Question

What is the basic relationship between the spectrum o f the real band pass signal x(t) and the spectrum of the equivalent low pass signal xl(t)?

(a) X (F) = $\frac{1}{2} [X_l (F-F_c)+X_l^* (F-F_c)]$

(b) X (F) = $\frac{1}{2} [X_l (F-F_c)+X_l^* (F+F_c)]$

(c) X (F) = $\frac{1}{2} [X_l (F+F_c)+X_l^* (F-F_c)]$

(d) X (F) = $\frac{1}{2} [X_l (F-F_c)+X_l^* (-F-F_c)]$

I had been asked this question during an interview.

My doubt stems from Sampling of Band Pass Signals topic in chapter Sampling and Reconstruction of Signals of Digital Signal Processing

Answer 1

The correct option is (d) X (F) = $\frac{1}{2} [X_l (F-F_c)+X_l^* (-F-F_c)]$

Best explanation: X(F) = $\frac{1}{2} [X_l (F-F_c)+X_l^* (-F-F_c)]$, where Xl(F) is the Fourier transform of xl(t). This is the basic relationship between the spectrum o f the real band pass signal x(t) and the spectrum of the equivalent low pass signal xl(t).