Volterra Series: Introduction & Application

ECEN 665 (ESS : RF Communication Circuits and Systems Volterra Series: Introduction & Application Prepared by: Heng Zhang Part of the material here provided is based on Dr. Chunyu Xin s dissertation

Outline Introduction: History, Basics Volterra Series in frequency domain Applications of Volterra Series: Example: Common Source Amplifier Example: Differential Pair Example3: Current Mirror Example4: Gilbert Mixer

Volterra Series: History In 887, Vito Volterra : Volterra Series as a model for nonlinear behavior In 94, Norbert Wiener: applied Volterra Series to nonlinear circuit analysis In 957, J. F. Barrett: systematically applied Volterra series to nonlinear system; Later, D.A. George: used the multidimensional Laplace transformation to study Volterra operators Nowadays: extensively used to calculate small, but nevertheless troublesome, distortion terms in transistor amplifiers and systems. 3

Why do we need Volterra Series? At high enough frequency, the assumption there s no memory effect due to capacitors and inductors NOT CORRECT! Taylor series analysis: NO memory effect, cannot calculate distortion at high frequency Low frequency analysis: high-frequency effect can degrade distortion performance by 00% more than predicted. eg. Fully differential circuits without mismatch: Low-frequency analysis predicts the HD to be zero Volterra series reveals an HD as high as -3dB (see the numerical example later 4

When Volterra Series Are Good? Can calculate the high-frequency-lowdistortion terms for weakly non-linear timeinvariant (NLTI system with memory effect Weakly nonlinear assumption: Input excitation is small use polynomials to model nonlinearities When small inputs, the st term dominates 5

When Volterra Series Are Bad? Results are a sum of infinite numbers of terms, possible to diverge Define: weak enough system (G>>G3>>G5; G, G3, G5 = st order, 3 rd order, and 5 th order Volterra kernels; G5 is small to be negligible, the infinite sums will converge and converge rapidly Strongly nonlinear system: sum will diverge, Volterra series becomes invalid Volterra series are impractical in strongly nonlinear problems 6

Volterra Series: basic Linear system without memory : yt ( = h xt ( Output y at instant t only depends on input x at that instant only. h: linear gain Linear, discrete, causal and time-invariant system with memory (described by summing all the effects of past inputs with proper weights : yn ( = h( τ xn ( τ i= 0 n n: time index h(τ: impulse response continuous time domain (the convolution sum becomes a convolution integral: t y( t = h( τ x( t τ dτ 0 i i 7

Volterra Series: basic System with nd order nonlinearity: Memory-less system: y t h x t ( = ( System with memory: (firstly: discrete and assume all the terms sum with equal weights yn ( = xn ( xn ( + xn ( xn ( +... + xn ( x(0 + xn ( xn ( + xn ( xn ( +... + xn ( x(0 n n n n +... + x(0 x(0 = x( n x( j + xn ( x( j +... = xi ( x( j i= 0 i= 0 j= 0 i= 0 Next: Add proper weights to make it a weighed double sum: n n j= 0 i= 0 ( yn ( = h p, p xn ( p xn ( p i j i j h : a function of time index p i, p j ( nd order impulse response Continuous time domain (the convolution sum becomes a convolution integral : t y ( t = h ( τ, τ x( t τ x( t τ dτ dτ 0 8

Volterra Series: basic n th -order nonlinear system : Memory-less system: System with memory: yt ( = h( τ xt ( τ dτ t 0 0 t 0 y t h x t h x t h x t n ( = ( + ( +... + n ( + h ( τ, τ x( t τ x( t τ dτ dτ t + h ( τ, τ, τ x( t τ x( t τ x( t τ dτ dτ dτ 3 3 3 3 t + L+... h ( τ, τ... τ x( t τ x( t τ... x( t τ dτ dτ... dτ h ( τ, L, τ n n 0 n n n n Volterra Series expansion: an infinite sum of multidimensional convolution integrals :n th order Volterra Kernels (n th order impulse response of the system 9

Volterra Series vs. Taylor Series Nonlinear memory-less system represented using Taylor series: n yt ( = K xt ( + K x( t + K x( t +... + K x( t +... 3 3 Nonlinear system with memory represented using Volterra series: n in which H [ xt ( ] =... h( τ, τ... τ xt ( τ xt ( τ... xt ( τ dτ dτ... dτ n n n n n : n th order Volterra operator. h ( τ, L, τ : n th order Volterra Kernels n n 0

Volterra kernel h ( τ, L, τ n h ( τ, L, τ = 0 n n n : n th order Volterra Kernels : for any τ j < 0, j =,,,n h ( τ, L, τ n n is not necessarily symmetrical to its variables Always possible to construct a symmetrical kernel from the asymmetrical kernel h ( τ, L, τ = h ( τ, L, τ ( s ( a n n n n n! p Summation runs through all possible permutations

Volterra series in frequency domain Why study the nonlinear system in frequency domain? Frequency domain Volterra kernels are needed to calculate the distortion. eg. HD, HD 3, IM 3, Fourier transform: time domain Volterra frequency domain Volterra eg. the n-dimensional Fourier transform for an n th order Volterra kernel h ( τ, L, τ is: n H n ( ω, L, ω = F{ h ( τ, L, τ } n n n n jωτ jωτ n n L h ( τ, L, τ e Le dτ... dτ = n n n H n is the frequency domain Volterra kernel.

Volterra series in frequency domain An input with m frequency components: X = A(cosω t+ cos ω t+... + cos ω t The output of the n th order nonlinear system can be denoted as: m Y = H ( jω ox + H ( jω, jω ox +... + H ( jω, jω,... jω ox p p p n p p pn n ω, ω,... ω where can be chosen from p p pn They can be equal or different, and have both the + and combination For each term, the frequency components in H n are the same as in X n The operator 〇 means: Multiply each frequency component in X n by: Shift phase by: (analogy to filtering operation 3

Volterra series in frequency domain X = A(cos wt+ cos w t eg, input with two frequency components: + w, + w wp, wp,... wpn can be chosen from: H ( jw, jw o X p p represents the following terms (eliminates all the overlap terms: H ( jw, jw X H ( jw, jw = H ( jw, jw A cos( wt + H( j w H ( jw, jw X H ( jw, jw = H( jw, jw A ( H ( jw, jw X H ( jw, jw = H ( jw, jw A cos( w + w t+ H ( j( w + w ( H ( jw, jw X H ( jw, jw = H ( jw, jw A cos( w w t+ H ( j( w w H( jw, jw X H( jw, jw = H( jw, jw A cos( wt+ H( j w 4

Definition of HD, HD 3 and IM 3 Volterra Series Taylor Series HD (, ( jw H jw jw A H a A a HD 3 H,, 3( jω jω jω A 4 H ( jω 4 a3 A a IM 3 3 H,, 3( jω jω jω 4 H ( jω A 3 4 a3 A Observation:. Volterra series incorporates the frequency dependent effects a H ( jw, jw, jw H ( jw, jw, jw 3HD. 3 3 3 3 IM 5

Example : Common Source Amplifier vo vi M RL CL MOS transistor: nonlinearity mainly introduced by transconductance; output impedance also contributes to distortion. At high frequency, load impedance dominated by C L memory effect cannot be neglected, need to use Volterra series to analyze nonlinearity. 6

Procedure:. Express output voltage Vout using Volterra series: (. Large signal transfer function using long channel device model: ( (3 where 3. Substitute ( into (3:, Z L : impedance of R and C in parallel (4 where 7

Procedure: 4. To find the Volterra kernel H k, equate same order terms of v i in both sides of (4 and use the following relationships: (5 (6 (7 (8 D0 It can be found that: H0 = λd (9 H H 3 3 where: ( w, w, w + λh H ( w = D ( w, w λ 0 λd0 0 ( D + λ DH + D H 0 = ( DH + DH λd = λd 0 0 (0 ( ( 8

Verification of Volterra Series M=5um/0.6um, K = 30uA/V,g o =00uA/V, gm = 500uA/V, Vod =.9V. Input at 0dBm(0.36V, f = GHz, f =.GHz Volterra Analysis Cadence Simulation HD -33.7dB -38dB HD 3-49dB -58dB IM3-34dB -4.9dB

Simulation Results

Example : Differential Pair Assume the only memory effect is introduced by the parasitic capacitance (Cs at node Vs, all other capacitance are ignored. DC bias current: (a The small signal voltages applied to the gates of M and M are V d / and V d / respectively The goal is to obtain the Volterra series expansion for the small signal drain current id of M (or M in terms of input differential voltage vd up to 3rd order

Procedure:. Determine the Volterra series expansion of vs: (b is the k-th order term of the Volterra series of vs. Apply KCL law at the common source node: (a where Vs, Vgs and Vgs can be written into DC and signal small signal terms: (b

Procedure: 3. Combining equations ( & ( yields: (3 where Substituting (b into (3 and taking phasor form of : (4 3

Procedure: 4. Keeping only the st order terms of (4: (5 5. Substituting (6 into (4, keeping only the nd order terms: (6 (7 Notice that w becomes w +w because we are interested in the nd order terms Because of the operator, (8 consists of four equations for four different cases: (8 Where w and w are symbols and w a, w b are the frequency components of input. 4

Procedure: 6. For the 3 rd order term: (9 Replace jw by j(w+w+w3 since we are interested in the 3rd order terms Notice that G always consists of one frequency component (w; G always consists of two (w, w, and G3 consists of three(w, w, w3. 7. Higher order kernels can be calculated by repeating the above steps. So: 5

Procedure: 8. Our final goal is to obtain the Volterra series expansion for id in terms of vd: Using the MOS device equation containing DC and time varying components: K vd vd K vd K I + i = V + v V = K V V v + v + V V ( ( d d GS s t GS t s s GS t vd K vd id = gm vs + vs Volterra series expansion of id becomes: K K K id+ id + id3+... = gm vd vs... + vd + vs +... vd vs +... 8 ( ( ( 6

Procedure: 9. Keeping the st order term of v d coefficients in (: 0. Isolating the nd order terms of v d coefficients in (:. Separating the 3 rd order terms of v d coefficients from (: 7

Summary of steps to determine Volterra Series: Step. Determine the Volterra series expansion of the intermediate variable vs in terms of input signal v d ( Step. Using KCL and MOS device equation to express output signal i d in terms of v d and v s ( Step3. Substitute vs determined in( into ( and derive: Hn becomes a function of Gn, and Gn has been determined in step. 8

Volterra series versus Taylor series of a MOS differential pair Note that for Taylor expansion, second order distortion is 0, while Volterra series reveals the high frequency second order distortion for a differential pair. 9

Verification of Volterra Series For MOS transistors with W/L=50um/0.6um, Volterra series and simulated HD3 result comparison. -5-30 HD3 (db -35-40 Simulation Volterra -45-50 0 3 4 5 Frequency (GHz d V H HD3 = 4H 3 30

Example 3: Current Mirror At high frequency, the parasitic capacitance of the transistors cannot be neglected. Therefore, Volterra Series is used to model this non-linear system with memory. The parasitic capacitance Cp seen at the gate of M, M affects the Vgs, which is related to the current mirror accuracy. The goal is to obtain the Volterra series expansion for the small signal drain current io of M in terms of input current Iin up to 3rd order. 3

Procedure: Follow the same steps as described in Example :. Determine the Volterra series of the gate voltage vi first: vi = G 〇 I in + G 〇 I in + G 3 〇 I 3in + = vi + vi + vi 3 + ( where vi k = G k 〇 I kin is the k-th order term of the Volterra series of vi. Appling KCL at the gate node of M: dv dt K i C p + ( Vi Vt = Iin + I BIAS ( where V i can be written into DC and small signal term: V i = V I + v i (3 Also, K ( VI V t = I BIAS (4 3

Procedure: 3. Substituting (3 and (4 into ( and simplifying it: dv dt K i C p + gmvi + vi = I in (5 where gm = K(VI Vt is the transconductance of M Substituting (5 into ( and taking the phasor form of dv i dt ( K jwc p + gm ( vi + vi + vi3 +... + 3... = ( vi + vi + vi + Iin (6 33

Procedure: 4. The Volterra kernel Gk can be obtained by equating the same order term of I in at both sides of (6. Keeping only the first order terms: ( ( w G g jwc m p + 〇 I in = I in (7 jwc p g m w G + = ( (8 5. Substituting into (6, and keeping only the second order terms:, ( ] ( [ w w G g C w w j m p + + ( G w K 〇 I in + 〇 I in = 0 (9 ]( ( [ ( ( ( (, ( 3 m p m p m m p m p m p g jw C g C w w j g K g C w w j g jwc K g C w w j w G K w w G + + + + = + + = (0 34

Procedure: 6. Factoring out the third order terms from (6: [ j ( w + w + w 3 C p + g m ] G 3 ( w, w, w 3 〇 I 3in + K ( w, w G ( w KG( w, w G( w G3( w, w, w3 = jw ( + w+ w C + g 3 G 〇 I 3in = 0 K C jwc C [ ( + ]( [ ( + + ] g g g g p p p jw 4 w jw w w3 m m m m 7. Now, vi has been expanded into Volterra series up to the third order. Our final goal is to obtain the Volterra series expansion for i o in terms of I in : i o = H 〇 I in + H 〇 I in + H 3 〇 I 3in + = i o + i o + i o3 + ( p m ( 35

Procedure: 8. The small signal output current io can be related to vi as: i o = g m v i (3 Substituting ( and ( into (3: io + io + io3 = g m G (w o I in + g m G(w,w o I in + g m G3(w,w,w3 o I 3in g C m p H( w = g mg ( w ( jw g g H ( w, w = g m G ( w m, w m Kg 3 g m m [ j( w + w C g p m ]( jw C g m p H ( w, w, w = g G ( w, w, w 3 3 m 3 3 C jwc C [ ( + ]( [ ( + + ] K gm p p p jw 4 w jw w w3 g m gm gm gm 36

Verification of Volterra Series Assume a current gain of 4, use long channel length to minimize channel length modulation effect. Choose M = um/um, M = 4*um/um, I BIAS = 84.4uA, Cp~90fF, gm = 76μA/V, gm = 705μA/V, I in = A cosw t + A cosw t, while w = 90 * pi MHz, w = 900 * pi MHz, A = A = I BIAS /0 = 8.44μA, we can get: IM 3A H ( jw, jw, jw = = 4 H ( jw 3 3 6 db IIP3 = P in + IM 3 / = -9.5dBm + 6/ = - 60.5dBm -db compression point = IIP3 0dB = -70.5dBm 37

Simulation Results 38

Comparison between Volterra analytical results and Cadence simulation Volterra analysis Cadence simulation IIP3-60.5dBm -53.7dBm IM3 6dB 76dB -db point -70.5dBm -74.7dBm 39

Example 4:Gilbert Mixer The high frequency linearity of Gilbert mixer is dominated by the V-I conversion. Transistors M3-M6 works as large signal current switches and are not limiting factor for linearity. Assume V LO is not switching ans so the mixer is a NLTI system 40

Volterra kernels: The Volterra series analysis of Gilbert mixer follows the Volterra series analysis of a differential pair The Volterra kernels for the V-I conversion transistors are: 4

Finding IM3 and HD3 the linearity expression for Gilbert mixer cell is: HD 3 H ( jω, jω, jω = 4 ( A 3 rf H jω IM 3 = 3 H ( jω, jω, jω 4 ( A 3 rf H jω IM3 is caused by adjacent channel interference w is defined to be at w - w such that the 3 rd order nonlinearity term w +w will generate a term at w - w A rf should be replaced by A interference 4

Low frequency approximation w = w = w3 = 0 K = 6( V V GS t K = V GS ( V t HD 3 H ( ω, ω, ω Arf K = A = 4 H ( j 3 I 3 rf ω SS For low frequency (no memory, HD 3 agrees with Taylor series expansion Do the same for IM3 and it would agree with Taylor series expansion 43

IM3 and HD3 For intermediate frequency: j(w +w Cs << K(V GS -V t ( ( ( K Arf j w Cs HD3= I 3 V V K V V SS GS t GS t 3 A int erference jw ( Cs IM 3= 3( V 3 K( VGS Vt GS V t IM 3HD 3 3 44

When V LO is switching The mixer is no longer a time-invariant system The high-frequency distortion still dominated by V-I conversion To incorporate switching effect, multiply the output by the Fourier series representation of a square wave Frequency of the st and 3 rd order products is shifted by f LO, Amplitude are reduced by /pi. HD3 and IM3 remain unchanged 45

Gilbert Mixer IM 3 Verification Assume a Gilbert Mixer has V-I conversion transistor M=M=50um/0.6um. Let f rf =GHz, two interference signal with 0dBm power (0.36V at GHz(w and.00ghz(w. Cs ~= 0.pF. Make V GS -V t = 0.387V, K=650 ua/v, thus gm =.4mA/V Theoretically: 3 H ( ω, ω, ω IM 3 = A = 4.0dB 4 ( 3 rf H jω 46

Simulation result Gilbert cell simulation result: with two tones at GHz and.00ghz, the input amplitude of each tone is 0dBm. IM3 is around -5.5 db from simulation. The deviation error is.5db(5.9% 47

HD Incorporating Memory: HD A rf = = H H 3dB Ignoring memory: HD Aa rf = = a 0 Volterra series analysis predicts that RF feedthrough will still be Present in a balance mixer with zero mismatch! 48

Conclusions Taylor series analysis cannot calculate the distortion correctly at high frequency due to memory effect. Volterra series can calculate the high-frequency-lowdistortion terms for any weakly non-linear time-invariant system with memory effect. Volterra series may diverge when nonlinearity is strong Volterra series are applied to four basic circuit examples and theoretical results agrees with simulation 49

Reference W. Sansen, Distortion in elementary transistor circuits, IEEE Trans. On Cir-cuits and Systems-II: Analog and Digital Signal Processing, vol. 46, pp. 35~35, March 999. B. Leung, VLSI for Wireless Communication. Upper Saddle River, N.J.: Prentice Hall, 00. B. Kim, J.-S. Ko, and K. Lee, Highly linear CMOS RF MMIC amplifier using multiple gated transistors and its volterra series analysis, IEEE MTT-S Int.Microwave Symp. Dig., vol., pp. 55~58, May 00. 50