Question

Derive an expression for the FT of e ʲᵃ⁽ᵗ−ᵗ ⁰​ x(at), in terms of X(jω) and other relevant variables. Assume that a is real and greater than zero.

Answer 1

The Fourier Transform of ejatx(at) is derived using the time scaling property and the modulation theorem of FT, resulting in the expression 1/|a|X(j(ω-a)).

Step-by-step explanation:

The student's question asks for the derivation of an expression for the Fourier Transform (FT) of ejatx(at), where x(at) is a function of time scaled by a real and positive constant a, and shifted by t0. Utilizing the properties of FT, the expression for the FT of ejatx(at) in terms of X(jω) and other relevant variables can be derived using the time scaling property and the modulation theorem.

First, the time scaling property states that the FT of x(at) where a is a scaling factor, is 1/|a|X(jω/a). Second, the modulation theorem indicates that the FT of ejω0tx(t) is equivalent to a shift in frequency domain by ω0, which in our equation is replaced by a, leading to X(j(ω-a)).

Therefore, combining these properties, the required FT is 1/|a|X(j(ω-a)), assuming a > 0. This represents a scaled and shifted version of the original FT of x(t) by the factor a, which includes time scaling and frequency shifting effects as per the given modulation and scaling.