Question

(3). Consider a periodic triangle waveform signal x(t) as shown below. x(t) 1 -2 -1 0 1 2 3 4 t -1 ... (a). Find the trigonometric-form Fourier representation of x(t). (b). Define $y(t) = \frac{d \ x(t)}{dt}$. Find the trigonometric-form Fourier series coefficients of y(t). Comment on your results. (c). Derive the complex-exponential Fourier series of y(t) based on the results obtained in (b). (d). Show that the Fourier series result of y(t) derived in (c) can be also obtained by properly shifting and scaling of the pulse train f(t) as described in the notes (page 82).

          (3). Consider a periodic triangle waveform signal x(t) as shown below.
x(t)
1
-2 -1  0  1  2  3  4 t
-1
...
(a). Find the trigonometric-form Fourier representation of x(t).
(b). Define $y(t) = \frac{d \ x(t)}{dt}$. Find the trigonometric-form Fourier series coefficients of y(t).
Comment on your results.
(c). Derive the complex-exponential Fourier series of y(t) based on the results obtained in (b).
(d). Show that the Fourier series result of y(t) derived in (c) can be also obtained by properly
shifting and scaling of the pulse train f(t) as described in the notes (page 82).

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(3). Consider a periodic triangle waveform signal x(t) as shown below.
x(t)
1
-2 -1 0 1 2 3 4 t
-1
...
(a). Find the trigonometric-form Fourier representation of x(t).
(b). Define y(t) = (d x(t))/(dt). Find the trigonometric-form Fourier series coefficients of y(t).
Comment on your results.
(c). Derive the complex-exponential Fourier series of y(t) based on the results obtained in (b).
(d). Show that the Fourier series result of y(t) derived in (c) can be also obtained by properly
shifting and scaling of the pulse train f(t) as described in the notes (page 82).

Added by Kelsey R.

Fundamentals of Electric Circuits

Charles K. Alexander, Matthew N.O.… 3rd Edition

Chapter 17

Instant Answer

Solved by Expert Stanley Enemuo

Step 1

From the graph, we can observe that x(t) has a period T = 4, with the following values: - x(t) increases linearly from 0 to 1 over [0, 1] - x(t) decreases linearly from 1 to -1 over [1, 3] - x(t) increases linearly from -1 to 0 over [3, 4] Show more…

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Consider the periodic triangle waveform signal x(t) as shown below: x(t). Find the trigonometric-form Fourier representation of x(t). Define Y(t) = ? Find the trigonometric-form Fourier series coefficients of y(t). Comment on your results. Derive the complex-exponential Fourier series of x(t) based on the results obtained in (6). Show that the Fourier series result of x(t) derived in (c) can also be obtained by properly shifting and scaling the pulse train f(t) as described in the notes (page 82).

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