Final answer:
The Fourier transform of a periodic signal is not typically defined; however, Fourier series are used for periodic signals. Differences in wave equations relate to their amplitude, frequency, and phase. The phase shift affects the starting point of the wave's cycle.
Step-by-step explanation:
The Fourier transform, x(jω), of a periodic signal x(t) = sin(2πt + π/4) is a mathematical expression that transforms the time domain signal into the frequency domain. The response to the given signal isn't straightforward because a Fourier transform typically applies to non-periodic signals. For periodic signals, we often use the Fourier series to express the signal as a sum of sinusoidal components. If we were to find the continuous-time Fourier series for x(t), we would have components at the signal's fundamental frequency and its harmonics. However, the student's question might be phrased incorrectly, as the Fourier transform of a periodic signal like a sine wave would theoretically consist of impulses at the fundamental frequency and its harmonics, not a continuous function of ω.
When dealing with sinusoidal wave equations, understanding similarities and differences involves analyzing amplitude, frequency, wavelength, and phase. Wave equations like Wave₁ = A₁sin(2ωt) and Wave₂ = A₂sin(4ωt) differ in frequency, indicating that Wave2 oscillates twice as fast as Wave₁. However, if the amplitudes are the same, they reach the same maximum and minimum values. Phase shifts, represented by phi in an equation like x(t) = A cos(ωt + ϕ), affect where a wave starts within its cycle.