Weakly Nonlinear Time-Varying Systems

Abstract

The theory of linear time-varying systems can be extended to weakly-nonlinear time-varying (WNTV) systems in a similar way as we did for linear time-invariant systems. In this chapter we first define WNTV systems mathematically and highlight some important differences from the theory of WNTI systems. We then discuss weakly-nonlinear periodically time-varying (WNPTV) systems. These type of systems generate a characteristic spectrum that is relatively easy to describe and is relevant, for example, in the study and design of communication systems.

The theory of linear time-varying systems can be extended to weakly-nonlinear time-varying (WNTV) systems in a similar way as we did for linear time-invariant systems. In this chapter we first define WNTV systems mathematically and highlight some important differences from the theory of WNTI systems. We then discuss weakly-nonlinear periodically time-varying (WNPTV) systems. These type of systems generate a characteristic spectrum that is relatively easy to describe and is relevant, for example, in the study and design of communication systems.

1 Weakly Nonlinear Time-Varying Systems

1.1 Definition

A Weakly-nonlinear time-varying system is defined as a system whose response to the input signal x can be described by

$$\begin{aligned} y(t) = \sum _{k=1}^\infty w_k(t, \tau _1,\dotsc ,\tau _k) \star x^{\otimes k}(\tau _1,\dotsc ,\tau _k)\,. \end{aligned}$$

(13.1)

The operator $\star$ is the extension of the operator introduced in Sect. 12.2.2 to higher dimensions. For causal systems described by regular distributions and driven by a right sided input x it is defined by

$$\begin{aligned} w_k(t, \tau _1,\dotsc ,\tau _k) \star & x^{\otimes k}(\tau _1,\dotsc ,\tau _k) :=\nonumber \\ & \int \limits _0^t\cdots \int \limits _0^t w_k(t, \tau _1,\dotsc ,\tau _k) x(\tau _1)\cdots x(\tau _k) \textrm{d}\tau _1\cdots \textrm{d}\tau _k\,. \end{aligned}$$

(13.2)

$w_k$ is the kth order fundamental kernel of the system. As for WNTI systems, to guarantee uniqueness, we require it to be symmetric in the variables $\tau _1,\dotsc ,\tau _k$.

Generalizations valid for a wider class of input signals can be done in the same way as was done for LTV systems. Note that $w_k\star x^{\otimes k}$ is a distribution of the single variable t and not a higher dimensional distribution as for WNTI systems. The reason for this is explained next.

Consider a WNTV system described by a differential equation of the form

$$\begin{aligned} L(t, D) y = N(t, D) x + c_2(t)y^2 + c_3(t)y^3 + \cdots \end{aligned}$$

with

$$\begin{aligned} L(t, D) &= D^{m} + a_{m-1}(t)D^{m-1} + \dots + a_0(t) \\ N(t, D) &= b_n(t)D^{n} + b_{n-1}(t)D^{n-1} + \dots + b_0(t) \end{aligned}$$

and where all coefficients $a_i, b_i$ and $c_i$ are indefinitely differentiable functions. The equation can be solved iteratively as in the case of WNTI systems. We first solve the linear part of the equation. The solution $y_1$ is then used in the nonlinear terms to compute “nonlinear sources” of second order. With them we solve the part of the equation consisting of terms of second order only, a linear equation, and so on.

There is an important difference compared to the case of WNTI systems: in the case of WNTI systems, to get around the lack of a general multiplication between arbitrary distributions, we made use of a direct product of distributions and introduced a multiplication based on the tensor product. Here the same method doesn’t work as the coefficients of the differential equation are functions of the single time variable t and it is unclear how to adapt them for use with higher order distributions. For this reason here the responses of all orders $y_k$ are distributions of the single variable t. To solve the equation we must therefore assume the existence of all appearing multiplications and powers $y^n, k=2,3,\dotsc$.

If we consider t as a fix parameter then the multiplication between components of y act as a tensor product like operation. Consider the product between $y_k$ and $y_l$

$$\begin{aligned} y_k(t) y_l(t) = \int \limits _0^t \cdots \int \limits _0^t w_k(t, \tau _1, & \dotsc , \tau _k) x^{\otimes k}(\tau _1,\cdots , \tau _k) \textrm{d}\tau _1 \cdots \textrm{d}\tau _k\nonumber \\ \cdot \int \limits _0^t \cdots \int \limits _0^t w_l(t, \tau _1, & \dotsc , \tau _l) x^{\otimes l}(\tau _1,\cdots , \tau _l) \textrm{d}\tau _1 \cdots \textrm{d}\tau _l\nonumber \\ = \int \limits _0^t \cdots \int \limits _0^t w_k(t, \tau _1, & \dotsc , \tau _k) w_l(t, \tau _{k+1}, \dotsc , \tau _{k+l})\nonumber \\ & \cdot x^{\otimes k+l}(\tau _1,\cdots , \tau _{k+l}) \textrm{d}\tau _1 \cdots \textrm{d}\tau _{k+l}\,. \end{aligned}$$

(13.3)

The result has the form of a response of order $k+l$ which can be interpreted as a “nonlinear source” generated by nonlinearities and lower order responses as desired.

To solve the equation we must be able to solve the equation for each order independently and verify that it has the desired form. Solving the equations is (in principle) simple as all equations are linear. The solution of the equation consisting of terms of order k is given by

$$\begin{aligned} y_k(t) = \int \limits _0^t\int \limits _0^\tau \cdots \int \limits _0^\tau v(t, \tau ) z_k(\tau , \tau _1, \dotsc , \tau _k) x^{\otimes k}(\tau _1,\cdots , \tau _k) \textrm{d}\tau _1 \cdots \textrm{d}\tau _k \textrm{d}\tau \end{aligned}$$

with $z_k\star x^{\otimes k}$ the nonlinear source and v the fundamental kernel of the equation.

To show that this expression can be transformed in the desire form, consider the integral

$$\begin{aligned} \int \limits _0^t \int \limits _0^\tau \int \limits _0^\tau f(\tau , \tau _1, \tau _2) \textrm{d}\tau _2 \textrm{d}\tau _1 \textrm{d}\tau \,. \end{aligned}$$

As a first step we exchange the order of integration between $\tau$ and $\tau _1$ and obtain

$$\begin{aligned} \int \limits _0^t \int \limits _{\tau _1}^t \int \limits _0^\tau f(\tau , \tau _1, \tau _2) \textrm{d}\tau _2 \textrm{d}\tau \textrm{d}\tau _1\,. \end{aligned}$$

We then perform a second exchange between $\tau _2$ and $\tau$ (refer to Fig. 13.1) which results in

$$\begin{aligned} \int \limits _0^t \int \limits _0^t \int \limits _{\max (\tau _1, \tau _2)}^t f(\tau , \tau _1, \tau _2) \textrm{d}\tau \textrm{d}\tau _2 \textrm{d}\tau _1\,. \end{aligned}$$

If the integral would involve more integrations between 0 and $\tau$ then we could repeat the last step more times giving

$$\begin{aligned} \int \limits _0^t \cdots \int \limits _0^t \int \limits _{\max (\tau _1, \dotsc , \tau _k)}^t f(\tau , \tau _1, \dotsc , \tau _k) \textrm{d}\tau \textrm{d}\tau _k \cdots \textrm{d}\tau _1\,. \end{aligned}$$

(13.4)

Fig. 13.1

Domain of integration (refer to text)

Using this result we can transform the above expression for $y_k(t)$ into

$$\begin{aligned} y_k(t) = \int \limits _0^t \cdots \int \limits _0^t \int \limits _{\max (\tau _1, \dotsc , \tau _k)}^t v(t, \tau ) z_k(\tau , \tau _1, \dotsc , \tau _k) & \textrm{d}\tau \\ \cdot x^{\otimes k} & (\tau _1,\dotsc , \tau _k) \textrm{d}\tau _1 \cdots \textrm{d}\tau _k \end{aligned}$$

which has the desired form $w_k \star x^{\otimes k}$.

1.2 Time-Varying Nonlinear Impulse Responses

As for LTV systems, the response of WNTV systems can also be expressed in terms of the  time-varying nonlinear impulse responses

$$\begin{aligned} h_k(t,\xi _1,\dotsc ,\xi _k) :=w_k(t, t - \xi _1, \dotsc , t-\xi _k) \end{aligned}$$

(13.5)

and the convolution operator $*_t$ for time varying systems

$$\begin{aligned} h_k(t,\xi _1, & \dotsc ,\xi _k) *_tx^{\otimes k}(\xi _1,\dotsc ,\xi _k) :=\nonumber \\ & \int \limits _0^t\cdots \int \limits _0^t h_k(t, t-\xi _1,\dotsc ,t-\xi _k) x(\xi _1)\cdots x(\xi _k) \textrm{d}\xi _1\cdots \textrm{d}\xi _k\,. \end{aligned}$$

(13.6)

Example 13.1

Consider a WNTV system described by the following differential equation

$$\begin{aligned} Dy + a(t) y = x + y^2\,. \end{aligned}$$

We are interested in the second order fundamental kernel of the system.

The fundamental kernel of the linearized equation is given by (12.26) which, taking into account the commutativity of the product of scalar functions simplifies to

$$\begin{aligned} w_1(t,\tau _1) = \textrm{e}^{-\int _{\tau _1}^t a(\lambda ) \textrm{d}\lambda }\,. \end{aligned}$$

With it the linear response of the system is

$$\begin{aligned} y_1(t) = w_1(t, \tau _1) \star x(t)\,. \end{aligned}$$

Given $y_1$ we can compute the “nonlinear source” of second order

$$\begin{aligned} y_1^2(t) = \int \limits _0^t \int \limits _0^t w_1(t, \tau _1) w_1(t, \tau _2) x(\tau _1) x(\tau _2) \textrm{d}\tau _1 \textrm{d}\tau _2\,. \end{aligned}$$

With it we can then solve the equation consisting of terms of second order only

$$\begin{aligned} \big (D+ a(t)\big ) y_2 = y_1^2\,. \end{aligned}$$

The fundamental kernel of this equation is the same as the one of the first order equation. The second order response of the system is therefore

$$\begin{aligned} \begin{aligned} y_2(t) &= \int \limits _0^t w_1(t, \tau ) y_1^2(\tau ) \textrm{d}\tau \\ &= \int \limits _0^t \int \limits _0^t \int \limits _{\max (\tau _1, \tau _2)}^t w_1(t, \tau ) w_1(\tau , \tau _1) w_1(\tau , \tau _2) \textrm{d}\tau \, x(\tau _1) x(\tau _2) \textrm{d}\tau _1 \textrm{d}\tau _2\,. \end{aligned} \end{aligned}$$

The second order fundamental kernel of the system can be found by comparing this expression with $y_2=w_2(t,\tau _1, \tau _2) \star x^{\otimes 2}(\tau _1, \tau _2)$ giving

$$\begin{aligned} w_2(t, \tau _1, \tau _2) = \int \limits _{\max (\tau _1, \tau _2)}^t \textrm{e}^{-\int _{\tau }^t a(\lambda ) \textrm{d}\lambda - \int _{\tau _1}^\tau a(\lambda ) \textrm{d}\lambda - \int _{\tau _2}^\tau a(\lambda ) \textrm{d}\lambda } \textrm{d}\tau \,. \end{aligned}$$

As a check we verify that in the special case in which a(t) is constant we obtain the same result as in Example 9.5. Evaluating the integrals gives

$$\begin{aligned} w_2(\tau _1, \tau _2) = \frac{1}{a}\Big ( \textrm{e}^{-a[t - \min (\tau _2, \tau _1)]} - \textrm{e}^{-a(2t - \tau _1 - \tau _2)} \Big ) \end{aligned}$$

and, after the variable substitutions $\xi _i = t - \tau _i, i=1,2$ we indeed obtain an expression equivalent to $h_2$ in Example 9.5.

1.3 Time-Varying Nonlinear Frequency Responses

Weakly-nonlinear time-varying systems can equivalently be characterised by  time-varying nonlinear frequency responses. The kth order one is defined as the Fourier transform with respect to $\xi _1,\dotsc ,\xi _k$ of the impulse response $h_k(t,\xi _1,\dotsc ,\xi _k)$. For regular distributions

$$\begin{aligned} \hat{h}_k(t,\omega _1,\dotsc ,\omega _k) :=\int \limits _{-\infty }^\infty \cdots \int \limits _{-\infty }^\infty h_k(t,\xi _1,\dotsc ,\xi _k) \textrm{e}^{-\jmath \left( \omega ,\xi \right) } \textrm{d}^k\xi \end{aligned}$$

(13.7)

with $\omega , \xi \in {\mathbb {R}}^k$.

The response of order k of a system can be calculated by

$$\begin{aligned} y_k(t) = \frac{1}{(2\pi )^k} \hat{h}_k(t,\omega _1,\dotsc ,\omega _k) \, \textrm{e}^{\jmath (\omega _1 + \cdots + \omega _k) t} \star \hat{x}^{\otimes k}(\omega _1,\dotsc ,\omega _k)\,. \end{aligned}$$

(13.8)

The derivation is entirely analogous to the one dimensional case carried out in Sect. 12.4.1.

2 Weakly Nonlinear Periodically Time-Varying Systems

Weakly nonlinear periodically time-varying (WNPTV) systems are weakly nonlinear systems whose characteristics vary periodically in time. In other words, their fundamental kernels, impulse responses and frequency responses are periodic functions of time and can therefore be expanded in Fourier series. For example, the kth order frequency response of a ${\mathcal {T}}$-periodic system can be represented by the series

$$\begin{aligned} \hat{h}_k(t,\omega _1,\dotsc ,\omega _k) = \sum _{n=-\infty }^\infty \hat{h}_{k,n}(\omega _1,\dotsc ,\omega _k) \textrm{e}^{\jmath n \omega _{\mathcal {T}}t} \end{aligned}$$

with $\omega _{\mathcal {T}}= 2\pi /{\mathcal {T}}$. This representation highlights the fact that such systems can be represented by a parallel connection of a countable set of weakly nonlinear time-invariant networks whose outputs are shifted in frequency by a multiple of $\omega _{\mathcal {T}}$ (see Fig. 13.2). Practical applications where this representation is particularly useful include the analysis and design of communication systems.

Fig. 13.2

Generic representation of a WNPTV system

In the rest of this section we focus on the special case in which weakly nonlinear periodically time-varying systems are driven by a set of tones. This will reveal a spectrum characteristic of this type of systems.

2.1 Discrete Convolution

Before turning to actually calculating the response of WNPTV systems driven by a set of tones, it’s convenient to introduce some notation that will simplify many expressions.

A series $\sum \limits _{n=-\infty }^\infty a_n$ is absolutely convergent if the sum of the absolute values of the terms converges

$$\begin{aligned} \sum _{n=-\infty }^\infty |a_n | < \infty \,. \end{aligned}$$

In this case the value of the series doesn’t depend on the order of the elements. The product of two absolutely convergent series $\sum _{n=-\infty }^\infty a_n$ and $\sum _{n=-\infty }^\infty b_n$ is also absolutely convergent

$$\begin{aligned} |{\sum _{n=-\infty }^\infty a_n \sum _{n=-\infty }^\infty b_n} | \le \sum _{n=-\infty }^\infty |a_n | \sum _{n=-\infty }^\infty |b_n | < \infty \end{aligned}$$

and can be expressed as

$$\begin{aligned} \sum _{n=-\infty }^\infty a_n \sum _{n=-\infty }^\infty b_n = \sum _{n=-\infty }^\infty \Big ( \sum _{q=-\infty }^\infty a_q b_{n-q} \Big )\,. \end{aligned}$$

The inner sum in the last expression is called discrete convolution (or Cauchy product). For convenience, we are going to denote it by

$$\begin{aligned} (a_. *_db_.)_n :=\sum _{q=-\infty }^\infty a_q b_{n-q} \end{aligned}$$

(13.9)

The discrete convolution is associative and commutative

$$\begin{aligned} \big ((a_. *_db_.) *_dc_.\big )_n &= \big (a_. *_d(b_. *_dc_.)\big )_n\\ (a_. *_db_.)_n &= (b_. *_da_.)_n \end{aligned}$$

and has a unit element, the Kronecker delta

$$\begin{aligned} \delta _n = {\left\{ \begin{array}{ll} 1 &{} n = 0\\ 0 &{} n \ne 0\,. \end{array}\right. } \end{aligned}$$

(13.10)

2.2 Product of Fourier Series

In the following we use the convention introduced in Sect. 4.5 of denoting the kth Fourier coefficient of a distribution f by $c_k(f)$.

It is well known that if $t \mapsto f(t)$ is a continuous ${\mathcal {T}}$-periodic function, its Fourier series is absolutely convergent for all values of t [23]. If f and g are two such functions then their product is well-defined and continuous. In addition, the Fourier coefficients of the product can be expressed in terms of the coefficients of the individual series

$$\begin{aligned} \sum _{n=-\infty }^\infty c_n(f) \textrm{e}^{\jmath n \omega _{\mathcal {T}}t} \sum _{n=-\infty }^\infty c_n(g) \textrm{e}^{\jmath n \omega _{\mathcal {T}}t} = \sum _{n=-\infty }^\infty \bigg ( \sum _{q=-\infty }^\infty c_q(f) \, c_{n-q}(g) \bigg ) \textrm{e}^{\jmath n \omega _{\mathcal {T}}t}\,. \end{aligned}$$

The coefficients of the product are evidently the convolution product of the coefficients of the two series

$$\begin{aligned} \big ( c_.(f) *_dc_.(g) \big )_n = \sum _{q=-\infty }^\infty c_q(f) \, c_{n-q}(g)\,. \end{aligned}$$

(13.11)

Let now f and g be two ${\mathcal {T}}$-periodic distributions. Let further introduce the sequences $(f_k)$ and $(g_k)$ defined by

$$\begin{aligned} f_k = f *\beta _k \qquad \text {and}\qquad g_k = g *\beta _k \end{aligned}$$

with $(\beta _k)$ a sequence of functions in ${\mathcal {D}}$ converging to $\delta$ (for example the sequence of Example 2.5). Then $(f_k)$ and $(g_k)$ are sequences of indefinitely differentiable functions converging as distributions to f and g respectively. The Fourier series of each member of each sequence is thus absolutely convergent.

If the product $f \, g$ exists, then it defines a ${\mathcal {T}}$-periodic distribution which must coincide with the limit of the sequence

$$\begin{aligned} f \, g = \lim _{k\rightarrow \infty } f_k \, g_k\,. \end{aligned}$$

The Fourier series of each member of the sequence can be written as

$$\begin{aligned} f_k \, g_k = \sum _{n=-\infty }^\infty \big (c_.(f_k) *_dc_.(g_k)\big )_n \textrm{e}^{\jmath n\omega _{\mathcal {T}}t}\,. \end{aligned}$$

Therefore, from the assumption of convergence and the uniqueness of the Fourier series representation of periodic distributions we conclude that the Fourier coefficients of $f \, g$ must be

$$\begin{aligned} c_n(f\,g) = \big ( c_.(f) *_dc_.(g) \big )_n :=\lim _{k\rightarrow \infty } \big ( c_.(f_k) *_dc_.(g_k) \big )_n\,. \end{aligned}$$

Example 13.2

Consider the regular ${\mathcal {T}}$-periodic distribution shown in Fig. 13.3 that we denote by f and whose Fourier coefficients are

$$\begin{aligned} c_n(f) = {\left\{ \begin{array}{ll} 0 &{} n\ \text {even}\\ \frac{2}{\pi n} (-1)^{\frac{n-1}{2}} &{} n\ \text {odd}\,. \end{array}\right. } \end{aligned}$$

From the graph it’s apparent that the product of f with itself is well-defined and produces the regular distribution with constant value 1. The Fourier coefficients are evidently all zero apart from the zeroth one whose value is one $c_0(f\,f) = 1$. We show that, despite the fact that the Fourier series of f is not absolutely convergent, $c_.(f) *_dc_.(f)$ produces the right answer.

First note that for n odd either $c_q(f)$ or $c_{n-q}(f)$ is zero for every value of q. Hence,

$$\begin{aligned} \big ( c_.(f) *_dc_.(f) \big )_n = 0 \qquad n\ \text {odd}\,. \end{aligned}$$

For n even the convolution product is

$$\begin{aligned} \begin{aligned} \big ( c_.(f) *_dc_.(f) \big )_n &= \sum _{k=-\infty }^\infty \frac{2}{\pi (2k + 1)} (-1)^k \frac{2}{\pi (n - (2k + 1))} (-1)^{\frac{n - 2(k+1)}{2}}\\ &= (-1)^{\frac{n}{2}-1} \Big (\frac{2}{\pi }\Big )^2 \sum _{k=-\infty }^\infty \frac{1}{(2k + 1)(n - (2k + 1))}\,. \end{aligned} \end{aligned}$$

For the particular case $n = 0$ the summation in the last expression can be written as

$$\begin{aligned} \sum _{k=-\infty }^\infty \frac{-1}{(2k + 1)^2} = -2 \sum _{k=0}^\infty \frac{1}{(2k + 1)^2} = -\frac{\pi ^2}{4}\,. \end{aligned}$$

The zeroth coefficient is therefore

$$\begin{aligned} \big ( c_.(f) *_dc_.(f) \big )_0 = \Big (\frac{2}{\pi }\Big )^2 \frac{\pi ^2}{4} = 1\,. \end{aligned}$$

To evaluate the Fourier coefficient for $n\ne 0$ it’s convenient to rewrite the summation as

$$\begin{aligned} \sum _{k=-\infty }^\infty \frac{1}{(2k + 1)(n - (2k + 1))} = \sum _{k=-\infty }^\infty \frac{1/n}{2k + 1} + \frac{1/n}{(n - (2k + 1))}\,. \end{aligned}$$

In this form it’s apparent that for each value of n all terms cancel in pair (the kth with the $(n/2 + k)$th), thus giving

$$\begin{aligned} \big ( c_.(f) *_dc_.(f) \big )_n = 0 \qquad n\ne 0\ \text {even}\,. \end{aligned}$$

Fig. 13.3

Square regular ${\mathcal {T}}$-periodic distribution

2.3 Response to Multi-tones

Consider a weakly nonlinear periodically time-varying system described by the differential equation

$$\begin{aligned} L(t, D) y = N(t, D) x + c_2(t)y^2 + c_3(t)y^3 + \cdots \end{aligned}$$

with

$$\begin{aligned} L(t, D) &= D^{m} + a_{m-1}(t)D^{m-1} + \dots + a_0(t) \\ N(t, D) &= b_n(t)D^{n} + b_{n-1}(t)D^{n-1} + \dots + b_0(t) \end{aligned}$$

and where all coefficients $a_i, b_i$ and $c_i$ are smooth ${\mathcal {T}}$-periodic functions. We assume that the system is driven by N complex tones

$$\begin{aligned} x(t) = A_1 \textrm{e}^{\jmath \omega _1t} + \cdots + A_N \textrm{e}^{\jmath \omega _Nt} \end{aligned}$$

with $A_1,\dotsc ,A_N$ the phasors of the tones.

In Sect. 12.4.2 we saw that the solution of the linear part of the equation is given by

$$\begin{aligned} y_1(t) = \sum _{n=1}^N A_n \hat{h}_1(t,\omega _n) \textrm{e}^{\jmath \omega _n t} \end{aligned}$$

with $\hat{h}_1$ the (first order) time-varying frequency response of the system. We also saw (Sect. 12.5.2) that $t \mapsto \hat{h}(t,\omega _1)$ is a ${\mathcal {T}}$-periodic function. Expanding it in a Fourier series, $y_1$ can be written as

$$\begin{aligned} y_1(t) = \sum _{n=1}^N A_n \textrm{e}^{\jmath \omega _n t} \sum _{q=-\infty }^\infty \hat{h}_{1,q}(\omega _n) \textrm{e}^{\jmath q \omega _{\mathcal {T}}t}\,. \end{aligned}$$

$y_1$ is therefore a sum of tones at $q\omega _{{\mathcal {T}}} + \omega _n$.

We now solve the nonlinear equation by adding terms to $y_1$ in a similar way as we did for weakly nonlinear time invariant systems in Sect. 9.5. As explained in Sect. 13.1 here we must assume the existence of the powers $y_1^k, k=2,3,\dotsc$ and the others that will appear below.

For the sake of solving the equation let’s assume that the frequencies $\omega _{{\mathcal {T}}}, \omega _1,\dotsc ,\omega _N$ are all incommensurate. Under this assumption, the only power resulting in terms proportional to $A_jA_l\textrm{e}^{\jmath (\omega _j + \omega _l)t}; j,l=1,\dotsc ,N$ is the second order one

$$\begin{aligned} c_2(t) y_1^2(t) = & \sum _{|m |=2} \frac{2!}{m!} A_1^{m_1} \cdots A_N^{m_N} \textrm{e}^{\jmath \omega _m t}\\ & \sum _{q=-\infty }^\infty \big ( c_.(c_2) *_d\hat{h}_{1,.}(\omega _1)^{*_dm_1} *_d\cdots *_d\hat{h}_{1,.}(\omega _N)^{*_dm_N} \big )_q \textrm{e}^{\jmath q \omega _{{\mathcal {T}}} t} \end{aligned}$$

with m the multi-index $m=(m_1,\dotsc ,m_N)$ whose elements range from 0 to k (=2) and $\omega _m$ as defined in (9.27) and repeated here for convenience

$$\begin{aligned} \omega _m = \sum _{n = 1}^N m_n \, \omega _n = m_1\omega _1 + \cdots + m_N\omega _N\,. \end{aligned}$$

Similarly to the time invariant case we can assume that the solution of the nonlinear differential equation includes a term of second order $y_2$ proportional to $A_jA_l\textrm{e}^{\jmath (\omega _j + \omega _l)t}; j,l=1,\dotsc ,N$. $y_2$ can be found by retaining only those terms in the equation that are proportional to $A_jA_l\textrm{e}^{\jmath (\omega _j + \omega _l)t}$. The resulting equation is linear with $c_2(t) y_1^2(t)$ playing the role of a source composed by tones. Exploiting linearity we can solve the equation for a single tone at $q\omega _{{\mathcal {T}}} + \omega _1 + \omega _2$ and combine the results at the end

$$\begin{aligned} L(t, D) \hat{g}_{2,q}(t, \omega _1, \omega _2) \textrm{e}^{\jmath (q\omega _{{\mathcal {T}}} + \omega _1 + \omega _2)t} = \textrm{e}^{\jmath (q\omega _{{\mathcal {T}}} + \omega _1 + \omega _2)t}\,. \end{aligned}$$

$t \mapsto \hat{g}_{2,q}(t, \omega _1, \omega _2)$ is also ${\mathcal {T}}$-periodic and can be expanded in a Fourier series

$$\begin{aligned} \hat{g}_{2,q}(t, \omega _1, \omega _2) \textrm{e}^{\jmath (q\omega _{{\mathcal {T}}} + \omega _1 + \omega _2)t} = \sum _{q_2=-\infty }^\infty c_{q_2}(\hat{g}_{2,q}) \textrm{e}^{\jmath \big ((q + q_2)\omega _{\mathcal {T}}+ \omega _1 + \omega _2 \big ) t}\,. \end{aligned}$$

With it the second order term $y_2$ is given by

$$\begin{aligned} y_2(t) = \sum _{|m |=2} \frac{2!}{m!} A_1^{m_1} \cdots A_N^{m_N} \textrm{e}^{\jmath \omega _m t} \qquad \qquad & \\ \cdot \sum _{q_1=-\infty }^\infty \big ( c_.(c_2) *_d\hat{h}_{1,.}(\omega _1)^{*_dm_1} *_d\cdots *_d\hat{h}_{1,.} & (\omega _N)^{*_dm_N} \big )_{q_1} \textrm{e}^{\jmath q_1 \omega _{{\mathcal {T}}} t}\\ & \cdot \sum _{q_2=-\infty }^\infty c_{q_2}(\hat{g}_{2,q_1}) \textrm{e}^{\jmath q_2 \omega _{\mathcal {T}}t} \end{aligned}$$

which, with the change of variable $l = q_1 + q_2$, can be rewritten as

$$\begin{aligned} y_2(t) = \sum _{|m |=2} \frac{2!}{m!} A_1^{m_1} \cdots A_N^{m_N} \sum _{l=-\infty }^\infty \hat{h}_{2,m,l} \textrm{e}^{\jmath (l \omega _{\mathcal {T}}+ \omega _m) t} \end{aligned}$$

with

$$\begin{aligned} \hat{h}_{2,m,l} :=\sum _{q_1=-\infty }^\infty \big ( c_.(c_2) *_d\hat{h}_{1,.}(\omega _1)^{*_dm_1} *_d\cdots *_d\hat{h}_{1,.}(\omega _N)^{*_dm_N} \big )_{q_1} c_{l-q_1}(\hat{g}_{2,q_1})\,. \end{aligned}$$

The second order response of the system thus consists of tones at all possible sums of two of the input tone frequencies at a time, around each of the harmonics of the fundamental frequency of the system.

The higher order responses can be calculated in a similar manner. The kth order response has the form

$$\begin{aligned} y_k(t) = \sum _{|m |=k} \frac{k!}{m!} A_1^{m_1} \cdots A_N^{m_N} \sum _{l=-\infty }^\infty \hat{h}_{k,m,l} \textrm{e}^{\jmath (l \omega _{\mathcal {T}}+ \omega _m) t} \end{aligned}$$

(13.12)

and is composed by tones at all possible sums of k input tone frequencies at a time, around each of the harmonics of the system fundamental frequency. A comparison of the typical two tones response of LTI-. WNTI-, LPTV- and WNPTV-systems is shown in Fig. 13.4.

Fig. 13.4

Comparison of typical two (real) tones spectral response of LTI-, LPTV-, WNTI- and WNPTV-systems

Note that the factor

$$\begin{aligned} \sum _{l=-\infty }^\infty \hat{h}_{k,m,l} \textrm{e}^{\jmath l \omega _{\mathcal {T}}t} \end{aligned}$$

appearing in the kth order response $y_k$ is the Fourier series of the time-varying kth order nonlinear frequency response of the system $\hat{h}_k(t, \omega _1,\dotsc ,\omega _k)$. It is related to $\hat{h}_{k,m,l}$ by

$$\begin{aligned} \hat{h}_{k,m,l} = c_l\big ( \hat{h}_k(t, \underbrace{\omega _1,\ldots ,\omega _1}_{m_1},\ldots , \underbrace{\omega _N,\ldots ,\omega _N}_{m_N}) \big )\,, \qquad |m |=k\,. \end{aligned}$$