-
70
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.
-
71
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
-
72
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)
-
73
(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.
-
74
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)
-
75
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
-
76
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
Coupling coefficient
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.
-
77
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
Coupling coefficient
angle ( nB) rad
Coupling coefficient
angle ( nC) rad
Phase shift value ( nA)
rad
Phase shift value ( nB)
rad
Phase shift value ( nC)
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.
-
78
Table 4.3 Circuit parameters of a 1 x 5 IIR optical filter for
ex =-0.0922 rad
Stage No
Coupling coefficient angle ( nA)
rad
Coupling coefficient
angle ( nB) rad
Coupling coefficient
angle ( nC) rad
Phase shift value ( nA)
rad
Phase shift value ( nB)
rad
Phase shift value ( nC)
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
-
79
Table 4.4 Circuit parameters of a 1 x 5 IIR optical filter for
ex =-0.0719 rad
Stage No
Coupling coefficient angle ( nA)
rad
Coupling coefficient
angle ( nB) rad
Coupling coefficient angle ( nC)
rad
Phase shift value ( nA)
rad
Phase shift value ( nB)
rad
Phase shift value ( nC)
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
-
80
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
-
81
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
-
82
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
-
83
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)
-
84
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)
Figure 4.5 Magnitude response of 1 x 5 filter ( ex = -0.0922
rad)
-
85
Figure 4.6 Magnitude response of 1 x 5 filter ( ex = -0.0719
rad)
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.
-
86
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)
Figure 4.8 Phase response of 1x3 filter ( ex = -0.1857 rad)
-
87
Figure 4.9 Phase response of 1x5 filter ( ex = -0.0922 rad)
Figure 4.10 Phase response of 1x5 filter ( ex = -0.0719 rad)
-
88
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
-
89
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
-
90
Figure 4.15 Variation of 3dB bandwidth with output ports for 1 x
3 IIR filter
Figure 4.16 Variation of 3dB bandwidth with output ports for 1 x
5 IIR filter
-
91
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
-
92
(a) Port 1 Output
(b) Port 2 Output
Figure 4.18 a and b Output ports corresponding to an optimized
filter
-
93
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.