2.w? -/6.455J Sonar, Radar and Seismic Signal Processing Review quiz The following problems are indicative of the aspects of linear systems, Fourier trans:forms and probability which are expected as prerequisites for this subject. Please complete them; if you have difficulty, please contact me. These problems are for your information and will not be part of your grade. 1) Linear Systems - impulse response, convolution and correlation A linear, time invariant system has an impulse response, h(t), given by T h(t) = ?; It- To[ < 2 0 otherwise An input signal x(t) has the form a) Find and sketch the system output, y(t). b) Find and sketch the correlation function %(T) = J'? y(t)y(t - r)dt. 2) Fourier transforms a) A signal x(t) has the form (U(t) is the unit step function.) Find and sketch the Fourier transform, X(f), of the signal. b) The signal x(t) is the input to an LTI system with transfer fucnction j27r f - a H(f) = j2af + a , forall f Find the energy spect-um S,( f) = [Z(f)l2 of the system output, r(t).3) Probability T.heory a) A random variable x has the probability density function 0 elsewhere i) Find the mean and variance of the random variable, y, iij Fi.nd the probability density, p,(Y). b) The random variables xl and x2 are statistically independent and both have the same probability density px(X) as described in part (a). Find the characteristic functron, Md(jv) = E(ejvd) of the difference d = XI -