Final answer:
The signal x(t) is decomposed into complex exponential signals using Euler's relation. Fundamental frequency f0 is 0 Hz, N is 250 Hz, and complex amplitudes for frequencies 100 Hz and 250 Hz are 10 and 5, respectively.
Step-by-step explanation:
The student's question involves expressing a given signal as a sum of complex exponential signals using the Fourier synthesis approach. The signal is x(t) = 10 + 20cos(2π(100)t + π/4) + 10cos(2π(250)t). We can use Euler's relation eiτ = cos(τ) + i sin (τ) to express the cosines as the sum of complex exponentials. To find αk, the complex amplitudes for the Fourier synthesis, we do not need to evaluate integrals as this is a straightforward application of Euler's formula.
The fundamental frequency (α0) of the signal is that of the first term, which is a DC term (0 Hz) and thus the complex amplitude α0 = 10. The next term has a frequency of 100 Hz, and the last has a frequency of 250 Hz, implying N to be 250 (the highest frequency). The signal can be decomposed using complex exponentials:
- 20cos(2π(100)t + π/4) = 10ei(2π(100)t + π/4) + 10e-i(2π(100)t + π/4)
- 10cos(2π(250)t) = 5ei(2π(250)t) + 5e-i(2π(250)t)
The complex amplitudes for α1, α-1, α2, and α-2 corresponding to 100 Hz and 250 Hz frequencies, respectively, would be 10, 10, 5, and 5.