Question

A signal x(t)has Fourier transform X(ω). Determine the Fourier transform Y(ω) in terms of X(ω) for each of the following signals y(t):

a). y(t) = 1/5x(-2t + 3)

Answer 1

The Fourier transform Y(ω) of the signal y(t) = 1/5x(-2t + 3), given X(ω) as the Fourier transform of x(t), is determined by applying scaling and shifting properties and is Y(ω) = (1/10)e^{j3ω/2}X(ω/2).

Step-by-step explanation:

The given problem concerns the computation of the Fourier transform of a modified signal y(t) when the Fourier transform X(ω) of the original signal x(t) is known. For the modified signal y(t) = 1/5x(-2t + 3), factors like time-scaling and time-shifting play a role in determining the new Fourier transform, which we will denote by Y(ω).

For a signal x(t) modified by scaling by a factor a, the Fourier transform scales by 1/|a| in the frequency domain (with a potential change in sign depending on the direction of time scaling). Time-shifting adds a linear phase term to the Fourier transform. Thus, for y(t) = 1/5x(-2t + 3), the Fourier transform is Y(ω) = (1/10)e^{j3ω/2}X(ω/2), considering the scaling and time-shifting properties of the Fourier transform.