Final answer:
The Fourier transform of an instantaneously sampled sinusoidal wave reveals the frequency content of the signal. Depending on the sampling period, it can show replicas of the original signal's spectrum at multiples of the sampling rate. The magnitude plot of this spectrum demonstrates the strength of various frequencies present.
Step-by-step explanation:
Finding the Fourier Transform of a Sampled Sinusoidal Wave
In the processing of signals, the Fourier transform is used to determine the frequency content of continuous-time signals. The function given is g(t) = sin(πt), which represents a sinusoidal wave. When this wave is sampled instantaneously at a particular sampling period, it creates a sequence of impulses that can be analyzed using the Fourier transform.
The sampling theorem, stated by Nyquist and Shannon, suggests that a signal can be perfectly reconstructed if it is sampled at a rate greater than twice its highest frequency. However, when the sampling rate is lower than this Nyquist rate, it results in an effect called aliasing.
Mathematically, the process of instantaneous sampling can be modeled as multiplying the sinusoidal signal by a Dirac delta function series corresponding to the sampling intervals. The resulting transform will show replicas of the signal's spectrum centered around integer multiples of the sampling frequency. Examining these spectra at different sampling periods would reveal how the Fourier transform changes, and the magnitude of the transform would indicate the strength of the various frequency components present in the sampled signal.
To illustrate this concept graphically, consider if the sampling period is an integral multiple of the signal period. In such a case, all samples would occur at the same phase points of the original signal, possibly resulting in a Fourier transform that has a non-zero value only at DC (0 Hz) or certain harmonic frequencies depending on the sampling instants. Conversely, if the sampling period is not synchronized with the signal period, the Fourier transform will exhibit a wider spread of frequencies.
The spectrum's magnitude can be plotted to visualize the strength of different frequency components induced by the sampling process. However, without specific sampling periods provided, we cannot carry out the precise calculations or graphs.
Understanding the concepts of amplitude, wavelength, period, and frequency is critical in signal processing and physics. These characteristics can be extracted from the wave function itself, as shown in various examples, including electromagnetic waves and mechanical waves on a string.