envelope spectra Um(s), i.e. the envelope spectra can have non-zero signal content over the entire frequency axis. However, in gaining understanding and giving interpre- tation to the theory that is developed, it is often helpful to think of the Um(s) as being bandlimited. 3.3 LAPLACE-DOMAIN MODELING OF LPTV SYSTEMS USING HARMONIC TRANSFER MATRICES U_{s) U0(s) U1(s) Figure 3.10: For a single wideband input signal, it is often artificial to break it up into different components. In this case, it is more convenient to choose U0(s) as on the right, i.e. in a way that does not strictly adhere to (3.36). 3.3.3 Harmonic transfer matrices: capturing transfer of signal content between carrier waves Linear periodically modulated waveforms capture the idea of different signal contents that are stored in different frequency bands. LPTV systems are claimed to transfer signal content from one frequency band to another. Hence, t is natural to ask how LPTV systems act upon linear periodically modulated waveforms: what kind of output signals result and how can we compute them? To answer this question, we substitute (3.31) into the input-output relation (3.24). This yields (3.37) Since At, h(t, T) is T-periodic in t, with T = 2/wo, we can expand the kernel using a Fourier series, or (3.38) 3.3.3 Harmonic transfer matrices: capturing transfer of signal content between carrier waves Substituting (3.38) into (3.37), we obtain (3.39) = yn(t)ejinwot n=-00 In obtaining the second equality, we changed summation indices using n =m+k. The result shows that an LPTV system acting on a linear periodically modulated signal produces another linear periodically modulated signal with equivalent baseband com- ponents that equal (3.40) Each term in the right-hand side of (3.40) represents a linear time-invariant filtering operation. Hence, (3.40) is readily rewritten in the Laplace domain as (3.41) whereUm(s)=L{um(t)}, Yn(s)=L{yn(t)} and = = Hk(s)=L{hk(t)} = By organizing the Um(s) in an o-dimensional column vector U(s)= [ U-2(s) U-1(s) U0(s) U1(s