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Problem 4 -π/3) in quadrature form. 2π A) Express the function Y1 = (2 m) sin(...
- Problem 6. Consider the function y2 m) sin ((10/s) t+π/4). lndicate whether each of the...
- Problem 6. Consider the function y2 m) sin ((10/s) t+π/4). lndicate whether each of the following waveforms is equivalent to y? Briefly justify your answers 1. (2 m) cos(10/s)t+/4) 2. (2 m) cos( (10/s)t+3 T/4) 3. (2m) cos( (10/s)t -/4) 4. (2 m) sin((10/ s) t + π/4 + 4 π) 5.-(2 m) sin ((10/s) t + π /4-3r) 6. (-2m) cos((10/s) t + 12n/4) 7. (v ฐ m) [cos((10/s) t) + sin((10/s) t)] 8. (2 m) cos ((-10/s)...
Problem 6. Consider the function y=(2m) sin ((10/s) t+π/4). Indicate whether each of the following waveforms...
Problem 6. Consider the function y=(2m) sin ((10/s) t+π/4). Indicate whether each of the following waveforms is equivalent to y? Briefly justify your answers. 1. (2 m) cos((10/s)1+π/4) 2. (2 m) cos((10/s) t +3r/4) 3. (2 m) cos((10/s) t-r/4) 4. (2 m) sin(10/s)t +/4+4T) 5.-(2 m) sin ((10/s)t+/4-3m) 6. (-2m) oos((10/s) t+12r/4) 7. (VZn) [cos((10/s) t) + sin((10/s)t)] 8. (2m) cos ((-10/s)t+r/4) m) 1-cos((10/s) t + π)-sin((10/s) t-π
Problem 5. (20 pts) In this problem, I wil lead you through a procedure that allows...
Problem 5. (20 pts) In this problem, I wil lead you through a procedure that allows you to write the function f(z) = (2 m) cos(kx-π / 5) + (3 m) sin(kx + π / 4) in cosine form. This method can be extended to write the sum of any frequency as a single cosine function inusoidal f unctions of the same spatia A) Write (2 m) cos(ka-π/5) in quadrature form B) write (3 m) sin(kx+π/4) in quadrature form. C)...
5. (a) (6) Carefully sketch the odd periodic extension, of period 2m, of the function f(x)1, 0 < x < π. (Only sketch over the interval z E [-2π, 2π). (b) (10) Find the Fourier sine series o...
5. (a) (6) Carefully sketch the odd periodic extension, of period 2m, of the function f(x)1, 0 < x < π. (Only sketch over the interval z E [-2π, 2π). (b) (10) Find the Fourier sine series of the function in part (a)
5. (a) (6) Carefully sketch the odd periodic extension, of period 2m, of the function f(x)1, 0
(1 point) Suppose you're given the following Fourier coefficients for a function on the interval [-π, π : ao = 2, ak = 0 for k 2 i, and for k > 1. Find the following Fourier approximations to...
(1 point) Suppose you're given the following Fourier coefficients for a function on the interval [-π, π : ao = 2, ak = 0 for k 2 i, and for k > 1. Find the following Fourier approximations to the Fourier series a0 + 〉 ,(an cos(nz) + bn sin(nx)) bk = F, (z) = F,(z) = Fs(x)
(1 point) Suppose you're given the following Fourier coefficients for a function on the interval [-π, π : ao = 2, ak...
Problem 2 x < π; f(x)-x-2π when π Function f(x) =-x when 0 f(x + 2π)...
Problem 2 x < π; f(x)-x-2π when π Function f(x) =-x when 0 f(x + 2π) = f(x). x < 2π. Also 1. draw the graph of f(x) 2. derive Fourier series
2. Set n 100 and generate and plot the time series xt 2 cos(2π.06t) + 3 sin(2π.06t) Ý,-4 cos(2n. ...
Time series analysis
2. Set n 100 and generate and plot the time series xt 2 cos(2π.06t) + 3 sin(2π.06t) Ý,-4 cos(2n. 10t) + 5 sin(2m10) z, 6 cos(2π·40t) + 7 sin(2π·40t) (a) Use the periodogram function in R to plot the periodogram of Vi. Can you explain the spikes? (b) Now let wi ~ N(0, 25) be iid and plot the periodogram of the series V +w. Does it still pick out the periodic components?
2. Set n 100...
Problem 7. (20 pts) A) Find all of the values of θ for which sin θ-1/4....
Problem 7. (20 pts) A) Find all of the values of θ for which sin θ-1/4. B) Find all of the values of θ for which cos θ-1/4. C) Find all of the values of x for which sin(うー) = 1 /4 D) Find all of the values of z at which the function y -(l m) sinl 2m" -T/3 ) has the value y-^. 2π 2 7T
2. Find the CTFT for the following PERIODIC signals: a. xdt) = sin(2t + π/4)) b....
2. Find the CTFT for the following PERIODIC signals: a. xdt) = sin(2t + π/4)) b. Xb(t) = 2 + cos(2π/3 t) + 4sin(5π/3 t)
Supposez1 =4 cos 3 +isin 3 andz2 =2 cos 6 +isin 6 . Computez1z2. (a) 8(cos?π?+isin?π?)...
Supposez1 =4 cos 3 +isin 3 andz2 =2 cos 6 +isin 6 .
Computez1z2.
(a) 8(cos?π?+isin?π?) 22
(b) 4(cos?4π?+isin?4π?) 66
(c) 2(cos?π?+isin?π?) 66
(d) cos(π)+isin(π)
(e) 8(cos?π?+isin?π?)
66
17. Suppose z1 = 4 (cos (1) + i sin (5)) and z2 = 2 (cos () + i sin (7)). Compute z122. (a) 8(cos (7) + i sin (7)) (b) 4(cos (4) + i sin (*)) (c) 2(cos (7) + i sin ()) (d) cos(T) + i sin(TT) (e) 8(cos (7)...
- Problem 6. Consider the function y2 m) sin ((10/s) t+π/4). lndicate whether each of the following waveforms is equivalent to y? Briefly justify your answers 1. (2 m) cos(10/s)t+/4) 2. (2 m) cos( (10/s)t+3 T/4) 3. (2m) cos( (10/s)t -/4) 4. (2 m) sin((10/ s) t + π/4 + 4 π) 5.-(2 m) sin ((10/s) t + π /4-3r) 6. (-2m) cos((10/s) t + 12n/4) 7. (v ฐ m) [cos((10/s) t) + sin((10/s) t)] 8. (2 m) cos ((-10/s)...
Problem 6. Consider the function y=(2m) sin ((10/s) t+π/4). Indicate whether each of the following waveforms is equivalent to y? Briefly justify your answers. 1. (2 m) cos((10/s)1+π/4) 2. (2 m) cos((10/s) t +3r/4) 3. (2 m) cos((10/s) t-r/4) 4. (2 m) sin(10/s)t +/4+4T) 5.-(2 m) sin ((10/s)t+/4-3m) 6. (-2m) oos((10/s) t+12r/4) 7. (VZn) [cos((10/s) t) + sin((10/s)t)] 8. (2m) cos ((-10/s)t+r/4) m) 1-cos((10/s) t + π)-sin((10/s) t-π
Problem 5. (20 pts) In this problem, I wil lead you through a procedure that allows you to write the function f(z) = (2 m) cos(kx-π / 5) + (3 m) sin(kx + π / 4) in cosine form. This method can be extended to write the sum of any frequency as a single cosine function inusoidal f unctions of the same spatia A) Write (2 m) cos(ka-π/5) in quadrature form B) write (3 m) sin(kx+π/4) in quadrature form. C)...
5. (a) (6) Carefully sketch the odd periodic extension, of period 2m, of the function f(x)1, 0 < x < π. (Only sketch over the interval z E [-2π, 2π). (b) (10) Find the Fourier sine series of the function in part (a)
5. (a) (6) Carefully sketch the odd periodic extension, of period 2m, of the function f(x)1, 0
(1 point) Suppose you're given the following Fourier coefficients for a function on the interval [-π, π : ao = 2, ak = 0 for k 2 i, and for k > 1. Find the following Fourier approximations to the Fourier series a0 + 〉 ,(an cos(nz) + bn sin(nx)) bk = F, (z) = F,(z) = Fs(x)
(1 point) Suppose you're given the following Fourier coefficients for a function on the interval [-π, π : ao = 2, ak...
Problem 2 x < π; f(x)-x-2π when π Function f(x) =-x when 0 f(x + 2π) = f(x). x < 2π. Also 1. draw the graph of f(x) 2. derive Fourier series
Time series analysis
2. Set n 100 and generate and plot the time series xt 2 cos(2π.06t) + 3 sin(2π.06t) Ý,-4 cos(2n. 10t) + 5 sin(2m10) z, 6 cos(2π·40t) + 7 sin(2π·40t) (a) Use the periodogram function in R to plot the periodogram of Vi. Can you explain the spikes? (b) Now let wi ~ N(0, 25) be iid and plot the periodogram of the series V +w. Does it still pick out the periodic components?
2. Set n 100...
Problem 7. (20 pts) A) Find all of the values of θ for which sin θ-1/4. B) Find all of the values of θ for which cos θ-1/4. C) Find all of the values of x for which sin(うー) = 1 /4 D) Find all of the values of z at which the function y -(l m) sinl 2m" -T/3 ) has the value y-^. 2π 2 7T
2. Find the CTFT for the following PERIODIC signals: a. xdt) = sin(2t + π/4)) b. Xb(t) = 2 + cos(2π/3 t) + 4sin(5π/3 t)
Supposez1 =4 cos 3 +isin 3 andz2 =2 cos 6 +isin 6 .
Computez1z2.
(a) 8(cos?π?+isin?π?) 22
(b) 4(cos?4π?+isin?4π?) 66
(c) 2(cos?π?+isin?π?) 66
(d) cos(π)+isin(π)
(e) 8(cos?π?+isin?π?)
66
17. Suppose z1 = 4 (cos (1) + i sin (5)) and z2 = 2 (cos () + i sin (7)). Compute z122. (a) 8(cos (7) + i sin (7)) (b) 4(cos (4) + i sin (*)) (c) 2(cos (7) + i sin ()) (d) cos(T) + i sin(TT) (e) 8(cos (7)...
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