Final answer:
The Fourier transform of the sequence πnx[n] = sin(πn/3) cannot be directly calculated because of the undefined nature at πn = 0. Assuming the sequence is sin(πn/3), the Fourier transform would yield impulses in the frequency domain representing the sinusoidal nature of the sequence.
Step-by-step explanation:
The sequence x[n] = sin(πn/3)πn, where n ranges from negative to positive infinity, represents a sinusoidal function. To find its Fourier transform, we must understand the properties of sine functions and Fourier transforms. The Fourier transform of a discrete-time signal involves decomposing the signal into its frequency components. However, the sequence as given appears to be ill-defined because there is a term πn in the denominator, which would be zero when n = 0, thus making the sequence undefined at n = 0. If we assume that this is a typographical error and ignore the πn term, the Fourier transform of sin(πn/3), where n is defined over all integers, is a pair of impulses in the frequency domain centered at ±3/3, since it is a purely sinusoidal function.
The actual calculation of the Fourier transform for discrete signals generally involves summation or integration over an infinite number of terms, which requires the application of the Dirac delta function in the frequency domain.
To illustrate this with sinusoidal wave functions, consider y1(x, t) = A sin(kx - wt) as a wave moving in the positive direction and y2(x, t) = A sin(kx + wt + p) for a wave moving in the opposite direction. When these two waves are superimposed, they create a standing wave form. For example, simple harmonic motion is observed at each point on a string oscillating sinusoidally, and the positions of nodes in a standing wave can be determined by finding the points where sin(kx) = 0.