It is the natural response of a Linear Time Invariant System, which again is the most beautiful system to look at. Possessing a transfer function and obeying superposition, this system is like a straight forward person. Eigen functions ( Natural responses) ofLTI systems are complex exponential signals. Complex exponential s are the most general exponential s of the form e(σ+jω)t . If the real part is absent, it is an everlasting periodic signal of constant amplitude. Two such signals, the positive rotating and negative rotating combine to form the real everlasting sinusoidal signal which is called the harmonic signal, true to the harmony such signals generate in our hearts. Sum of such sinusoids of many harmonics build up all musical tones, all repetitive signals. So the music in our lives has the roots at the exponential signal.
With a real part which is negative also combined, the sinusoid is exponentially decayed, reducing to 36.8% of it’s initial value in one time constant, given by 1/σ in the above equation, emphasizing again the fact that σ is negative. Obviously, if σ is positive, the signal grows without bounds, and so is useless from our point of view. Unbounded outputs from a system which have bounded inputs applied to them makes the system unstable and hence useless.
If we plot the poles of the transfer function of an LTI system, in the complex ‘s’ plane, we get all the information about the natural, exponential responses of the LTI system. The damped harmonic frequencies of such systems and the rate at which they diminish are known. The inverse of the x- coordinate distance to the pole is the time constant of that frequency or ‘mode’.
The Laplace Transform and the Fourier Transform tools help us to understand the modes of the system, hence enabling us to determine what tunes or exponential modes are natural to the system.
By
Professor P. Dinakaran
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