Bài giảng Signal & Systems - Chapter 3: Fourier series representation of periodic signals

Nội dung text: Bài giảng Signal & Systems - Chapter 3: Fourier series representation of periodic signals

1. 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 2 Ef=c Ex+E e. 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+E y. 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 c 1 and c 2, the energies of c 1x(t)+c 2y(t) and c1x(t)-c2y(t) are identical, given by: 2 2 c1 E x +c 2y E Signal & Systems - FEEE, HCMUT – Semester: 02/10-11 1
2. 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: D 1=D -1=2, D 3=(D -3)*=j4. Express f(t) in the form: +∞ t f(t)=∑ Cn cos(ωk+ ϕ n ) n=0 P3.10 . Using the Fourier series analysis to calculate the coefficients Dn for the continuous-time periodic signal 1.5; 0≤ t<1 f(t)=  -1.5; 1≤ t<2 with fundamential frequency ω0=π Signal & Systems - FEEE, HCMUT – Semester: 02/10-11 Ch-3: Fourier series representation of periodic signals P3.11 . Determine the Fourier series representation for each of the periodic signals depicted in Figure P3.11. Signal & Systems - FEEE, HCMUT – Semester: 02/10-11 3
3. Ch-3: Fourier series representation of periodic signals P3.15 . Consider an LTI system with impulse response h(t)=e -4t u(t). Find the Fourier series representation of the output y(t) for each of the following inputs: (a) f(t)=cos(2π t) (b) f(t)=sin(4π t)+cos(6 π t+ π /4) +∞ (c) f(t)=∑ δ (t− n) n= −∞ +∞ (d) f(t)=∑ (− 1)n δ (t − n) n= −∞ (e) f(t) is periodic square wave dipicte d in Figure P3.15 f(t) Figure P3.15 Signal & Systems - FEEE, HCMUT – Semester: 02/10-11 5