Define the linearity property of fourier transform

Question

Assignment:

1) Compute the Fourier transform for x(t) = te-tu(t).

• X(ω) =1 / 1+ω2-j2ωw
• X(ω) = 1 / jw+1
• X(ω) = 1 / 1-ω2+j2ω
• X(ω) = 1 / (jw-1)2

2) The linearity property of the Fourier transform is defined as:

• X(t) ↔ 2πx(-ω)
• x(t-c) ↔ X(ω)e-jωc
• Ax(t) + bv(t) ↔ aX(ω) + bV(ω)
• ∫t-∞x(λ)dλ ↔ 1 / jω X(ω) + πX(0)δ(ω)

3) Determine the exponential Fourier series for:

x(t) = _k=-∞ Σ^∞δ( t- kT )

• x(t) = _k=-∞ Σ^∞ e ^{j 2 πk t/ T}
• x(t) = _k=-∞ Σ^{∞1e/ T jt}
• x(t) = k=-∞ Σ^{∞ 1e/ T j 2πk t/ T}      
• x(t) = _k=0 Σ^{t 1e/ t j 2πk t/ T}    

4) Using complex notation, combine the expressions to form a single sinusoid for:

 cos (10t + π/ 2) + 2cos ( 10t - π/ 3)

• 2 cos(10t - π/6)
• 1.239 cos(10t-0.6319)
• 1.475 cos(10t-0.5231)
• 0.6319 cos(10t+ 1.239)

5) The polar notation for the function 1 + ej4 is:

• cos(2) + 1
• e^j2[2cos(2)]
• e-^j4sin(2)
• e^j4sin(4)

6 ) The duality property of the Fourier transform is defined as:

• X(t) ↔ 2πx(-ω)
• x(t-c) ↔ X(ω)e-^jωc
• ax(t) + bv(t) ↔ aX(ω) + bV(ω)
• ^∫t-∞^x(λ)dλ ↔ ^{1 x(ω)}/ jω + πX(0)δ(ω)

7 ) A continuous time signal x(t) has the Fourier transform:

X(ω) = 1 / jωw+b

where b is a constant. Determine the Fourier transform for v(t) = t²x(t).

• V(ω) = 2/ (jω+b)³
• V(ω) = -∞ Σ^{∞ 1 e/ T jω}
• V(ω) = 1/ (jω+b)²
• V(ω = ₀Σ^{n e j 2πk}/ T

8 ) Compute the inverse Fourier transform for X(ω) = cos4ω.

• x(t) = π[δ(t + 4) - δ(t - 4)]
• x(t) = π/2[δ(t + 4) - δ(t - 4)]
• x(t) = 1/2[δ(t + 4) - δ(t - 4)]
• x(t) = 1/4[δ(t + 4) - δ(t - 4)]

9 ) A continuous time signal x(t) has the Fourier transform:

X(ω) = 1/ jω+b

where b is a constant. Determine the Fourier transform for v(t) = x(t) * x(t).

• V(ω) = -2ω/ jω+2b
• V(ω) = 1/ (b+jω)²
• V(ω) = 1/ (b-jω)²
• V(ω) = -ω²/ jω+b

10 )The polar notation for the function 1 + e^j4 + e^j2 is:

• cos(2) + 1
• ej²[2cos(2)]
• e-^j4sin(2)
• e^j2(1+2cos(2))