Whenever I am given a system type with a certain transfer function, I am always asked to proceed to find out the 1. impulse response and 2. step response of the system. My question is, what is so special about these two response types that other responses, like for example parabolic response, is not what I am asked to find out?
Answer
Actually, when you apply impulse signal to any LTI system, the output you get is 'impulse response'. Similarly, when input to LTI system is step signal, the output it produces is known as 'step response'.
Now, sampled version of any signal can be represented as the product of original continuous time signal with shifted version of unit impulse signal (Sifting Property). Hence, the response of LTI system to any input signal is nothing but convolution of input signal & impulse response of LTI system. Hence, impulse response is very important practically. That's why they usually ask about finding impulse response of LTI system.
For mathematical details, you may want to look at this.
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