Question

Let x(t) be a band-limited baseband signal with absolute bandwidth ω abs =B(rad/sec) Determine if each of the following signals is band-limited or not; if it is band-limited, find its absolute bandwidth.

(a) x(t+1)+x(t−1)
(b) dt /dx(t)
(c) x 2 (t−5)
(d) x(t)cos(ω 0t+ 3/π )

Answer 1

The question seeks the determination of the band-limited nature and absolute bandwidth for transformed signals. Time-shifting a band-limited signal does not change its bandwidth, while differentiation and squaring operations can result in signals that are not band-limited. Modulating a band-limited signal results in sidebands and an increased bandwidth.

Step-by-step explanation:

The question involves analyzing different transformations of a band-limited baseband signal to determine if the transformed signals are band-limited, and if so, calculate their absolute bandwidths:

• a. x(t+1) + x(t-1) is band-limited because time shifting does not affect the bandwidth of a signal. The absolute bandwidth remains ωabs = B (rad/sec).
• b. d/dt [x(t)] (dx/dt) is not generally band-limited because differentiation can introduce higher frequency components, potentially extending the bandwidth indefinitely.
• c. x2(t-5) is not band-limited since the squaring operation can create frequencies that are up to twice the original signal's highest frequency, thus exceeding the original bandwidth.
• d. x(t)cos(ω0t + 3/π) is band-limited because the modulation of a band-limited signal with a cosine function translates the signal in frequency, resulting in two sidebands. The absolute bandwidth would then be the sum of the bandwidth of x(t) and the frequency of the cosine function, giving 2B + ω0 (rad/sec).