Question

Find and sketch the Fourier Transform (or the magnitude of the FT) of each signal. Determine if each signal is baseband (low pass), high pass or pass band. Specify the bandwidth and cut off frequencies of the signals. (a) x(t)=4sinc² (100t)

Answer 1

The signal x(t)=4sinc² (100t) is a baseband (low pass) signal with a Fourier Transform that exhibits a triangular shape in the frequency domain. Its bandwidth is 200 Hz, with cutoff frequencies at ± 100 Hz.

Step-by-step explanation:

To find and sketch the Fourier Transform (FT) of the signal x(t)=4sinc² (100t), we must recognize that the sinc function in the time domain corresponds to a rectangular function in the frequency domain. The squared term indicates a convolution of two rect functions in the frequency domain, which results in a triangular function.

The signal is baseband (low pass) since the sinc function is centered at zero frequency. The bandwidth of the original sinc function is twice the first null of the sinc, which is at 200 Hz, since the sinc function has a null at 1/(pi*T). The cutoff frequencies are therefore at ± 100 Hz.

The magnitude of the FT, due to the square term, will be highest at zero frequency and will decay linearly to zero at the cutoff frequencies. Beyond these points, the magnitude will be zero. There is no high-pass or pass-band behavior in this signal.