Question

(a) Determine the Fourier transformation of the signal A(t) defined as $\Lambda(t) = \begin{cases} t+1, & -1 \le t \le 0; \\ -t+1, & 0 \le t \le 1; \\ 0, & \text{otherwise.} \end{cases}$ (b) (i) State the scaling property of Fourier transformation. (ii) Hence or otherwise find the Fourier transformation of the function $\sin(8\pi f_0 t)$ given that Fourier transformation of $\sin(\pi t) = \frac{j}{2} \left[ \delta(f + \frac{1}{2}) - \delta(f - \frac{1}{2}) \right]$ where $\delta$ denote the Unit Impulse function:

          (a) Determine the Fourier transformation of the signal A(t) defined
as
$\Lambda(t) = \begin{cases} t+1, & -1 \le t \le 0; \\ -t+1, & 0 \le t \le 1; \\ 0, & \text{otherwise.} \end{cases}$
(b) (i) State the scaling property of Fourier transformation.
(ii) Hence or otherwise find the Fourier transformation of the
function $\sin(8\pi f_0 t)$ given that Fourier transformation of
$\sin(\pi t) = \frac{j}{2} \left[ \delta(f + \frac{1}{2}) - \delta(f - \frac{1}{2}) \right]$ where $\delta$ denote the Unit
Impulse function:

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(a) Determine the Fourier transformation of the signal A(t) defined
as
Λ(t) = t+1, -1 ≤ t ≤ 0;
-t+1, 0 ≤ t ≤ 1;
0, otherwise.
(b) (i) State the scaling property of Fourier transformation.
(ii) Hence or otherwise find the Fourier transformation of the
function sin(8π f0 t) given that Fourier transformation of
sin(π t) = (j)/(2)[ δ(f + (1)/(2)) - δ(f - (1)/(2)) ] where δ denote the Unit
Impulse function:

Added by Brooke W.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sri K

08/01/2023

Step 1

75 - |t|, t ∈ [-1, 1] otherwise. To find the Fourier transformation of A(t), we need to evaluate the integral: F(A(t)) = ∫[from -∞ to ∞] A(t) * e^(-i2πft) dt Since A(t) is defined piecewise, we need to split the integral into two parts: F(A(t)) = ∫[from -1 to Show more…

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Texts: a Determine the Fourier transformation of the signal At defined as A(t) = 0.75 - |t|, t ∈ [-1, 1] otherwise. (b) i State the scaling property of Fourier transformation. ii Hence or otherwise, find the Fourier transformation of the function sin(8πft) given that the Fourier transformation of sin(xt) is δ(f - x) - δ(f + x), where δ denotes the Dirac Delta function.

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