Volterra/Wiener Representation of Non-Linear Systems

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1 Berkeley Volterra/Wiener Representation of Non-Linear Systems Prof. Ali M. U.C. Berkeley Copyright c 2014 by Ali M.

2 Linear Input/Output Representation A linear (LTI) system is completely characterized by its impulse response function: y(t) = causality h(t) = 0; t < 0 y(t) has memory since it depends on h(τ)x(t τ)dτ x(t τ); τ [, ]

3 Non-Linear Order-N Convolution Consider a degree-n system: y n (t) =... h n (τ 1, τ 2,..., τ n )x(t τ 1 )...x(t τ n )dτ 1...dτ n If x (t) = αx(t) y n (t) = α n y n (t) Change of variables - α j = t τ j τ j = t α j dα j = dτ j

4 Generalized Convolution Generalization of convolution integral of order n: y n (t) =... h n (τ 1, τ 2,..., τ n )x(t τ 1 )...x(t τ n )dτ 1...dτ n This describes a system that depends not only on the input x(t) and all past values of the input x(t τ), but also on powers of x(t τ) where we take products between past values at different times.

5 Non-Linear Example y j (t) = h j (t τ)x(τ)dτ y(t) = y 1 (t)y 2 (t)y 3 (t) = h 1 (τ 1 )x(t τ 1 )dτ 1 2 dτ 2 3 dτ 3

6 Non-Linear Example (cont) = h 1 (τ 1 )x(t τ 1 )dτ 1 2 dτ 2 3 dτ 3 = h 1 (τ 1 )h 2 (τ 2 )h 3 (τ 3 )x(t τ 1 )x(t τ 2 )x(t τ 3 )dτ 1 dτ 2 dτ 3 h(t 1, t 2, t 3 ) = h 1 (t 1 )h 2 (t 2 )h 3 (t 3 ) h s ( ) = 1 6 {h(t 1, t 2, t 3 ) + h(t 2, t 1, t 3 ) + h(t 2, t 3, t 1 ) +...} Kernel is not in unique. We can define a unique symmetric kernel as above.

7 Symmetry Kernel h can be expressed as a symmetric function of its arguments: Consider output of a system where we permute any number of indices of h: = h(τ 2, τ 1 )x(t τ 2 )x(t τ 1 )dτ 2 dτ 1 h(τ 1, τ 2 )x(t τ 1 )x(t τ 2 )dτ 1 dτ 2 h(τ 1, τ 2 ) h(τ 2, τ 1 ) For n arguments, n! permutations

8 Symmetric kernel We create a symmetric kernel by h sym (t 1,..., t n ) = 1 ( ) h tπ(1),..., t n! π(n) System output identical to original unsymmetrical kernel Note that since the kernel is not unique, we are free to choose any valid kernel. The symmetric choice is one way to do it and it simplifies some of our calculations down the line.

9 Volterra Series y(t) = N n=1 h n (τ 1,..., τ n )x(t τ 1 )...x(t τ n )dτ 1...dτ n Volterra Series: Polynomial of degree N If h n (t 1,..., t n ) = a n δ(t 1 )δ(t 2 )...δ(t n ), we get an ordinary power series: y(t) = a 1 x(t) + a 2 x(t) a N x(t) N It can be rigorously shown by the Stone-Weierstrass theorem that the above polynomial approximates a non-linear system to any desired precision if N is made sufficiently large.

10 Non-rigorous Proof Say y(t) is a non-linear function of x(t τ) for all τ > 0 (all past input) Fix time t and say that x(t τ) can be characterized by the set {x 1 (t),..., x n (t),...} so that y(t) is some non-linear function: y(t) = f (x 1 (t), x 2 (t),...)

11 Non-Rigorous Proof (cont) Let {ϕ 1 (t), ϕ 2 (t),...} be an orthonormal basis for the space ϕ i (τ)ϕ j (τ)dτ = δ ij Thus inner product x(t τ) = x i (t)ϕ i (τ) i=1 x i (t) = x(t τ)ϕ i (τ)dτ

12 Non-Rigorous Proof (cont) Expand f into a Taylor series : f (x 1 (t), x 2 (t),...) y(t) = a o + a i x i (t) + a i a j x i (t)x j (t) +... = a o + 0 i=1 i=1 i=1 j=1 a i ϕ(τ 1 )x(t τ 1 )dτ 1 + a ij ϕ i (τ 1 )ϕ j (τ 2 )x(t 0 0 i=1 j=1 This is the Volterra/Wiener representation for a non-linear system Sifting Property: x(σ) = δ(t σ)x(t)dt

13 Interconnection of Non-Linear Systems Sum: f n (t 1, t 2,..., t n ) = h n (t 1,..., t n ) + g m (t 1,..., t m )

14 Product Interconnection y(t) = h n (τ 1,..., τ n )x(t τ 1 )...x(t τ n )dτ 1...dτ n g m (τ 1,..., τ m )x(t τ 1 )...x(t τ m )dτ 1...dτ m = h n (τ 1,..., τ n )g m (τ n+1,..., τ n+m )x(t τ 1 )...x(t τ n+m )dτ 1...dτ n+m f n+m (t 1,..., t n+m ) = h n (t 1,..., t n )g m (t n+1,..., t n+m )

15 Volterra Series Laplace Domain Transform domain input/output representation Linear systems in time domain F (s) = L[f (t)] = f (t)e st dt Define Generalized Laplace Transform: 0 F (s 1,..., s n ) = L[f (t 1,..., t n )] = f (t 1,..., t n )e s 1t1 e sntn dt 1 dt n 0

16 Volterra Series Example Generalized transform of a function of two variables: f (t 1, t 2 ) = t 1 t 1 e t 2 t 1, t 2 0 F (s 1, s 2 ) = 0 0 F (s 1, s 2 ) = 1 s 1 2 t 1 e s 1t 1 e s 2t 2 dt 1 dt 2 0 e s 2t 2 dt t 1 e t 2 e s 1t 1 e s 2t 2 dt 1 dt 2 e t 2 e s 2t 2 dt 2 = 1 s 1 2 s 2 (s 2 + 1)

17 Properties of Transform Property 1: L is linear Property 2: f (t 1,..., t n ) = h(t 1,..., t k )g(t k+1,..., t n ) F (s 1,..., s n ) = H(s 1,..., s k )G(s k+1,..., s n ) Property 3: Convolution form #1 f (t 1,..., t n ) = h(τ)g(t 1 τ,..., t n τ)dτ 0 F (s 1,..., s n ) = H(s s n )G(s 1,..., s n )

18 Properties of Generalized Transform Property 4: Convolution Form #2: f (t 1,..., t n ) = h(t 1 τ 1,..., t n τ n )g(τ 1,..., τ n ) dτ 1...dτ n 0 F (s 1,..., s n ) = H(s 1,..., s n )G(s 1,..., s n ) Property 5: Time delay τ j > 0 L[f (t 1 τ 1,..., t n τ n )] = F (s 1,..., s n )e s 1τ 1...s nτ n

19 Cascades of Systems Cascade #1: F n (s 1,..., s n ) = H n (s 1, s n )G 1 (s s n ) Cascade #2: F n (s 1,..., s n ) = H 1 (s 1 ) H 1 (s n )G n (s 1,..., s n )

20 Cascade Example F (s 1, s 2 ) = H 1 (s 1 )H 2 (s 2 )H 3 (s 1 + s 2 ) By property #1 and property #2 Note that H is not symmetric

21 Impulse Response of Non-Linear Volterra System Suppose we apply two impulses into an n th order system y(t) = h n (τ 1, τ 2, )x(t τ 1 )x(t τ 2 ) dτ 1 dτ 2 Using the sifting property of the delta functions, we have y(t) = h n (t, t, )

22 Two Impulse Response of Non-Linear Volterra System Suppose we apply two impulses into a second order system x(t) = δ(t) + δ(t T ) Taking the product of x(t 1 ) x(t 2 ) gives four products δ(t 1 )δ(t 2 )+δ(t 1 T )δ(t 2 T )+δ(t 1 T )δ(t 2 )+δ(t 1 )δ(t 2 T ) After integration, we have an interesting result h 2 (τ 1, τ 2 )x(t τ 2 )x(t τ 1 )dτ 1 dτ 2 = h 2 (t, t) + h 2 (t T, t) + h 2 (t, t T ) + h 2 (t T, t T ) With two delta inputs, and by varying their relative delay, we can find the second order non-linearity of a system

23 Exponential Response of n-th Order System y n (t) = h n (τ 1,..., τ n )x(t τ 1 ) x(t τ n )dτ 1...dτ n x(t) = N α i e λ i t i=1 y n (t) = h n (τ 1,..., τ n ) j=1 n [α 1 e λ 1(t τ j ) + + α p e λp(t τ j ) ] dτ 1 dτ n N N α k1 e λ k 1 (t τ 1 )... α kn e λ kn (t τn) k 1 =1 k n=1

24 Exponential Response (cont) N N n... k 1 =1 k n=1 j=1 α kj exp{λ k1 (t τ 1 ) λ kn (t τ n )} n exp{ λ kj (t τ j )} j=1 N N n y n(t) =... α kj k 1 =1 k n=1 j=1 j=1 n n exp{ λ kj t} h n(τ 1,...τ n) exp{ λ kj τ j }dτ 1...dτ n j=1 H n (λ k1,..., λ kn )

25 Collecting Terms y n (t) = N... k 1 =1 N n k n=1 j=1 α kj exp{ n λ kl t}h n (λ k1,..., λ kn ) l=1 We ve seen this before... A particular frequency mix m 1 λ 1 + m 2 λ m p λ p has response α 1 m 1...α p m p G m1,...,m p (λ 1,..., λ n )e (m 1λ m pλ p)t where the function G takes into account how many times a particular mix occurs See the lecture module on multi-tone input into a memoryless non-linear power series for a review.

26 Frequency Mix Vector k y n (t) = k m α 1 m 1...α p p G ( λ) exp{ m x } k Sum over all vectors k such that 0 k i n and j k j = N If H n (s 1,..., s n ) is symmetric, then we can group the terms as before G ( λ) = (n; k )H n,sym (λ 1,..., λ 1,..., λ k }{{} p,..., λ p ) }{{} k 1 k p

27 Important special case N = n G k (λ 1,..., λ n ) = n!h n,sym (λ 1,..., λ n ) To derive H n,sym (λ 1,..., λ n ), we can apply n exponentials to a degree n system and the symmetric transfer function is given by 1 n! times the coefficient of e λ 1t+...+λ nt We call this the Growing Exponential Method

28 Example 3 Excite system with two-tones: ( ) v(t) = H 1 (λ 1 )e λ1t + H 1 (λ 2 )e λ 2t (H ) 2 (λ 1 )e λ1t + H 2 (λ 2 )e λ 2t = H 1 (λ 1 )H 2 (λ 1 )e 2λ 1t + H 1 (λ 2 )H 2 (λ 2 )e 2λ 2t +H 1 (λ 1 )H 2 (λ 2 )e (λ 1+λ 2 )t + H 1 (λ 2 )H 2 (λ 1 )e (λ 1+λ 2 )t

29 Example 3 (cont) y(t) = H 1 (λ 1 )H 2 (λ 1 )H 3 (2λ 1 )e 2λ 1t + H 1 (λ 2 )H 2 (λ 2 )H 3 (2λ 2 )e 2λ 2t +[H 1 (λ 1 )H 2 (λ 2 ) + H 1 (λ 2 )H 2 (λ 1 )] H 3 (λ 1 + λ 2 )e (λ 1+λ 2 )t 2! H sym (s 1, s 2 ) H sym (s 1, s 2 ) = 1 2 [H 1(s 1 )H 2 (s 2 ) + H 1 (s 2 )H 2 (s 1 )]H 3 (s 1 + s 2 )

30 Example 4 Non-linear system in parallel with linear system: y 1 = h 1 (τ 1 )x(t τ 1 )dτ 1 y 2 = h 2 (τ 2, τ 3 )x(t τ 2 )x(t τ 3 )dτ 2 dτ 3 y 1 y 2 = h 1 (τ 1 )h 2 (τ 2, τ 3 )x(t τ 1 ) x(t τ 3 )dτ 1...dτ 3

31 Example 4 (cont) H sym (s 1, s 2, s 3 ) = 1 3 {H 1(s 1 )H 2 (s 2, s 3 ) + H 1 (s 2 )H 2 (s 1, s 3 ) +H 1 (s 3 )H 2 (s 1, s 2 )} assuming H 2 is symmetric Notation: H sym (s 1, s 2, s 3 ) = H 1 (s 1 )H 2 (s 2, s 3 )

32 Example 4 Again Redo example with growing exponential method Overall system is third order, so apply sum of 3 exponentials to system e λ 1t + e λ 2t + e λ 3t

33 Example 4 Again (cont) We can drop terms that we don t care about We only care about the final term e λ 1t + e λ 2t + e λ 3t, so for now ignore terms except e (λ j +λ k )t where j k Focus on terms in y 2 first: 2e (λ 1+λ 2 )t H 2s (λ 1, λ 2 ) 2e (λ 1+λ 2 )t H 2s (λ 1, λ 3 ) 2e (λ 2+λ 3 )t H 2s (λ 2, λ 3 ) H 2s (λ 1, λ 2 ) = H 2s (λ 2, λ 1 )

34 Example 4 again (cont) Now the product of y 1 (t) and y 2 (t) produces terms like e (λ 1+λ 2 +λ 3 )t 2H 2s (λ 1, λ 2 )H 1 (λ 3 )e (λ 1+λ 2 +λ 3 )t +2H 2s (λ 1, λ 3 )H 1 (λ 2 )e (λ 1+λ 2 +λ 3 )t +2H 2s (λ 2, λ 3 )H 1 (λ 1 )e (λ 1+λ 2 +λ 3 )t = 3! H 3s (λ 1, λ 2, λ 3 ) H 3s (s 1, s 2, s 3 ) = 2 3! ( )

35 Singe-Tone Sinusoidal Response Let s warm up by calculating the n th order response to a sinusoidal input y(t) = n h sym(σ1,,σ n) j=1 e jω(t σ j ) + e jω(t σ j ) dσ 1 dσ n 2 Let s group the exponentials by using the following notation: λ 1 = jω and λ 2 = jω. Expanding the product as sums n = 2 n exp(λ kj (t σ j )) k 1 =1 k n=1 j=1

36 Sinusoidal Response (invoke Laplace) We can now simplify the expression by noting that the n h sym(σ 1,, σ n) exp( λ kj σ j )dσ 1 dσ n = H(λ k1,, λ kn ) j=1 = n k 1 =1 2 n exp( λ kj t)h(λ k1,, λ kn ) k n=1 A particular term has a frequency given by kλ 1 + (n k)λ 2 = (2k n)ω = 1 n ( ) n 2 n k e j(2k n)ωt H(ω,, ω, ω,, ω) }{{}}{{} k=0 k n k j=1 = 1 n 2 n G k,n k (jω, jω)e j(2k n)ωt k=0 Just as expected, an nth order system generates the n th harmonic and every other harmonic down to either DC (n is even) or fundamental (n is odd).

37 Grouping Terms We can group positive and negative frequency terms together by noting that ( ) n G k,n k (jω, jω) = H k sym (ω,, ω, ω,, ω) G n k,k ( jω, jω) = ( n n k ( n k ) = } {{ } k } {{ } n k ) H sym ( ω,, ω, ω,, ω) }{{}}{{} n k k ( n n k By the symmetry of the kernel and the binomial coefficient, we have G n k,k ( jω, jω) = G k,n k (jω, jω) )

38 Sinusoidal Response Summary We can now write the total sinusoidal response as y n(t) = 1 2 n 1 G n,0 (jω, jω) cos(nωt + G n,0 (jω, jω))+ 1 2 n 1 G n 1,1 (jω, jω) cos((n 2)ωt + G n 1,1 (jω, jω)) + + { 1 Gn/2,n/2 2 n 1 (jω, jω) n even 1 G 2 n 1 n+1 2, n+1 (jω, jω) cos(ωt + G n+1 2 2, n+1 (jω, jω)) n odd 2 We need to form the above some for each power in the Volterra series.

39 Sinusoidal Multi-Tone Response, n th power only Calculation is very similar to exponential response, but now we just need to keep track of complex conjugate frequency. Using our shorthand notation ω k = ω k and A 0 0 y(t) = 1 2 n x(t) = N k 1 = N N A k cos(ω k t) = 1 2 k=0 h sym (σ q,, σ n ) ( n N i=1 A k1 e jω k 1 (t σ 1 ) k= N N k 2 = N N k= N A k e jω kt A k 2 ejω k(t σ i ) ) dσ 1 dσ n A k2 e jω k2(t σ 2 )

40 Product of Sums as Sum of Products Expanding the product of sums, we can sum over all possible vectors k = k G k e j k ωt e j k σt The term G k is used to collect all frequency products that sum to the same frequency, determined by the vector k: G k = 1 2 n (n; k)a k 1 1 Ak 2 2 Akn n Where we have already met the multi-nomial coefficient (n; k) to account for the number of times a particular frequency product occurs.

41 Sinusoidal Response (cont) Performing the integration, we observe that h sym (σ 1,, σ n )e j(ω k 1 σ 1 + +ω kn σ n) dσ 1 dσ n = H(ω k1,, ω kn ) Keep in mind the symmetry of the multi-nomial coefficient and the fact that every positive frequency term is accompanied by a negative counterpart obtained by inverting the k vector. This allows us to write the final sinusoidal sum as. (n; k) A k 1 1 Akn n 2 n 1 H(ω k1,, ω kn ) cos( k ωt+ H(ω k1,, ω kn )) k

42 Capacitive non-linearity Non-linear capacitors: BJT: C µ and C cs FET: C db and C sb Small signal (incremental) capacitance (n 2 3) C j = dq dv j = K (Φ + V j ) 1 n Let V j = V Q + v C j = K (Φ + V j ) 1 n (1 + v Φ+V Q ) 1 n C µo + C µ1 v + C µ2 v

43 Cap Non-Linearity (cont) i = dq dt = dq dv dv dt = C j(v) dv dt Model: i = C µ0 dv dt + C µ 1 v dv dv = C µ0 dt + Cµ 1 2 dv dt dt + C µ 2 v 2 dv dt C µ2 3 dv dt

44 Overall Model i = dq dt = dq dv dv dt = C j(v) dv dt C j (v) = C µo + C µ1 v + C µ2 v dv i = C µo dt C dv 2 µ dt 3 C dv 3 µ dt

45 Cap Model Decomposition Let v = H 1 (s 1 ) x v 2 = H 1 (s 1 ) H 1 (s 2 ) x 2 i 2 = (s 1 + s 2 ) H 1 (s 1 ) H 1 (s 2 ) 1 2 C µ 1 x 2 i n = (s s n ) H 1 (s 1 )...H 1 (s n ) 1 n C µ,n 1x n

46 Volterra Operator Notation The operation will be used as a shorthand notation H(jω 1, jω 2, ) v k The above equation implies that the coefficient H must be evaluated at the distortion product(s) of the signal v k. Thus, the generalized power series is written in this form v 0 = H 1 (jω) v i +H 2 (jω 1, jω 2 ) v 2 i + +H k (jω 1, jω 2, ) v k

47 A Real Circuit Example Find distortion in v o for sinusoidal steady state response v o = B 1 (jω 1 ) v i + B 2 (jω 1, jω 2 ) v 2 i +... Need to also find v 1 = A 1 (jω 1 ) v i + A 2 (jω 1, jω 2 ) v 2 i +...

48 Circuit Example (cont) Setup non-linearities Diode: Capacitor: i d = I S e (vo+v Q)/V T I Q = I S e V Q/V T e vo/v T I Q = I Q ( e v o/v T 1 ) = g 1 v o + g 2 v o C x = dq dv 1 = C o + C 1 v 1 + C 2 v dv 1 i cx = C o dt + C 1 dv C 2 dv dt 3 dt

49 Second-Order Terms 0 = A 2 R 1 + j (ω a + ω b ) C (A 2 B 2 ) + j (ω a + ω b ) C o A 2 + j (ω a + ω b ) C 1 2 A 1 (jω o ) A 1 (jω b ) j (ω a + ω b ) C (A 2 B 2 ) + g 1 B 2 (jω a, jω b ) + g 2 B 1 (jω a ) B 1 (jω b ) = 0 Solve for A and B

50 Third-Order Terms A 3 R 1 + j (ω a + ω b + ω c ) C (A 3 B 3 ) + j (ω a + ω b + ω c ) C o A 3 + j (ω a + ω b + ω c ) C 1 2 2A 1 (jω a ) A 2 (jω a, jω b )+ j (ω a + ω b + ω c ) C 2 3 A 1 (jω a ) A 1 (jω b ) A 1 (jω c ) = 0 j (ω a + ω b + ω c ) C (A 3 B 3 ) + g 1 B 3 + g 2 2B 1 B 2 + g 3 B 1 B 1 B 1 = 0 Solve for A 3 and B 3

51 Distortion Calc at High Freq s o = H 1 (jω a ) s i + H 2 (jω a, jω b ) s i Compute IM 3 at ω 2 ω 1 only generated by n = 3 k I M3 = ( ) H 3 is symmetric so we can group all terms producing this frequency mix by H 3 (3; k I M3 ) = 3! 2! 4 = 3 4 I M 3 = 3 s 1 s 2 2 H 3 (jω 2, jω 2, jω 1 ) 4 H 1 (jω 1 ) s H 3 (jω 2, jω 2, jω 1 ) s 1 s 2 2 For equal amp o/p signal, we adjust each input amp so that: s o = H 1 (jω 1 ) s 1 = H 1 (jω 2 ) s 2

52 Disto Calc at High Freq (2) At low frequency: I M 3 = 3 H 3 (jω 2, jω 2, jω 1 ) 4 H 1 (jω 1 ) H 1 (jω 2 ) 2 s o 2 a 3 I M 3 = 3 4 a 3 s o 2 1 Conclude that at high frequency all third order distortion (fractional) (signal level) 2 for small distortion. All second order (signal level)

53 Disto Calc at High Freq (3) Similarly HD 3 = s H 3 (jω 1, jω 1, jω 1 ) s o s o = H 1 (jω 1 ) s 1 Low Freq. HD 3 = 1 H 3 (jω 1, jω 1, jω 1 ) 2 4 H 1 (jω 1 ) 3 s o a 3 HD 3 = 1 4 a 3 s o 2 1 No fixed relation between HD 3 and IM 3

54 High Freq Distortion & Feedback Let s o = A 1 (jω a ) s 2 + A 2 (jω a, jω b ) s ε s fb = β (jω a ) s o Look for s ε = s i s fb s o = B 1 (jω a ) s i + B 2 (jω a, jω b ) s i B 1 (jω a) s i +B 2 s i = A 1 ( si β (jω a) ( B 1 (jω a) s i + B 2 s i ) +... ) +A 2 ()

55 High Freq Disto & FB (2) First order: B 1 (jω a ) = Second order: A 1 (jω a) 1+A 1 (jω a)β(jω a) B 2 = A 1 (jω a + jω b ) β (jω a + jω b ) B 2 (jω a, jω b ) + A 2 (jω a, jω b ) B 1 (jω a ) B 1 (jω b ) B 2 (jω a, jω b ) = B 2 (jω a, jω b ) = A 2 (jω a, jω b ) B 1 (jω a ) B 1 (jω b ) (1 + A 1 (jω a + jω b ) β (jω a + jω b ) B 2 (jω a, jω b )) A 2 (jω a, jω b ) [1 + A 1 (jω a + jω b ) β (jω a + jω b )] [1 + A 1 (jω a ) β (jω a )] [1 + A 1 (jω b ) β (jω b )]

56 Comments about HF/LF Disto Feedback reduces distortion at low frequency and high frequency 1 1+T for a fixed output signal level 1 True at high frequency if we use 1+T (jω) where ω is evaluated at the frequency of the distortion product While IM/HD no longer related, CM, TB, P 1dB, P BL are related since frequencies close together Most circuits (90%) can be analyzed with a power series

57 References Nonlinear System Theory: The Volterra/Wiener Approach, Wilson J. Rugh. Baltimore : Johns Hopkins University Press, c1981. UCB EECS 242 Class Notes, Robert G. Meyer, Spring 1995

58 More References Piet Wambacq and Willy M.C. Sansen, Distortion Analysis of Analog Integrated Circuits (The International Series in Engineering and Computer Science) (Hardcover) M. Schetzen, The Volterra and Wiener theories of nonlinear systems, New York: Wiley, L. O. Chua and N. C.-Y., Frequency-domain analysis of nonlinear systems: formulation of transfer functions, IEE Journal on Electronic Circuits and Systems, vol. 3, pp , J. J. Bussgang, L. Ehrman, and J. W. Graham, Analysis of Nonlinear Systems with Multiple Inputs, Proceedings of the IEEE, vol. 62, pp , J. Engberg and T. Larsen, Noise Theory of Linear and Nonlinear Circuits, New Yory: John Wiley and Sons, 1995.