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0.0013 Tc 0.1554 Tw 2.1733 0 Td
( functional. The output is still an )Tj
0.0009 Tc 0.2491 Tw -12.68 -1.1733 Td
[(instantaneous )-7(function of the )]TJ
0.0007 Tc 0.2493 Tw 12.86 0 Td
(input waveform, but it does not )Tj
0.001 Tc 0.2823 Tw -12.86 -1.1733 Td
[(depend only)-12( on the values of the input signal, )]TJ
/TT1 1 Tf
0.0025 Tc 0 Tw 21.04 0 Td
[(V\()9(t)]TJ
/TT0 1 Tf
-0.0003 Tc 0.2836 Tw 1.22 0 Td
[(\). A )7(more )]TJ
0.0006 Tc 0.1561 Tw -22.26 -1.1733 Td
[(complicated )-7(class )-7(of models is given by)-13( the )]TJ
/TT1 1 Tf
0.0015 Tc 0 Tw 18.9667 0 Td
(feedback)Tj
/TT0 1 Tf
0.0004 Tc 0.1563 Tw 3.5667 0 Td
( model of )Tj
0.0013 Tc 0.002 Tw -22.5333 -1.1733 Td
(Equation 1.3. )Tj
0 Tc 0 Tw 0 -2.1333 TD
( )Tj
0.337 Tc 12 0 0 12 122.64 536.76 Tm
[(\(\))-1380(\()-360(\()-5(\))345(,)-463(\(\))340(,)-273(\(\))340(\))]TJ
/TT1 1 Tf
0.447 Tc -0.455 -0.0047 Td
[(It)-1150(f)-60(V)-32(t)-230(V)-37(t)-355(It)]TJ
/C2_0 1 Tf
0 Tc 1.69 -0.0003 Td
<0020>Tj
/C2_1 1 Tf
1.622 Tc 4.135 0.27 Td
<00050005>Tj
/TT0 1 Tf
0.0007 Tc 4.6026 Tw 9 0 0 9 233.04 536.7003 Tm
( \(1.3\) )Tj
0.0005 Tc 0.0295 Tw -19.8933 -1.9333 Td
[(Feedback models embody)-13( the notion )]TJ
0.001 Tc 0.029 Tw 15.1467 0 Td
[(of memory)-12(, or internal )7(state. )]TJ
0.0007 Tc 0.2826 Tw -15.1467 -1.1733 Td
[(Static )-7(m)5(odels )-7(such )-7(as Equation 1.2 will produce AM to PM )]TJ
0.0008 Tc 0.0292 Tw 0 -1.1733 TD
[(distortion, but there is no notion of state or )7(true )7(memory)-12( )7(in )7(such )7(a )]TJ
0.0005 Tc 0.3695 Tw T*
[(model. It is the incorporation of )7(long-term )7(memory)-13( )7(bey)-13(ond )]TJ
0.0012 Tc 0.0088 Tw T*
(otherwise static nonlinear behavioral)Tj
0.0009 Tc 0.0024 Tw 14.8133 0 Td
[( )-7(models that distinguishes the )]TJ
0.001 Tc 0.0023 Tw -14.8133 -1.1733 Td
(newer behavioral modeling approaches. )Tj
/TT2 1 Tf
0.0005 Tc 0.3495 Tw 12 0 0 12 54 446.88 Tm
(2.4 Excitations )Tj
/TT0 1 Tf
0.0011 Tc 0.1222 Tw 9 0 0 9 54 435.9003 Tm
(Excitation design refers to the set )Tj
0.0016 Tc 0.1151 Tw 14.3733 0 Td
[(of stim)6(uli needed to elicit the )]TJ
0.0003 Tc 0.2564 Tw -14.3733 -1.1733 Td
[(full dy)-13(namics of the sy)-13(stem re)]TJ
0.0009 Tc 0.2558 Tw 13.4 0 Td
[(sponse in order to identify)-12( )7(the )]TJ
0.0007 Tc 0.116 Tw -13.4 -1.1733 Td
(behavioral models. It is important)Tj
0.0012 Tc 0.1155 Tw 14.0133 0 Td
[( to m)6(a)-2(ke sure the excitation is )]TJ
0.502 Tw -14.0133 -1.1733 Td
[(sufficiently)-12( rich to construct a behavioral )7(model )7(that )7(can )]TJ
0.0016 Tc 0.5484 Tw T*
[(accurately)-12( )-7(repres)4(ent s)4(i)-1(gnals)4( \223)5(c)-1(los)4(e\224 to thos)4(e us)4(ed in the )]TJ
0.1485 Tw T*
(characterization or identification.)Tj
0.0008 Tc 0.1425 Tw 13.6533 0 Td
[( Models identified from)5( sim)5(p)1(le )]TJ
0.0692 Tw -13.6533 -1.1733 Td
[(excitations m)5(a)-2(y)-12( be )7(less )7(transporta)]TJ
0.0007 Tc 0.0626 Tw 13.8 0 Td
(ble than those constructed from )Tj
0.0014 Tc 0.0286 Tw -13.8 -1.1733 Td
[(a varied set of )7(excitations. )7(This )7(is)]TJ
0.0015 Tc 0.0218 Tw 13.6067 0 Td
[( es)4(pecially)-12( true for s)4(y)-12(s)4(t)-1(em)6(s)4( with )]TJ
-0.0022 Tc 0 Tw -13.6067 -1.1733 Td
[(me)-5(mory)-15(.)-5( )]TJ
/TT2 1 Tf
-0.0005 Tc 0.0005 Tw 12 0 0 12 54 331.8 Tm
[(3. )-350(TIME DOMAIN TECHNIQUES )]TJ
0.0005 Tc -0.0005 Tw 0 -1.335 TD
[(3.1 )-350(Dynamic Neural Netw)8(orks )]TJ
/TT0 1 Tf
0.0007 Tc 0.3826 Tw 9 0 0 9 54 304.8003 Tm
[(Artificial Neural Networks \(ANNs\) [7,8])7( represent nonlinear )]TJ
0.0011 Tc 0.1356 Tw 0 -1.1733 TD
[(m)6(u)1(ltivariate function fitting m)6(e)-2(thods patterned )7(after )7(the )7(nervous )]TJ
-0.0005 Tc 0.0038 Tw T*
[(sy)-14(ste)-3(m)4(.)-4( )-7(The)-3( ba)-3(sis se)-3(t c)-3(onsists of )]TJ
0.0007 Tc 0.0026 Tw 13.24 0 Td
(nested sigmoidal functions which )Tj
0.0009 Tc 0.7424 Tw -13.24 -1.1733 Td
[(can universally)-12( and )7(uniformly)]TJ
0.0012 Tc 0.7355 Tw 14.2933 0 Td
( approximate differentiable )Tj
0.0014 Tc 0.1153 Tw -14.2933 -1.1733 Td
[(functions)4( )-7(in )-7(a com)6(p)1(act s)4(ubs)4(et of n-dim)6(e)-1(ns)4(ional Euclidian s)4(p)1(ace. )]TJ
0.0011 Tc 0.6689 Tw T*
[(Until very)-12( recently)-12(, ANNs for high-frequency)-12( electronic )]TJ
0.0008 Tc 0.1625 Tw T*
(modeling were applied almost excl)Tj
0.0009 Tc 14.6733 0 Td
[(usively)-12( to static )7(functions )7(of )]TJ
0.0014 Tc 0.0486 Tw -14.6733 -1.1733 Td
[(the )-7(input )-7(variables alone. The app)]TJ
0.0006 Tc 0.0494 Tw 13.8533 0 Td
(lication of ANNs to functionals )Tj
0.001 Tc 0.0823 Tw -13.8533 -1.1733 Td
[(of the inputs, that is, to functions )7(of )7(the )7(input )7(variables )7(and )7(their )]TJ
0.1956 Tw T*
[(tim)6(e )-7(derivatives, was first consider)]TJ
0.0012 Tc 0.1955 Tw 14.7867 0 Td
[(ed in [9])8(.)-2( This is called the )]TJ
0.0005 Tc 0.0028 Tw -14.7867 -1.1733 Td
[(method of dy)-13(namic neural networks \(DNNs\). )]TJ
0.0013 Tc 0.0354 Tw 0 -1.62 TD
[(Excellent )-7(results have been )]TJ
0.0001 Tc 0.0366 Tw 11.3467 0 Td
(obtained from the DNN method for rf )Tj
0.0013 Tc 0.3287 Tw -11.3467 -1.1733 Td
[(am)6(plifiers and m)6(i)-1(xers. Sim)6(p)1(le ex)]TJ
0.0011 Tc 0.3222 Tw 14.58 0 Td
[(citations )-7(have been used to )]TJ
0.0222 Tw -14.58 -1.1733 Td
[(identify)-12( these models, )]TJ
0.0002 Tc 0.0231 Tw 9.1533 0 Td
[(such as continuous wave \(CW\) tones )7(swept )]TJ
0.0011 Tc 0.0156 Tw -9.1533 -1.1733 Td
[(in amplitude and frequency)-12(.)-2( In fact)]TJ
0.0089 Tw 14.18 0 Td
(, simple single-tone excitations )Tj
0.3157 Tw -14.18 -1.1733 Td
[(were )-7(shown to predict the in)]TJ
0.0007 Tc 0.316 Tw 12.9667 0 Td
(termodulation response of these )Tj
0.0014 Tc 0.0019 Tw -12.9667 -1.1733 Td
(circuits to two tone excitations. )Tj
0.0005 Tc 0.1028 Tw T*
(DNN models are nonlinear ordinary)Tj
0.0011 Tc 0.1022 Tw 14.88 0 Td
[( differential equations )7(which )]TJ
0.0016 Tc 0.0217 Tw -14.88 -1.1733 Td
[(can )-7(therefore be im)6(plem)6(ented in generic com)6(m)6(e)-1(rcial tim)6(e-dom)6(ain )]TJ
0.0011 Tc 0.1089 Tw 0 -1.1733 TD
[(simulators. The solution of the circuit )7(equations )7(can )7(be )7(obtained )]TJ
0.0008 Tc 0.0025 Tw T*
[(using )-7(any)-12( )-7(of )-7(the simulator algorithms including harmonic balance, )]TJ
0.001 Tc 0.0023 Tw T*
(transient envelope, and shooting )Tj
13.2333 0 Td
(methods in the time domain. )Tj
0.0008 Tc 0.2492 Tw 16.0867 70.7333 Td
[(Training, which is )7(the )7(process )]TJ
0.0006 Tc 0.2427 Tw 13.5267 0 Td
(of determining the weights \(or )Tj
0.1561 Tw -13.5267 -1.1733 Td
[(coefficients\) in the basis set expansion for the ANNs,)-3( has been )]TJ
0.0009 Tc 0.0558 Tw T*
(done in the time domain and in )Tj
0.0008 Tc 0.0492 Tw 13.1067 0 Td
[(the )-7(frequency)-12( domain. Combined )]TJ
0.0025 Tw -13.1067 -1.1733 Td
[(time and frequency)-12( domain traini)]TJ
0.0009 Tc 13.38 0 Td
(ng has also been reported. )Tj
0.0012 Tc 0.1221 Tw -13.38 -1.62 Td
[(Proprietary)-12( )-7(software )-7(im)6(plem)6(ents the training and the attachm)6(e)-2(nt )]TJ
0.122 Tw T*
[(of )-7(the )-7(neural network m)6(odel to a com)6(m)6(e)-1(rcial sim)6(u)1(lator)]TJ
/TT1 1 Tf
0.0016 Tc 0.1217 Tw 23.08 0 Td
[( \()8(A)-1(gilent )]TJ
0.0002 Tc 0 Tw -23.08 -1.1733 Td
(ADS)Tj
/TT0 1 Tf
0.0018 Tc 1.8333 0 Td
(\). )Tj
0.0009 Tc 0.1758 Tw -1.8333 -1.62 Td
[(Limitations of the DNN approach include the ad hoc )7(nature )7(of )]TJ
0.4091 Tw T*
[(selecting the number )7(of )7(independe)]TJ
0.0016 Tc 0.4017 Tw 15.4267 0 Td
(nt variables. That is, the )Tj
0.0008 Tc 0.0225 Tw -15.4267 -1.1733 Td
(number of time derivatives of the )Tj
0.0012 Tc 0.0221 Tw 13.7667 0 Td
(candidate independent variables )Tj
0.001 Tc 0.0423 Tw -13.7667 -1.1733 Td
[(to include as dependent )7(variables )]TJ
0.0357 Tw 13.76 0 Td
(is not an algorithmic procedure. )Tj
0.0003 Tc 0.183 Tw -13.76 -1.1733 Td
(The DNN method depends on standard)Tj
0.001 Tc 0.1757 Tw 16.56 0 Td
[( )-7(issues related to neural )]TJ
0.4689 Tw -16.56 -1.1733 Td
[(network function fitting, including the neural net )7(topology)-12( )]TJ
0.0002 Tc 0.0698 Tw T*
[(\(number of hidden lay)-13(e)-3(rs\), numbe)]TJ
0.0008 Tc 0.0692 Tw 13.94 0 Td
[(r and ty)-12(pes of neurons, )7(and )7(the )]TJ
0.0012 Tc 0.0021 Tw -13.94 -1.1733 Td
(potential issues associated with overtraining. )Tj
/TT2 1 Tf
0.0005 Tc -0.0005 Tw 12 0 0 12 317.88 536.1 Tm
[(3.2 )-350(Nonlinear Time Series )]TJ
/TT0 1 Tf
0.0006 Tc 0.5427 Tw 9 0 0 9 317.88 525.1203 Tm
(Nonlienar time series \(NLTS\) )Tj
0.0008 Tc 0.5359 Tw 14.4067 0 Td
[([10-14])7( )-7(is a fundamentally)-12( )]TJ
0.0004 Tc 0.5096 Tw -14.4067 -1.1733 Td
[(nonlinear )-7(method based on concepts and procedures from )]TJ
0.0015 Tc 0.0485 Tw T*
[(nonlinear dy)-12(nam)6(i)-1(cs)4(. The theory)-12( is)4( )]TJ
0.0011 Tc 0.0489 Tw 14.1867 0 Td
[(routed in differential geometry)-12( )]TJ
0.0006 Tc 0.1694 Tw -14.1867 -1.1733 Td
[(and )-7(is )-7(related )-7(to )-7(the Whitney)-13( embedding theorem of manifolds )]TJ
0.4227 Tw T*
[([14, 15])7(. The method assumes that the unknown nonlinear )]TJ
0.001 Tc 0.3757 Tw T*
[(component of arbitrary)-12( complexity)-12( can be )7(represented )7(by)-12( )7(a )]TJ
0.0003 Tc 0.063 Tw T*
[(dy)-13(namical sy)-13(stem, governed )7(by)-13( )7(st)]TJ
0.0011 Tc 0.0556 Tw 13.8733 0 Td
(ate equations in canonical form )Tj
0.0009 Tc 0.0024 Tw -13.8733 -1.1733 Td
(as shown in Equations 1.4 and 1.5. )Tj
0 Tc 0 Tw 0 -2.1333 TD
( )Tj
0.337 Tc 12 0 0 12 400.32 432 Tm
[(\(\))-1300(\()-240(\(\))340(,)-338(\(\))340(\))]TJ
/TT1 1 Tf
0.3862 Tc -0.5 0.0003 Td
[(xt)-1071(g)-99(xt)-351(u)-24(t)]TJ
/C2_0 1 Tf
0 Tc 1.735 -0.0003 Td
<0020>Tj
/C2_1 1 Tf
2.21 Tc -1.69 0.58 Td
[<004B004B>1690<004B>695<004B>]TJ
0 Tc 0.04 -0.33 Td
<0005>Tj
/TT0 1 Tf
0.0007 Tc 6.0493 Tw 9 0 0 9 483.9 432.0003 Tm
( \(1.4\) )Tj
0 Tc 0 Tw -18.4467 -2.2267 Td
( )Tj
0.3395 Tc 12 0 0 12 401.16 411.96 Tm
[(\(\))-1187(\()-237(\()-2(\))348(,)-335(\(\))348(\))]TJ
/TT1 1 Tf
0.3712 Tc -0.485 0.0003 Td
[(yt)-1036(h)-54(x)-15(t)-366(u)-39(t)]TJ
/C2_0 1 Tf
0 Tc 1.72 -0.0003 Td
<0020>Tj
/C2_1 1 Tf
1.515 Tc 1.75 0.58 Td
<004B004B>Tj
/TT0 1 Tf
0.0007 Tc 6.1026 Tw 9 0 0 9 483.42 411.9603 Tm
( \(1.5\) )Tj
0.0009 Tc 0 Tw -18.3933 -2.2267 Td
(Here )Tj
0.337 Tc 12 0 0 12 346.62 391.92 Tm
(\(\))Tj
/TT1 1 Tf
0.3862 Tc -0.5 0.0053 Td
(xt)Tj
/C2_1 1 Tf
0 Tc 0.045 0.5747 Td
<004B>Tj
/TT0 1 Tf
0.0017 Tc 0.0616 Tw 9 0 0 9 361.2 391.9203 Tm
[( is)4( the vector of s)4(tate variables)4(,)-2( )]TJ
0.337 Tc 0 Tw 12 0 0 12 487.86 391.92 Tm
(\(\))Tj
/TT1 1 Tf
0.41 Tc -0.58 0.0053 Td
(ut)Tj
/C2_1 1 Tf
0 Tc 0.125 0.5747 Td
<004B>Tj
/TT0 1 Tf
0.0011 Tc 0.0622 Tw 9 0 0 9 502.44 391.9203 Tm
(is the vector of )Tj
0.0005 Tc 0.0495 Tw -20.5067 -1.78 Td
[(measured inputs, )7(and )]TJ
0.337 Tc 0 Tw 12 0 0 12 406.44 375.9 Tm
(\(\))Tj
/TT1 1 Tf
0.3661 Tc -0.485 0.0053 Td
(yt)Tj
/TT0 1 Tf
0.0008 Tc 0.0425 Tw 9 0 0 9 420.66 375.9003 Tm
(is a generic measured output sampled )Tj
0.001 Tc 0.1757 Tw -11.42 -1.4867 Td
[(in )-7(time \(time series\). The ope)]TJ
0.0008 Tc 0.1759 Tw 12.6867 0 Td
[(rational methodology)-12( of apply)-12(i)-1(ng )]TJ
0.0012 Tc 0.0021 Tw -12.6867 -1.1733 Td
(nonlinear time series is given in Figure 2. )Tj
/TT2 1 Tf
-0.0011 Tc 0.0044 Tw 1.9467 -21.56 Td
[(F)-4(i)-3(gur)-4(e)-4( 2 Non-Line)-4(ar)-4( Time)-4( Se)-4(r)-4(i)-3(e)-4(s)1( e)-4(m)-1(be)-4(dding pr)-4(oc)-4(e)-4(ss )]TJ
/TT0 1 Tf
0.0011 Tc 0.0422 Tw -1.9467 -1.6 Td
[(The )-7(outputs, )-7(the )-7(inputs, )-7(and their time derivatives are unfolded in )]TJ
0.0009 Tc 0.0891 Tw 0 -1.1733 TD
[(higher and higher dim)5(e)-2(nsional spaces until )7(the )7(output )7(becom)5(e)-2(s )7(a )]TJ
0.1224 Tw T*
[(single-valued )-7(function )-7(of these variables. Algorithms to find the )]TJ
0.5224 Tw T*
[(proper choice of )7(variables )7(\(t)]TJ
0.0007 Tc 0.516 Tw 13.38 0 Td
(he embedding\) are based on )Tj
0.0014 Tc 0.7019 Tw -13.38 -1.1733 Td
[(inform)6(ation-theoretic principles)4( [16])8(. W)5(h)1(en the trajectory)-12( )]TJ
0.0011 Tc 0.4422 Tw T*
[(becomes a single valued function, the output )7(can )7(then )7(be )]TJ
0.0012 Tc 0.0021 Tw T*
[(des)4(c)-2(ribed according to Equation 1.6. )]TJ
ET
1 1 1 scn
/CS0 CS 0 0 0 SCN
0.72 w 10 M 0 j 0 J []0 d
316.8 338.4 244.62 -164.76 re
B
0 0 0 scn
354.78 216.96 1.68 30.24 re
f
345.12 219.84 73.32 1.68 re
f
0.388 0.675 1 scn
354.9 222.18 m
356.46 221.82 l
356.7 222.96 l
356.7 223.2 l
356.76 224.52 l
356.94 225.84 l
356.1 225.84 l
356.94 225.54 l
357.24 226.68 l
356.46 227.04 l
357.24 226.56 l
357.6 227.22 l
358.08 227.88 l
358.38 228.36 l
358.44 228.54 l
358.44 228.48 l
357.72 228.96 l
357.72 228.06 l
357.96 228.18 l
358.26 228.36 l
357.54 228.06 l
357.9 228 l
358.38 227.82 l
359.1 227.46 l
359.4 228.18 l
358.86 227.64 l
359.46 227.04 l
359.88 226.8 l
360.42 226.38 l
360.9 227.04 l
360.12 226.68 l
360.72 225.72 l
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361.74 223.86 l
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362.94 223.2 l
362.1 223.14 l
362.28 221.82 l
362.28 221.52 l
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362.88 219.48 l
363.24 218.7 l
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365.88 214.86 l
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368.28 216 l
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372.54 229.68 l
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374.76 237.36 l
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376.02 238.86 l
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392.94 221.82 l
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[(y\()107(t)]TJ
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352.92 273.72 1.68 26.16 re
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344.76 276.18 62.82 1.68 re
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(\))Tj
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(t)Tj
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(i)Tj
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501 257.7 m
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491.4 224.34 m
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0.0009 Tc 0.0091 Tw T*
[(static models [9])7(,)-2( although there )]TJ
0.0007 Tc 0.0093 Tw 13.2333 0 Td
(is nothing fundamental about this )Tj
0.0008 Tc 0.4425 Tw -13.2333 -1.1733 Td
(restriction. Moreover, NLTS ha)Tj
0.0014 Tc 0.4353 Tw 14.0467 0 Td
[(s)4( )-7(as)4(s)4(o)1(ciated algorithm)6(s)4( with )]TJ
0.0005 Tc 0.2895 Tw -14.0467 -1.1733 Td
[(which )-7(to define, more sy)-13(stema)]TJ
0.0019 Tc 0.2881 Tw 13.44 0 Td
[(tically)-11(, the choice of variables)4( )]TJ
0.0006 Tc 0.3561 Tw 15.88 70.74 Td
[(\(how many)-13( time derivatives of )]TJ
0.0011 Tc 0.3489 Tw 14.3267 0 Td
(the variables to use for the )Tj
0.0015 Tc 0.1618 Tw -14.3267 -1.1733 Td
[(em)6(bedding\). )-7(The )-7(notion of phas)4(e)-1( s)4(p)2(ace dens)4(ity)-12(, inherent in the )]TJ
0.6084 Tw T*
[(nonlinear tim)6(e s)4(e)-1(ries)4( approach, is)4( a )7(way)-12( )7(to )7(gauge )7(the )]TJ
0.0011 Tc 0.5689 Tw T*
[(com)6(p)1(leteness )-7(of )-7(the excitations with im)6(plications for the )]TJ
0.0008 Tc 0.3759 Tw T*
[(transportability)-12( of the m)5(odel. While sy)-12(stem)5(s with )7(long-term)5( )]TJ
0.0004 Tc 0.0629 Tw T*
[(memory)-13( can be modeled )7(with )7(Equa)]TJ
0.0006 Tc 0.0561 Tw 14.52 0 Td
(tion 1.6, the process of NLTS )Tj
0.0012 Tc 0.5421 Tw -14.52 -1.1733 Td
[(m)6(odel identification can get cum)6(b)1(ersom)6(e)-2( if )7(the )7(problem)6(s )]TJ
0.0015 Tc 0.0085 Tw T*
[(cons)4(idered have tim)6(e-s)4(cales)4( of )7(ve)]TJ
0.0011 Tc 0.0022 Tw 13.5533 0 Td
[(ry)-12( different orders of magnitude. )]TJ
0.0007 Tc 0.0026 Tw -13.5533 -1.1733 Td
(A new technique is proposed to ame)Tj
0.0015 Tc 0.0018 Tw 14.5867 0 Td
[(liorate this difficulty)-12( in [18])8(. )]TJ
/TT2 1 Tf
-0.0009 Tc 0.0009 Tw 12 0 0 12 317.88 607.5 Tm
[(4. )-350(FREQUENCY DOMAIN TECHNIQUES )]TJ
/TT0 1 Tf
0.0011 Tc 0.1889 Tw 9 0 0 9 317.88 596.5203 Tm
[(Traditionally)-12(, microwave modeling techniques in the )7(frequency)-12( )]TJ
0.0008 Tc 0.4092 Tw T*
[(domain )-7(have been limited to )]TJ
0.0009 Tc 0.4091 Tw 13.7133 0 Td
(narrow-band models, that are )Tj
0 Tc 0.05 Tw -13.7133 -1.1733 Td
[(described as memory)-13(less or quasi)]TJ
0.0001 Tc 0.0432 Tw 13.66 0 Td
[(-memory)-13(l)-2(ess \(high frequency)-13( or )]TJ
0.0005 Tc 0.0962 Tw -13.66 -1.1733 Td
[(short-term memory)-13(\). These te)]TJ
0.0003 Tc 0.0897 Tw 12.2533 0 Td
[(rms )-7(correspond to the narrow-band )]TJ
0.0011 Tc 0.3489 Tw -12.2533 -1.1733 Td
[(AM)4( to AM)4( and AM)4( to P)4(M)4( )7(characteris)4(tics)4( )7(of )7(the )7(s)4(y)-12(s)4(t)-1(em)6(, )]TJ
0.0004 Tc 0.4496 Tw T*
[(respectively)-13(.)-3( Such modeling me)]TJ
0.001 Tc 0.4423 Tw 14.2133 0 Td
[(thods )-7(have been applied to )]TJ
0.129 Tw -14.2133 -1.1733 Td
[(wireless power am)6(plifiers. Excitation designs are usually)-12( sim)6(p)1(le )]TJ
0.0005 Tc 0.3362 Tw T*
(single-tone measurements of th)Tj
0.0016 Tc 0.3351 Tw 13.5467 0 Td
[(e am)6(plifier, and the )7(m)6(odel )7(is )]TJ
0.0009 Tc 0.3624 Tw -13.5467 -1.1733 Td
[(identified by)-12( fitting odd-order poly)-12(nom)5(ial functions )7(to )7(these )]TJ
0.002 Tc 0 Tw T*
[(characteris)5(tics)5(.)-1( )]TJ
0.0005 Tc 0.1628 Tw 0 -1.62 TD
[(Several )-7(extensions to such mode)]TJ
0.0006 Tc 0.1627 Tw 13.7267 0 Td
[(ls to account for ')-6(l)-2(ong-term')-6( or )]TJ
0.0004 Tc 0.2763 Tw -13.7267 -1.1733 Td
[(low-frequency)-13( memory)-13( effects ha)]TJ
0.0006 Tc 0.2694 Tw 14.3933 0 Td
[(ve )-7(been proposed in [19,20])7( )]TJ
0.0008 Tc 0.0159 Tw -14.3933 -1.1733 Td
[(based )-7(on Wiener-Volterra theory)-12(, though mostly)-12( these approaches )]TJ
0.001 Tc 0.0023 Tw 0 -1.1733 TD
[(rem)6(a)-2(in lim)6(ited to narrow-band sy)-12(stem)6(s and weak nonlinearities. )]TJ
0.1423 Tw 0 -1.62 TD
[(An extension to the Volterra technique [21])7( allows the )7(Volterra )]TJ
0.0008 Tc 0.5025 Tw 0 -1.1733 TD
[(kernels in the nonlinear frequency)-12( domain solution to )7(be )]TJ
0.0016 Tc 0.0751 Tw T*
[(calculated as)4( a function of the )7(in)]TJ
0.0011 Tc 0.0689 Tw 13.4867 0 Td
(stantaneous envelope amplitude. )Tj
0.0014 Tc 0.2686 Tw -13.4867 -1.1733 Td
[(The kernel functions)4( them)6(s)4(e)-1(lves)4( are calculated from)6( two-tone )]TJ
0.0005 Tc 0.0028 Tw T*
[(measurements )-7(using a single fixed )]TJ
0.0012 Tc 0.0021 Tw 14.06 0 Td
(carrier tone and a second swept )Tj
0.3287 Tw -14.06 -1.1733 Td
[(tone to describe the kernel as a function )7(of )7(the )7(distance )7(in )]TJ
0.0007 Tc 0.2426 Tw T*
[(frequency)-13( )-7(from the carrier. Good results for power amplifier )]TJ
0.001 Tc 0.809 Tw T*
[(m)6(odeling )-7(across )-7(a m)6(u)1(lti-channel bandwidth have been )]TJ
0.0009 Tc 0.0024 Tw T*
(demonstrated. )Tj
0.0013 Tc 0.042 Tw 0 -1.62 TD
[(An attem)6(p)1(t to incorporate this )]TJ
0.0421 Tw 12.4867 0 Td
[(nonlinear m)6(odeling into a )7(transient )]TJ
0.0011 Tc 0.3422 Tw -12.4867 -1.1733 Td
[(sim)6(u)1(lator has been dem)6(onstrated in [22])7(. Here the )7(am)6(plifier )]TJ
0.0008 Tc 0.1759 Tw 0 -1.1733 TD
[(model itself computes )7(the )7(narro)]TJ
0.0009 Tc 0.1691 Tw 13.3733 0 Td
(w-band spectral response to the )Tj
0.001 Tc 0.149 Tw -13.3733 -1.1733 Td
[(tim)6(e-dom)6(ain m)6(odulated input. This approach is )7(successful )7(for )7(a )]TJ
0.0014 Tc 0.0553 Tw T*
[(single )-7(com)6(ponent )-7(instance, )-7(but )-7(m)6(a)-1(y)-12( be difficult to scale to m)6(u)1(lti-)]TJ
0 Tc 0.6767 Tw T*
[(component sy)-13(stem simulation,)]TJ
0.0007 Tc 0.6693 Tw 13.5667 0 Td
[( )-7(where it would be more )]TJ
0.001 Tc 0.3223 Tw -13.5667 -1.1733 Td
(appropriate to perform the simulation in a harmonic balance )Tj
0.0009 Tc 0.0024 Tw T*
[(sim)5(u)1(lator )]TJ
/TT2 1 Tf
0.0003 Tc 0.3497 Tw 12 0 0 12 317.88 248.04 Tm
[(4.1 Describing )350(Functions )]TJ
/TT0 1 Tf
0.0011 Tc 0.3556 Tw 9 0 0 9 317.88 237.0603 Tm
[(The describing function technique is )7(an )7(inherently)-12( )7(nonlinear )]TJ
0.0007 Tc 0.196 Tw T*
(formulation based in the freque)Tj
0.0008 Tc 0.1959 Tw 13.36 0 Td
[(ncy)-12( domain [23])7(. The )7(model )7(is )]TJ
0.1692 Tw -13.36 -1.1733 Td
[(based )-7(on the assumption of a harmonic balance solution of the )]TJ
0.0013 Tc 0.0087 Tw T*
[(circuit equations, which )7(can )7(therefor)]TJ
0.001 Tc 0.0023 Tw 14.7 0 Td
(e be written as Fourier Series )Tj
0.0009 Tc 0.1824 Tw -14.7 -1.1733 Td
[(according to Equations)3( 1.7 )7(and )7(1.)]TJ
0.0014 Tc 0.1753 Tw 14.3467 0 Td
[(8. The expans)4(ion coefficients)4( )]TJ
0.0007 Tc 0.016 Tw -14.3467 -1.1733 Td
(are complex number representing )Tj
0.0093 Tw 13.74 0 Td
(the harmonic components of the )Tj
0.0011 Tc 0 Tw -13.74 -1.1733 Td
(signal. )Tj
0 Tc 0 -2.1333 TD
( )Tj
0.0004 Tc 12 0 0 12 385.32 154.5 Tm
[(\()-337(\))-1012(Re)-2346(exp\()-1847(\))]TJ
/TT1 1 Tf
5.5049 Tc 7.02 0 0 7.02 444.6 151.4999 Tm
(kk)Tj
0 Tc -2.2222 -1.2051 Td
(k)Tj
0.4842 Tc 12 0 0 12 376.08 154.5036 Tm
[(Vt)-3273(V)-2000(j)-528(t)]TJ
/C2_0 1 Tf
0 Tc 12 0 3.899 12 477.5995 154.5 Tm
<005A>Tj
12 0 0 12 400.08 154.5 Tm
<0020>Tj
18 0 0 18 424.38 151.743 Tm
<00A6>Tj
/TT0 1 Tf
0.0007 Tc 4.216 Tw 9 0 0 9 500.4 154.5003 Tm
( \(1.7\) )Tj
0 Tc 0 Tw -20.28 -3.4467 Td
( )Tj
0.0012 Tc 12 0 0 12 384.54 123.54 Tm
[(\()-336(\))-1011(Re)-2285(e)10(xp\()-1846(\))]TJ
/TT1 1 Tf
5.5049 Tc 7.02 0 0 7.02 443.1 120.4799 Tm
(kk)Tj
0 Tc -2.1197 -1.1966 Td
(k)Tj
0.447 Tc 12 0 0 12 379.08 123.4836 Tm
[(It)-3440(I)-2125(j)-565(t)]TJ
/C2_0 1 Tf
0 Tc 12 0 3.899 12 476.0995 123.48 Tm
<005A>Tj
12 0 0 12 399.36 123.48 Tm
<0020>Tj
18 0 0 18 423.66 120.783 Tm
<00A6>Tj
/TT0 1 Tf
0.0007 Tc 4.3826 Tw 9 0 0 9 498.9 123.4803 Tm
( \(1.8\) )Tj
0.2961 Tw -20.1133 -2.9333 Td
(A describing function model can )Tj
0.0009 Tc 0.2958 Tw 14.86 0 Td
[(be defined by)-12( the )7(complex )]TJ
0.001 Tc 0.009 Tw -14.86 -1.1733 Td
[(mapping relating )7(the )]TJ
/TT1 1 Tf
0 Tc 0 Tw 8.56 0 Td
(I)Tj
6 0 0 6 397.92 85.02 Tm
(k)Tj
/TT0 1 Tf
0.001 Tc 0.0023 Tw 9 0 0 9 400.56 86.5203 Tm
(s as nonlinear functions of the )Tj
/TT1 1 Tf
0 Tc 0 Tw 12.3133 0 Td
(V)Tj
6 0 0 6 516.9 85.02 Tm
(k)Tj
/TT0 1 Tf
0.0013 Tc 0.002 Tw 9 0 0 9 519.54 86.5203 Tm
[(s. Usually)-12(, )]TJ
0.001 Tc 0.389 Tw -22.4067 -1.1733 Td
[(however, )-7(a )-7(change )-7(of variables is done and the describing )]TJ
ET
1 1 1 scn
/CS0 CS 0 0 0 SCN
0.72 w 10 M 0 j 0 J []0 d
50.4 489.6 237.96 -284.04 re
B
0.54 w
58.32 485.28 222.18 -275.7 re
B
0.988 0.004 0.157 scn
82.8 445.62 m
81.54 445.86 l
81.72 445.86 l
81.72 446.16 l
81.96 446.16 l
81.96 446.34 l
82.14 446.34 l
82.14 446.64 l
82.32 446.64 l
82.32 446.82 l
82.98 446.82 l
82.98 447.12 l
84.3 447.12 l
84.3 446.82 l
84.9 446.82 l
84.9 446.64 l
85.14 446.64 l
85.14 446.34 l
85.38 446.34 l
85.38 444.72 l
85.14 444.72 l
85.14 443.52 l
84.9 443.52 l
84.9 442.32 l
85.14 442.32 l
83.82 442.32 l
83.82 442.86 l
83.64 442.86 l
83.64 442.56 l
83.46 442.56 l
83.46 442.32 l
82.8 442.32 l
82.8 442.14 l
81.96 442.14 l
81.96 442.32 l
81.54 442.32 l
81.54 442.56 l
81.3 442.56 l
81.3 443.04 l
81.06 443.04 l
81.06 444.06 l
81.3 444.24 l
81.3 444.42 l
81.54 444.42 l
81.54 444.72 l
81.96 444.72 l
81.96 444.96 l
82.8 444.96 l
82.8 445.14 l
84.12 445.14 l
84.12 444.42 l
83.64 444.42 l
83.64 444.24 l
82.8 444.24 l
82.62 444.06 l
82.32 444.06 l
82.32 443.52 l
82.14 443.52 l
82.32 443.34 l
82.32 443.16 l
83.46 443.16 l
83.46 443.34 l
83.64 443.34 l
83.64 443.52 l
83.82 443.52 l
83.82 444.24 l
84.12 444.42 l
84.12 445.32 l
84.3 445.32 l
84.3 445.86 l
84.12 445.86 l
84.12 445.92 l
83.82 445.92 l
83.82 446.16 l
83.46 446.16 l
83.46 445.92 l
82.98 445.92 l
82.98 445.86 l
82.8 445.86 l
82.8 445.62 l
f*
87.24 442.02 m
87.42 442.74 l
87.66 442.74 l
87.66 443.04 l
88.02 443.04 l
88.02 443.22 l
88.38 443.22 l
88.38 443.46 l
88.56 443.46 l
88.56 443.64 l
88.8 443.64 l
88.8 443.94 l
89.34 443.94 l
88.56 439.44 l
87.84 439.44 l
88.38 442.44 l
88.02 442.44 l
88.02 442.26 l
87.66 442.26 l
87.42 442.02 l
87.24 442.02 l
f*
0.988 0.004 0.157 SCN
0.78 w 1 j 1 J
82.5 437.04 m
92.4 437.04 l
S
95.58 437.34 m
91.32 439.74 l
95.58 437.34 l
91.32 435.06 l
91.32 439.74 l
95.58 437.34 l
f*
103.5 458.22 m
103.5 475.32 l
S
103.5 479.28 m
101.04 475.38 l
103.5 479.28 l
105.84 475.38 l
101.04 475.38 l
103.5 479.28 l
f*
112.62 467.1 m
112.62 467.52 l
111.36 469.98 l
110.34 467.52 l
112.62 467.52 l
112.62 467.1 l
110.34 467.1 l
109.86 466.32 l
109.56 466.08 l
109.56 465.6 l
109.86 465.6 l
109.86 465.42 l
110.34 465.42 l
110.34 465.12 l
108.42 465.12 l
108.42 465.42 l
108.78 465.42 l
108.78 465.6 l
109.08 465.84 l
109.08 466.08 l
109.38 466.08 l
109.38 466.5 l
111.54 471.42 l
113.82 466.32 l
114 466.08 l
114 465.84 l
114.3 465.84 l
114.3 465.6 l
114.6 465.6 l
114.6 465.42 l
115.08 465.42 l
115.08 465.12 l
112.62 465.12 l
112.62 465.42 l
112.98 465.42 l
112.98 465.6 l
113.28 465.6 l
113.28 466.08 l
112.98 466.08 l
112.98 466.32 l
112.62 467.1 l
f*
120.12 463.5 m
120.12 463.74 l
120.18 463.74 l
120.18 463.8 l
120.36 463.8 l
120.36 464.04 l
120.54 464.22 l
120.54 464.82 l
120.12 464.82 l
120.12 465.3 l
120.18 465.3 l
120.18 465.54 l
120.78 465.54 l
120.78 465.12 l
120.84 465.12 l
120.84 464.22 l
120.78 464.22 l
120.78 464.04 l
120.54 464.04 l
120.54 463.8 l
120.36 463.8 l
120.36 463.74 l
120.18 463.74 l
120.18 463.5 l
120.12 463.5 l
f*
124.74 469.38 m
125.34 469.38 l
125.34 469.2 l
125.76 469.2 l
125.76 468.9 l
126 468.9 l
126 468.72 l
126.18 468.72 l
126.18 468.18 l
126.36 468.18 l
126.36 467.52 l
126.54 467.52 l
126.54 466.8 l
126.36 466.8 l
126.36 466.2 l
126.18 466.2 l
126.18 465.72 l
126 465.72 l
126 465.54 l
125.76 465.54 l
125.76 465.24 l
125.58 465.24 l
125.58 465 l
124.74 465 l
124.74 465.24 l
125.1 465.24 l
125.1 465.54 l
125.34 465.54 l
125.34 465.72 l
125.58 465.72 l
125.58 466.2 l
125.76 466.2 l
125.76 468 l
125.58 468 l
125.58 468.72 l
125.34 468.72 l
125.34 468.9 l
125.1 468.9 l
125.1 469.2 l
124.74 469.2 l
124.74 463.02 l
124.26 463.02 l
124.26 465 l
123.6 465 l
123.42 465.24 l
123.24 465.24 l
123.24 465.54 l
123.06 465.54 l
123.06 465.72 l
122.76 465.72 l
122.76 466.2 l
122.58 466.2 l
122.58 468.18 l
122.76 468.18 l
122.76 468.72 l
123.06 468.72 l
123.06 468.9 l
123.24 468.9 l
123.24 469.2 l
123.6 469.2 l
123.6 469.38 l
124.26 469.38 l
124.26 469.2 l
123.78 469.2 l
123.78 468.9 l
123.6 468.9 l
123.6 468.72 l
123.42 468.48 l
123.42 467.7 l
123.24 467.7 l
123.24 466.8 l
123.42 466.68 l
123.42 466.02 l
123.6 466.02 l
123.6 465.54 l
123.78 465.54 l
123.78 465.24 l
124.26 465.24 l
124.26 471.72 l
124.74 471.72 l
124.74 469.38 l
f*
116.76 466.08 m
117.36 466.38 l
117.54 466.38 l
117.54 462.42 l
117.84 462.42 l
117.84 462.24 l
116.76 462.24 l
116.76 462.42 l
117.06 462.42 l
117.06 466.08 l
116.76 466.08 l
116.76 465.84 l
116.76 466.08 l
f*
128.34 466.08 m
129.12 466.38 l
129.3 466.38 l
129.3 462.42 l
129.6 462.42 l
129.6 462.24 l
128.52 462.24 l
128.52 462.42 l
128.82 462.42 l
128.82 466.08 l
128.52 466.08 l
128.52 465.84 l
128.34 466.08 l
f*
88.98 458.4 m
131.58 458.4 l
S
103.5 455.04 m
103.5 454.74 l
102.84 454.74 l
102.36 452.7 l
102.36 452.34 l
102.06 452.34 l
102.06 451.62 l
101.94 451.62 l
101.94 451.02 l
101.76 451.02 l
101.76 450.54 l
101.46 450.54 l
101.46 450.06 l
101.28 450.06 l
101.28 449.82 l
101.04 449.52 l
101.04 449.34 l
100.8 449.34 l
100.62 449.16 l
100.38 449.16 l
100.38 448.86 l
99.42 448.86 l
99.42 449.16 l
99.3 449.16 l
99.3 449.52 l
99.42 449.52 l
99.42 449.82 l
99.6 449.82 l
99.6 449.52 l
99.9 449.52 l
99.9 449.34 l
99.6 449.34 l
99.6 449.16 l
100.38 449.16 l
100.38 449.34 l
100.62 449.34 l
100.62 449.52 l
100.8 449.52 l
100.8 450.06 l
101.04 450.06 l
101.04 450.72 l
101.28 450.72 l
101.28 451.62 l
102.06 454.74 l
101.28 454.74 l
101.46 455.04 l
102.06 455.04 l
102.06 455.22 l
102.36 455.22 l
102.36 455.7 l
102.6 455.7 l
102.6 456.24 l
102.84 456.24 l
102.84 456.6 l
103.08 456.6 l
103.08 456.9 l
103.32 456.9 l
103.32 457.14 l
103.5 457.14 l
103.5 457.2 l
103.98 457.2 l
103.98 457.44 l
104.82 457.44 l
104.82 457.2 l
105 457.2 l
105 457.14 l
105.3 457.14 l
105.3 456.6 l
104.7 456.6 l
104.7 457.2 l
103.98 457.2 l
103.98 457.14 l
103.74 457.14 l
103.74 456.9 l
103.5 456.9 l
103.5 456.42 l
103.32 456.42 l
103.32 455.7 l
103.08 455.7 l
103.08 455.04 l
102.84 455.04 l
103.5 455.04 l
f*
106.02 452.46 m
106.02 452.22 l
105.6 452.22 l
105.6 452.04 l
105.36 452.04 l
105.36 451.56 l
105.18 451.56 l
105.18 451.14 l
105.12 451.14 l
105.12 450.36 l
104.88 450.36 l
104.88 448.32 l
105.6 448.32 l
105.6 448.86 l
105.84 448.86 l
105.84 449.22 l
106.02 449.22 l
106.02 449.76 l
106.26 449.94 l
106.26 450.84 l
106.44 450.84 l
106.44 452.04 l
106.26 452.04 l
106.26 452.22 l
106.02 452.22 l
106.02 452.46 l
106.44 452.46 l
106.44 452.22 l
106.62 452.22 l
106.62 451.86 l
106.92 451.86 l
106.92 450.12 l
106.62 450.12 l
106.62 449.46 l
106.44 449.46 l
106.44 449.04 l
106.26 449.04 l
106.26 448.86 l
106.02 448.86 l
106.02 448.56 l
105.84 448.56 l
105.84 448.32 l
105.6 448.32 l
105.6 448.26 l
104.88 448.26 l
104.88 448.32 l
104.64 448.32 l
104.64 448.56 l
104.46 448.56 l
104.46 449.22 l
104.22 449.22 l
104.22 449.94 l
104.46 449.94 l
104.46 450.84 l
104.64 450.84 l
104.64 451.32 l
104.88 451.32 l
104.88 451.56 l
105.12 451.86 l
105.12 452.04 l
105.18 452.04 l
105.18 452.22 l
105.6 452.22 l
105.6 452.46 l
106.02 452.46 l
f*
0 1 0 scn
82.68 412.8 m
83.94 419.28 l
85.38 419.28 l
84.9 417.06 l
84.9 417.36 l
85.38 417.36 l
85.38 417.6 l
85.92 417.6 l
85.92 417.78 l
86.4 417.78 l
86.4 417.6 l
86.94 417.6 l
86.94 417.36 l
87.24 417.36 l
87.24 417.06 l
87.42 417.06 l
87.42 416.7 l
87.72 416.7 l
87.72 414.48 l
87.42 414.48 l
87.42 414.06 l
87.24 414.06 l
87.24 413.58 l
86.94 413.58 l
86.94 413.4 l
86.76 413.4 l
86.76 413.1 l
86.1 413.1 l
86.1 412.8 l
84.9 412.8 l
84.9 413.1 l
84.42 413.1 l
84.42 414.78 l
84.42 414.06 l
84.6 414.06 l
84.6 413.76 l
85.08 413.76 l
85.08 413.58 l
85.56 413.58 l
85.56 413.76 l
85.92 413.76 l
85.92 414.06 l
86.1 414.06 l
86.1 414.48 l
86.4 414.48 l
86.4 415.2 l
86.58 415.2 l
86.58 415.98 l
86.4 415.98 l
86.4 416.4 l
86.1 416.4 l
86.1 416.7 l
85.08 416.7 l
85.08 416.4 l
84.9 416.4 l
84.9 416.28 l
84.6 416.28 l
84.6 415.86 l
84.42 415.86 l
84.42 413.1 l
84.12 413.1 l
84.12 413.4 l
83.94 413.4 l
83.94 413.58 l
83.76 412.8 l
82.68 412.8 l
f*
89.22 412.68 m
89.46 413.22 l
89.46 413.4 l
89.88 413.4 l
89.88 413.7 l
90.3 413.7 l
90.3 413.88 l
90.48 413.88 l
90.78 414.06 l
90.96 414.06 l
90.96 414.3 l
91.14 414.3 l
91.14 414.48 l
91.62 414.48 l
90.78 409.98 l
89.88 409.98 l
90.48 412.98 l
90.3 412.98 l
90.3 412.8 l
89.88 412.8 l
89.7 412.68 l
89.22 412.68 l
f*
82.14 424.26 m
86.22 426.66 l
82.14 424.26 l
86.22 421.98 l
86.22 426.66 l
82.14 424.26 l
f*
0 1 0 SCN
86.1 423.96 m
96 423.96 l
101.46 387.84 m
101.46 402.9 l
S
101.4 407.28 m
98.76 403.02 l
101.4 407.28 l
103.56 403.02 l
98.76 403.02 l
101.4 407.28 l
f*
112.98 388.14 m
112.98 394.2 l
S
112.68 398.22 m
110.22 394.32 l
112.68 398.22 l
115.14 394.32 l
110.22 394.32 l
112.68 398.22 l
f*
123.6 388.14 m
123.6 392.1 l
S
123.3 396.3 m
120.84 392.1 l
123.3 396.3 l
125.7 392.1 l
120.84 392.1 l
123.3 396.3 l
f*
87.84 388.32 m
130.44 388.32 l
S
102.36 384.84 m
102.36 384.66 l
101.58 384.66 l
101.22 382.56 l
101.22 382.44 l
100.92 382.14 l
100.92 381.42 l
100.74 381.42 l
100.74 380.76 l
100.56 380.76 l
100.56 380.22 l
100.38 380.22 l
100.38 379.86 l
100.14 379.86 l
100.14 379.56 l
99.9 379.56 l
99.9 379.44 l
99.42 378.96 l
99.12 378.96 l
99.12 378.72 l
98.28 378.72 l
98.28 378.96 l
97.98 378.96 l
97.98 379.44 l
98.28 379.44 l
98.28 379.56 l
98.46 379.56 l
98.46 379.44 l
98.76 379.44 l
98.76 379.14 l
98.46 379.14 l
98.46 378.96 l
99.12 378.96 l
99.12 379.14 l
99.42 379.14 l
99.42 379.44 l
99.6 379.44 l
99.6 379.86 l
99.9 379.86 l
99.9 380.76 l
100.14 380.76 l
100.14 381.42 l
100.92 384.66 l
100.14 384.66 l
100.38 384.84 l
100.92 384.84 l
100.92 385.14 l
101.22 385.14 l
101.22 385.44 l
101.4 385.44 l
101.4 385.92 l
101.58 385.92 l
101.58 386.34 l
101.88 386.34 l
101.88 386.64 l
102.06 386.64 l
102.06 386.82 l
102.36 386.82 l
102.36 387 l
102.84 387 l
102.84 387.3 l
103.68 387.3 l
103.68 387 l
103.86 387 l
103.86 386.82 l
104.04 386.82 l
104.04 386.64 l
103.86 386.64 l
103.86 386.34 l
103.38 386.34 l
103.38 387 l
102.84 387 l
102.84 386.82 l
102.6 386.82 l
102.6 386.64 l
102.36 386.64 l
102.36 386.16 l
102.06 386.16 l
102.06 385.44 l
101.88 385.44 l
101.88 384.84 l
101.58 384.84 l
102.36 384.84 l
f*
104.82 382.14 m
104.82 382.02 l
104.46 382.02 l
104.46 381.84 l
104.22 381.84 l
104.22 381.42 l
103.98 381.42 l
103.98 380.94 l
103.74 380.94 l
103.74 380.04 l
103.68 380.04 l
103.68 378.12 l
104.22 378.12 l
104.46 378.42 l
104.46 378.66 l
104.64 378.66 l
104.64 379.02 l
104.82 379.02 l
104.82 379.62 l
105 379.62 l
105 380.52 l
105.3 380.52 l
105.3 381.84 l
105 381.84 l
105 382.02 l
104.82 382.02 l
104.82 382.14 l
105.3 382.14 l
105.3 382.02 l
105.48 382.02 l
105.48 381.66 l
105.66 381.66 l
105.66 379.86 l
105.48 379.86 l
105.48 379.32 l
105.3 379.32 l
105.3 378.84 l
105 378.84 l
105 378.66 l
104.82 378.66 l
104.82 378.42 l
104.64 378.42 l
104.64 378.12 l
104.46 378.12 l
104.46 377.94 l
103.68 377.94 l
103.68 378.12 l
103.5 378.12 l
103.5 378.42 l
103.2 378.42 l
103.2 378.84 l
103.02 378.84 l
103.02 379.62 l
103.2 379.86 l
103.2 380.52 l
103.5 380.76 l
103.5 381.12 l
103.68 381.12 l
103.68 381.42 l
103.74 381.66 l
103.74 381.84 l
103.98 381.84 l
103.98 382.02 l
104.22 382.02 l
104.46 382.14 l
104.82 382.14 l
f*
247.74 407.22 m
249.06 413.88 l
250.5 413.88 l
250.02 411.6 l
250.2 411.78 l
250.5 411.78 l
250.68 412.08 l
251.76 412.08 l
251.94 411.78 l
252.12 411.78 l
252.12 411.6 l
252.42 411.6 l
252.42 411.42 l
252.6 411.42 l
252.6 411 l
252.9 411 l
252.9 409.08 l
252.6 409.08 l
252.6 408.42 l
252.42 408.42 l
252.42 408.12 l
252.12 408.12 l
252.12 407.88 l
251.94 407.88 l
251.94 407.58 l
251.76 407.58 l
251.76 407.4 l
251.46 407.4 l
251.28 407.22 l
250.68 407.22 l
250.68 406.92 l
250.5 406.92 l
250.2 407.22 l
249.54 407.22 l
249.54 407.4 l
249.24 407.4 l
249.24 407.58 l
249.06 407.58 l
249.06 408.12 l
249.24 409.08 l
249.24 408.9 l
249.54 408.72 l
249.54 408.18 l
249.72 408.18 l
250.02 408.12 l
250.86 408.12 l
250.86 408.18 l
251.04 408.18 l
251.04 408.42 l
251.28 408.42 l
251.28 408.9 l
251.46 408.9 l
251.46 410.82 l
251.28 410.82 l
251.28 411 l
251.04 411 l
251.04 411.18 l
250.5 411.18 l
250.5 411 l
250.02 411 l
250.02 410.82 l
249.72 410.82 l
249.72 410.28 l
249.54 410.28 l
249.54 409.62 l
249.24 409.62 l
249.24 409.08 l
249.06 408.12 l
248.82 407.22 l
247.74 407.22 l
f*
255.12 407.52 m
254.28 407.52 l
254.28 407.82 l
254.46 407.82 l
254.46 408.18 l
254.76 408.18 l
254.76 408.42 l
254.94 408.42 l
255.12 408.72 l
256.5 408.72 l
256.5 408.42 l
256.74 408.42 l
256.74 408 l
256.98 408 l
256.98 406.98 l
256.74 406.98 l
256.74 406.62 l
256.68 406.62 l
256.68 406.32 l
256.5 406.32 l
256.5 406.2 l
256.26 406.2 l
256.26 405.9 l
256.02 405.9 l
256.02 405.72 l
255.84 405.72 l
255.84 405.48 l
255.6 405.48 l
255.6 405.3 l
255.36 405.3 l
255.36 405 l
255.12 405 l
256.74 405 l
256.68 404.28 l
253.92 404.28 l
253.92 404.7 l
254.1 404.7 l
254.1 405 l
254.28 405 l
254.28 405.3 l
254.46 405.3 l
254.46 405.48 l
254.76 405.72 l
254.94 405.9 l
255.12 406.2 l
255.36 406.32 l
255.6 406.32 l
255.6 406.62 l
255.84 406.62 l
255.84 406.8 l
256.02 406.8 l
256.02 407.22 l
256.26 407.22 l
256.26 407.82 l
256.02 407.82 l
256.02 408 l
255.36 408 l
255.36 407.7 l
255.12 407.7 l
255.12 407.52 l
f*
262.56 417.78 m
258.18 415.38 l
262.56 417.78 l
258.18 420.3 l
258.18 415.38 l
262.56 417.78 l
f*
258.96 417.48 m
248.04 417.48 l
229.14 442.26 m
229.14 468.42 l
S
228.78 472.62 m
226.08 468.42 l
228.78 472.62 l
231.24 468.42 l
226.08 468.42 l
228.78 472.62 l
f*
240.6 442.32 m
240.6 455.34 l
S
240.24 459.42 m
237.9 455.22 l
240.24 459.42 l
242.7 455.4 l
237.9 455.22 l
240.24 459.42 l
f*
251.04 442.32 m
251.04 450.24 l
S
250.68 454.32 m
248.34 450.06 l
250.68 454.32 l
253.14 450.06 l
248.34 450.06 l
250.68 454.32 l
f*
215.22 442.62 m
257.82 442.62 l
S
229.8 439.14 m
229.8 438.96 l
229.14 438.96 l
228.78 436.86 l
228.78 436.56 l
228.48 436.56 l
228.48 435.78 l
228.3 435.78 l
228.3 435.24 l
228.12 435.06 l
228.12 434.58 l
227.7 434.58 l
227.7 434.16 l
227.52 434.16 l
227.52 433.86 l
227.22 433.86 l
227.22 433.68 l
227.04 433.44 l
226.86 433.44 l
226.86 433.38 l
226.56 433.38 l
226.56 433.08 l
225.84 433.08 l
225.84 433.38 l
225.54 433.38 l
225.54 433.68 l
225.84 433.68 l
225.84 433.86 l
226.02 433.86 l
226.02 433.68 l
226.08 433.68 l
226.08 433.44 l
226.02 433.44 l
226.02 433.38 l
226.56 433.38 l
226.56 433.44 l
226.86 433.44 l
226.86 433.68 l
227.04 433.68 l
227.04 434.16 l
227.22 434.34 l
227.22 435.06 l
227.52 435.06 l
227.52 435.78 l
228.48 438.96 l
227.52 438.96 l
227.7 439.14 l
228.48 439.14 l
228.48 439.44 l
228.78 439.56 l
228.78 439.74 l
228.84 439.74 l
228.84 440.22 l
229.14 440.22 l
229.14 440.76 l
229.32 440.76 l
229.32 441.06 l
229.5 441.06 l
229.5 441.24 l
229.8 441.24 l
229.8 441.42 l
230.28 441.42 l
230.28 441.72 l
231.24 441.72 l
231.24 441.42 l
231.42 441.42 l
231.42 441.24 l
231.72 441.24 l
231.72 441.06 l
231.42 441.06 l
231.42 440.76 l
230.94 440.76 l
230.94 441.42 l
230.28 441.42 l
230.28 441.24 l
230.1 441.24 l
230.1 441.06 l
229.8 441.06 l
229.8 440.46 l
229.5 440.46 l
229.5 439.92 l
229.32 439.92 l
229.32 439.14 l
229.14 439.14 l
229.8 439.14 l
f*
232.26 436.44 m
232.26 436.38 l
232.02 436.38 l
232.02 436.14 l
231.78 436.14 l
231.78 435.96 l
231.6 435.78 l
231.6 435.24 l
231.42 435.24 l
231.42 434.58 l
231.24 434.58 l
231.24 433.38 l
230.94 433.38 l
230.94 432.96 l
231.24 432.78 l
231.24 432.48 l
231.78 432.48 l
232.02 432.78 l
232.02 432.96 l
232.08 432.96 l
232.08 433.38 l
232.26 433.38 l
232.26 433.98 l
232.44 434.28 l
232.44 435.06 l
232.74 435.06 l
232.74 436.14 l
232.44 436.14 l
232.44 436.38 l
232.26 436.38 l
232.26 436.44 l
232.74 436.44 l
232.74 436.38 l
232.92 436.38 l
232.92 435.96 l
233.1 435.96 l
233.1 434.34 l
232.92 434.34 l
232.92 433.74 l
232.74 433.56 l
232.74 433.38 l
232.44 433.14 l
232.44 432.96 l
232.26 432.96 l
232.26 432.78 l
232.08 432.78 l
232.08 432.48 l
232.02 432.48 l
232.02 432.36 l
231.24 432.36 l
231.24 432.48 l
230.94 432.48 l
230.76 432.78 l
230.76 433.14 l
230.58 433.14 l
230.58 434.28 l
230.76 434.28 l
230.76 435.06 l
230.94 435.06 l
230.94 435.48 l
231.24 435.48 l
231.24 435.96 l
231.42 435.96 l
231.42 436.14 l
231.6 436.14 l
231.6 436.38 l
231.78 436.38 l
232.02 436.44 l
232.26 436.44 l
f*
1 1 1 scn
0 0 0 SCN
1.14 w 0 j 0 J
134.4 447.84 67.68 -40.02 re
B
0 0 0 scn
117.6 431.34 17.88 1.2 re
f
117.6 422.88 17.88 1.2 re
f
202.74 432.06 17.94 1.2 re
f
202.74 423.6 17.94 1.2 re
f
1 1 1 scn
138.78 413.22 59.64 28.62 re
f
0 0 0 scn
BT
/TT0 1 Tf
-0.0314 Tc 0 Tw 8.28 0 0 8.28 152.76 431.4 Tm
[(No)-17(nl)-14(i)-29(n)-2(e)-8(a)-15(r)]TJ
-0.0103 Tc 0.0652 -1.1884 Td
[(Tw)52(o-)11(P)10(o)4(rt)]TJ
ET
q
209.9711 0 0 -15.96 64.02 359.46 cm
/Im0 Do
Q
q
209.9711 0 0 -15.96 64.02 343.5 cm
/Im1 Do
Q
q
209.9711 0 0 -16.02 64.02 327.54 cm
/Im2 Do
Q
q
209.9711 0 0 -15.96 64.02 311.52 cm
/Im3 Do
Q
q
209.9711 0 0 -15.96 64.02 295.56 cm
/Im4 Do
Q
q
209.9711 0 0 -15.96 64.02 279.6 cm
/Im5 Do
Q
q
209.9711 0 0 -16.02 64.02 263.64 cm
/Im6 Do
Q
q
209.9711 0 0 -15.96 64.02 247.62 cm
/Im7 Do
Q
q
209.9711 0 0 -5.7 64.02 231.66 cm
/Im8 Do
Q
q /GS1 gs
/Fm0 Do
Q
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BT
/TT0 1 Tf
0.0006 Tc 0.2094 Tw 9 0 0 9 54 711.6003 Tm
(functions are defined in terms of)Tj
0.001 Tc 0.209 Tw 14.0933 0 Td
( incident and scattered waves )Tj
0.0008 Tc 0.0825 Tw -14.0933 -1.1733 Td
(rather than voltages and currents.)Tj
0.0012 Tc 0.0821 Tw 13.6667 0 Td
[( The com)6(p)1(lex scattered wave )7(is )]TJ
0.0014 Tc 0.1886 Tw -13.6667 -1.1733 Td
[(therefore repres)4(ented as)4( a nonlinear com)6(p)1(lex \(but )7(not )7(analy)-12(tic\) )]TJ
0.0011 Tc 0.0022 Tw 0 -1.1733 TD
(function of the incident wa)Tj
0.0009 Tc 0.0024 Tw 10.8133 0 Td
[(ves given by)-12( Equation 1.9. )]TJ
0 Tc 0 Tw -10.8133 -2.1333 Td
( )Tj
0.582 Tc 12 0 0 12 180.06 660.72 Tm
(\(\))Tj
/TT1 1 Tf
1.285 Tc -2.3 0.0003 Td
[(bf)688(a)]TJ
/C2_0 1 Tf
0 Tc 0.815 -0.0003 Td
<0020>Tj
/C2_1 1 Tf
1.265 Tc -0.67 0.83 Td
<004B004B>Tj
0 Tc 2.62 -0.25 Td
<004B>Tj
/TT0 1 Tf
0.0007 Tc 8.6026 Tw 9 0 0 9 197.04 660.7203 Tm
( \(1.9\) )Tj
0.0008 Tc 0.0292 Tw -15.8933 -1.9333 Td
[(The components of the )7(vector )7(re)]TJ
0.0006 Tc 0.0227 Tw 13.1267 0 Td
(present the harmonic components )Tj
0.0011 Tc 0.1289 Tw -13.1267 -1.1733 Td
[(of )-7(the )-7(waves )-7(in )-7(the frequency)-12( domain. The length of the vector )]TJ
0.0004 Tc 0.1229 Tw T*
[(corresponds )-7(to the number of fre)]TJ
0.0007 Tc 0.1226 Tw 13.7133 0 Td
[(quency)-13( components kept in the )]TJ
0.0009 Tc 0.1958 Tw -13.7133 -1.1733 Td
[(approximation of the signal. In this theory)-12(, the waves have )7(an )]TJ
0.0018 Tc 0.1015 Tw T*
[(abs)4(o)2(lute s)4(ize as)4(s)4(o)2(ciated with th)]TJ
0.0949 Tw 12.9933 0 Td
[(em)6(, unlike the cas)4(e of \(linear\) S)5(-)]TJ
0.0009 Tc 0.0891 Tw -12.9933 -1.1733 Td
[(parameters which are defined )7(only)-12( )]TJ
0.0007 Tc 0.0826 Tw 14.5533 0 Td
[(in term)5(s of ratios of scattered )]TJ
0.0009 Tc 0.2558 Tw -14.5533 -1.1733 Td
[(to )-7(incident waves. Describing )]TJ
0.0007 Tc 0.256 Tw 13.1933 0 Td
(functions reduce to standard S-)Tj
0.0014 Tc 0.0019 Tw -13.1933 -1.1733 Td
[(param)6(e)-1(ters in the lim)6(it of sm)6(all signals. )]TJ
0.001 Tc 0.0557 Tw 0 -1.62 TD
(The function f in Equation 1.9 is )Tj
0.049 Tw 13.7 0 Td
[(expanded )-7(in )-7(a suitable bases set )]TJ
0.4823 Tw -13.7 -1.1733 Td
[(and fitted from)6( a set of data )7(obtained )7(by)-12( )7(m)6(easuring )7(\(or )]TJ
0.0005 Tc 0.2828 Tw 0 -1.1733 TD
[(simulating\) )-7(the )-7(response of the sy)-13(stems to incident waves of )]TJ
0.0006 Tc 0.0494 Tw T*
[(vary)-13(ing amplitudes and recording )]TJ
0.001 Tc 0.049 Tw 13.9533 0 Td
[(the magnitude and phase )7(of )7(the )]TJ
0.0006 Tc 0.2294 Tw -13.9533 -1.1733 Td
[(response. Applications based )7(on )]TJ
0.0011 Tc 0.2222 Tw 14.06 0 Td
(real microwave )Tj
0 Tc 0 Tw 6.9 0 Td
(measurements )Tj
0.0004 Tc 0.0163 Tw -20.96 -1.1733 Td
(require the VNNA instrument describe)Tj
0.0013 Tc 0.0154 Tw 15.6333 0 Td
[(d earlier,)-2( in order to get the )]TJ
-0.0002 Tc 0.0769 Tw -15.6333 -1.1733 Td
[(re)-3(la)-3(tive)-3( pha)-3(se)-3(s be)-3(twe)-3(e)-3(n suc)-3(c)-3(e)-3(ssive)]TJ
0.001 Tc 0.0757 Tw 14.2267 0 Td
( harmonics. The model can be )Tj
0.0009 Tc 0.1358 Tw -14.2267 -1.1733 Td
[(directly)-12( implemented in a harmoni)]TJ
0.001 Tc 0.1357 Tw 14.4133 0 Td
[(c balance simulator by)-12( )7(using, )]TJ
0.2357 Tw -14.4133 -1.1733 Td
[(for example, a frequency)-12( domai)]TJ
0.0009 Tc 0.2358 Tw 13.7267 0 Td
(n nonlinear modeling element, )Tj
0.0013 Tc 0.002 Tw -13.7267 -1.1733 Td
[(s)4(u)1(ch as)4( the F)4(r)1(equency)-12( Dom)6(a)-1(in Device \(F)4(DD\) in )]TJ
/TT1 1 Tf
0.0016 Tc 0.0017 Tw 19.66 0 Td
(Agilent ADS)Tj
/TT0 1 Tf
0.0033 Tc 0 Tw 4.9867 0 Td
(. )Tj
0.0015 Tc 0.3418 Tw -24.6467 -1.62 Td
[(A us)4(eful interm)6(ediate cas)4(e ex)]TJ
0.0011 Tc 0.3422 Tw 13.2267 0 Td
(ists between the full nonlinear )Tj
0.109 Tw -13.2267 -1.1733 Td
(describing functions and the linear )Tj
0.0009 Tc 0.1091 Tw 14.66 0 Td
[(S-parameter case. In )7([24])7( )7(the )]TJ
0.0011 Tc 0.5156 Tw -14.66 -1.1733 Td
[(describing functions are linearized )7(around )7(the )7(fundamental )]TJ
0.0012 Tc 0.4088 Tw T*
[(response to a large am)6(plitude signal, which can be )7(highly)-12( )]TJ
0.001 Tc 0.0757 Tw T*
[(nonlinear. The resulting approximate model has )7(the )7(property)-12( )7(that )]TJ
0.0012 Tc 0.1955 Tw T*
(it is linear in the harmonics ar)Tj
0.0005 Tc 0.1895 Tw 13.1467 0 Td
[(ound )-7(this fundamental response. )]TJ
0.0004 Tc 0.6029 Tw -13.1467 -1.1733 Td
(This \223harmonic superposition\224 )Tj
0.0005 Tc 0.5962 Tw 14.4 0 Td
[(approximation )-7(is useful in )]TJ
0.001 Tc 0.0023 Tw -14.4 -1.1733 Td
[(simplify)-12(i)-1(ng )-7(the excitation design and the subsequent identification )]TJ
T*
(of the approximate model for a power amplifier. )Tj
0.0001 Tc 0.0899 Tw 0 -1.62 TD
[(Unlike DNN and NLTS models,)-3( )7(whic)]TJ
0.0006 Tc 0.0827 Tw 15.6133 0 Td
(h can be used in all modes )Tj
0.0008 Tc 0.2225 Tw -15.6133 -1.1733 Td
(of simulation, describing functi)Tj
0.0005 Tc 0.2162 Tw 13.2467 0 Td
[(on models can be used only)-13( in )]TJ
0.0016 Tc 0.1951 Tw -13.2467 -1.1733 Td
[(harm)6(onic )-7(balance and linear anal)]TJ
0.001 Tc 0.1957 Tw 14.0467 0 Td
[(y)-12(s)4(is in the frequency)-12( domain. )]TJ
0.0014 Tc 0.0086 Tw -14.0467 -1.1733 Td
[(They)-12( can not be used directly)-12( to )]TJ
0.0006 Tc 0.0094 Tw 13.2067 0 Td
[(simulate the response of )7(a )7(sy)-13(stem )]TJ
0.0027 Tw -13.2067 -1.1733 Td
(to modulated signals such as W-CDMA. )Tj
/TT2 1 Tf
-0.0007 Tc 0.3507 Tw 12 0 0 12 54 284.28 Tm
[(5. ENVELOPE )350(METHODS )]TJ
/TT0 1 Tf
0.0017 Tc 0.335 Tw 9 0 0 9 54 273.3003 Tm
[(Thes)4(e )-7(techniques)4( )-7(com)6(b)2(ine )-7(s)4(o)2(m)6(e)-1( as)4(pects)4( of the tim)6(e dom)6(ain )]TJ
0.0003 Tc 0.5697 Tw 0 -1.1733 TD
[(formulations with some )7(as)]TJ
0.0008 Tc 0.5625 Tw 12.2867 0 Td
[(pects of the frequency)-12( domain )]TJ
0.0012 Tc 0.0221 Tw -12.2867 -1.1733 Td
[(techniques. The basic assum)6(p)1(tion is that the solution )7(of )7(the )7(circuit )]TJ
0.001 Tc 0.329 Tw T*
(equations can be represented by)Tj
14.1333 0 Td
[( a Fourier Series, sim)6(ilar )7(to )]TJ
0.0009 Tc 0.3291 Tw -14.1333 -1.1733 Td
(Equations 1.7 and 1.8, but this )Tj
0.0018 Tc 0.3215 Tw 14.42 0 Td
[(tim)6(e the coefficients)4( of this)4( )]TJ
0.0013 Tc 0.182 Tw -14.42 -1.1733 Td
[(expansion )-7(can )-7(vary)-12( in tim)6(e. That is, the solution of the circuit )]TJ
0.0007 Tc 0.0026 Tw T*
(equations is assumed to be gi)Tj
0.0009 Tc 0.0024 Tw 11.72 0 Td
[(ven by)-12( equations 1.10 and 1.11: )]TJ
0 Tc 0 Tw -11.72 -2.1333 Td
( )Tj
0.0004 Tc 12 0 0 12 115.44 190.8 Tm
[(\()-337(\))-1012(Re)-2236(\()-342(\))-117(exp\()-1847(\))]TJ
/TT1 1 Tf
7.2399 Tc 7.02 0 0 7.02 174.72 187.7399 Tm
(kk)Tj
0 Tc -2.2222 -1.1966 Td
(k)Tj
0.4842 Tc 12 0 0 12 106.2 190.7436 Tm
[(Vt)-3273(V)-235(t)-2018(j)-528(t)]TJ
/C2_0 1 Tf
0 Tc 12 0 3.899 12 219.8995 190.74 Tm
<005A>Tj
12 0 0 12 130.2 190.74 Tm
<0020>Tj
18 0 0 18 154.5 188.043 Tm
<00A6>Tj
/TT0 1 Tf
0.0007 Tc 3.0493 Tw 9 0 0 9 242.52 190.7403 Tm
( \(1.10\) )Tj
0 Tc 0 Tw -20.9467 -3.4467 Td
( )Tj
0.337 Tc 12 0 0 12 114.66 159.72 Tm
[(\(\))-675(R)334(e)-1839(\(\))215(e)346(x)337(p)337(\()-1515(\))]TJ
/TT1 1 Tf
7.2399 Tc 7.02 0 0 7.02 173.22 156.7199 Tm
(kk)Tj
0 Tc -2.1197 -1.2051 Td
(k)Tj
0.447 Tc 12 0 0 12 109.2 159.7236 Tm
[(It)-3440(I)-360(t)-2055(j)-565(t)]TJ
/C2_0 1 Tf
0 Tc 12 0 3.899 12 218.3995 159.72 Tm
<005A>Tj
12 0 0 12 129.48 159.72 Tm
<0020>Tj
18 0 0 18 153.78 157.023 Tm
<00A6>Tj
/TT0 1 Tf
0.0007 Tc 3.216 Tw 9 0 0 9 241.02 159.7203 Tm
( \(1.11\) )Tj
0.0017 Tc 0.2816 Tw -20.78 -2.9333 Td
[(Thes)4(e )-7(tim)6(e-dependent )-7(coefficients)4( are the com)6(p)2(lex envelopes)4(.)-2( )]TJ
0.0012 Tc 0.4621 Tw T*
[(The transient envelope solution technique [2,3])8( is efficient )]TJ
0.0554 Tw T*
[(provided that the tim)6(e vary)-12(ing coefficients)4( )7(vary)-12( )7(s)4(l)-1(owly)-12( )7(com)6(p)1(ared )]TJ
0.0011 Tc 0.3956 Tw T*
[(with )-7(the carriers. This is )]TJ
0.0009 Tc 0.3958 Tw 11.9533 0 Td
[(usually)-12( the case for narrow-band )]TJ
0.001 Tc 0.0023 Tw -11.9533 -1.1733 Td
(communication circuits excited by)Tj
0.0013 Tc 0.002 Tw 13.8533 0 Td
[( digitally)-12( modulated signals. )]TJ
0.001 Tc 0.2823 Tw 15.4667 68.9467 Td
[(By)-12( addressing the envelope behavior in the time-domain, the )]TJ
0.0008 Tc 0.0159 Tw T*
[(long-term memory)-12( effects can be included in )7(the )7(nonlinear )7(model. )]TJ
-0.0005 Tc 0.5705 Tw T*
[(Sy)-14(stems with long-term memory)]TJ
0.0013 Tc 0.5687 Tw 14.8667 0 Td
[( can have very)-12( different )]TJ
0.0007 Tc 0.016 Tw -14.8667 -1.1733 Td
[(behaviors depending on the )7(dy)-13(namics)]TJ
0.0094 Tw 15.2 0 Td
( of the envelopes It becomes )Tj
0.0017 Tc 0.0083 Tw -15.2 -1.1733 Td
[(im)6(portant )-7(to s)4(tim)6(ulate the s)4(y)-12(s)4(tem)6( in way)-12(s)4( that affect the dy)-12(nam)6(ics)4( )]TJ
0.0013 Tc 0.082 Tw T*
(of the envelopes. Excitation de)Tj
0.0007 Tc 0.0826 Tw 12.7267 0 Td
[(signs will need to )7(go )7(well )7(bey)-13(ond )]TJ
0.0012 Tc 0.1155 Tw -12.7267 -1.1733 Td
(the conventional two-tone techni)Tj
0.0008 Tc 0.1159 Tw 13.52 0 Td
[(ques. Using digitally)-12( modulated )]TJ
0.0002 Tc 0.3765 Tw -13.52 -1.1733 Td
(signals or band-limited noise ma)Tj
0.0021 Tc 0.3679 Tw 14.5733 0 Td
[(y)-11( be neces)5(s)5(a)-1(ry)-11(. S)5(p)2(ecialized )]TJ
0.0005 Tc 0.0028 Tw -14.5733 -1.1733 Td
[(hardware sy)-13(stems have been de)]TJ
0.0011 Tc 0.0022 Tw 12.6467 0 Td
[(veloped to do this [17, 21])7(. )]TJ
0.0012 Tc 0.0621 Tw -12.6467 -1.62 Td
[(A key)-12( requirem)6(e)-2(nt is the ability)-12( for the m)6(odeler to interface )7(with )]TJ
0.3155 Tw T*
[(the envelope sim)6(u)1(lator. Som)6(e)-2( )]TJ
0.0006 Tc 0.3094 Tw 13.42 0 Td
[(models )-7(are based on variables )]TJ
0.0012 Tc 0.3221 Tw -13.42 -1.1733 Td
[(characteristic )-7(of )-7(the envelope and RF carriers independently)-12(. )]TJ
0.0015 Tc 0.1285 Tw T*
[(M)4(o)2(s)4(t)-1( )-7(s)4(i)-1(m)6(u)2(lators)4( )-7(do )-7(not )-7(y)-12(e)-1(t have the interface to directly)-12( enable )]TJ
0.0008 Tc 0.0025 Tw T*
(modelers to exploit su)Tj
0.0012 Tc 0.0021 Tw 8.88 0 Td
(ch behavioral models. )Tj
/TT2 1 Tf
-0.0007 Tc 0.3507 Tw 12 0 0 12 317.88 550.68 Tm
(6. CONCLUSIONS )Tj
/TT0 1 Tf
0.0014 Tc 0.0753 Tw 9 0 0 9 317.88 539.7003 Tm
[(Behavioral m)6(odeling techniques)4( have recently)-12( been increas)4(ing )7(in )]TJ
0.0012 Tc 0.0355 Tw T*
[(sophistication and capability)-12(. Severa)]TJ
0.0288 Tw 14.8267 0 Td
[(l techniques, notably)-12( those in )]TJ
0.001 Tc 0.1957 Tw -14.8267 -1.1733 Td
[(the )-7(frequency)-12( domain and the enve)]TJ
0.0017 Tc 0.195 Tw 15.0533 0 Td
[(lope dom)6(ain, are intim)6(ately)-12( )]TJ
0.0008 Tc 0.4292 Tw -15.0533 -1.1733 Td
[(related )-7(to )-7(the )-7(underly)-12(i)-1(ng simulator algorithm used in their )]TJ
0.0009 Tc 0.8758 Tw T*
(solution. Successful applica)Tj
0.0008 Tc 0.8692 Tw 12.9133 0 Td
(tion of behavioral modeling )Tj
0.1492 Tw -12.9133 -1.1733 Td
[(techniques to im)5(portant desi)]TJ
0.0009 Tc 0.1491 Tw 11.7733 0 Td
[(gn challenges will depend upon )7(the )]TJ
0.0011 Tc 0.2556 Tw -11.7733 -1.1733 Td
[(ease of interfacing with the underly)-12(i)-1(ng )7(simulation )7(engine, )7(the )]TJ
0.0013 Tc 0.4354 Tw T*
[(ability)-12( to sy)-12(stem)6(atically)-12( apply)-12( the techniques )7(to )7(significant )]TJ
0.182 Tw T*
[(quantities )-7(of data, and the ability)-12( to distribute these behavioral )]TJ
0.0011 Tc 0.2156 Tw T*
[(m)6(odels )-7(to custom)6(ers. Finally)-12(, the ability)-12( to perform)6( waveform)6( )]TJ
0.0015 Tc 0.1485 Tw T*
[(m)6(eas)4(urem)6(ents)4( at m)6(i)-1(crowave frequencies)4( in order to apply)-12( )7(thes)4(e )]TJ
0.0013 Tc 0.2354 Tw T*
[(techniques to real data will be a big advantage to get )7(around )]TJ
0.2221 Tw T*
[(accuracy)-12( lim)6(itations)4( of building behavioral m)6(odels)4( )7(from)6( )7(lower )]TJ
0.0016 Tc 0.0017 Tw T*
[(level m)6(odels. )]TJ
/TT2 1 Tf
-0.0007 Tc 0.3507 Tw 12 0 0 12 317.88 378.78 Tm
(7. ACKNOWLEDGMENTS )Tj
/TT0 1 Tf
0.0004 Tc 0.0163 Tw 9 0 0 9 317.88 367.8003 Tm
(Thanks to Hassan Tanbakuchi, Domi)Tj
0.0008 Tc 0.0092 Tw 14.94 0 Td
(nique Schreurs, Alex Pekker, )Tj
0.0011 Tc 0.0422 Tw -14.94 -1.1733 Td
[(Darry)-12(l )-7(Okahata, Alex Cognata, Br)]TJ
0.0006 Tc 0.0427 Tw 13.94 0 Td
[(ian Hughes, Derry)-13( Hornbuckle, )]TJ
0.0009 Tc 0.4358 Tw -13.94 -1.1733 Td
[(Jerry)-12( Gladstone, and Agilent )]TJ
0.0008 Tc 0.4292 Tw 13.5267 0 Td
(Technologies Management for )Tj
0.0006 Tc 0.0027 Tw -13.5267 -1.1733 Td
(supporting this work. )Tj
/TT2 1 Tf
-0.0014 Tc 0.3514 Tw 12 0 0 12 317.88 310.5 Tm
[(8. REFERENCES )350( )]TJ
/TT0 1 Tf
0.0027 Tc 0 Tw 10.02 0 0 10.02 317.88 298.5 Tm
[([1])6( )]TJ
0.0012 Tc 0.1821 Tw 9 0 0 9 335.88 298.5003 Tm
[(Kundert, )-7(K. S., Sangiovanni-Vincentelli. A., Sim)6(u)1(lation of )]TJ
0.6754 Tw T*
[(nonlinear circuits in the frequency)-12( dom)6(ain. IEEE )]TJ
0.0006 Tc 0.5561 Tw T*
[(Transactions )-7(on )-7(Computer-Aided Design of Integrated )]TJ
0.0002 Tc 0.0031 Tw T*
[(Circuits and Sy)-13(stems CAD-5\(4\) pp.521-535 \(1986\). )]TJ
0.0027 Tc 0 Tw 10.02 0 0 10.02 317.88 251.22 Tm
[([2])6( )]TJ
0.0002 Tc 0.2765 Tw 9 0 0 9 335.88 251.2203 Tm
(Sharrit, D. A new method of)Tj
0.0006 Tc 0.2761 Tw 12.8067 0 Td
[( analy)-13(s)3(is of )7(communication )]TJ
0.0002 Tc 0.3165 Tw -12.8067 -1.1733 Td
[(sy)-13(stem)5(s. IEEE MTTS')-6(96 W)4(orkshop on )7(Nonlinear )7(CAD )]TJ
0.0008 Tc 0.0025 Tw T*
(\(June 1996\). )Tj
0.0027 Tc 0 Tw 10.02 0 0 10.02 317.88 214.5 Tm
[([3])6( )]TJ
0.0014 Tc 0.0419 Tw 9 0 0 9 335.88 214.5003 Tm
[(Ngoy)-12(a, E, and Larcheveque, R. )7(Envelop )7(transient )7(analy)-12(s)4(is: )7(a )]TJ
0.0009 Tc 0.2224 Tw T*
[(new method for the transient and )7(steady)-12( )7(state )7(analy)-12(s)3(is )7(of )]TJ
0.0011 Tc 0.4156 Tw T*
[(m)6(i)-1(crowave )-7(com)6(m)6(unication circuits and sy)-12(stem)6(s. IEEE )]TJ
0.0004 Tc 0.0029 Tw T*
[(MTTS')-6(96 \(June 1996\) pp1365-68. )]TJ
0.0027 Tc 0 Tw 10.02 0 0 10.02 317.88 167.22 Tm
[([4])6( )]TJ
0.0013 Tc 0.002 Tw 9 0 0 9 335.88 167.2203 Tm
[(Kundert, )-7(K. )-7(S., )-7(White, J. K., Sangiovanni-Vincentelli, A. An )]TJ
0.001 Tc 0.6757 Tw T*
[(envelope-following )-7(method for the efficient transient )]TJ
0.3022 Tw T*
[(sim)6(u)1(lation )-7(of )-7(switching )-7(power and filter circuits. IEEE )]TJ
0.0007 Tc 0.4293 Tw T*
[(ICCAD-88 International Conference on )7(Computer-Aided )]TJ
0.0005 Tc 0.0028 Tw T*
(Design \(November 1988\) pp. 446-449. )Tj
0.0027 Tc 0 Tw 10.02 0 0 10.02 317.88 109.38 Tm
[([5])6( )]TJ
0.0012 Tc 0.2488 Tw 9 0 0 9 335.88 109.3803 Tm
[(Kundert, )-7(K. )-7(S., )-7(White, J. K., Sangiovanni-Vincentelli, A. )]TJ
0.0007 Tc 0.1093 Tw T*
[(Steady)-13(-State Methods for simulating )7(analog )7(and )7(microwave )]TJ
0.0013 Tc 0.002 Tw T*
(circuits. Kluwer Acad)Tj
0.0005 Tc 0.0028 Tw 8.8267 0 Td
(emic Publishers 1990 )Tj
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12.2333 0 Td
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[(sy)-13(stems, Vol 22, No. 2, \(February)-13(,)-3( 2003\) )]TJ
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0.0009 Tc 0.1291 Tw 9 0 0 9 72 637.1403 Tm
[(Hay)-12(k)1(in, )-7(S. Neural Networks: a comprehensive foundation. )127( )]TJ
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(2nd ed. Prentice-Hall \(1999\). )Tj
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[([9])6( )]TJ
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(R., Zhang, Q. J. Neural )Tj
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[(based )-7(dy)-12(namic modeling of nonlinear microwave circuits. )227( )]TJ
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(IEEE MTT-S 2002 \(June 2002\) pp. 1101-1104. )Tj
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[( )-7(time series analy)-12(s)4(is. )327( )327( )]TJ
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(IEEE IMS2003 \(June 2003\). )Tj
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[([11])7( )]TJ
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[(A. Sy)-12(stematic behavioral )]TJ
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[(linear dy)-11(nam)6(ical s)4(y)-11(s)4(tem)6(s)4(.)-1( )60( )]TJ
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(IEEE BMAS 2002 \(October 2002\). )Tj
0.0028 Tc 0 Tw 10.02 0 0 10.02 54 490.26 Tm
[([12])7( )]TJ
0.0011 Tc 0.1022 Tw 9 0 0 9 72 490.2603 Tm
[(Schreurs, )-7(D., Wood, J., Tufilla)]TJ
0.0014 Tc 0.1019 Tw 12.72 0 Td
(ro, N., Barford, L. and Root, )Tj
0.0007 Tc 0.2626 Tw -12.72 -1.1733 Td
[(D. )-7(E. Construction of behavioural models for microwave )]TJ
0.0015 Tc 0.4285 Tw T*
[(devices)4( from)6( tim)6(e-dom)6(ain large-s)4(i)-1(gnal )7(m)6(eas)4(urem)6(ents)4( )7(to )]TJ
0.0006 Tc 0.0494 Tw T*
[(speed up high-level design )7(simula)]TJ
0.0012 Tc 0.0421 Tw 13.82 0 Td
(tions. International Journal )Tj
0.001 Tc 0.0357 Tw -13.82 -1.1733 Td
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[(Issue )-7(on )-7(')-6(M)3(easurement Techniques for Effective CAD')-6( 13\(1\) )]TJ
0.0008 Tc 0.0025 Tw T*
(pp. 54-61 \(2003\). )Tj
0.0028 Tc 0 Tw 10.02 0 0 10.02 54 411.3 Tm
[([13])7( )]TJ
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[(Schreurs, D.,Wood, J., )7(Tufillaro)]TJ
0.0014 Tc 0.0553 Tw 13.0667 0 Td
(, N., Usikov, D., Barford, L. )Tj
0.0011 Tc 0.4422 Tw -13.0667 -1.1733 Td
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0.0008 Tc 0.2559 Tw 11.64 0 Td
(owave devices based on time-)Tj
0.0015 Tc 0.3618 Tw -11.64 -1.1733 Td
[(dom)6(ain large-s)4(i)-1(gnal m)6(eas)4(urem)6(en)]TJ
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[(ts. International )7(Electron )]TJ
0.001 Tc 0.1757 Tw -14.0133 -1.1733 Td
[(Devices )-7(Meeting 2000, \(Decem)6(ber 2000\) IEDM Technical )]TJ
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(Digest pp. 819 -822. )Tj
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[([14])7( )]TJ
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[(Cambridge University)-13( Press \(1997\) )]TJ
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