Question

Suppose you have two periodic signals x(t) and y(t), of periods t₁ and t₂ respectively. Let xₖ and yₖ be the Fourier series coefficients of x(t) and y(t). If t₁ = t₂, then xₖ = yₖ.

a. True
b. False

Answer 1

It is false that two signals with the same period have the same Fourier series coefficients, as the coefficients also depend on the waveform's shape. The true principle of superposition allows waves of different frequencies to interact and form a resultant wave.

Step-by-step explanation:

The statement that if two periodic signals x(t) and y(t) have the same period t1 = t2, then their Fourier series coefficients xk and yk must be equal, is false. Having the same period simply means that the signals repeat themselves with the same frequency; however, the shapes or waveforms of the signals could be different, leading to different Fourier coefficients. Fourier series coefficients are dependent on the shape of the waveform, not just the period. Consider two wave functions y1 (x, t) and y2 (x, t) that differ only by a phase shift; the resulting waveform upon superposition will have the same frequency but may have a different amplitude and phase, which will affect the coefficients.

Also, the statement that waves can superimpose if their frequencies are different is true. This illustrates the principle of superposition, where multiple waves can interact within the same medium to produce a resultant wave that is the sum of the individual waves. The amplitudes may affect each other in the case of constructive or destructive interference, but only when the waves are precisely aligned, which determines the resultant amplitude.