Section 5.1

5.1. Use the definition of the Fourier transform (5.1) to find the transform of the following time signals:

(a) f(t) = (1 - e^-bt)u(t)
(b) f(t) = A cos (ω₀t + Φ)
(c) f(t) = e^atu(-t), a > 0
(d) f(t) = Cδ (t + t₀)

5.2 Find the Fourier transform for each of the following signals, using the Fourier integral:

(a) x(t) = A[u(t) - u(t - b)]
(b) x(t) = e^-t[u(t) - u(t - 5)]
(c) x(t) At[u(t) - u(t - b)]
(d) x(t) = 3 cos (3Πt - rect(t/3)

Section 5.2

5.3. Use the table of Fourier transforms (Table 5.2) and the table of properties (Table 5.1) to find the Fourier transform of each of the signals listed in Problem 5.1. Do not use th Fourier integral (5.1).

5.4.. Use the table of Fourier transforms (Table 5.2) and the table of properties (Table 5.1) to find the Fourier transforms of each of the signals in Problem 5.2.

5.5 The Fourier transform of 13 cos(coot) is given in Equation (5.11). Derive the Fourier transform of 2 sin( lot) using:
(a) The differentiation property.
(b) The time-shift property.

5.6. Prove mathematically that the following properties of the Fourier transform described in Table 5.1 are valid:
(a) Linearity
(b) Time shifting
(c) Duality
(d) Frequency shifting
(e) Time differentiation

5.7. Find the Fourier transform of the following signals:

(a) x(t) = sinc²t
(b) x(t) = e^-4|t|
(c) x(t) = sinc(at/2)
(d) x(t) = 4/(4-jt)²

5.9. Find and sketch the Fourier transform of the following time-domain signals.

(a) Ae^-βt cos ( ω₀t)u(t), R_e{β} > 0
(b) 6 sinc(0.5t)
(c) A sin(ω₁t) + B cos(ω₂t)
(d) 3 rect[(t - 2)/6]
(e) 4 sin(50t)[u(t) - u(t - 2)]
(f) 6tri[(t - 3)/4]
(g) 9 sinc²(12t)

5.11 Given

e-|t| ↔^ƒ 2/(ω² +1)

find the Fourier transform of the following:

(a) d/dt.e^-|t|

(b)1/2Π(t² + 1)

(c) 4 cos(2t)/(t² +1)

Attachment:- fourier 5.11.rar