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Zou, L. h oeo -S ytmi hsr eiethat device a phaser, a is system R-ASP a of core The from inspired [1], (R-ASP) processing signal Radio-analog ne Terms Index Abstract osGi qaie eofiual hsrfor Phaser Reconfigurable Equalized Loss-Gain inegZou, Lianfeng ∆ τ W rsn osgi qaie reconfigurable equalized loss-gain a present —We rvd etrfeunydsrmnto ntime, in discrimination frequency better provide Ls-anpi,eulzdpae,alps system, all-pass phaser, equalized pair, —Loss-gain .I I. tdn ebr IEEE, member, Student NTRODUCTION cl oyehiu eMontr´eal, de Polytechnique Ecole ´ ∆ τ plmlc) C. @polymtl.ca). “ hlb Gupta, Shulabh rba(email: Arabia τ max n the and n clEngi- ical ´ ogram pair n τ ance, ASP) trum τ asily p rate Re- ally min ω ays nal log 0], ill se le q ll y - - , , , , rbepae.Fnly hr ocuini ie nSc V Sec. in given the is and conclusion short pair a reco loss-gain Finally, V all-pass phaser. resulting equalized Sec. urable the and an of IV of demonstration Sec. experimental presents design Then subsequently the phaser. III pair describe Section loss-gain gain. combined or a and loss distributed with lumped first with C-sections, gain or loss separate eosrtstepooe osgi areulzdphaser equalized pair experiment loss-gain and proposed withi analytically the will dissipation, demonstrates paper increased This hence structure. longer and to the group due trapping, loss higher wave higher where time with accompanied phasers, always passive are th (a purely delays response flat in a exhibiting delay, impossible phaser, group is controllable pai perfect and a loss-gain magnitude in pass) equalized results using C-sections of Furthermore, magnit its [27]. reversing while phaser response the of delay enhances group equally overall [24]–[26] phaser C-section conventional onl but enhancement, delay compensation. group loss to feedb contribute the desig not in does amplification and loop the system, synchronization that external In complication. requires which switch, where ea wn ( swing delay hr sdet h qiaetcsaeof cascade equivalent the to enhancement due delay is group increa there the to However, phaser swing. C-section ampli delay a an group to the [23], added In is [1]. loop feedback distortion cation induces undesira this generally since an R-ASP is in mimic which respons to dependent, amplitude amplifier frequency obtained the distributed strongly but a [22], mi- [5], using structure at EBG [21] practical, in phase not reported reconfigurable is microwave hence A and [20]. frequencies long, crowave excessively be coupler, would (co-directional) resona a forward-wave ring a amplitude However, requires optical transmission [18] swing. the in the Moreover, delay in frequency. resonance notch group the a higher introduces [19] also in resonator wh loss tuning, ring results delay fiber group loss in single results a higher ring the in of that, loss use the shown varying each is real-ti to it assigned require [18], codes In users dispersion the of of pairs reconfiguration communicatio different dynamic are between channels hence switching communication and the time-variant, where essentially [12], DCMA is tions hne nefrne n ec oe i ro rate. error bit lower hence and interference, channel hl nDM 1] increasing [12], DCMA in while t,iera-ietnbegopdlyrsos ( response delay group tunable real-time reconfigura i.e ity, delay group require typically would vironments h ae sognzda olw.ScinI analyzes II Section follows. as organized is paper The ehv hw htitouigblne osadgi na in gain and loss balanced introducing that shown have We plctosrqiigra-ieaatto odnmcen dynamic to adaptation real-time requiring Applications ebr IEEE, member, N steieaincut otoldb nSD RF SPDT an by controlled counts iteration the is ∆ τ nRAP n xml fsc applica- such of example One R-ASP. in ) hitpeCaloz, Christophe ∆ τ eut nrdcdmulti- reduced in results elw IEEE Fellow, N dnia phasers, identical τ p ω q rgroup or ) which nfig- then ally ude ack is e bil- s. the ere me ble tor ll- fi- se at I. r. n y n n 1 r r - t , 2

II. SEPARATE LOSS OR GAIN C-SECTIONS gain profiles would be difficult to engineer, and we therefore A. Distributed Loss or Gain C-Sections now turn to lumped loss and gain inclusions, where the same conclusions will be shown to hold. A C-section phaser is formed by shorting the end of 1.5 15 a backward-wave (contra-directional) coupler, with isolated αℓ 0.4 αℓ 0.4 10 “´ transmission line propagation constant γ β jα [1]. C- “˘ αℓ 0.3 “ ´ αℓ 0.3 “´ section phasers reported to date are composed of purely 1 “˘ 5 1 0 T passive transmission lines, and therefore α 0 . We will (dB) { 0 | αℓ 0

ą 21 21 consider here also active transmission line C-section phasers, τ αℓ 0 “

0.5 S −5

“ | for which α 0. Such structures might be engineerable using αℓ 0.3 ă −10 “ traveling-wave tube structures or active artificial transmission αℓ 0.4 0 −15 “ lines. Figure 1 shows the general concept of a gain or loss 0 0.5 1 0 0.5 1 C-section phaser. βℓ (π) βℓ (π) (a) (b)

k Fig. 2: (a) Normalized group delays, with respect to the period, T0, of the quarter wavelength frequency and (b) amplitudes, jα jα with varying αℓ of a distributed loss or gain C-section, where ´ ´ the coupling factor k 0.5. β β ℓ “ “ “ γ γ

B. Lumped Loss or Gain C-Sections 1k 2 We thus now consider the lumped loss or gain C-section Fig. 1: C-section with physical length ℓ, maximum coupling phaser shown in Fig. 3, where the coupler is assumed, in first coefficient k, and isolated transmission line propagation con- approximation, to be lossless and terminated at one end by stant γ β jα. a load. The coupler lossless approximation is very reasonable “ ´ since dispersion loss is typically much larger than transmission The transfer function of such a C-section structure takes the line conductive or dielectric losses [1]. Assuming perfect load general form [26] 2 2 1 jκ tan γℓ TL 1 Load 2 S21 ´ S “1 jκ tan γℓ L 2 2 ` (1) Load r s ˆ 1 κ tanh αℓ j κ tanh αℓ tan βℓ 21 31 ´ ´ p ´ q , TC “1 κ tanh αℓ j κ tanh αℓ tan βℓ B ` ` p ` q C where κ 1 k 1 k , corresponding to the amplitude S “ p ´ q{p ` q C 4ˆ4 and the phase BC r s a TC 2 2 2 1 κ tanh αℓ κ tanh αℓ tan βℓ 1 2 1 1 S21 p ´ q2 ` p ´ q2 2 , (2a) 1 4 | |“ d 1 κ tanh αℓ κ tanh αℓ tan βℓ p ` q ` p ` q Fig. 3: Loaded C-section consisting of a coupled-line coupler, κ tanh αℓ =S21 arctan ´ tan βℓ represented by a 4 4 scattering matrix, SC 4ˆ4, and a load “´ 1 κ tanh αℓ represented by a 2 ˆ 2 scattering matrix, rS s . ˆ ´ ˙ (2b) L 2ˆ2 κ tanh αℓ ˆ r s arctan ` tan βℓ , ´ 1 κ tanh αℓ matching, the scattering matrix of the load is ˆ ` ˙ respectively. Inspecting (2a) and (2b) reveals that reversing the S 0 TL load 2ˆ2 , (4a) sign of α reverses S21 and keeps φ21 unchanged, i.e. r s “ TL 0 | | „  1 S21 α , (3a) where jφLpωq | p´ q| “ S21 α TL ω AL ω e , (4b) | p q| p q“ p q =S21 α =S21 α , (3b) p´ q“ p q is the load transfer function. Further assuming coupler perfect respectively. Equations (3a) and (3b) state that equalized dis- matching and isolation, the scattering matrix of the coupler is tributed loss ( α ) and gain ( α ) yields 0 dB-symmetric am- | | ´| | 0 TC 0 BC plitudes, 20 log S21 α 20 log S21 α , and identi- p| p´ q|q “ ´ p| p q|q TC 0 BC 0 cal group delays, τ21 α τ21 α , from τ21 =S21 ω, S , (5a) C 4ˆ4 » 0 B 0 T fi as plotted in Fig. 2.p´ Theq“ tuningp q effect of“ distributed ´B {B loss r s “ C C — BC 0 TC 0 ffi and gain on group delay will be later shown to allow real- — ffi time dispersion engineering. However, distributed loss and where – fl jk sin θ BC ω , (5b) 1The time dependence e`jωt is assumed throughout the paper. p q“ ?1 k2 cos θ j sin θ ´ ` 3

2 ?1 k Inspecting (10b) and (10c) reveals that reversing the load TC ω 2 ´ , (5c) p q“ ?1 k cos θ j sin θ transmission amplitude, i.e. AL 1 AL, reverses the loaded ´ ` Ñ { are the backward coupling and through transfer functions, C-section transfer function amplitude, S21 , while keeping the = | | respectively [20], with parameters k and θ πω 2ω0 being transmission phase S21, and hence also the group delay, the maximum coupling, occurring at the quarter-wavelength“ { τ21, unchanged. Therefore, the lumped loss or gain loaded frequency ω0, and the electrical length of the coupler at that C-section indeed exhibits the same fundamental properties as frequency, respectively. its distributed counterpart. For simplicity, we next assume that the load transmission is of constant amplitude and lin- The backward coupling introduces a feedback loop from ear phase, or is non-dispersive, i.e. AL ω G G 1 or the load output port 22 to the load input port 12 (see Fig. 3), p q“ p ą q AL ω L L 1 , so that leading to multiple scattering at port 2 of the loaded C-section. p q“ p ă q Assuming unity excitation to the loaded C-section port 1 and φL τLω, (11) “´ accounting for the multiple scattering events, one finds where τL is the load transmission delay. we may now paramet- S21 S12 rically study the behavior of the loaded C-section. Assuming “ 0 BC TcTLTC TCTLBCTLTC the first resonance frequency, ωp ω , the maximum of the “ ` ` loaded C-section group delay corresponds“ then to TCTLBCTLBCTLTC ... (6) ` ` 8 max φ21 2 n τ21 AL,ω τ21 AL,ω0 B BC T TL BCTL . C p q“ p q“´ ω ω“ω0 “ ` 0p q B ˇ n“ 2 ˇ ÿ T0?1 k 1 ˇ 1 If BCTL 1, the geometric series (6) diverges, corre- ´ ˇ “ 4 1 k AL ` 1 kAL (12) sponding| | to ě an oscillatory (unstable) regime. Therefore, the ˆ ´ { ´ ˙ 1 AL AL condition BCTL 1 must be enforced for stability, in which τLk { | |ă ` 1 k AL ` 1 kAL case (6) reduces to ˆ ´ { ´ ˙ 2 τL, TC TL ` S21 S12 B , (7) C 0 = 0 = 0 “ “ ` 1 BCTL where φL ω 2nπ BC ω 2nπ, with BC ω ´ 0 [20] hasp beenq“ used.´ Furtherp assumingq“ τ 0 reducesp q (12) “ with L 1 to “ T ω A ω . (8) L L B ω 2 | p q| “ p qă C max T0?1 k 1 1 | p q| τ21 AL,ω ´ , (13) Moreover, according to [28], the maximum of the group delay p q“ 4 1 k A ` 1 kA ˆ L L ˙ occurs at the resonance frequency, ω , where the multiple scat- ´ { ´ p with T0 1 f0 being the period at the quarter-wavelength fre- tered waves add in phase. This is equivalent to the following quency. The“ { minimum loaded C-section group delay occurring phase condition at the first anti-resonance frequency, 2ωp 2ω0, is “ = TL ωp BC ωp φL ωp =BC ωp 2nπ, (9) min φ21 r p q p qs “ p q` p q“ τ21 AL,ω τ21 AL, 2ω0 B where n is an integer. Equation (9) suggests that, given a p q“ p q“´ ω 2 0 B ˇω“ ω0,τL“ (14) coupler with known =Bc ω , the resonance frequency, ωp, T0 ˇ k p q 2 kA ˇ , may be tuned by varying the load transmission phase φ . 2 L ˇ L “4?1 k ´ ´ AL ´ ˆ ˙ Inserting (4b), (5b) and (5c) into (6) yields The difference of the last two relations corresponds to the 2 group delay swing ?1 k cot θ k AL sin φL j 1 k AL cos φL 21 S TL ´ 2 ` { ´ p ´ { q max min “ ?1 k cot θ kAL sin φL j 1 kAL cos φL ∆τ21 AL τ21 AL,ω τ21 AL,ω jφ´21 ` ` p ´ q p q“ p q´ p q S21 e , T0k AL 1 AL 2k 1 1 “ | | p ` { 2 ´ q . (10a) “ 4?1 k 1 k AL ` 1 kAL ´ ˆ ´ { ´ (15)˙ where

S21 We may at this point define the loaded C-section transmission | |“ group delay swing and amplitude tuning factors 2 2 2 ?1 k cot θ k AL sin φL 1 k AL cos φL ∆τ21 AL AL ´ ` { 2 ` p ´ { q , g 2 2 σ∆τ AL p q f` ?1 k cot θ kAL sin φL˘ 1 kAL cos φL p q“ ∆τ21 AL 1 f ´ ` ` p ´ q p “ q e (10b) AL 1 AL 2k 1 1 ` ˘ ` { ´ “ 4 1 k A ` 1 kA 1 k AL cos φL L L = 21 ˆ ´ { ´ ˙ S arctan 2´ { (16a) “´ ?1 k cot θ k AL sin φL ´ ` { 1 kAL cos φL (10c) arctan 2´ ´ ?1 k cot θ kAL sin φL ´ ` φL, ` 4

˝ ˝ ˝ ˝ ˝ ˝ φL 20 0 20 φL 20 0 20 and 2 “´ 20 “´ 4.6 dB 15 4.6 dB ˘ ` S21 AL,ω0 AL k 1.5 10 σA AL | p q| S21 AL,ω0 ´ . 5 p q“ S21 AL 1,ω0 “ | p q| “ 1 kAL 0 T ˇ ˇ (dB) | p “ q| ´ { ˇ (16b)ˇ 1 | 0 21 21 ˇ ˇ τ −5 Figure 4 shows σ∆τ AL and σA AL with three different S ˇ ˇ | p q p q 0.5 −10 coupling factors k. We see that σ∆τ AL and σA AL (in dB) p q p q −15 4.6 dB are even and odd function of AL (in dB), respectively, which 0 −20 ´ means that a balanced pair of load loss and load gain have 0 0.5 1 1.5 2 0 0.5 1 1.5 2 ω ω0 ω ω0 same tuning effect on group delay swing, while opposite { { tuning effect on amplitude. Moreover, smaller k gives higher (a) (b) group delay swing tuning range but at the cost of using higher Fig. 6: Loaded C-section (a) normalized group delays τ21 T0 { loss or gain load and hence consuming more power. Also note and (b) transmission amplitudes S21 , with maximum cou- | | that AL going above upper limit 1 k leads to oscillation, which pling factor k 6 dB (k 0.5) at ω0 and AL 4.6 dB, { “ “ ˝“˝ ˘ ˝ should be avoided, while going below lower limit k results in and varying load transmission phase φL 20 , 0 , 20 . negative group delay, which has been presented in [28]. “ t´ u

k 0.20 ( 14 dB) k 0.35 ( 9 dB) k 0.50 ( 6 dB) “ ´ “ ´ “ ´ III. COMBINED LOSS-GAIN EQUALIZED PAIR 25 40 An all-pass loss-gain equalized pair is formed by serially 20 20 connecting a loss C-section and a gain C-section, as shown

15 in Fig. 7, with appropriately tune gain, G, and loss, L, such τ (dB) ∆ loss gain 0 that G 1 L. The group delay of the resulting loss-gain pair σ 10 A “ { σ is twice that of a single loss or gain loaded C-section phaser, as −20 5 loss gain shown in Fig. 8(a), while the transmission amplitude becomes all-pass [Fig. 8(b)]. 0 −40 −20 −10 0 10 20 −15 −10 −5 0 5 10 15 1 AL (dB) AL (dB) G “ L (a) (b) G L Fig. 4: Tuning effect of the load transmission amplitude, AL in dB, on (a) the loaded C-section group delay swing and (b) the transmission amplitude, respectively, with tuning range of AL defined by k as the lower limit (left dashed line) and 1 k as the upper limit (right dashed line). Range: { AL k 0.5 dB, 1 k 0.5 dB . 2 Pr ` { ´ s 1 Fig. 7: Proposed all-pass loss-gain equalized pair formed by Figure 5 shows the loaded C-section transmission group serially connecting a loss loaded C-section and a gain loaded 0 5 delays and amplitudes for coupling coefficient k . . We see C-section, where the gain is the reverse of the loss, G 1 L. that the equalized loss and gain pair exhibits identical“ group “ { delay and symmetric amplitudes about S21 0 dB. | |“ 7 3.5 30 5.3 dB 6 ˘ 3 5.3 dB 20 ˘ 5.3 dB 5

2.5 ` 0 T 10 4 4.6 dB (dB) { 0 2 ˘ | 0

T 4.6 dB (dB) 21 3 21 {

0 τ

˘ | 3.5 dB S

21 1.5 ˘ | 21 2

τ 3.5 dB

S −10

˘ | 1 1 5.3 dB 0 dB 0.5 −20 0 0 dB ´ 0 0.5 1 1.5 2 0 0.5 1 1.5 2 0 −30 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 ω ω ω ω (a){ (b){ ω ω0 ω ω0 (a){ (b){ Fig. 8: Loss-gain equalized pair (a) normalized group delay Fig. 5: Loaded C-section (a) normalized group delays τ21 T0 τ21 T0 and (b) all-pass transmission amplitude S21 , with { { | | and (b) transmission amplitudes S21 , with maximum cou- maximum coupling factor k 0.5 at ω0 and different load | | “ pling factor k 6 dB (k 0.5) at ω0, and varying load transmission amplitudes AL 0, 3.5, 4.6, 5.3 dB. “ ´ “ “t ˘ ˘ ˘ u transmission amplitudes AL 0, 3.5, 4.6, 5.3 dB. “t ˘ ˘ ˘ u The analysis performed so far assumes an ideal (lossless, Apart from tuning the height of group delay peak, the perfectly matched and perfectly isolated) system. As a result, position of group delay peak, ωp, or resonance frequency, can the loaded C-section is stable as long as condition (8) is be also tuned by varying the load transmission phase, φL, as satisfied. In reality, the loaded C-section phaser may still shown in Fig. 6. The value of ωp is determined by using (9). become unstable due to non-ideal factors, such as coupler 5 forward-wave coupling (imperfect isolation) and load mis- internally matched variable loss-gain (VLG) chip, whose loss- match, since such non-idealities create wave paths that have gain is tuned by varying the control bias voltage, and a fixed not been accounted for in (8). A detailed stability analysis of stabilization and matching enhancement attenuator (ATT). a real C-section phaser is beyond the scope of this paper, and Both the VLG internal matching and the ATT matching and will be presented elsewhere. However, maximizing isolation attenuation contribute to the stability of the overall device and matching clearly appears to represent important design discussed in Sec. III. considerations from the viewpoint of stability.

IV. DESIGN OF LOSS-GAIN PAIR A. Microstrip Coupler The C-section phaser is implemented here in microstrip technology for easiest fabrication and testing. Unfortunately, due to their imperfect transverse electromagnetic nature, and corresponding unequal even and odd mode phase velocities (vo ve), microstrip couplers suffer from relatively poor isolationą [20]. To minimize the aforementioned subsequent Fig. 11: Fabricated variable loss or gain load (same substrate risks of instability, one should thus increase the natural as in Fig. 9) composed a VLG (Avago VMMK-3503 [32]: isolation of the coupler. Corresponding equalization of even internal matching, 0.5 18 GHz operation, 10 dB gain to 13 dB ´ and odd velocities may be achieved by different approaches, loss tuning range (23 dB interval) by varying Vc from 1.8 V such as using wiggly transmission lines [29], inductive com- to 0 V) and a fixed stabilization and matching enhancement pensation [30] or capacitive compensation [31], etc. We use attenuator (Minicircuit GAT-4+: 4 dB attenuation). here capacitive compensation, which consists in inserting a coupling enhancing capacitance in the gap between the two Figure 10 shows the corresponding measured response. The transmission lines. maximum measured gain is limited by the attenuator to the The fabricated microstrip coupler is shown in Fig. 9 while level of 6.7 dB, which lies in the stability region prescribed max Fig. 10 shows the corresponding measured response, with best by (8), namely TLF 2.5 GHz 1 Bc 2.5 GHz matching and isolation reached near 2.5 GHz and correspond- 10 dB. | p q| “ {| p q| “ ing 10 dB coupling. Therefore we will choose operation S21 S11 S22 S12 ´ | | | | | | | | frequency around 2.5 GHz in the design of phaser later. 10 10 0 0 6.7 dB -10 -10 0 dB -20 -20

-30 -30

-40 -40 -50 -50

S-parameters (dB) -60 S-parameters (dB) -60 2.0 2.2 2.4 2.6 2.8 3.0 2.0 2.2 2.4 2.6 2.8 3.0 Frequency (GHz) Frequency (GHz) (a) (b) Fig. 9: Fabricated microstrip coupler (Rogers RO6010, Fig. 12: Measured amplitude responses of the load shown ǫr 10.2, 0.5 oz cladding, 50 mil substrate). “ in Fig. 11 for (a) TL 2.5 GHz 6.7 dB and | p q| “ (b) TL 2.5 GHz 0 dB. | p |“ 10

-10 C. Numerical and Experimental Demonstration of Loss-Gain C-Sections -30

S11 The performance of the loss or gain C-section phaser S31 -50 may be predicted using a commercial RF circuit simulator Amplitude (dB) S21 S41 by importation of the measured coupler and load responses -70 into two-port and four-port scattering models, respectively. 2.00 2.25 2.50 2.75 3.00 Frequency (GHz) Figure 13(a) shows the corresponding responses. The relatively high reflection S22 is not a concern, since propagation is from Fig. 10: Measured S-parameters of the microstrip coupler port 1 to port 2|, as| long as good matching is achieved at port shown in Fig. 9. 1 of loaded C-sections (see Fig. 3). The forward transmission S21 2.5 GHz 16 dB, | p q| « with TL 6.7 dB, corresponding to the enhanced | | “ B. Variable Loss-Gain Load group delay τ21 2.5 GHz 3 ns [Fig. 13(b)], while p q « The fabricated loss-gain load is shown in Fig. 11 with parts S21 2.5 GHz 0 dB, with TL 0 dB, correspond- specifications. It is composed of the series connection of an ing| p to conventionalq| « C-section| (all| pass) “ with group delay 6

0 20 τ21 2.5 GHz 1.7 ns. The loss C-section performance is p q « 6.7 dB not simulated here because it is less possible to oscillate than −10 ´ 10 6.7 dB (dB) gain C-section, but it will be shown later in the experiment. −20 (dB) 0 | | 11 21 S 0 20 S | −30 6.7 dB | −10 6.7 dB -5 TL 2.5 GHz 6.7 dB ´ | p q| “ 15 -10 TL 2.5 GHz 0 dB −40 −20 | p q| “ 10 (dB) (dB) 0 0 -15 | | 6.7 dB 5 −10 11 -20 21 S S | | −20 0 -25 −5 −30 (dB) (dB) | -30 -5 | −40 6.7 dB 12 0 0 −10 22 ´ S S

| −50 -5 | −60 -5 -10 −15 −70 (dB) (dB) 2 2.2 2.4 2.6 2.8 3 2 2.2 2.4 2.6 2.8 3 -15 | | Frequency (GHz) Frequency (GHz) 12 -10 22 -20 S S | | (a) -25

-15 -30 3.5 2.0 2.2 2.4 2.6 2.8 3.0 2.0 2.2 2.4 2.6 2.8 3.0 Loss Frequency (GHz) Frequency (GHz) 3 6.7 dB Gain ˘ (a) (ns) 2.5 5 dB 21

τ ˘3.7 dB 3 2 ˘ 2.5 1.5 0 dB

(ns) 1 2 21 τ Group delay 0.5 1.5 0 2 2.2 2.4 2.6 2.8 3 1 Frequency (GHz) T 2.5 GHz 6.7 dB

Group delay L 0.5 | p q| “ (b) TL 2.5 GHz 0 dB 0 | p q| “ 2 2.2 2.4 2.6 2.8 3 Fig. 15: Measured (a) amplitudes and (b) group delays of the Frequency (GHz) loss or gain C-section in Fig. 14, with varying gains and losses (b) TL 0, 3.7, 5, 6.7 dB. | |“t ˘ ˘ ˘ u Fig. 13: Simulated (a) amplitudes and (b) group delays of a loaded C-section phasers with imported coupler and load experimental models corresponding to Fig. 10 and Fig. 12, respectively.

Figure 14 shows the fabricated loaded C-section. The corre- sponding measured amplitudes and group delays, with varied load loss and gain, are plotted in Fig. 15. Consistently with the analysis presented in Sec. II-B, the forward transmission Fig. 16: Fabricated equalized loss-gain pair phaser. amplitudes are symmetric about 0 dB while the group delays are identical.

6 10

5 S21 5 (ns) 0 | | (dB) 21 −5 τ 4 | G2 6.7 dB 21 −10 G1 “˘5 dB S

| 11 3 S “˘ , −15 G0 0 dB

| | | “ 11 −20

2 S | −25 Fig. 14: Fabricated loaded C-section phaser. Group delay 1 −30 2 2.2 2.4 2.6 2.8 3 2 2.2 2.4 2.6 2.8 3 Frequency (GHz) Frequency (GHz) Figure 16 shows the fabricated equalized loss-gain pair (a) (b) phaser. The corresponding measured amplitudes and group Fig. 17: Measured (a) group delays and (b) amplitudes of the delays, with varied load loss and gain, are plotted in Fig. 17. loss-gain pair phaser in Fig. 16. Consistently with the analysis presented in Sec. III, the combined pair transmission amplitudes are nearly flat while combined pair group delays are twice of the loss or gain C- section [compare to Fig. 15(b)]. 7

V. RECONFIGURABLE CASCADED LOSS-GAIN PAIR PHASER

A. Constitutive Loss-Gain Pairs Cascading loss-gain equalized pairs tuned at different res- onance frequencies allows to synthesize reconfigure group delay responses in real time with all-pass amplitude response over a certain bandwidth. To demonstrate this, we fabricated three loss-gain equalized pairs, shown in Fig. 18, tuned at ωp1 2.35, ωp2 2.45 and ωp3 2.6 GHz, respectively. “ “ “

(a)

Fig. 18: Fabricated loss-gain equalized pairs, tuned at 2.35, 2.45 and 2.6 GHz, respectively, by varying the length of the loads. The incorporated couplers are all identical. (b)

Fig. 19: (a) Experimental setup and (b) device under test, where a 3-section loss-gain reconfigurable phaser is formed B. Experimental Results by cascading three loss-gain equalized pairs.

The experimental setup and complete device under test are 11 5 0 shown in Fig. 19(a) and Fig. 19(b), respectively. The 3-section 10.5 reconfigurable phaser is formed by cascading the three loss- −5 Up chirp S21 10 (dB) | | gain equalized pairs, which were clipped on a copper plate | −10 Down chirp 9.5 21 −15 S placed underneath the pairs. The scattering matrix and group |

, −20

9 | delay were measured on a vector network analyzer. The two 11 −25 S11 S

8.5 | Group delay (ns) −30 | | power supplies provide a source voltage of Vs 3 V and a “ 8 −35 control voltage of Vc 2 V, respectively. 2.35 2.4 2.45 2.5 2.55 2.6 2.35 2.4 2.45 2.5 2.55 2.6 “ Frequency(GHz) Frequency(GHz) In Fig. 20 the control voltages (Vc) of the 3 loss-gain pairs (b) (6 voltages in all) are tuned to produce up-chirp and down- (a) chirp linear group delay responses [1]. This demonstrates the Fig. 20: (a) Group delays and (b) Transmission and reflection central point of the paper: the group delay response of the amplitudes of the cascade 3-section reconfigurable phaser. phaser can be reconfigured in real time with essentially all- pass transmission. The reconfigurability shown here, between positive and negative chirp responses, is naturally only an il- REFERENCES lustrative choice, other group delay responses being achievable [1] C. Caloz, S. Gupta, Q. Zhang, and B. Nikfal, “Analog signal process- with this phaser. ing: A possible alternative or complement to dominantly digital radio schemes,” IEEE Microw. Mag., vol. 14, no. 6, pp. 87–103, Sep. 2013. [2] B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics 2nd Ed. Hoboken, NJ: John Wiley, 2007. VI. CONCLUSION [3] J. Aza˜na, “Ultrafast analog all-optical signal processors based on fiber- grating devices,” IEEE Photon. J., vol. 2, no. 3, pp. 359–386, Jun. 2010. A loss-gain equalized reconfigurable phaser has been pro- [4] S. Abielmona, S. Gupta, and C. Caloz, “Compressive receiver using a posed, analyzed and demonstrated. Experimental results have CRLH-based dispersive delay line for analog signal processing,” IEEE Trans. Microw. Theory Tech., vol. 57, no. 11, pp. 2617–2626, Nov. 2009. confirmed that such a device provides real-time group delay [5] M. A. G. Laso, T. Lopetegi, M. J. Erro, D. Benito, M. J. Garde, M. A. reconfigurability while exhibiting an all-pass response. It will Muriel, M. Sorolla, and M. Guglielmi, “Real-time spectrum analysis enable radio analog signal processing (R-ASP) systems re- in microstrip technology,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 3, pp. 705–717, Mar. 2003. quiring dynamic adaptability, as for instance dispersion code [6] S. Gupta, S. Abielmona, and C. Caloz, “Microwave analog real-time multiple access (DCMA). spectrum analyzer (RTSA) based on the spectral-spatial decomposition 8

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