Consider a causal continuous-time LTI system with transfer function
where . Note that we have two poles (at and ), but no finite zeros. We can view , a 2nd order system, as the cascade connection of two causal 1st order LTI systems
where is the Heaviside step function. If , then the impulse response will eventually decay to zero. However, note that the exponential is multiplied by , which means that there is a transient “peak”. If is positive but “close” to zero, this transient could be quite “wild”! This phenomenon is somewhat similar to resonance, although we are dealing with decaying exponential excitations, not sinusoidal ones. If we apply a Dirac delta impulse to the system depicted above, the output of the first system in the cascade will be the impulse response of
which happens to be also the impulse response of , the second system in the cascade. The lesson is the following: cascading identical LTI systems leads to resonant-like behavior.
If , then we have a simple pole at and another simple pole at . Via partial fraction expansion of , we obtain
and, taking the inverse Laplace transform, we get the impulse response
Note that these are valid only if . If we make , we get pesky indeterminate stuff: and .
Let us see what happens if and are “almost identical”. To be precise, let us investigate the impulse response of the cascade when the (simple) poles at and are almost “on top of each other”. Let , where is “small”. Then, the impulse response of can be written as follows
For convenience, let us define
so that . Taking the Taylor expansion of about , we get
and, taking the limit as approaches zero, we obtain
and, therefore, we finally get
This shows, albeit in a sinfully non-rigorous manner, that the impulse response of approaches the impulse response of as the pole at gets “closer” and “closer” to the pole at . I concede that this result is not terribly exciting…
Let us now perform a numerical experiment! Let , and let , so that approaches as . The plot below illustrates the impulse responses for various values of
where corresponds to the cascade of two identical systems (which creates a double pole at ). Note that, as expected, as approaches zero, the impulse response of the cascade approaches the resonant one, .
Last but not least, here’s the MATLAB script that generates the plot:
% build transfer function of H1 a = 1; H1 = tf(1,[1 a]); % create time grid t = [0:0.01:5]; % compute impulse responses of the cascade h = ; for epsilon = [1 0.5 0.1 0]; % build transfer function of H2 b = a + epsilon; H2 = tf(1,[1 b]); % compute and store impulse response h = [h, impulse(H2 * H1, t)]; end % plot impulse responses figure; plot(t, h(:,1), 'r-', t, h(:,2), 'm-', t, h(:,3), 'b-', t, h(:,4), 'k--'); legend('\epsilon = 1', '\epsilon = 0.5', '\epsilon = 0.1', '\epsilon = 0'); xlabel('t (seconds)'); title('Impulse responses of the cascade');