Ch-3: Fourier series representation of periodic signals P3.1 For the signals f(t) and x(t) depicted in Figure P3.1, find the component of the form x(t) contained in f(t) In other words find the optimum value of c in the approximation f(t)≈cx(t) so that the error signal energy is minimum Find the error signal e(t) and it energy Ee Show that the error signal is orthogonal to x(t), and that Ef=c2Ex+Ee Can you explain this result in terms of vector? Signal & Systems - FEEE, HCMUT – Semester: 02/10-11 Ch-3: Fourier series representation of periodic signals P3.2 Repeat P3.1 if x(t) is sinusoid pulse shown in Figure P3.2 P3.3 If x(t) and y(t) are orthogonal, then show that the energy of the signal x(t)+y(t) is identical to the energy of the signal x(t)-y(t) and is given by Ex+Ey Explain this result using vector concepts In general, show that for orthogonal signal x(t) and y(t) and for any pair of arbitrary constant c1 and c2, the energies of c1x(t)+c2y(t) and c1x(t)-c2y(t) are identical, given by: c12 E x +c 22 E y Signal & Systems - FEEE, HCMUT – Semester: 02/10-11 Ch-3: Fourier series representation of periodic signals P3.4 Figure P3.4(a) shows the first eight functions in Walsh function set Represent f(t) in Figure P3.4(b) over interval [0,1] using a Walsh Fourier series using basis functions Compute the energy of e(t), the error in the approximation using the first N non-zero terms in the series for N=1, 2, and Signal & Systems - FEEE, HCMUT – Semester: 02/10-11 Ch-3: Fourier series representation of periodic signals P3.5 Determine the cross-correlation of the following signals: x(t)=rect( 2Tt ), and y(t)=e-at u(t);a>0 P3.6 Determine the cross-correlation of the following signals: x(t)=e-t u(t), and y(t)=e-2t u(t); a > P3.7 Consider the signal f(t)=rect(t-1/2) Determine its autocorrelation function and its energy using this function P3.8 Find the autocorrelation of the signal f(t)=cos(ω0t)rect(t/T) Signal & Systems - FEEE, HCMUT – Semester: 02/10-11 Ch-3: Fourier series representation of periodic signals P3.9 A continuous-time periodic signal f(t) is real valued and has a fundamental period T=8 The nonzero Fourier series coefficients fo f(t) are: D1=D-1=2, D3=(D-3)*=j4 Express f(t) in the form: +∞ f(t)= ∑ C n cos(ωk t + ϕn ) n =0 P3.10 Using the Fourier series analysis to calculate the coefficients Dn for the continuous-time periodic signal 1.5; ≤ t