is noted that the circuit synthesis method presented is also

applicable to FIR optical filters, since the standard transfer matrix of IIR

optical filters is equal to that of FIR optical filters when Q(z) =1.

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The synthesis algorithm for 1 x M optical delay line circuit is

presented in this section. This gives all unknown circuit parameters of the

proposed circuit when the desired filter characteristics is specified by a given

transmission spectrum. The three step synthesis process calculates the

unknown expansion coefficients ak, bk, ck (k = 0 ~ N), and dk (k=0 ~ N),

coupling coefficient angles ka, kb (k = 0 ~ N), kc (k = 1 ~ N) of directional

couplers, phase shift values ka, kb (k = 0 ~ N), kc (k = 1 ~ N) of phase

shifters and one external phase shifter ( ex).

The expansion coefficients , , and are found

using the following equations:

dkn-1= dk+1

n-1 -dk+1n-1

n

ckn-1=

ck+1n-1 -j sin kbe nbbk+1

n -cos nbck+1n

n

bkn-1=

bkn-1-j sin nae naak+1n - cos na cos nbe nbbk+1

n -j sin nb cos nack+1n

n

akn-1 n*ak+1n-1 -e na cos na e na ak+1n +j sin na cos nbe nbbk+1n - sin na sin nb ck+1n

A set of recursion equations to find the coupling coefficients of

directional couplers and phase shift values of phase shifters are already

discussed in chapter 3.

The external phase shifter value can be obtained as follows:

ex=-arg a00e 0a cos 0a +jb0

0e 0b cos 0b sin 0a - c00 sin 0a sin 0b (4.9)

Table 4.1 shows the calculated angles of directional couplers and

the phase shift values of the phase shifters( nA, nB, nC, nA, nB and nC) of 3

(4.8)

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output IIR optical filter with number of stages N = 18. It has been observed

that the output response using CLS algorithm is better in terms of the number

of stages compared to the method used by Azam et al (2008).

Figure 4.2 Flow chart of the algorithm

Yes No

From the desired periodic frequency, constant delay time difference is calculated.

Obtain the complex coefficients ak,bk,ck and dk using constrained least square algorithm

n > 0?

Compute all the poles n, coupling angles nc and phase values nc

Set initial value: = , = , = and = with n=N, k=0~N and d0=1

Calculate the circuit parameters using equations

n = n - 1

Calculate the expansion coefficients

Calculate the circuit parameters of the first stage

All the circuit parameters are obtained

Input to the CLS algorithm – Order of the filter, Pass Band and Stop Band Edge frequencies

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Table 4.1 Expansion coefficients and circuit parameters of a 1 x 3 IIR optical filter ( ex =-0.2117 rad)

Stage No

Coupling coefficient angle ( nA)

rad

Coupling coefficient

angle ( nB) rad


angle ( nC) rad

Phase shift value ( nA)

rad

Phase shift value ( nB)

rad

Phase shift value ( nC)

rad

1 0.9284 13.535 -0.2659 0.7483 1.4912 1.3015

2 1.9698 0.4271 -0.1319 1.101 0.4404 1.4384

3 1.4472 0.6909 -0.0003 0.9661 0.0087 1.5704

4 1.0935 0.6541 -0.0659 0.83 0.5376 1.5047

5 1.6558 2.8499 -0.0556 1.0275 1.3683 1.5151

6 0.6244 1.1242 -0.0096 0.5582 0.4616 1.5611

7 0.8869 1.65 -0.023 0.7255 0.8193 1.5477

8 4.2179 0.2125 -0.0216 1.338 0.4109 1.5491

9 1.5172 0.6591 -0.0004 0.988 0.0181 1.5703

10 0.4772 0.9275 -0.018 0.4452 0.744 1.5527

11 7.3305 0.7762 -0.0185 1.4352 1.4026 1.5522

12 0.1171 1.361 -0.0071 0.1166 0.7608 1.5636

13 1.3827 0.7327 -0.0032 0.9446 0.2582 1.5675

14 29.427 0.0309 -0.0056 1.5368 0.4897 1.5651

15 1.5376 0.6184 -0.0023 0.9941 0.2146 1.5684

16 0.5441 1.5907 -0.0012 0.4983 0.2982 1.5695

17 0.7306 0.8613 -0.0021 0.6309 0.5632 1.5686

18 3.8189 2.1998 -0.0011 1.3146 1.4518 1.5696

The calculation is repeated for different values of ex and an

optimum value corresponding to minimum band overlap is identified as

ex = -0.1857 rad. The circuit parameters corresponding to optimum value of

ex = -0.1857 are provided in table 4.2.

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Table 4.2 Expansion coefficients and circuit parameters of a 1 x 3 IIR optical ( ex =-0.1857 rad)

Stage No

Coupling coefficien

t angle nA) rad


angle ( nB) rad


angle ( nC) rad


rad


rad


rad

1 0.8592 15.308 -0.214 0.710 1.495 1.355

2 0.9376 0.694 -0.150 0.753 0.599 1.421

3 1.0502 1.135 -0.069 0.810 1.071 1.502

4 4.1846 0.239 -0.002 1.336 0.013 1.569

5 0.2252 1.631 -0.043 0.222 0.900 1.528

6 1.9673 0.603 -0.051 1.101 0.726 1.520

7 31.363 0.081 -0.034 1.539 1.211 1.537

8 0.4088 35.784 -0.010 0.388 1.502 1.561

9 0.2725 2.646 -0.009 0.266 0.781 1.562

10 2.5663 0.412 -0.015 1.199 0.633 1.556

11 2.4264 0.415 -0.009 1.180 0.466 1.562

12 1.2416 0.806 -0.001 0.893 0.056 1.570

13 1.7637 0.608 -0.009 1.055 0.557 1.562

14 25.623 0.058 -0.012 1.532 1.029 1.559

15 1.1607 0.808 -0.010 0.860 0.985 1.560

16 0.886 4.485 -0.006 0.725 1.343 1.564

17 0.604 3.064 -0.003 0.543 1.101 1.568

18 0.261 13.357 0.000 0.255 1.286 1.570

Similar procedure is repeated for 1x5 structure also. Table 4.3

shows the calculated angles of directional couplers and the phase shift values

of the phase shifters( Na, Nb, Nc, Na, Nb and Nc) of 5 port IIR lattice filter

with number of stages N = 35, using constrained least square algorithm. The

same approach is used to determine the optimum value of ex which is equal

to -0.0719 and the parameters of the filter are shown in Table 4.4.

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Table 4.3 Circuit parameters of a 1 x 5 IIR optical filter for ex =-0.0922 rad

Stage No


rad


angle ( nB) rad


angle ( nC) rad


rad


rad


rad

1 0.5234 0.0616 1.5652 0.5771 1.7359 -0.00552 1.1429 1.1212 1.4007 2.1929 1.0258 -0.16913 1.5416 0.0938 1.5559 34.3375 0.029 -0.01474 0.8963 1.0755 1.4302 1.2506 1.0032 -0.145 0.4937 0.331 1.5516 0.5381 1.9515 -0.0196 0.3923 0.7759 1.4714 0.4137 0.9825 -0.09917 0.5093 0.369 1.5535 0.5584 1.4234 -0.01728 1.0729 1.1541 1.5154 1.8398 0.9522 -0.05539 1.5005 0.2714 1.5605 14.2142 0.0676 -0.0102

10 1.1871 1.0098 1.5536 2.4772 0.7021 -0.01711 0.2450 0.7240 1.5700 0.2500 2.5295 -0.000712 0.7867 0.7509 1.561 1.0028 1.3562 -0.009713 0.6767 0.5708 1.5629 0.8033 1.4779 -0.007814 1.1743 1.1769 1.5475 2.3887 1.0743 -0.023215 1.4173 0.4239 1.5578 6.4671 0.1479 -0.012916 0.6258 0.9332 1.5457 0.7227 0.9553 -0.02517 0.5522 0.9733 1.5573 0.616 2.5779 -0.013418 0.4871 0.6989 1.5517 0.5297 0.9025 -0.019919 0.4104 0.6945 1.5608 0.4351 0.9949 -0.009920 1.1672 1.2003 1.5608 2.3419 0.9079 -0.009821 1.4642 0.5585 1.5665 9.3535 0.0941 -0.004222 1.5013 0.8413 1.5692 14.3726 0.077 -0.001523 0.4215 1.0427 1.5694 0.4484 4.3792 -0.001324 0.8417 0.6039 1.5669 1.1196 1.0536 -0.003825 0.9292 0.7481 1.5655 1.3387 0.9744 -0.005226 1.2096 1.2052 1.5648 2.6471 1.0406 -0.005927 1.3661 0.6367 1.564 4.818 0.2059 -0.006728 0.6127 0.7021 1.5652 0.703 0.7755 -0.005529 0.7745 1.2747 1.5645 0.9784 3.2006 -0.006230 0.7583 0.5182 1.567 0.9472 0.7471 -0.003731 0.7565 0.7768 1.5665 0.9438 0.6562 -0.004232 1.1635 1.3877 1.5686 2.3179 2.2304 -0.002133 1.4584 0.7895 1.5682 8.8614 0.107 -0.002534 0.971 1.3388 1.5699 1.4626 2.8638 -0.000835 1.053 0.9994 1.5701 1.7558 1.0205 -0.0006

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Table 4.4 Circuit parameters of a 1 x 5 IIR optical filter for ex =-0.0719 rad

Stage No


rad


angle ( nB) rad

Coupling coefficient angle ( nC)

rad


rad


rad


rad

1 0.4911 0.0165 1.5698 0.5348 1.869 -0.00092 1.0426 1.142 1.4636 1.7138 1.4006 -0.10693 1.3625 0.0256 1.5681 4.7312 0.2112 -0.00264 1.2460 1.0636 1.4713 2.97 0.4772 -0.09895 0.5394 0.1013 1.5668 0.5691 1.6445 -0.00396 0.4245 0.7906 1.4841 0.4519 0.9545 -0.08657 0.5453 0.0957 1.5661 0.6067 1.6556 -0.00468 0.7754 1.218 1.4995 0.9803 2.3671 -0.07119 1.0669 0.0749 1.566 1.814 0.5513 -0.0047

10 1.4335 0.9768 1.5163 7.2378 0.1998 -0.054411 0.8439 0.2773 1.5665 1.1244 0.9111 -0.004212 1.1044 0.7681 1.5329 1.9865 0.6228 -0.037813 0.7172 0.1448 1.5674 0.8721 1.1129 -0.003314 0.8652 1.2948 1.5478 1.1739 2.948 -0.022915 1.0283 0.116 1.5687 1.6591 0.5941 -0.00216 1.1557 0.8572 1.5601 2.2692 0.3846 -0.010617 1.4833 0.5178 1.5699 11.4054 0.0782 -0.000818 1.3401 0.5712 1.5692 4.2582 0.1983 -0.001519 0.3152 0.2638 1.5705 0.3261 2.9984 -0.000220 0.6149 1.4234 1.5664 0.7062 9.4435 -0.004321 0.7048 0.172 1.5696 0.8505 1.1909 -0.001122 0.9644 0.816 1.5633 1.4419 0.9665 -0.007423 0.2472 1.0694 1.5692 0.2523 4.4981 -0.001524 1.5586 0.7245 1.5625 8.3263 0.0116 -0.008125 1.1954 0.1937 1.569 2.5376 0.3814 -0.001726 0.9394 1.5269 1.5633 1.3677 16.6136 -0.007427 0.9349 0.1878 1.5692 1.3547 0.7084 -0.001528 0.7719 0.665 1.5649 0.9734 0.6457 -0.005829 0.3835 1.4267 1.5695 0.4034 16.6225 -0.001230 1.3233 0.6329 1.5668 3.9577 0.2102 -0.003931 1.5221 0.1773 1.5699 2.5294 0.048 -0.000832 1.197 1.429 1.5684 2.55 2.7429 -0.002333 1.2178 0.1905 1.5702 2.7148 0.3667 -0.000534 1.3152 0.4276 1.5695 3.8276 0.2546 -0.001235 0.7227 0.5518 1.5706 0.882 1.3233 -0.0001

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Table 4.5 Bandwidth of overlap region for two different ex values

Ports Bandwidth of overlap region at -50 dB level

ex =-0.2117 ex =-0.1857

1 & 2 94 MHz 29 MHz 2 & 3 96 MHz 28 MHz

From the table it is seen that the overlap region between the

adjacent ports can be reduced by changing the external phase shifter value.

For ex =-0.2117, the overlap bandwidth between the adjacent ports is around

94 MHz and for ex =-0.1857, the overlap bandwidth is very much reduced

and is around 28 MHz.

Table 4.6 Comparison of CLS algorithm with existing algorithm for 1 x 3 filter

Parameters

1x3 using REMEZ algorithm by

Shafiul Azam et al., 2008 [1]

1x3 using CLS algorithm

Number of stages 20 18

Stop band Attenuation in dB 60 65

Bandwidth of overlap region in rad/sample 0.08 0.07

Attenuation level corresponding to intersection

of two bands (dB) 3 25

Table 4.6 shows the comparison of CLS algorithm for 1 x 3 filter

with the existing REMEZ algorithm. From the table, it is observed that by

using CLS algorithm, better magnitude response filters can be obtained with a

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minimum number of stages (18). The stopband attenuation is also increased.

The attenuation level corresponding to intersection of two bands is around 25

dB.

Table 4.7 Comparison of 1 x 5 filter for two different external phase shift values

Parameters ex=-0.0922 rad ex=-0.0719 rad (optimum)

Bandwidth of overlap region at -50 dB level 50 MHz 6 MHz

Attenuation level corresponding to

intersection of two bands (dB)

3 35

Table 4.7 shows the comparison of overlap region and attenuation

level corresponding to intersection of two bands of 1 x 5 IIR filter for two

different external phase shift values. The bandwidth of overlap region at -50

dB level is very much reduced (6MHz) thus leading to better performance

cross talk.

Table 4.8 Performance comparison of 1 x 3 and 1 x 5 IIR filter

Parameters 1 x 3 IIR filter

1 x 5 IIR filter

Number of stages 18 35

Stop band Attenuation in dB 65 50

Bandwidth of overlap region at -50 dB in MHz

23 6

Minimum attenuation in the overlap region in dB

26 35

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Table 4.8 shows the performance comparison of 1 x 3 and 1 x 5

filter in terms of number of stages, stop band attenuation and overlap region.

4.4 RESULTS AND DISCUSSION

The results obtained from two schemes of lattice form IIR optical

delay line filters, viz three port (1 x 3) and five port (1 x 5), for RF filter

approach are presented.

The synthesis of 1x3 and 1x5 lattice form band pass IIR optical

delay line filters is performed using constraint least square algorithm

containing 2x(N+1) directional couplers and 2x(N+1) phase shifters and one

external phase shifter. The synthesis of these filters have been carried out

using constrained least square (CLS) method and compared with the existing

method. The synthesis methods are compared in terms of the number of

stages, coupling coefficient values, phase angles, pass band and stop band

attenuation levels.

The output magnitude response using constraint least square

algorithm for 1x3 and 1x5 are shown in figures 4.3-4.4 and 4.5-4.6

respectively.

The results obtained shows that the maximum number of stages

used to design 1 x 3 filter is 18 (k=18). This value is less than that reported by

Shafiul Azam et al (2008), where the number of stages used was 21 and the

stop band attenuation was about 60 dB. The maximum number of stages used

to design the multi-channel filter (1 x 5) is 35 (k=35).

Figure 4.4 shows the magnitude response of the filter with better

separation in pass bands, which is obtained by tuning the external phase

shifter value and corresponding circuit parameters. The bandwidth

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corresponding to overlap region of port 1 & 2 of the filter is around 35 MHz

which is less compared to the bandwidth reported in the figure 4.3 (105

MHz). Further, this overlap region 0.06 rad/sample is less (0.08rad/sample)

compared to the literature reported by Shafiul Azam et al. A maximum value

of attenuation of 33 dB is found to be occur for a value of ex = -0.1272.

From the Figure 4.6 it is observed that the bandwidth corresponding

to the overlap region of different output ports is around 10 MHz, which is

very much less (50 MHz) compared to the bandwidth shown in the figure 4.5.

By tuning the external phase shifter, it is observed that nearly 50% of overlap

region bandwidth is reduced. A maximum attenuation of 35 dB is found for a

value of ex = -0.0719.

Figure 4.3 Magnitude response of 1 x 3 filter ( ex =-0.2117 rad)

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Table 4.9 Bandwidth of overlap region for two different ex values

PortsBandwidth of overlap region at -50dB level

ex =-0.0902 ex =-0.1272

1 & 2 109MHz 35 MHz

2 & 3 105 MHz 35 MHz

Figure 4.4 Magnitude response of 1 x 3 filter ( ex = -0.1857 rad)


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The phase response of the lattice form 1 x 3 and 1 x 5 band pass

delay line filters are shown in figures 4.7- 4.10. The Figure 4.11 shows the

magnitude response of 1 x 3 filter for different orders. It is observed that the 3

dB bandwidth is constant for higher orders of the filter and it is found that the

minimum order is 18, for a 3 dB bandwidth of 0.27 rad/sample. The variation

of 3 dB bandwidth for different order is shown in Figure 4.12. The Figure

4.13 shows the magnitude response of 1 x 3 filter for two different external

phase shifter values. The optimized ex (-0.1857 rad) provides better

performance in the overlap region compared to the other value, with a

compromise in the 3 dB Bandwidth. For application requiring sharp roll off

characteristics with minimum band overlap, the optimized value of ex and its

corresponding parameters can be used in the structure.

Figure 4.14 shows the variation of 3 dB bandwidth with different

number of stages for 1 x 5 IIR filter. The 3 dB bandwidth obtained from the

filter shows almost constant bandwidth of 0.27 dB(x rad/sample) and 0.089

dB (x rad/sample) for 1 x 3 and 1 x 5 Structures respectively.

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The variation of 3dB bandwidth for different output ports are

shown in Figure 4.15 and 4.16. It is observed that the 3 dB bandwidth

obtained from the filter shows almost constant bandwidth of 43 MHz for 1 x 3

and 38 MHz for 1 x 5 filter.

Figure 4.7 Phase response of 1x3 filter ( ex = -0.2117 rad)


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Figure 4.11 Magnitude response of 1 x 3 IIR filter for different orders

Figure 4.12 Variation of 3dB bandwidth with number of stages for 1 x 3 IIR filter

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Figure 4.13 Magnitude response of 1 x 3 IIR filter for two different exvalues

Figure 4.14 Variation of 3dB bandwidth with output ports for 1 x 5 IIR filter

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Figure 4.17 (a) RF Input signal at 110 MHz before optimization filter

Figure 4.17 (b) RF Output at port 1 without optimization

4.17 (c) RF Output at port 2 without optimization

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(a) Port 1 Output

(b) Port 2 Output

Figure 4.18 a and b Output ports corresponding to an optimized filter

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A System level simulation of the designed IIR filter was done using

Optisystem software. The system comprises of a 1550nm Laser diode and a

Mach Zehnder external modulator, driven by a RF signal source. The optical

filter output is fed to a photo detector and a RF spectrum analyzer. A RF

amplifier is also added to the photo detector to provide an amplified RF

signal. The input RF signal modulates the optical signal and was send through

the 1 x 3 IIR filter before and after optimization. The spectrum of the signal

was viewed in the output ports. The results are shown in figure 4.17 and 4.18

From the output it is observed that before optimization, the signal

in port 1 interfere with the signal in the output port 2(-55 dB). After

optimization, the interference is greatly reduced.

chapter 4 synthesis of lattice form iir optical...

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