I've come across this question in my Signals and Systems class but I can't seem to understand what the answer might be.
Here is the entire question:
Consider a signal x(t) which has its Laplace transform X(s). X(s) has four poles. The signal x(t) contains an impulse at t = 0. What is the minimum number of zeros of X(s)?
What logic should I be using to find the answer?
If the system has 4 poles, the denominator degree is at least 4. It is only at least 4 because the poles may have a multiplicity greater than one.
If the impulse response has an impulse at t=0, the denominator degree needs to be the same as the numerator degree. Otherwise the signal does not have an "proportional part". i.e. no delta at t=0, indicating integrating behavior with numerator degree smaller than denominator degree. Alternatively it has the derivative of the delta at t=0 which makes it differentiating behavior (for numerator degree > denominator degree).
So the numerator degree needs to be at least 4 as well. This means it needs at least one zero (of multiplicity of at least 4) or several zeros of according multiplicity.
If one sticks to all zeros and poles with multiplicity of 1, then it needs exactly 4 zeros (because it has exactly 4 poles).