Center of Gravity

In this problem, the following are Fourier transform pairs:

$$\begin{array} {c} s(t) \leftrightarrow S(f) \\ h(t) \leftrightarrow H(f) \\ y(t) \leftrightarrow Y(f)\end{array}$$

1.

Show that the following is a Fourier transform pair

$$t \cdot s(t) \leftrightarrow \frac{\mbox{j}}{2\pi} S^{\prime}(f) = \frac{\mbox{j}}{2\pi} \frac{dS(f)}{df}.$$

2.

The ``center of gravity'' for a signal s(t) is defined as

$$t_s = \frac{\int_{-\infty}^{\infty} t \cdot s(t)\, dt} {\int_{-\infty}^{\infty} s(t)\, dt}.$$

Express t_s in terms of S(f) (and it's derivative).

3.

Let s(t) be the input to a linear, time-invariant system with impulse response h(t). The output of the system is y(t). The ``centers of gravity'' of s(t), h(t), and y(t) are t_s, t_h, and t_y, respectively. Find an expression for t_y in terms of t_s and t_h.

Hint: Remember, $\int_{-\infty}^{\infty} s(t)\,dt = S(0)$.