Final answer:
To sketch the waveforms for h(t) = 1/2sinc(2t) and x(t) = 3sinc(t/3), draw two sinc functions: h(t) is a half-amplitude, compressed sinc function, and x(t) is a tripled-amplitude, stretched sinc function. Mark the time axis (t) and amplitude on the vertical axis, showing the central peaks and zero-crossings based on the scaling and stretching factors.
Step-by-step explanation:
The student's question involves sketching the time domain waveforms of two signals, specifically sinc-pulses. The two waveforms to sketch are h(t) = 1/2sinc(2t) and x(t) = 3sinc(t/3), where sinc(t) is defined as sin(πt)/πt. In the time domain, sinc functions have a central peak at t = 0 and minor oscillations that slowly decrease in amplitude as t moves away from zero.
To sketch h(t), plot a sinc function that is scaled by a factor of 1/2 and horizontally compressed by a factor of 2. This means the central peak will have a height of 1/2 and the zero-crossings will occur at intervals of 1/2, 3/2, 5/2, etc.
For x(t), plot a sinc function that is scaled by a factor of 3 and horizontally stretched by a factor of 3. The central peak will have a height of 3, and the zero-crossings will be at intervals of 3, 9, 15, etc.
The axes should be clearly labeled with 't' (time) along the horizontal axis and amplitude along the vertical axis. Note that the sinc function is the Fourier Transform pair of the rectangular function, and these sinc-pulses correspond to the bandwidth-limited signals in time domain which have infinite extent with their energy concentrated around the central peak.