Abstract
In this chapter we consider linear time-varying (LTV) systems. These are systems whose behaviour depends on the particular moment in time at which they are used. The change with time may arise for example due to the sensitivity of system components to environmental changes. Examples of systems suffering from this type of sensitivity include wireless communication systems in which the communication channel between the transmitter- and the receiver-antennas is highly dependent on the environment in between and around the antennas.
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1 Linear Time-Varying Systems
In this chapter we consider linear time-varying (LTV) systems. These are systems whose behaviour depends on the particular moment in time at which they are used. The change with time may arise for example due to the sensitivity of system components to environmental changes. Examples of systems suffering from this type of sensitivity include wireless communication systems in which the communication channel between the transmitter- and the receiver-antennas is highly dependent on the environment in between and around the antennas. The variation in time may also be imposed intentionally by design to achieve functions that can’t be realised with LTI-systems. This is the case for example in communication mixers whose function is to shift in frequency the spectrum of a signal.
In this section we introduce a definition of linear time-varying systems valid under the assumption that all signals are regular distributions. A generalisation will be given in Sect. 12.3. The assumption of linearity means that the superposition principle must hold. In addition, as for LTI-systems, we require that LTV-systems depend continuously on the input signal. We therefore define
Definition 12.1
(LTV-system) A single-input, single-output, linear time-varying system is a system that when driven by the input signal x produces a response y that can be expressed by
\(h(t,\xi )\) is the time-varying impulse response of the system.
The meaning of the variable \(\xi \) is best illustrated by anticipating somewhat the results of Sect. 12.3 and apply as input signal a Dirac impulse at time \(t_0\)
Thus \(\xi \) represents the time lapsed since the application of the input impulse.
For causal systems the output must vanish before the input is applied. This implies that the impulse response must vanish for negative values of \(\xi \)
Therefore, the response of a causal system driven by a regular distributions \(x \in {\mathcal {D_+'}}\) is given by
2 Linear Ordinary Differential Equations
An important class of LTV-systems is the one of systems described by differential equations with variable coefficients of the form
with
time-dependent differential operators. It’s easy to verify that every such system with \(n < m\) can be represented in a state-space representation with time dependent matrices
with u the system state. Given an input signal x, the system response y is fully determined if one can find a state u satisfying the first equation and suitable initial conditions. The study of the dynamics of the system can therefore be reduced to the study of a system of n differential equations of first order.
2.1 Fundamental Solution
Consider the initial value problem described by the system of n differential equations
and initial conditions
with A(.) an \(n \times n\) matrix of complex valued functions of time \(a_{ij}(.)\). If the functions forming A(.) are bounded and continuous, then the right-hand side of the equation is Lipschitz continuous and, as discussed in Sect. 9.1, the equation has a unique solution. By choosing the initial value equal to the unit vector \(e_j \in {\mathbb {C}}^n\) pointing in direction j, for \(j=1,\dotsc ,n\) we can thus obtain n independent solutions \(y_j\) of the equation. The matrix formed by the column vectors \(y_j\)
is called principal fundamental matrix of the system and satisfies the matrix equation
Knowing Y, the solution of the initial value problem is thus given by
In addition, since the columns of Y are independent at all times, \(\det (Y(t)) \ne 0\) at all times. The inverse of Y, \(Y^{-1}\), is thus well-defined as is the evolution operator (also called state transition matrix)
Note that the evolution operator satisfies
and
and is thus the principal fundamental matrix of the system at time \(\tau \). From (12.9) we also immediately obtain
and
The initial value problem described by (12.5) and initial conditions \(y(t_0) = y_0\) can be translated in the language of distributions by extending the functions by zero for \(t < t_0\) and by replacing the differential operator by the distributional one
as usual. Differently from the case where A(.) is constant, this equation can not be written as a convolution equation. For this reason and since for arbitrary distributions multiplication is only well-defined with smooth functions, for the equation to be well-defined the functions \(a_{ij}(.)\) must belong to \({\mathcal {E}}\). This may seem like a very serious limitation, but remember that any distribution can be approximated to arbitrary accuracy by such a function (see Sect. 3.3). In this case the fundamental (or elementary) solution of the equation relative to time \(\tau \) is defined as the solution of the matrix equation
with L the differential operator
If \(U(t,\tau )\) is the evolution operator of the original differential equation (12.5) then
shows that
is the fundamental solution relative to \(\tau \) of the above distributional equation and
is the solution of (12.10).
2.2 Formal Solution
We now look for an explicit formal solution in \({\mathcal {D_+'}}({\mathbb {R}},{\mathbb {C}}^n)\) of the equation
with A(.) a matrix of functions in \({\mathcal {E}}\) as before. As a first step we rewrite the equation as an integral equation. To do this we write \(Dy\) as \(D\delta *y\) and convolve both sides of the equation with \(\textsf{1}_{+}\) to obtain
Thus, if x is a bounded regular distribution then the equation can be written as
Instead of solving this equation directly, we consider a more general integral equation and then specialise to this case.
A Volterra integral equation of the second kind is an equation of the form
with x a given regular distribution in \({\mathcal {D_+'}}({\mathbb {R}},{\mathbb {C}}^n)\), k an \(n \times n\) matrix of continuous functions \([k_{ij}]\), \(i,j=1,\dotsc ,n\) and y the required unknown in \({\mathcal {D_+'}}({\mathbb {R}},{\mathbb {C}}^n)\). This equation can be solved by an algebraic method based on a group [33, 34].
Definition 12.2
(Group) A group is a pair \(({\mathcal {G}}, \bullet )\) consisting of a non-empty set of objects \({\mathcal {G}}\) and a binary operation \(\bullet \), usually called the group multiplication, satisfying the following properties
-
1.
\(\bullet \) is associative: \((g_1 \bullet g_2) \bullet g_3 = g_1 \bullet (g_2 \bullet g_3)\).
-
2.
\(\bullet \) has an identity element e: \(g \bullet e = e \bullet g = g\).
-
3.
Every element \(g \in {\mathcal {G}}\) has an inverse element \(g^{-1} \in {\mathcal {G}}\):
$$\begin{aligned} g \bullet g^{-1} = g^{-1} \bullet g = e. \end{aligned}$$
Note that the unit element is unique, since if \(e'\) is a second unit \(e = e \bullet e' = e'\) shows that it must be equal to the first one. A group \({\mathcal {G}}\) acts (from the left) on a non-empty set X if there is a function
such that the following hold:
-
1.
\(e \cdot x = x\) for all \(x \in X\).
-
2.
\(g_1 \cdot (g_2 \cdot x) = (g_1 \bullet g_2) \cdot x\) for all \(g_1,g_2 \in {\mathcal {G}}\) and \(x \in X\).
Let now \(k(t,\tau )\) be an \(n \times n\) matrix of functions \([k_{ij}]\), \(i,j=1,\dotsc ,n\) continuous in the two variables \(t, \tau \), with \(0 \le \tau \le t\) and x a locally bounded, locally integrable function in \({\mathcal {D_+'}}({\mathbb {R}},{\mathbb {C}}^n)\). That x is locally bounded means that it is bounded on every finite interval. We define the operation of \(I + k\) on x by
The resulting function is again a locally bounded, locally integrable function in \({\mathcal {D_+'}}({\mathbb {R}},{\mathbb {C}}^n)\) as x and the elements \(I + k\) can be made to form a group. A suitable group multiplication can be found by writing
and noting that
Since the inner integral on the right-hand side results in a matrix of continuous functions, we can define the group multiplication by
with
For convenience we also put
so that we can write
The unit of the group is readily seen to be I.
It remains to show that every element of the group \(I + k\) has an inverse \((I + k)^{-1}\). From the similarity with the geometric series we infer that the inverse is given by
and show that this series converges in every interval \(0 \le \tau \le t \le T\). By definition, for every locally bounded function \(x=(x_1,\dotsc ,x_n)\) and every finite interval \(0 \le t \le T\) we can find an upper bound given by
so that, given the linearity of k,
with
Thus
and by induction
This upper bound is the nth term of a convergent series and implies the converges of (12.19) for every value of T. Having established convergence one immediately verifies that indeed
and
With this group the Volterra equation (12.16) can be written as
and is solved by multiplying on the left with \((I - k)^{-1}\)
The matrix function w is called the resolvent kernel of the equation.
The group can’t be extended to a ring or an algebra with the natural addition as these would include the elements k. These elements pose two problems. First, the inverse of these elements are not necessarily functions. For example, the inverse of \((t - \tau )^{m-1}/(m-1)!\) is \(D^m\delta \) and for singular distributions multiplication is only defined with functions in \({\mathcal {E}}\). Second, such a ring includes zero divisors. From now on we will generally drop the symbols \(\bullet \) of group multiplication and \(\cdot \) of group operation as is commonly done with multiplication symbols.
We now come back to the special case of (12.15) for which
The solution is given by (12.20) with
These expressions can be written more compactly by introducing the notion of a time-ordered product of operators. We define \(T \{A_1(\tau _1) \cdots A_n(\tau _n)\}\) as the product with factors arranged from left to right in order of decreasing times. For example
With this meta-operator we can now write
and more generally
because there are n! possible orderings of the n times \(\tau _1,\dotsc ,\tau _n\). Using these expressions we have
and
The solution of (12.15) can thus be written in the simple form
with
the fundamental solution of the equation relative to \(\tau \) and where we have made explicit the fact that for \(t < \tau \) it is zero.
In the special case in which A(.) commutes with \(\int _\tau ^t A(\lambda )\,\text {d}\lambda \) the time ordering operator has no effect and the solution of the equation is a direct generalisation of the solution obtained using the method of separation of the variables for the scalar equation
In particular this is the case if A(.) is constant, in which case the fundamental solution becomes
and the expression for the solution y becomes a convolution identical to (8.11).
For this particular case it is interesting to observe that, for a small-time increment \(\Delta t\), the evolution from an initial state \(y_0\) can be approximated (to first order) by
so that, by iteration
Now if we set \(\Delta t = t/n\) we obtain that, in the limit as n tends to infinity
The fundamental solution of (12.14) given by (12.26) can also be interpreted as a matrix function of the two variables t and \(\tau \)
As every element of the matrix is locally integrable, it is also a regular distribution that can be applied to test functions \(\phi \in {\mathcal {D}}({\mathbb {R}}^2)\). In particular, we can choose test functions of the form \(\psi (t)x_j(\tau )\) with \(\psi ,x_j\in {\mathcal {D}}({\mathbb {R}})\), \(j=1,\dotsc n\) in which case we obtain
with \(x = (x_1,\dotsc ,x_n)\). The inner integral on the right-hand side evaluates to a matrix of indefinitely differentiable functions in \({\mathcal {E}}\) [16, 35]. For this reason and remembering that every distribution f is the limit of a sequence of indefinitely differentiable functions (for example \(f_m = f *\beta _m\) with \(\beta _m\) the test functions of Example (2.4)) we can extend \(W \star x\) by continuity to operate on vector valued distributions in \({\mathcal {E'}}({\mathbb {R}},{\mathbb {C}}^n)\) by defining it as the distribution satisfying the system of equations
The thus extended linear map \(W \star \) is a distribution valued continuous function
With this definition we obtain for example that the solution of the equation with an input signal
is
or
This shows that the matrix W plays a similar role as the fundamental solution \(E_\tau (t)\) and is called the (two-sided) fundamental kernel (or elementary kernel) of the differential operator \(D- A(t)\). It also shows that, as with LTI systems, the initial conditions can be absorbed in the input vector signal x.
Example 12.1: Oscillator with Increasing Resonance [33]
Consider an ideal oscillator with a resonance frequency increasing with the square root of time
to which we apply an input signal
corresponding to initial conditions \(y(0) = y_0\) and \(y'(0) = y_1\).
The equation can be rewritten in the state-space form by defining the state
to obtain
with
In essence we need to calculate W(t, 0). Using (12.23) and remembering (12.22) we have, for n even
and for n odd
where \((a_k)_{k \ge 1}\) and \((b_k)_{k \ge 1}\) are the following sequences of integers
The fundamental kernel at (t, 0) is thus
The series can be recognised as linear combinations of the Airy \(\textsf{Ai}\) and \(\textsf{Bi}\) functions and their derivatives \(\mathsf {Ai'}\) and \(\mathsf {Bi'}\)
with
and
The signal of interest y is thus given by
Specifically, for \(y_0 = 1\) and \(y_1 = 0\) (see Fig. 12.1)
The full fundamental kernel \(W(t,\tau )\) can be obtained using Eqs. (12.9) and (12.12) and computing the inverse of W(t, 0)
Solutions of (12.28) for \(y_0=1\), \(y_1=0\) and \(\omega _0=2\pi \)
2.3 Perturbation Theory
The solution of (12.14) presented above is of great theoretical value. However, when it comes to solving practical problems it is in general very difficult to find a closed form for the fundamental kernel \(W(t,\tau )\). In many situations the problem at hand looks similar to a solvable problem, but with additional terms. If those terms are small in comparison to the ones of the solvable problem then one can obtain a good approximation to the solution of the problem by the following perturbative method.
Suppose that the matrix A(.) can be split in two parts: one that leads to a solvable problem and that we denote by \(A_0(.)\) and one with relatively small elements, the perturbation term, that makes the equation unsolvable and that we denote by \(\tilde{A}(.)\)
Let \(W_0(t,\tau )\) be the fundamental kernel of the solvable part of the equation, \(Y_0(.)\) its principal fundamental matrix, that is, the solution of the matrix equation
and let express y in terms of a new vector \(\tilde{y}\) defined by
Then the equation becomes
which reduces to
with
This equation has the same form as the original one. Its solution is therefore given by
The advantage that we gain is the fact that, if the elements of \(\tilde{A}(.)\) are small, then the series expansion of this solution converges very quickly. Differently from this, to obtain a good approximation using the series of the original formulation of the problem requires a large number of terms (compare with Example 12.1).
If x is composed by regular distributions then the first terms of the solution of the equation are given by
The first term that we denote by \(y_0\) is the solution of the unperturbed equation. In general, it is given by
The next term is the first order perturbation term that we denote by \(y_1\). Note that it can be expressed as the action of the unperturbed system on an input signal \(x_1\) constructed by multiplying \(y_0\) by the perturbation \(\tilde{A}\)
Similarly, the nth order perturbation term can be represented as the action of the unperturbed system on an input signal obtained by multiplying the perturbation term of order \(n-1\) by \(\tilde{A}\)
The output of the system
can thus be calculated iteratively starting from the response of the unperturbed system where each successive term is the result of multiplying the output of the previous term by \(\tilde{A}\) and feeding it back as input of the unperturbed system. This reminds of a feedback system with the unperturbed system playing the role of the forward path and \(\tilde{A}\) of the feedback one.
2.4 Non-smooth Coefficients
For several applications the requirement of differential operators with indefinitely differentiable coefficients is too restrictive. In those situations it’s useful to work in the subspace of \({\mathcal {D}}'\) constituted by distributions that are m times differentiable and denoted by \({\mathcal {D}}'^m\). These distributions are said to be of order m and are the continuous linear functionals on the set of m times continuously differentiable functions with compact support \({\mathcal {D}}^m\). Convergence is defined in a similar way as for distributions in \({\mathcal {D}}'\).
Given a distribution \(f \in {\mathcal {D}}'^m\) the product of f with an m times continuously differentiable function g is well-defined
since if \(\phi \in {\mathcal {D}}^m\) then \(g \phi \) is also in \({\mathcal {D}}^m\). Note that we can exchange the roles of f and g and still obtain a well-defined multiplication. Thus if f is an m times continuously differentiable function, it can be multiplied by a distribution of order m.
Example 12.2: Dirac Distribution
The Dirac distribution \(\delta \) belongs to \({\mathcal {D}}'^0\) and its multiplication with continuous functions is well-defined as long as one restricts considerations to \({\mathcal {D}}'^m\).
3 Impulse Response Generalisation
In the previous section we saw that differential equations describing LTV systems aren’t convolution equations. In spite of this we found that the solution of the equation can be written with the help of the operator \(\star \) acting on a (matrix) function characterising the system (the fundamental kernel) and the input vector x. In particular, for a system described by the state-space representation (12.3)–(12.4) with \(D(t)=0\), the input-output characteristic is given by
This expression highlights how in LTV systems the operator \(\star \) is the natural operator taking the place of convolution in LTI systems. However, because the Fourier- and Laplace-transform convert convolutions into products and because for many purposes the frequency domain characteristics of a system are more interesting than the time domain ones, in engineering circles it is common to express the input-output characteristic of LTV systems in terms of a convolution like operator as we did in Sect. 12.1. This is easily done by the change of variable
and by defining the time-varying impulse response \(h(t, \xi )\) as a function of the variables t and \(\xi \)
where we have assumed x to be a regular distribution in \({\mathcal {D_+'}}\). Note that, while the above integral looks very similar to a convolution, it differs from the convolution that we defined in Sect. 3.2. A generalisation of the above convolution like operation for LTV systems is obtained by adapting (12.27) and defining it as the distribution satisfying the following equality
where we have generalised the inner integral of (12.27) to the application of a parameterised distribution to the test function \(\phi \). Since this operation shares several properties with convolution, the operator \(*_t\) is called the convolution product for time-varying systems. In the technical literature it is most often called convolution and denoted by the same symbol as the one used for convolution. In the following we will also often simply call it convolution, but maintain the use of the symbol \(*_t\) to make it clear that it is not the operation defined by (3.6).
In the special case in which the time-varying impulse response is the product of an indefinitely differentiable function f and a distribution g
the convolution product for time-varying systems can be expressed in terms of a proper convolution by
In the previous section we discussed the fact that, for systems described by a differential equation, the application \(\langle h(t,t - \xi ), \phi (t)\rangle \) appearing on the right-hand side of (12.29), regarded as a function of the parameter \(\xi \), is a function belonging to \({\mathcal {E}}\). For this reason, for the equation to have a meaning, x must be restricted to distributions in \({\mathcal {E}}'\). However, if we define a function \(\gamma \in {\mathcal {E}}\) bounded from the left with \(\gamma (t) = 1\) in a neighbourhood of \([0,\infty )\) and assume h to be such that
is a Schwartz function for every \(\phi \in {\mathcal {S}}\), then (12.29) remains valid for right-sided tempered distributions
Note the similarity with the definition of the Laplace transform and the fact that, as for the Laplace transform, the value of the distribution does not depend on the choice of \(\gamma \). For this reason and as is commonly done for the Laplace-transform, we will generally not write \(\gamma \) explicitly.
Before concluding this section we note some properties of the operator \(*_t\). The first is that it is associative
with \(h_A\) and \(h_B\) the time-varying impulse responses of two systems. This is a direct consequence of the fact that \(*_t\) is related to \(\star \) by a simple variable transformation and by the definition of the latter (see Eqs. (12.17) and (12.18)).
A second important property, or rather the lack of it, is that, \(*_t\) is not commutative. Therefore, differently from LTI systems, the order of LTV systems is important. As an example consider the cascade of a low-pass filter with a 3 dB cut-off frequency of \(\omega _{3dB}\) and the frequency shifting system of Example 12.4 with \(w_0 \gg \omega _{3dB}\). Suppose that the system is driven by a signal with a frequency falling in the pass-band of the LPF. Then if the signal passes first through the LPF and then into the frequency shifting system, the output will have a large magnitude. Differently from this, if the input signal first passes through the frequency translating system then the signal at the input of the LPF will lie in the stop-band of the latter and will appear much attenuated at its output.
4 Time-Varying Frequency Response
4.1 Definition
Consider a system described by the time-varying impulse response \(h(t,\xi )\). Under the assumption that the input signal x is a right-sided tempered distribution the system response y can be written as
or
where \(\hat{h}(t,\omega )\) is the Fourier transform with respect to \(\xi \) of \(h(t,\xi )\) and is called thetime-varying frequency response of the system
In particular, for regular distributions we have
It’s easy to check that for real systems the time-varying frequency response at \(-\omega \) is equal to the conjugate complex of the value at \(\omega \)
for each value of t.
To obtain a physical interpretation for \(\hat{h}(t,\omega )\) we apply a complex tone \(\text {e}^{\jmath \omega _0t}\) as input signal. This is allowed because periodic distributions are isomorphic to distributions with compact support (see Sect. 3.4). With this input signal the output of the system is found with the help of (12.30) to be
and suggests the interpretation for the time-varying frequency response \(\hat{h}(t, \omega )\) as the complex envelope at \(\omega _0\) of the output signal (see Fig. 12.2).
Illustrative representation of the response of a real LTV system to an input tone
If the output signal y is a tempered distribution it can be Fourier transformed. A useful expression relating the spectrum of y and the one of the input signal x can be obtained by expressing y with the help of (12.30)
or
with
The function \(\hat{\hat{h}}\) is the two-dimensional Fourier transform of the time-varying impulse response h and, in the context of communication systems, is called the doppler-spread function. Equation (12.32) shows that the input and output spectra of an LTV system are related by a convolution like operation. In particular, for regular distributions they are related by the following integral
For tempered distributions the time-varying impulse response h, the time-varying frequency response \(\hat{h}\) and the doppler-spread function \(\hat{\hat{h}}\) are isomorphic to each other. For this reason an LTV system with a tempered time-varying impulse response can be described by any of these functions.
Example 12.3
In this example we investigate the relationship between an LTI system to which we apply a right-sided input tone and an LTV system activated at \(t=0\) s and driven by a tone, resulting in equal output signals.
Consider an LTI system described by the differential equation
to which we apply the signal \(x(t) = \textsf{1}_{+}(t) \text {e}^{\jmath \omega t} \in {\mathcal {D_+'}}\). The response of the system can be calculated with the help of the Laplace transform. The transfer function of the system and the Laplace transformed of the input signals are
and
respectively. The system response is thus found by inverse Laplace transforming
which gives
We now re-interpret the system as a time-variable one consisting of the above LTI system and an ideal switch at its input. For \(t < 0\) the input is disconnected from the system (switch open) which therefore produces the constant output signal \(y(t) = 0\). At \(t = 0\) the input signal is connected to the input of the LTI system by closing the switch. The full system is therefore described by the differential equation
The input signal is now the complex tone \(x(t) = \text {e}^{\jmath \omega t}\).
To obtain the system response we first compute the time evolution operator \(U(t,\tau )\) which is the solution of
and given by
With it the response of the system to the input \(x(t) = \text {e}^{\jmath \omega t}\) is calculated to be
which of course agrees with the calculation through Laplace transform. However, with the new interpretation we see that the system posses a time-varying frequency response \(\hat{h}(t, \omega )\). The easiest way to calculate it is through the relation \(y(t) = \hat{h}(t, \omega ) \text {e}^{\jmath \omega t}\) and we obtain
This shows the relationship between \(\hat{h}(t, \omega )\) and the LTI frequency response \(H(\jmath \omega ) = 1/(a + \jmath \omega )\). Differently from the latter, \(\hat{h}(t, \omega )\) includes the full information about the variation in time of the system. In this particular example, about when the switch is closed.
Example 12.4: Frequency Translation
Consider a system described by the doppler-spread function
According to (12.32) the spectrum of the output signal is given by
Therefore, the effect of the system described by the above doppler-spread function is to shift in frequency the spectrum of the input signal by \(w_0\). Such a device is referred to as a mixer.
The time-varying frequency response and the time-varying impulse response corresponding to this delay-spread function are easily calculated to be
and
respectively. If we apply a complex tone \(\text {e}^{\jmath \omega _0t}\) as input signal we can calculate the output signal from the former and (12.30) as
or from the latter and (12.29) as
In both cases the angular frequency of the input tone is shifted by \(w_0\) as expected.
The time-varying impulse response shows clearly that the system is memory-less, that is, the value of the output signal at time t only depends on the input signal at time t. The effect of the system is to simply multiply the input signal by the complex tone \(\text {e}^{\jmath w_0t}\) as illustrated in Fig. 12.3.
Block diagram of a frequency-translating LTV system
Ideal sample and hold
Example 12.5: Sample and Hold
In this example we consider an ideal sample and hold: the output of the system is constructed by sampling the input signal x at regular intervals \({\mathcal {T}}\) and by holding the value of each sample constant for the duration of a period \({\mathcal {T}}\). Sample and hold blocks are used for example at the input of analog-to-digital converters (ADC) to give the converter enough time to compare the value of a sample with one or more reference signal levels. The operation of a sample and hold is illustrated in Fig. 12.4.
The ideal sample and hold is characterised by the following time-varying impulse response
with
Note that in this case (12.29) doesn’t make sense as h is a singular distribution and in the right-hand side expression x is not applied to a smooth function. To give a meaning to
we have to restrict the input signal x to belong to \({\mathcal {E}}\). Then we can write
or
and we obtain the desired system response. The system response can also be written as a (proper) convolution
From this expression, assuming x to be Fourier transformable, it’s easy to compute the output spectrum. From (4.14) we read that the Fourier transform of \({\mathcal {T}}\delta _{\mathcal {T}}\, x\) is the convolution of the transforms of the factors divided by \(2\pi \)
with \(\omega _s\) the sampling angular frequency \(2\pi /{\mathcal {T}}\). Thus, the output spectrum is
with
This expression shows clearly the effects of sampling and of holding in the frequency domain. The operation of sampling is represented by the factor in square brackets. Its effect is to produce an infinite number of copies of the spectrum of the input signal shifted by multiples of \(\omega _s\)
If the original signal has to be recovered from the samples then one must avoid (or reduce to negligible levels) overlapping between the copies. This amount to saying that the power of the input signal residing outside the frequency range \((-\omega _s/2,\omega _s/2)\) must be negligible. Or, in other words, the sampling frequency must be at least twice the frequency of the highest component of the input signal spectrum containing a non-negligible amount of power. This is the statement of the famous sampling theorem. If this condition is satisfied then the input signal can be recovered with the help of a low-pass-filter eliminating the copies with \(n\ne 0\). When the copies of the input signal do overlap one says that sampling causes aliasing. Note that, if the spectrum of the input signal x only occupies a small fraction of the frequency range \((-\omega _s/2,\omega _s/2)\) then one may find a sampling frequency lower than \(\omega _s\) not causing aliasing.
The effect of holding act as an LTI filter introducing a delay of \({\mathcal {T}}/2\). The filter has a low-pass characteristic with notches at multiples of \(\omega _s\). The effects of sampling and of holding on the spectrum on a signal are illustrated in Fig. 12.5.
The need to restrict x to being an indefinitely differentiable function may seem like excess of rigor. Note however that if x is not continuous at the sample instants \(n{\mathcal {T}}\) then the problem is not “merely” a mathematical one, but any physical implementation will fail to work properly. This is so because if the input signal varies very rapidly compared to the actual speed of the physical sampling switch, then the value of the sample will be affected by many implementation details and in particular by noise. The result is a system producing unpredictable sample values.
From a mathematical point of view one may enlarge the type of allowed input signals to the class of continuous functions. Then the system response is mathematically well-defined, but it’s not a distribution anymore. In fact, the value of a Dirac impulse is defined as the value of the test function at zero. If we multiply the test function with a continuous function, the value is still well-defined. However, we can’t expect to be able to compute the derivatives of the output signal. Compare also with Sect. 12.2.4.
Illustration of the effect of sampling and of a sample and hold on the spectrum of a signal
We started this section by performing a calculation leading to the definition of the time-varying frequency response of a system and a relation expressing the output of the system in terms of it. If we assume Laplace transformable, right-sided signals and redo a similar calculation replacing the Fourier transform by the Laplace one we obtain the time-varying transfer function of the system
With it the output of the system is given by
4.2 Differential Equation
Consider again a linear time-varying system whose state u is described by the system of differential equations
and assume that it is driven by a complex tone
From (12.30) we know that the components of the state u can be represented by
Inserting this representation for u and the complex tone for x in the equation we obtain
The left-hand side can be written as
so that we obtain an equation for \(\hat{u}(t,\omega )\)
With \(\hat{u}(t,\omega )\) we can directly obtain the time-varying frequency response of the system without having to first compute the fundamental kernel
In particular, if the system is described by a (possibly) higher-order differential equation
with
we can directly obtain an equation for the time-varying frequency response of the system by replacing the differential operator \(D\) in L by the operator \(\jmath \omega + D\) and in N by \(\jmath \omega \) [36]
Note that this formulation in terms of distributions and distributional derivatives takes care of the initial conditions automatically. If one works with functions and the standard derivative then the initial conditions for the problem are obtained from
Example 12.6
Consider a system that is switched off up to time \(t=0\) (\(y(t) = 0, t < 0\)) at which point it is turned on and is then described by the differential equation
We are interested in the time-varying frequency response of the system. We compute it in three different ways.
First we compute it via the time evolution operator U. For \(t \ge \tau > 0\) it is found by solving the differential equation
As can be verified by inserting it into the equation, it is given by
To obtain the time-varying frequency response we apply the input \(x(t) = \text {e}^{\jmath \omega t}\) and obtain
From this and
we deduce that
The time-varying frequency response of the system can also be obtained by Fourier transforming the time-varying impulse response. The latter is obtained from the time evolution operator using the variable substitution \(\xi = t - \tau \)
where we made explicit that for \(t < 0\) the response of the system vanishes. The time-varying frequency response is thus
which matches the one obtained with the previous method.
A third method to compute the time-varying frequency response is by solving the corresponding differential equation
The solution is
as is verified by inserting it in the equation and where we made explicit that for \(t < 0\) it is zero.
5 Linear Periodically Time-Varying Systems
5.1 Floquet Theory
In this section we consider in more details linear periodically time-varying (LPTV) systems. In particular, we study systems that can be described by a state-space representation with matrices A(.), B(.), C(.) and D(.) having periodic smooth functions as elements. These include systems described by differential equations with periodic, indefinitely differentiable coefficients.
Consider the differential equation
with A(.) an \(n\times n\)-matrix and B(.) an \(n\times 1\) one, both with \({\mathcal {T}}\)-periodic indefinitely differentiable elements and where, for brevity, we denote by \(\dot{y}\) the (distributional) derivative of y and similarly for other quantities. Let further Y(.) be the principal fundamental matrix of the equation and
the evolution operator. From the periodicity of A(.) we obtain
from which, with \(U(t, t) = I\) and the uniqueness of the solution of the equation we deduce
and
Let now introduce
with F an \(n\times n\) matrix with constant coefficients and define the variable transformation
In terms of z the equation becomes
or
To simplify this equation we calculate the derivative of P
Using this result in the previous expression we finally obtain
This equation is similar to the original one, but with the important difference that the periodically time-varying matrix A(.) of the original equation has been replaced by a constant matrix F. This shows that the evolution operator of any system of differential equations with A(.) a \({\mathcal {T}}\)-periodic smooth matrix can be represented in the form
This is called the Floquet representation of the evolution operator.
Let \(y_0 \in {\mathbb {C}}^n\), from the analysis of LTI-systems we know that \(\text {e}^{t F} y_0\) is a linear combination of functions of the form
with \(\lambda _i\) an eigenvalue of F and \(p_i\) a polynomial of degree lower than the algebraic multiplicity of \(\lambda _i\). The Floquet representation tells us that the solution of (12.38) is a linear combination of functions of the form
where \(\tilde{p}_i\) are again polynomials, but in this case with \({\mathcal {T}}\)-periodic smooth coefficients.
Example 12.7
In this example we look for the solution of the equation
with
In particular we are interested in the evolution operator of the equation as it allows us to calculate the solution for an arbitrary input x.
First observe that A(.) can be written as a sum of two matrices
the first of which is constant, and we denote it by F. To find the principal fundamental matrix we make the ansatz
with p an indefinitely differentiable periodic function with period \(2\pi /\omega _m\). Inserting this ansatz in the equation we find
From this expression we see that it satisfies the equation if
The function p can be calculated from this equation and the condition \(Y(0) = I\) using the method of the separation of variables from which we obtain
The principal fundamental matrix is thus
With Y and using the results of Example 8.2 for \(\text {e}^{t F}\) the evolution operator is found to be
5.2 Time-Varying Frequency Response
Consider a SISO linear periodically time-varying system described by the state-space representation
with A(.), B(.), C(.) and D(.) indefinitely differentiable \({\mathcal {T}}\)-periodic matrix functions. Thanks to linearity we can analyse the response of the system for \(D(t) = 0\) and add the contribution of D(t)x at the end.
In the previous section we established that the evolution operator of (12.40) can be expressed in the form
with P(t) an invertible, indefinitely differentiable \({\mathcal {T}}\)-periodic matrix function and F a constant matrix. Using this representation for the response of the system we obtain
or, in terms of the time-varying impulse response
with
Representation of a stable LPTV system
If we now add the contribution to the output from D(t)x we finally find
with
The fist term \(h_C\) is a regular distribution growing at most exponentially with respect to \(\xi \) while the second has bounded support. The impulse response h is therefore Laplace transformable with respect to \(\xi \). This implies that the system possess a time-varying transfer function H(t, s). H(t, s) is a function in the variables t and s and the above expression makes it clear that it is periodic in t. Therefore, with respect to t, we can expand it in a Fourier series
with \(\omega _{\mathcal {T}}= 2\pi /{\mathcal {T}}\) and \(H_n(s)\) functions of the variable s alone.
The last expression shows that LPTV systems can be regarded as the parallel connection of LTI subsystems with transfer functions \(H_n\) whose outputs are shifted in frequency by \(n \omega _{\mathcal {T}}\) (see Fig. 12.6 and Example 12.4). This is best seen by applying a complex tone to a stable system. Thus, assume that all the eigenvalues of F have a negative real part, then the time-varying frequency response \(\hat{h}(t,\omega )\) does also exist and is also a regular distribution that can be identified with a function in the variables t and \(\omega \). Proceeding as above we can write it as
with \(\hat{h}_n(\omega ) = H_n(\jmath \omega )\). If we now apply a complex input tone \(\text {e}^{\jmath \omega _0 t}\) to the system and use (12.30) to calculate the system response we obtain
The output is thus seen to consist of a sum of tones at \(\omega _0 + n\omega _{\mathcal {T}}\), \(n\in {\mathbb {Z}}\), each one weighted by \(\hat{h}_n(\omega _0)\). It is readily seen that for a real system the following relation must hold
RC low-pass filter with a PTV resistor
Example 12.8: LPTV LPF
In this example we examine a series RC low-pass filter (LPF) where, to reduce the physical area occupied by the circuit, the series resistor is implemented with a MOSFET. While this will produce some distortion, here we are interested in what happens if the gate bias voltage is disturbed by a periodic signal (see Fig. 12.7). This could happen for example if in a mixed-signal system (both analog and digital signals present) a line distributing the system clock is in proximity of the gate bias line, and the two are not properly isolated. The system is described by the following differential equation
with y the voltage across the capacitor, x the source voltage and R(.) a periodic function. Given the periodicity of R(.), \(\omega _{3dB}(.)\) is also a periodic function with the same period that we assume to be smooth. \(\omega _{3dB}(.)\) can therefore be expanded in a Fourier series that, for simplicity of analysis, we assume to be given by
with \(\Delta \omega \ll \omega _0\). We are interested in characterising the frequency response of the filter.
The equation describing the system separates into a differential equation with constant coefficients and a small perturbation term
We can therefore solve the problem using the perturbation theory that we developed in Sect. 12.2.3. In addition, instead of solving for the time-varying impulse response and obtain the time-varying frequency response by Fourier transformation, it is convenient to solve directly the equation for the latter. Proceeding as in Sect. 12.4.2 we obtain
and we can identify \(-\Delta \omega \cos (\omega _m t)\) with the perturbation term \(\tilde{A}\) and \(-(\jmath \omega + \omega _0)\) with the matrix \(A_0\) of the unperturbed system.
We start by computing the time-varying frequency response of the unperturbed system that we denote by \(\hat{h}_0(t, \omega )\) and which has to satisfy
where we have represented \(\cos \omega _mt\) by complex tones. Note that the variation in R(.) results in additional input tones to an otherwise time invariant system. The solution of the equation is readily calculated to be
with
the frequency response of the RC filter without disturbances (that is for \(\omega _{3dB}(t) = \omega _0\)). \(H(\omega )/\omega _0\) plays the role of the fundamental kernel \(W_0\) of Sect. 12.2.3. However, because \(A_0\) is time invariant we can work in the convolution algebra of periodic distributions and instead of the fundamental kernel, the system can be characterised by the fundamental solution of the equation. In this example the kth Fourier coefficient of the fundamental solution of the equation is given by (see Example 7.5)
We now calculate the first order perturbation term. The first step consists in calculating the new “input signal” produced by the perturbation \(\tilde{A}\)
The first order perturbation term of the frequency response \(\hat{h}_1(t, \omega )\) is then obtained by applying this signal to the unperturbed system
The solution of the equation is given by
Note that both \(\hat{h}_0\) and \(\hat{h}_1\) include terms of order \(\Delta \omega \). Since \(\tilde{A}\) is proportional to \(\Delta \omega \) and all terms of \(\hat{h}_1\) are proportional to powers of this quantity, no higher perturbation term will include a contribution of order \(\Delta \omega \). The first two terms \(\hat{h}_0\) and \(\hat{h}_1\) are therefore enough to establish the effects of the perturbation of order \(\Delta \omega \). To obtain an estimate to second order in \(\Delta \omega \) we would need to calculate \(\hat{h}_2\) as well.
With these results the first order response of the system when driven by a tone at \(\omega \) is given by
It is comprised by tones at \(\omega + n\omega _m, n=-2,-1,0,1,2\). It’s not difficult to see that if we would calculate higher order terms we would obtain similar tones for larger values of \(|n |\) and in the limit, when including all perturbation terms, for all \(n\in {\mathbb {Z}}\).
Let’s consider more closely the component at \(\omega + \omega _m\)
and assume that \(\omega _m \gg \omega _0\). If the filter is part of a transmitter and used to suppress noise outside the channel allocated to the user or service, then a \(2\pi /\omega _m\)-periodic perturbation is seen to create spurious emissions that can fall in frequency ranges reserved for other users or services and violate the maximum allowed emission levels. From the above expression we note that a wide nominal filter bandwidth \(\omega _0\) helps in reducing the emission level caused by the perturbation. This can be interpreted intuitively as follows. If the input signal frequency is much smaller than the 3 dB cut-off frequency of the filter, then it produces a very small current flowing through the filer components and, in the limit of zero current, the output signal doesn’t depend on the value of the filter components.
If the input tone is well above the nominal 3 dB cut-off frequency of the filter \(|\omega | \gg \omega _0\) then \(|H(\omega ) | \ll 1\) and the output tone at \(\omega + \omega _m\) can be approximated by
If the frequency of the input tone is such that \(|\omega + \omega _m | < \omega _0\) then the tone falls in a spurious pass band of the filter and for \(|\omega + \omega _m | \ll \omega _0\) it can be approximated by
If the filter is part of a communication receiver responsible to suppress interfering signals (the channel filter) then we see that \(2\pi /\omega _m\)-periodic perturbations introduce spurious responses in the stop band of the filter at multiples of \(\omega _m\) that down-convert interfering signals in band, possibly masking the wanted signal. The amplitude of the dominant spurious response is proportional to the perturbation magnitude \(\Delta \omega \) relative to the nominal 3 dB cut-off frequency of the filter.
Example 12.9: Quadrature (De-)Modulator
Consider the frequency translating system of Example 12.4 with time-varying impulse response
It is a complex system in the sense that if we apply a real valued input signal its response is complex valued. In this example we show that the system can be implemented using two real sub-systems.
Let’s decompose the input signal into its real and imaginary parts
The system response is given by
and can be written as
In this form the system response is seen to be the sum of the responses of two real systems driven by correlated signals (see Fig. 12.8). By linearity, if the two systems are driven by the real part only of the input signals, that is by r and q respectively, then the response of the system is
The combination of the two real systems is called a quadrature modulator. Each of the two real subsystems is called mixer and effectively multiply the input signal with a second real valued signal l called the local oscillator (LO) signal. A mixer can therefore be considered a system having two input ports.
Consider now a system that shifts the spectrum of the input signal in the opposite direction
We would like to find a real system implementation that when driven by the signal
allows us to recover the signals used at the input of the quadrature modulator used to generate it. Such a system is readily found by observing that
and similarly
Thus, if the signals r and q are band-limited to frequencies smaller than \(w_0\), the original signals can be recovered (up to a fixed scaling factor) by use of two mixers driven by quadrature (orthogonal) local oscillator signals followed by low-pass filters (see Fig. 12.9). Such a system is called a quadrature demodulator. By linearity, if the system is driven by the real signal
the two output signals are the real parts of what we found above, that is r/2 and q/2 respectively.
Quadrature modulator
Quadrature demodulator
a Typical local oscillator unipolar waveform b Typical local oscillator bipolar waveform
Generic N-path receiver
Quadrature N-path demodulator
Example 12.10: Harmonic-Reject Mixer
We saw in Example 12.9that a mixer is a system multiplying the input signal with a \({\mathcal {T}}\)-periodic signal called the local oscillator signal
In practical implementations, to minimise the signal-to-noise degradation caused by the circuit, the local oscillator signal is not a pure sinusoidal. Instead, it is most often designed to approach a rectangular waveform as depicted in Fig. 12.10b. Being periodic the signal l can be represented by a Fourier series
with
for the waveform in Fig. 12.10a and
for the one in Fig. 12.10b. Therefore, a mixer driven by an input tone
produces an output tone at \(\omega _1\) for every value of n for which \(a_n \ne 0\). When the mixer is part of a receiver designed to down-convert a signal at \(\omega _{\mathcal {T}}+ \omega _1\) to \(\omega _1\) for further processing and detection, the spurious responses (\(n \ne -1\)) are undesired as they could cause an interfering signal to overlap in frequency with the desired signal and prevent reception of the latter. The spurious responses are most often suppressed by preceding the mixer with a suitable filter. However, in some situations such a filter is undesired. In the following we present a method to suppress the dominant spurious responses of a mixer without the need for filters and still using rectangular waveforms as local oscillator signals.
Note that, while the idealised local oscillator waveforms shown in Figs. 12.10a and 12.10b are discontinuous, their Fourier series representations truncated at an arbitrarily high value of \(|n |\) are indefinitely differentiable functions. Suitably truncated Fourier series are adequate representations of practical signals and do not cause any mathematical difficulty.
Consider the generic N-path receiver shown in Fig. 12.11. It is composed by N subsystems that are equal apart from the fact that the local oscillator signal of path \(k, k=0,\dotsc ,N-1\) is delayed by \({\mathcal {T}}k/N\) with respect to path 0. The blocks preceding the output signals \(y_k\) represent LTI subsystem with impulse response h. Let the input signal be as in (12.42). Then, due to the tone at \(-n\omega _{\mathcal {T}}\) in the Fourier series of l, the kth output signal includes a tone at \(\omega _1\) given by
with H the Laplace transform of h. This shows that the output components of interest (at \(\omega _1\)) are the product of the signal \(y_{0,-n}\) and the constants \(\text {e}^{\jmath n 2\pi \frac{k}{N}}, k=0,\dotsc ,N-1\). By exploiting the properties of trigonometric functions we can form weighted sums of the outputs \(y_k\) such that the resulting tone at \(\omega _1\) vanishes for some values of n
Note that the sum on the right-hand side corresponds to a discrete Fourier transform of the weighting coefficients. For example, by choosing
we obtain
The sums are geometric series that evaluate to
and therefore the signal \(z_{-n}\) is
For example, for \(N=8\) all harmonics below the 15th except for the 7th and the 9th are suppressed. A mixer with no spurious responses at some odd harmonics is called a harmonic-reject mixer.
The weighting factors of (12.43) are not the only possible choice. For example, any rotation of the indexes \(w_{(k + m) \mod N}\) produces a similar result with the addition of a phase factor to the output signal. For N even we can thus construct a full quadrature demodulator by building two weighted sums, one with weighting factors as given by (12.43) and the other by factors rotated by N/2 (\(w_{k+N/2}\); see Fig. 12.12). The case with \(N=4\) corresponds to classical situation with differential output signals. Further choices of weighting factors allow isolating responses at values of n different from 1.
While we discussed summing the signals after the LTI systems characterised by h, the same results apply if the signals are summed right after the mixers. The rejection obtained in practice is limited by mismatch between the paths. The place where the summation is implemented plays a role in this respect.
If we revert the direction of the signals in the system of Fig. 12.11 we obtain an N-path transmitter. This is a generalisation of the classic case with \(N=4\) with the 4 input signals being differential versions of the r and q modulator input signals. As with the receiver, a larger value of N allow suppressing spurious emissions at harmonics of the local oscillator signal without the use of filters.
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Beffa, F. (2024). Linear Time-Varying Systems. In: Weakly Nonlinear Systems. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-031-40681-2_12
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