Example Problem Set 2

We compute the representation of some continuous function f(t)in a space spanned by an orthonormal family $\{\Psi_{j}(t)\}$ of continuous basis functions by projecting f(t) onto them. We express these projections in bracket notation $<f(t), \Psi_{j}(t)>$denoting ${\displaystyle \int_{-\infty}^{\infty}f(t) \Psi_{j}^{*}(t) dt}$, where the ^* denotes complex conjugate; and f(t) is assumed to be square-integrable.

1.

Give an expression for computing f(t) if we know only its projections $<f(t), \Psi_{j}(t)>$ onto this set of basis functions $\{\Psi_{j}(t)\}$. Explain what is happening.

2.

Now give an expression for computing f⁽ⁿ⁾(t), the nth derivative of f(t) with respect to t, in terms of the same projections and continuous basis set. (You may assume the existence of all derivatives.) Explain your answer.

3.

Now consider a linear, time-invariant system with impulse-response function h(t), having time-varying input s(t) and time-varying output r(t):

$s(t) \longrightarrow \; \;$ $\; \; \longrightarrow r(t)$

In the case that the input is the complex exponential $s(t)=e^{i\mu_{j}t}$(where $i = \sqrt{-1}$ and $\mu_{j}$ is a constant), what can you say about the output r(t) of such a system?

4.

If the input s(t) has been represented in terms of a set of complex exponentials $\Psi_{j}(t)=e^{i\mu_{j}t}$ as described at the beginning of these problems, is it possible for different complex exponentials (not included in this set) to appear in the output r(t) when it too is represented in terms of complex exponentials? Justify your answer.