Question

A signal x(t) is bandlimited to B Hz. Show that the signal y(t) = xⁿ(t), i.e. multiplying x(t) n times, is bandlimited to n B Hz.

Answer 1

The signal y(t) = xⁿ(t) is bandlimited to nB Hz.

Step-by-step explanation:

To show that the signal y(t) = xⁿ(t) is bandlimited to nB Hz, we can use the concept of Fourier transforms. The Fourier transform of x(t) is X(f), where f is the frequency. The signal y(t) = xⁿ(t) can be written as y(t) = x(t) × x(t) × ... × x(t), n times. Taking the Fourier transform of y(t) and using the property that the Fourier transform of a product of functions is the convolution of their Fourier transforms, we can show that Y(f) = X(f) * X(f) * ... * X(f), n times. Since the Fourier transform of x(t) is bandlimited to B Hz, each repeated convolution will result in a widening of the frequency range by a factor of B Hz. Therefore, the signal y(t) = xⁿ(t) is bandlimited to nB Hz.