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Consider a continuous-time, real-valued signal f(t) whose Fourier transform F(ω) = ∫_(-∞)^(∞) f(t) exp(−j ω t) dt exists. Which one of the following statements is always TRUE?

1. |F(ω)| ≤ ∫_(-∞)^(∞) |f(t)| dt
2. |F(ω)| > ∫_(-∞)^(∞) |f(t)| dt
3. |F(ω)| ≤ ∫_(-∞)^(∞) f(t) dt
4. |F(ω)| ≥ ∫_(-∞)^(∞) f(t) dt

Correct answer: |F(ω)| ≤ ∫_(-∞)^(∞) |f(t)| dt

Solution

The correct option is true due to the triangle inequality in the context of integrals, which states that the absolute value of the integral of a function is less than or equal to the integral of the absolute value of that function. This means that the magnitude of the Fourier transform |F(ω)| cannot exceed the total integral of the absolute value of the signal f(t).

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