System Impulse Response of a Sinusoidal Input

When a sinusoidal input signal of the form X(t) = sin(pi*t) is applied to a system, the output signal Y(t) can be determined by convolving the input signal with the impulse response of the system. The impulse response of a system is defined as the output of the system when an impulse signal (i.e. a signal that is zero everywhere except at t=0 where it has an infinitesimal value) is applied to the system.

Convolution of the Input Signal with the Impulse Response

To determine the impulse response of the system for the given input signal, we need to convolve X(t) with the system's impulse response h(t). The convolution integral can be written as:

Y(t) = X(t) * h(t) = integral from -infinity to infinity of X(tau)*h(t-tau) d(tau)

Substituting the given input signal and the impulse response of the system, we get:

Y(t) = sin(pi*t) * sin(pi*t) = 0.5 - 0.5*cos(2*pi*t)

Explanation of the Impulse Response

The resulting impulse response of the system is a cosine signal with a frequency of 2*pi, which is the double of the frequency of the input signal. The cosine signal has a DC offset of 0.5, which means that the system has a non-zero output even when there is no input signal. This is due to the fact that the input signal is not symmetric around t=0, and thus the system responds differently to positive and negative parts of the input signal.

Conclusion

In summary, when a sinusoidal input signal of the form X(t) = sin(pi*t) is applied to a system with an impulse response h(t) = sin(pi*t), the resulting output signal Y(t) is given by Y(t) = 0.5 - 0.5*cos(2*pi*t). The impulse response of the system is a cosine signal with a frequency of 2*pi and a DC offset of 0.5.