Question

solve example 3.1 in detail with step by step

of how broad a class of signals could be represented as a linear combination of complex exponentials. In the next few sections we examine this question for in continuous time and then in discrete time, and in Chapters 4 and 3 we consider the extension of these representations to aperiodic signals. Although in general, the variables s and: in eqs (3.1-3.16) may be arbitrary complex numbers. Fourier analysis involves restricting cur attention to particular forms for these variables. In particular, in continuous time we focus on purely imaginary values of s-ie., j-and thus, we consider only complex exponentials of the form Similarly, in discrete time we restrict the range of values of to those of unit magnitude-ie..:-e'"--so that we focus on complex exponentials of the forme first Example 3.1 As an illustration of eqs. (3.3) and (3.6), consider an LTI system for which the input ata) and output v(r) are related by a time shifı of 3, ie.. v -x-3) (3.17) If the input to this system is the complex exponential signal xtn - e, then, from eq. (3.17), (3.18) Equation (3.18) is in the form of eq (3.5), as we would expect, since ef is an eigen- function. The associated eigenvalue is H2) is straightforward to confirm eq. (3.6) for this example. Specifically. from eq. (3.17), the impulse response of the sys- tem is h(,)-δ(,-3). Substituting into eq. (36), we oban so that H( j2) = e As a second example, in this case illustrating eqs. (3.11) and (3.12), consider the input signal xt) cost4r) + cos(7). From eq (3.17). y) will of course be 13 19 To see that this will also result from eq. (3.12), we first expand xin using Euler 's relaion From eqs. (3.11) and (3.12). Fourier Series Representation oft Periodi Signals Charp 3 or cos(4( 3)) + cos(7u 3)). For this simple example, multiplication of each periodic exponential component of x(1)-for example, e-by the corresponding eigenvalue-e.g. H4 effectively causes the input component to shift in time by 3. Obviously, in this case we can determine y() in eq. (3.19) by inspection rather than by empl and (3.12). However, as we will see, the general property embodied in eqs. 3.11) and (3.12) not only allows us to calculate the responses of more complex LTI systems but also provides the basis for the frequency domain representation and analysis of LTI systems oying eas. 3.11)

Answer 1

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