interested in causal systems, i.e. systems that have no knowledge about the future. The behavior of a causal linear time-varying system is captured by the following definition: Definition (causal linear time-varying system): A causal linear time-varying system H[u(t)] computes the output at a time t as a linear combination of the input samples u(t) that it has received until that time. The coefficients used for computing this linear combination may depend upon the time instance t at which the output is computed. The most natural way to translate this definition into mathematics goes as (3.24) Here, u(t) represents the input signal(s) and y(t) the output signal(s)². In (3.24) the integration indicates a summation over a continuous range of (past) input samples. The quantity h(t,T) is called the kernel characterizing the linear system. This kernel can be interpreted as a (possibly time-varying) impulse response. Since causal LPTV systems are a subclass of causal LTV systems, linearity has taught us that their behavior can be captured using (3.24). 3.3.1.2 Periodic time variance Not all kernels h(t,T) represent periodically time-varying systems. To do so, the input- output behavior described by (3.24) must satisfy: Definition (Periodically time-varying): A T -periodic time-varying system is a system whose response depends in a periodic manner on time. More precisely, equal input signals applied at instances of time that are a period T apart result in equal output signals. 2For notational convenience, we denote u(t) and y(t) as scalars; they can however also be considered as vectors in which case h(t,T) is a matrix. LAPLACE-DOMAIN MODELING OF LPTV SYSTEMS USING HARMONIC TRANSFER MATRICES Figure 3.7: A T-periodic time-varying system produces similarly shaped out- put signals when a Dirac impulse is applied at time instances that are a period T apart (compare first and last input-output combination). However, when the Dirac impulses are applied at time instances that are not a period T apart, e.g. when they are half a 