Fourier transform and steady state solution?

[hanson]

Hi all!
I am asking about a question about Fourier transform.
I can only roughly remember things about Fourier transform.
I am told that Fourier transform gives the steady state solution, is it?
I can hardly relate these two concepts.
Can someone try to explain?
Many thanks.

 

[matematikawan]

May be http://users.rowan.edu/~polikar/WAVELETS/WTpart1.html" might give a clue in understanding.

Let say that we have a signal
$$f(t) = cos(20\pi t) + cos(50\pi t) + cos(100\pi t) + cos(200\pi t)$$
which is a stationary signal (steady state I presume). The Fourier transform of this signal will identify that this signal has frequencies of 10, 25, 50, and 100 Hz at any given time instant.
Next consider another signal of period 1s,

$$g(t)=\left\{\begin{array}{cc}cos(200\pi t),&\mbox{ if } 0\leq t < 0.3 \\ cos(100\pi t), & \mbox{ if } 0.3 \leq t < 0.6 \\ cos(50\pi t), & \mbox{ if } 0.6 \leq t < 0.8 \\ cos(20\pi t), & \mbox{ if } 0.8 \leq t < 1 \end{array}\right$$

Signal g(t) is a transient signal. But the Fourier transforms of g(t) and f(t) are almost identical. So given some coeffients, the Fourier transform will identify the signal as f(t) the stationary signal.
To analyse the second signal we use other transform e.g. wavelet transform.

What am I writing here?