3 Fourier series (FS) 1.x(t) = 1+cos(2nt) + sin(int), 2。2(t) T=1 T-2. 2-t 1Kts2
Find the trigonometric Fourier series (FS) and the exponential FS of the following: x(t) TT Ana -3т -2n -TT 2TT d) x(t) πι -no -TT 0 TE 2TT exponential FS f(t) = En=-- Cnejnwot Where (n = +S40+" f(t)e-inwot dt trigonometric 30 f(t)=a, + a, cos(no),t)+b, sin(no,t n-1 ao 1 T. 2 to a. So f(t)dt -5° f(t)cos(no),1)dt Sº f(t)sin(no,t)dt oy b 2 T
Find the trigonometric Fourier series (FS) and the exponential FS of the following: 2 *(1) 1.5 1 0.5 O -0.5 1 -1 2 -1 0 2 b) 2 3 4 6 exponential FS Cnejnwot f(t) = En=-00 Where Cn 7Se+ f(t)e-inwot dt trigonometric f(t)= a, +Ža, cos(n6,t)+b, sin(n0,1 ao 1 T. 2 to an S f(t)dt sº f(t)cos(n0,1)dt f(t)sin (no,t)dt To 2 pt b,
(1 point) Find the Fourier series expansion, i.e., f(x) [an cos(170) + by sin(t, x)] n1 J1 0< for the function f(1) = 30 < <3 <0 on - SIST ao = 1 an = cos npix bn = Thus the Fourier series can be written as f() = 1/2
Signal system
8. Consider the periodic signal x(t) = cos(2nt) + cos(2nt) I. a. Find the Fourier series coefficients for this signal. (4 points). b. If this signal passes through a LTI system with the impulse response h(t)=e* u(t), particularly, would the output signal also be a periodic signal ? If so, what would be the Fourier coefficients of the output signal ? (4 points). c. Give the mathematical expression for the output signal y(t). (2 points).
Calculate the Fourier Series coefficients of x(t) = cos(2*pi*1*t) + 2*sin(2*pi*4*t). Based on your results which set of FS coefficients corresponding to the positive side of the spectra is correct. a0=0, a1=1/2, a2=1/j, a3 = 0 a1=1, a2=2, a3 = 0 a1=1/2j, a2=1/2, a3 = 0 a1=1/2, a2=2/2j, a3 = 0
Find the Fourier Transform of the following signals: (a) x(t) = Sin (t). Cos (5 t) (b) x(t) = Sin (t + /3). Cos(5t-5) (c) a periodic delta function (comb signal) is given x(t) = (-OS (t-n · T). Express x(t) in Fourier Series. (d) Find X(w) by taking Fourier Transform of the Fourier Series you found in (a). No credit will be given for nlugging into the formula in the formula sheet.
(30 pts) Consider the following sampling system where the input is x(t) = sin 2nt + cos 3nt r(t) Cp (t) (a) (10 pts) Find and plot the Fourier Transform of x(t) (b) (10 pts) What is the Nyquist frequency and period for sampling? (c) (10 pts) Find and plot the Fourier Transform of xp(t) using the Nyquist rate.
(30 pts) Consider the following sampling system where the input is x(t) = sin 2nt + cos 3nt r(t) Cp (t)...
1. (45 pts) DT FS. Find the fundamental period and the Fourier Series (FS) spectral coefficients for these periodic signals. Sketch the spectrum in magnitude and phase. Express each x[n] as the sum of the spectral coefficients for k = [0, N-1]. a. ?1[?] = ???( ? 3 ?) b. ?2 [?] = cos ( ? 3 ?) + sin( ? 4 ?) c. ?3[?]
Example 2 Find the Fourier Series coefficients for y(t) = sin? 20,1 + cos² ept =
Determine the trigonometric Fourier series coefficients an and bn for signal x(t) sin(3m + 1) + 2 cos(7m-2) . Determine also the signal's fundamental radian frequency wo. No integration is required to solve this cos( ( ? problem