#### cnh1995

Homework Helper

Gold Member

- 3,257

- 1,029

As per my understanding from my textbook and notes, the s in Laplace transform is called complex frequency and is given by s=σ+jω, where the real part σ tells how fast the signal decays(exponential part) and jw corresponds to the frequency of oscillations (sinusoidal part) of the signal.

So, if f(t)=1, F(s) is 1/s or 1/(σ+jω).

If σ= -1 and w= ±1, we get |F(s)|= 0.707 and phase angle Φ= -45°. This means the signal f(t)=1 contains a an exponentially decaying sinusoidal signal of time constant 1 whose peak magnitude is 0.707 and frequency of oscillations of this decaying signal is 1 Hz. Is this correct?

If yes, does this mean if I take "all" the possible values of σ and w and compute the magnitudes and phase angles for each 1/s term (God knows how), and add all of them, will that addition result in f(t)=1?

Please correct me if I am wrong. Thanks a lot in advance.