Nonlinear Microwave Circuit Design phần 2 docx

28 NONLINEAR ANALYSIS METHODS 1 2 3 4 5 6 7 8 9 10 11 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 Figure 1.18 Currents and voltages in the example circuit at the sampling times only, as computed from Fourier transform Linear subnetwork Nonlinear subnetwork m = M m = 1 m = 1 Figure 1.19 A general nonlinear network as partitioned for the harmonic balance analysis harmonics) and a real phasor at DC (n = 0). The continuous curves in the previous figures have been plotted by means of eq. (1.69) and eq. (1.73) once the values of the phasors are known. In general terms, a nonlinear circuit is divided into two parts connected by M ports (Figure 1.19): a part including only linear elements and a part including only nonlinear ones; the voltages at the connecting ports are expressed by Fourier series expansions: v m (t) = Re  N  n=0 V m,n · e jnω 0 t  = V m,0 + Re  N  n=1 V m,n · e jnω 0 t  = V m,0 + N  n=1 {V r m,n cos(nω 0 t) − V i m,n sin(nω 0 t)} m = 1, ,M (1.81) SOLUTION THROUGH SERIES EXPANSION 29 The voltages at the connecting ports are the unknowns of Kirchhoff’s node equa- tions. In our formulation, the unknowns are actually the phasors that appear in their Fourier series expansion; since the series is truncated, they are M ·(2N + 1). In vector form,  V =  V 1,0 V M,0 ,V r 1,1 V i 1,1 V r M,1 V i M,1 ,V r 1,N V i 1,N V r M,N V i M,N  T (1.82) The linear part of the circuit is replaced by its Norton equivalent; the currents flowing into it are computed by simple multiplication of the (still unknown) vector of the voltage phasors by the Norton equivalent admittance matrix, plus the (known) Norton equivalent current sources due to the input signal (Figure 1.20).  I L = ↔ Y ·  V +  I L,0 (1.83) where  I =  I 1,0,L I M,0,L ,I r 1,1,L I i 1,1,L I r M,1,L I i M,1,L ,I r 1,N,L I i 1,N,L I r M,N,L I i M,N,L  T (1.84) When the voltages and currents are ordered as in eqs. (1.82) and (1.84), the admit- tance matrix, relative to the linear subcircuit, is block-diagonal ↔ Y =     ↔ Y L (0) 00 0 0 ↔ Y L (ω 0 ) 00 00 0 000 ↔ Y L (Nω 0 )     (1.85) where ↔ Y L (ω) =   y 11 (ω) . y 1M (ω) . y M1 (ω) . y MM (ω)   (1.86) is the m × m standard linear admittance matrix of the linear subnetwork at frequency ω. Linear subnetwork Nonlinear subnetwork I L 1,1 I NL 1,1 I NL 2,1 I L 2,1 I 0,L 1,1 n = 1 m = 1 I 0,L 2,1 n = 2 m = 1 I NL N,M I L N,M I 0,L N,M n = N m = M Figure 1.20 Currents and voltages for the harmonic balance analysis 30 NONLINEAR ANALYSIS METHODS + − V C g i s i L i NL Linear subnetwork Nonlinear subnetwork Figure 1.21 The example circuit partitioned for the harmonic balance analysis In the case of our example circuit, the linear part is already a one-port Norton equivalent network (Figure 1.21). The currents flowing into the nonlinear part of the circuit are computed as stated above. The time-domain currents are first computed at each connecting port i m,NL (t) = G m (v(t)) m = 1, ,M (1.87) where G m (v) is the nonlinear current–voltage characteristic of the nonlinear subnetwork at port m, and the voltage vector is v(t) =   v 1 (t) v M (t)   (1.88) that is, the vector of the time-domain voltages at all ports. The latter is computed from the voltage phasors by means of an inverse Fourier transform: v(t) = −1 (  V) (1.89) The phasors of the currents flowing into the nonlinear subnetwork are then com- puted by means of a Fourier transform: I m,n,NL =(i m,NL (t)) (1.90) In an actual harmonic balance algorithm, the time-domain voltages and currents are sampled at a set of time instant satisfying Nykvist’s theorem, for calculation of the phasors by means of a DFT. A detailed description is not given here, but it can easily be deduced by generalisation to an M-port problem from the formulae reported in the Appendix A.4 for M = 1. The solving system is now written at each connecting port and for each harmonic: I r m,n,L + I r m,n,NL = 0 (1.90a) I i m,n,L + I i m,n,NL = 0 m = 1, ,M n= 0, ,N (1.90b) SOLUTION THROUGH SERIES EXPANSION 31 The unknowns of the system are the voltages, or more exactly the phasors of the trun- cated Fourier series expansions of the voltages at the ports connecting the linear and the nonlinear subnetworks. The values of the phasors are found by an iterative numerical algorithm, given the nonlinearity of the equations. The real and imaginary parts must be equated separately because of the non-analyticity of the dependence of currents on voltages, as stated above. For the numerical analysis, the nonlinear equation system (1.90) is written as I L (V)+ I NL (V)= F(V)= 0 (1.91) This system is usually solved by means of the zero-searching iterative algorithm known as Newton–Raphson’s method [1, 2, 41]. A first guess for the value of the voltage phasors must be given; let us call it  V first guess =  V (0) (1.92) Obviously, this will not be the exact solution of eq. (1.91). An improved value will be found by the recursive formula V (k+1) = V (k) −  J (k)  −1 · F  V (k)  (1.93) which is the vector form of the well-known Newton–Raphson’s tangent method. The J matrix is the Jacobian matrix of eq. (1.91), corresponding to the derivative of the scalar function in a scalar Newton–Raphson’s method: ↔ J(  V)= ∂  F(  V) ∂  V (1.94) The Jacobian matrix can be computed analytically, if the nonlinear function is known in analytical form, or numerically by incremental ratio, if the nonlinearity is avail- able as a look-up table or if analytical derivation is unpractical. The analytical derivation, however, has better numerical properties, and it is advisable when available. A more detailed description of the Jacobian matrix is given in the Appendix A.6. The inver- sion of the Jacobian matrix is a computationally heavy step of the algorithm; several approaches have been developed to improve its efficiency [42–44]. The algorithm will hopefully converge towards the correct solution, and will be stopped when the error decreases below a limit value. The error is actually the vector of the error currents, real and imaginary parts, at each node and for each harmonic frequency; convergence will be assumed to be reached when its norm will be lower than a desired accuracy level in the currents: | F (k) | <ε (1.95) The actual value of ε will normally vary with the current levels in the circuit: a value below 100 µA will probably be satisfactory in most cases. Alternatively, the algorithm is stopped when the solution does not vary any more: | V (k+1) − V (k) | <δ (1.96) 32 NONLINEAR ANALYSIS METHODS Once more, a reasonable value for δ depends on the voltage levels in the circuit, but a value below 1 mV will probably be adequate in most cases. Another critical point in the algorithm is the choice of the first guess. A well- chosen first guess will considerably ease the convergence of the algorithm to the correct solution. If the circuit is mildly nonlinear, the linear solution, obtained for a low-level input, will probably be a good first guess. If the circuit is driven into strong nonlinearity, a continuation method will probably be the best approach. The level of the input signal is first reduced to a quasi-linear excitation and a mildly nonlinear analysis is performed; then, the input level is increased stepwise, using the result of the previous step as a first guess. In most cases the intermediate results will also be of practical interest, as in the case of a power amplifier driven from small-signal level into compression. Most commer- cially available CAD programmes automatically enforce this method when convergence becomes difficult or when it is not reached at all. The described nodal formulation is based on Kirchhoff’s voltage law: the unknown is the voltage, and the circuit elements are described as admittances. Alternatively, Kirch- hoff’s current law can be used, with the current being the unknown, and the circuit elements described as impedances. While no problem usually arises for the linear ele- ments, the nonlinear elements are usually voltage-controlled nonlinear conductances (e.g. a junction, or the output characteristics of a transistor) or capacitances (e.g. junction capacitances in a diode or in a transistor). This is why the nodal formulation (KVL) is the standard form. However, any alternative form of Kirchhoff’s equations is allowed as a basis for the harmonic balance algorithm in the cases in which the nonlinear elements have a different representation. Another alternative formulation is obtained when the nonlinear equation (1.13) is rewritten as C · dv(t) dt     t=t k + i max · tgh  g ·v(t k ) i max  + i s (t k ) = 0 k = 0, 1, ,2N(1.97) where the unknowns are the time-domain voltage samples at the 2N +1 sampling instants: v k = v(t k )k= 0, 1, ,2N(1.98) In this formulation, eq. (1.13) must be satisfied only at 2N + 1 time instants. The formulation is similar to that of the time-domain solution (Section 1.2), but in this case the derivative with respect to time is expressed as dv(t) dt     t=t k = d N  n=−N V n · e jnω 0 t dt        t=t k = N  n=−N jnω 0 V n · e jnω 0 t k (1.99) according to the assumption of a periodic solution with limited bandwidth. The voltage phasors are computed from the time-domain voltage samples by means of a DFT: SOLUTION THROUGH SERIES EXPANSION 33 v k = v(t k ) ⇒⇒V n (1.100) The nonlinear equation system (1.97) is again solved by an iterative numerical method. This formulation of the nonlinear problem is known as waveform balance,since in eq. (1.97) the current waveforms of the linear and nonlinear subcircuits must be bal- anced at a finite set of time instants. In fact, it can easily be seen that the standard harmonic balance formulation also satisfies eq. (1.13) only at the sampling instants where the DFT is computed. There are two other formulations of the kind: in the first, Kirchhoff’s equations are written in the time domain (eq. (1.97) above), but the unknowns are the voltage phasors; in the second, Kirchhoff’s equations are written in the frequency domain (eq. (1.77) or eq. (1.90) above), and the unknowns are the time-domain voltage samples. The four formulations are actually completely equivalent, at least in principle; one or the other may be more convenient in some cases, when special problems must be dealt with. 1.3.2.2 Multi-tone analysis So far, only strictly periodic excitation and steady-state have been considered. In the real world, however, many important phenomena occur when two or more periodic signals with different periods excite a nonlinear circuit, as shown in Section 1.3.1 on the Volterra series. In some cases a single-tone analysis gives enough information to the designer, but in many other cases a more realistic picture is needed, especially when distortion or intermodulation is a critical issue. Moreover, the behaviour of circuits like mixers can by no means be reduced to a simply periodic one. A first step towards a more realistic picture is the introduction of a more complex Fourier series for the signal, composed of two tones: v(t) = ∞  n 1 =−∞ ∞  n 2 =−∞ V n 1 ,n 2 · e j(n 1 ω 1 +n 2 ω 2 )t = ∞  n 1 =−∞ ∞  n 2 =−∞ V n 1 ,n 2 · e jω n 1 ,n 2 t (1.101) where the two frequencies ω 1 and ω 2 are the frequencies of the input signal or signals: for instance, two equal tones at closely spaced frequencies in the case of intermodulation analysis in a power amplifier; or the local oscillator and the RF signal in the case of a mixer. The unknowns of the problem are still the phasors of the voltage, but now they are not relative to the harmonics of a periodic signal: they rather represent a complex spectrum, as shown in the case of Volterra series analysis. The series in eq. (1.101) must be truncated so that only important terms are retained: a proper choice increases the accuracy of the analysis while limiting the numerical effort. If the two basic frequencies ω 1 and ω 2 are incommensurate, the signal is said to be quasi-periodic. On the other hand, when the two basic frequencies ω 1 and ω 2 are commensurate, they can be considered as the harmonics of a dummy fundamental frequency ω 0 , and the problem can formally be taken back to the single-tone case [45]. However, if the two basic frequencies are closely spaced or are very different from one another, a very large number of harmonics must be included in the analysis. For instance, 34 NONLINEAR ANALYSIS METHODS when two input tones at 1 GHz and 1.01 GHz are applied for intermodulation analysis of an amplifier, at least 300 harmonics of the signal at the dummy 10 MHz fundamental frequency must be included for third-order analysis of the signal. This redundance can be reduced by retaining only the meaningful terms in the Fourier series expansion; in this case, however, the DFT described above experiences the same severe numerical problems as in the quasi-periodic case, as explained in the following. A special two-tones form of the harmonic balance algorithm is therefore usually adopted also in these cases. The formalism for two-tone analysis is easily extended to multi-tone analysis, when more than two periodic signals at different frequencies are present in the circuit; however, the computational burden increases quickly, usually limiting the effective analysis capa- bilities to no more than three tones. For more complex signals, different techniques are used to extend the algorithm, which are shortly described in the following paragraphs. The harmonic balance method requires some adjustments for two-tone analysis. First of all, a suitable truncation of the Fourier series must be defined [11]. The expan- sion of a single-tone signal is truncated so that the neglected harmonics have negligible amplitude. The same principle holds for a multi-tone analysis. The frequency spectrum includes all the frequencies that are combinations of the two basic frequencies, or more than two for multi-tone analysis: ω n 1 ,n 2 = n 1 ω 1 + n 2 ω 2 (1.102) The sum of the absolute values of the two indices n =|n 1 |+|n 2 | is the order of the harmonic component. Not all the lines of the spectrum, however, have significant amplitude. It is a reasonable assumption that the amplitude of a spectral component decreases as its order increases. However, the picture can vary for different cases. A typical example is given in Figure 1.12, where two equal-amplitude signals at closely spaced frequencies are fed to a power amplifier, generating distortion. A partially different situation occurs when a mixer is considered. Typically, the local oscillator has a much higher amplitude than the input signal (e.g. at RF) or the output signal (e.g. at IF), and the situation is rather as in Figure 1.22. A first example of truncation of the expansion is the so-called box truncation scheme: all the combinations of the two indices n 1 and n 2 are retained for values of the indices less than N 1 and N 2 respectively. This truncation can be illustrated graphically as shown in Figure 1.23. Since real signals have Hermitean spectral coefficients, only half of them are actually needed. This is obtained, for instance, by retaining only the terms whose indices satisfy the following conditions: n 1 ≥ 0; n 2 ≥ 0ifn 1 = 0 (1.103) The resulting reduced spectrum is shown in Figure 1.24. In this case the terms with maximum order are those with n 1 = N 1 , n 2 =±N 2 and n max = N 1 + N 2 . The number of terms retained in the Fourier series expansion for a real signal is 2 · N 1 · N 2 + N 1 + N 2 + 1. SOLUTION THROUGH SERIES EXPANSION 35 f IF RF LO Figure 1.22 Typical spectrum of the electrical quantities in a mixer n 2 n 1 Figure 1.23 Box truncation scheme for two-tone Fourier series expansion n 2 n 1 Figure 1.24 Reduced box truncation scheme for real signals An example of the spectrum resulting from a box truncation scheme with N 1 = 3 and N 2 = 2 is shown in Figure 1.25, where frequency f 1 is much larger than frequency f 2 ; both the complete (light) and reduced (dark) spectra are depicted. The choice of a box truncation is very simple, but not necessarily the most effective one. As said above, a reasonable assumption is that the amplitude of a spectral component 36 NONLINEAR ANALYSIS METHODS −3 f 1 −2 f 2 −3 f 1 +2 f 2 −3 f 1 − f 2 −3 f 1 + f 2 −3 f 1 3 f 1 −2 f 2 3 f 1 +2 f 2 3 f 1 − f 2 3 f 1 + f 2 3 f 1 −2 f 1 −2 f 2 −2 f 1 +2 f 2 −2 f 1 − f 2 −2 f 1 + f 2 −2 f 1 2 f 1 −2 f 2 2 f 1 +2 f 2 2 f 1 − f 2 2 f 1 + f 2 2 f 1 − f 1 −2 f 2 − f 1 +2 f 2 − f 1 − f 2 − f 1 + f 2 − f 1 f 1 −2 f 2 f 1 +2 f 2 f 1 − f 2 f 1 + f 2 f 1 −2 f 2 2 f 2 − f 2 f 2 DC Figure 1.25 Complete (light) and reduced (dark) spectra of a two-tone signal after box truncation with N 1 = 3andN 2 = 2 SOLUTION THROUGH SERIES EXPANSION 37 decays with its order. A reasonable truncation scheme therefore drops all terms with n>n max , retaining all those with n ≤ n max . This is called diamond truncation scheme, as apparent from Figure 1.26; the scheme for real signals is also indicated. The number of terms retained in the Fourier series expansion is approximately one half that of the box truncation scheme. An example of the spectrum resulting from a diamond truncation scheme with N 1 = N 2 = 3isshowninFigure1.27. A further variation of the truncation scheme is illustrated in Figure 1.28. This scheme allows an independent choice of the number of harmonics of the two input tones, and of the maximum number of intermodulation products as in the diamond truncation scheme. An example of the spectrum resulting from a mixed truncation scheme is shown in Figure 1.29. The general structure of the harmonic balance algorithm, as described above, still holds. The main modification is related to the Fourier transform that becomes severely inaccurate unless special schemes are used. The main difficulty is related to the choice of the sampling time instants. In principle, a number of time instants equal to the num- ber of variables to be determined (the coefficients in the Fourier series expansion in eq. (1.101)) always allows for a Fourier transformation from time to frequency domain, by inversion of a suitable matrix similar to that described in the Appendix A.4. How- ever, the matrix becomes very ill-conditioned unless the sampling time instants are n 2 n 1 n 2 n 1 Figure 1.26 Diamond truncation scheme for general (a) and real signals (b), and N 1 = N 2 f DC −3 f 1 −2 f 1 −2 f 1 − f 2 −2 f 1 + f 2 − f 1 −2 f 2 − f 1 − f 2 − f 1 + f 2 − f 1 +2 f 2 −3 f 2 −2 f 2 − f 2 f 2 2 f 2 3 f 2 f 1 +2 f 2 f 1 − f 2 f 1 + f 2 f 1 −2 f 2 f 1 2 f 1 − f 2 2 f 1 + f 2 2 f 1 3 f 1 − f 1 Figure 1.27 Spectrum relative to the diamond truncation scheme with N 1 = N 2 = 3 [...]... frequencies f (Hz) n1 n2 f (MHz) f n1 n2 f f n1 n2 f 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 −3 2 −1 0 1 2 3 0 0 0 0 1 1 1 1 1 1 1 0 100 20 0 300 1700 1800 1900 20 00 21 00 22 00 23 00 11 12 13 14 15 16 17 18 19 20 21 −3 2 −1 0 1 2 3 −3 2 −1 0 2 2 2 2 2 2 2 3 3 3 3 3700 3800 3900 4000 4100 420 0 4300 5700 5800 5900 6000 22 23 24 25 26 27 28 29 30 31 32 1 2 3 −3 2 −1 0 1 2 3 −3 3 3 3 4 4 4 4 4 4 4 5 6100 620 0 6300 7700...38 NONLINEAR ANALYSIS METHODS n2 n2 n1 n1 Figure 1 .28 Mixed truncation scheme general (a) and real signals (b) − f1 f1 −2f1 2 f1 −3f1 −f1−f2 −f +f 1 2 −f2 f2 −2f2 DC 2f2 3 f1 f1−f2 f1+f2 f Figure 1 .29 Spectrum relative to the mixed truncation scheme properly chosen Several schemes have been proposed... Trans Microwave Theory Tech., MTT-40(1), 12 28 , 19 92 58 NONLINEAR ANALYSIS METHODS [41] C.P Silva, ‘Efficient and reliable numerical algorithms for nonlinear microwave computeraided design , IEEE Microwave Theory Tech Symp Dig., 19 92, pp 422 – 428 [ 42] H.G Brachtendorf, G Welsch, R Laur, ‘Fast simulation of the steady-state of circuits by the harmonic balance technique’, Proc IEEE Int Symp on Circuits... small-signal regimes’, IEEE Trans Microwave Theory Tech., MTT-46( 12) , 20 16 20 24, 1998 [71] S Egami, Nonlinear- linear analysis and computer-aided design of resistive mixers’, IEEE Trans Microwave Theory Tech., MTT -22 , 27 0 27 5, 1974 [ 72] D.N Held, A.R Kerr, ‘Conversion loss and noise of microwave abd``nd millimetre-wave ıı mixers: part 1 - theory’, IEEE Trans Microwave Theory Tech., MTT -26 , 49–55, 1978 [73] A.R... 10 8 6 4 2 0 2 −4 −6 −8 −10 0 0 .2 0.4 0.6 0.8 1 1 .2 1.4 1.6 1.8 2 ×10−8 t (s) Figure 1. 32 Nonlinear current waveform with uniform samples, oversampling, and optimum samples after orthonormalisation 0 .2 0.15 0.1 0.05 0 −0.05 −0.1 −0.15 −0 .2 0 0.5 1 1.5 2 t (s) 2. 5 3 3.5 4 × 10−8 Figure 1.33 Voltages and currents in the example circuit for a two-tone input signal and a time-domain analysis 42 NONLINEAR. .. in the study of nonlinear active microwave circuits’, IEEE Trans Microwave Theory Tech., MTT -20 , 4 02 409, 19 72 [64] G.L Heiter, ‘Characterisation of nonlinearities in microwave devices and systems’, IEEE Trans Microwave Theory Tech., MTT -21 , 797–805, 1973 [65] R.S Tucker, ‘Third-order intermodulation distortion and gain compression of GaAs FET’s’, IEEE Trans Microwave Theory Tech., MTT -27 , 400–408, 1979... performances of a device or circuit under nonlinear regime and of the corresponding operating conditions Nonlinear network vector analysis mainly aims at the direct measurement or optimisation of a nonlinear circuit, or at the identification of a nonlinear model; pulsed measurements Nonlinear Microwave Circuit Design F Giannini and G Leuzzi  20 04 John Wiley & Sons, Ltd ISBN: 0-470-84701-8 62 NONLINEAR MEASUREMENTS... investigation of nonlinear microwave circuits’, IEE Proc Part H, 133(5), 351–3 62, 1986 [54] P.J.C Rodrigues, ‘A general mapping technique for Fourier transform computation in nonlinear circuit analysis’, IEEE Microwave Guided Wave Lett., 7(11), 374–376, 1997 [55] R.J Gilmore, Nonlinear circuit design using the modified harmonic-balance algorithm’, IEEE Trans Microwave Theory Tech., MTT-34, 129 4–1307, 1986... analysis of nonlinear systems’, IEEE Trans Microwave Theory Tech., MTT -24 (7), 422 –433, 1976 [25 ] R.A Minasian, ‘Intermodulation distortion analysis of MESFET amplifiers using Volterra series representation’, IEEE Trans Microwave Theory Tech., MTT -28 (1), 1–8, 1980 [26 ] D.D Weiner, J.F Spina, Sinusoidal Analysis and Modelling of Weakly Nonlinear Circuit, Van Nostrand Reinhold, New York (NY), 1980 [27 ] M Schetzen,... (NY), 1980 [27 ] M Schetzen, ‘Multilinear theory of nonlinear networks’, J Franklin Inst., 320 (5), 22 1 22 8, 1985 [28 ] C.L law, C.S Aitchison, ‘Prediction of wideband power performance of MESFET distributed amplifiers using the Volterra series representation’, IEEE Trans Microwave Theory Tech., MTT-34, 1038–1317, 1986 [29 ] S.A Maas, Nonlinear Microwave Circuits, Artech House, New York (NY), 1988 [30] S.A . METHODS −3 f 1 2 f 2 −3 f 1 +2 f 2 −3 f 1 − f 2 −3 f 1 + f 2 −3 f 1 3 f 1 2 f 2 3 f 1 +2 f 2 3 f 1 − f 2 3 f 1 + f 2 3 f 1 2 f 1 2 f 2 2 f 1 +2 f 2 2 f 1 − f 2 2 f 1 + f 2 2 f 1 2 f 1 2 f 2 2 f 1 +2 f 2 2 f 1 − f 2 2 f 1 + f 2 2 f 1 − f 1 2 f 2 − f 1 +2 f 2 − f 1 − f 2 − f 1 + f 2 − f 1 f 1 2 f 2 f 1 +2 f 2 f 1 − f 2 f 1 + f 2 f 1 2 f 2 2 f 2 − f 2 f 2 DC Figure. component 36 NONLINEAR ANALYSIS METHODS −3 f 1 2 f 2 −3 f 1 +2 f 2 −3 f 1 − f 2 −3 f 1 + f 2 −3 f 1 3 f 1 2 f 2 3 f 1 +2 f 2 3 f 1 − f 2 3 f 1 + f 2 3 f 1 2 f 1 2 f 2 2 f 1 +2 f 2 2 f 1 − f 2 2 f 1 + f 2 2 f 1 2 f 1 2 f 2 2 f 1 +2 f 2 2 f 1 − f 2 2 f 1 + f 2 2 f 1 − f 1 2 f 2 − f 1 +2 f 2 − f 1 − f 2 − f 1 + f 2 − f 1 f 1 2 f 2 f 1 +2 f 2 f 1 − f 2 f 1 + f 2 f 1 2 f 2 2 f 2 − f 2 f 2 DC Figure. N 2 f DC −3 f 1 2 f 1 2 f 1 − f 2 2 f 1 + f 2 − f 1 2 f 2 − f 1 − f 2 − f 1 + f 2 − f 1 +2 f 2 −3 f 2 2 f 2 − f 2 f 2 2 f 2 3 f 2 f 1 +2 f 2 f 1 − f 2 f 1 + f 2 f 1 2 f 2 f 1 2 f 1 − f 2 2 f 1 + f 2 2 f 1 3 f 1 − f 1 Figure 1 .27 Spectrum relative to the