A complex sinusoid is denoted: y(t) = e(o+jw)t First compute a vector representing time from 0...

 

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A complex sinusoid is denoted: y(t) = e(o+jw)t First compute a vector representing time from 0 to 10 seconds in about 500 steps (You can use np.linspace). Use this vector to compute a complex sinusoid with a period of 2 seconds, and a decay rate that reduces the signal level at 10 seconds to 1/3 its original value. What o and w did you choose? Evaluate (y(t) as shown above i.e. y(t) = e(o+jw)t Hint: to define complex number e(5+6j) y = np.exp (5 + 1j*6) (b) (7 points) Task 2 Use the np.real (y) and np. imag (y) Python functions to extract the real and imagi- nary parts of the complex exponential. i. (5 points) Plot them as a function of time (plot them separately, you can use subplot for this task). This should look more reasonable. Label your axes, and check that your signal has the required period and decay rate. ii. (2 points) Plot the imaginary component of y(t) as a function of real component of y(t). What can be inferred from the plot? Hint: Comment on the shape of the plot and what we can infer about the envelope of y(t) from the shape? (c) (5 points) Task 3 Use the np.abs() and np.angle() functions to plot the magnitude and phase angle of the complex exponential (plot them in the same figure). Scale the np. angle() plot by dividing it by 2*pi so that it fits well on the same plot as the np.abs() plot (i.e. plot the angle in cycles, instead of radians, the function np. angle (x) returns the angle in radians). Feel free to also explore and visualize the change in the wave-forms for different sigma and omega values. A complex sinusoid is denoted: y(t) = e(o+jw)t First compute a vector representing time from 0 to 10 seconds in about 500 steps (You can use np.linspace). Use this vector to compute a complex sinusoid with a period of 2 seconds, and a decay rate that reduces the signal level at 10 seconds to 1/3 its original value. What o and w did you choose? Evaluate (y(t) as shown above i.e. y(t) = e(o+jw)t Hint: to define complex number e(5+6j) y = np.exp (5 + 1j*6) (b) (7 points) Task 2 Use the np.real (y) and np. imag (y) Python functions to extract the real and imagi- nary parts of the complex exponential. i. (5 points) Plot them as a function of time (plot them separately, you can use subplot for this task). This should look more reasonable. Label your axes, and check that your signal has the required period and decay rate. ii. (2 points) Plot the imaginary component of y(t) as a function of real component of y(t). What can be inferred from the plot? Hint: Comment on the shape of the plot and what we can infer about the envelope of y(t) from the shape? (c) (5 points) Task 3 Use the np.abs() and np.angle() functions to plot the magnitude and phase angle of the complex exponential (plot them in the same figure). Scale the np. angle() plot by dividing it by 2*pi so that it fits well on the same plot as the np.abs() plot (i.e. plot the angle in cycles, instead of radians, the function np. angle (x) returns the angle in radians). Feel free to also explore and visualize the change in the wave-forms for different sigma and omega values.

Answer rating: 100% (QA)

Answer 1 given time period T2 therefore for decay rat... View the full answer