Final answer:
The continuous-time signals for the transforms given are obtained using the inverse Fourier transform, leading to two cosine functions with different frequencies for the second part of the transform.
Step-by-step explanation:
To determine the continuous-time signal corresponding to each of the given transforms, we need to use the inverse Fourier transform. The first part of the transform, X(jw) = cos(4w + π/3), does not correspond to a standard Fourier transform pair and may be a typo or an irrelevant part of the question. However, the second part X(jw) = 2[δ(w−1)−δ(w+1)] + 3[δ(w−2π)+δ(w+2π)] indicates the presence of impulses in the frequency domain. Transforms involving delta functions (δ) correspond to sine or cosine functions in the time domain
The inverse transform of the delta functions can be represented as sinusoidal functions. For the given transform:
- X(jw) = 2[δ(w−1)−δ(w+1)] corresponds to a continuous-time signal x1(t) = 2cos(t), since cosine is an even function and the delta functions are symmetrically located at ±1.
- X(jw) = 3[δ(w−2π)+δ(w+2π)] corresponds to x2(t) = 3cos(2πt), again since cosine is even and the deltas are at multiples of 2π.
Thus, the continuous-time signal for the given transform is a combination of two cosine functions with different frequencies represented by x1(t) and x2(t).