1. (50 points) Consider the following RC circuit: Vin(t) ME Vow It) Let the input voltage be denoted Vin(t) and the output voltage vout(t). The impulse response of this circuit (which we will derive later in the course) is 1 h() = (1) RC where the product RC is known as the time constant of the system. In the following, please compute the output voltage Vout (t) as a function of the input voltage Vin(). Hint: use properties of LTI systems when you can to avoid having to make all the calculations for each part. Be lazy and efficient! (a) Compute Nout (t) for Vin() = cos(wt) where w is a given frequency in rad/s. Hint: Eulers will actually make the convolution ier. (b) Compute vout () for Vin(t) = u(t). This is also called the step response. (c) Compute Vout(t) for vin(t) = 2ut - 1). This is also called the step response. (d) Compute vout(t) for vin(t) = et/2u(1 t). (e) Compute Vout(t) for vin(t) = 10rect(t-1/2) where the rectangle function is defined as rect(t) = 1 for t e [-1/2,1/2] (f) Do you prefer convolving with cosine functions or rectangle functions and why? (g) Compute Vout() for the square wave 00 Vin(t) = rect(t 2RCk 1/2). (2) k-00 (h) Compute vout () for Win(t) = cos(250ml) + rect(t). (i) Plot each of the above output voltages in MATLAB. You may use a subplot arranged in a 2 grid. Assume that w = 100 rad/s for the first part