Question

If we substitute the equation \(x_l (t)= u_c (t)+j u_s (t)\) in equation x (t) + j ẋ (t) = xl(t) e^j2πFct and equate real and imaginary parts on side, then what are the relations that we obtain?

(a) x(t)=\(u_c (t) \,cos⁡2π \,F_c \,t+u_s (t) \,sin⁡2π \,F_c \,t\); ẋ(t)=\(u_s (t) \,cos⁡2π \,F_c \,t-u_c \,(t) \,sin⁡2π \,F_c \,t\)

(b) x(t)=\(u_c (t) \,cos⁡2π \,F_c \,t-u_s (t) \,sin⁡2π \,F_c \,t\); ẋ(t)=\(u_s (t) \,cos⁡2π \,F_c t+u_c (t) \,sin⁡2π \,F_c \,t\)

(c) x(t)=\(u_c (t) \,cos⁡2π \,F_c t+u_s (t) \,sin⁡2π \,F_c \,t\); ẋ(t)=\(u_s (t) \,cos⁡2π \,F_c t+u_c (t) \,sin⁡2π \,F_c \,t\)

(d) x(t)=\(u_c (t) \,cos⁡2π \,F_c \,t-u_s (t) \,sin⁡2π \,F_c \,t\); ẋ(t)=\(u_s (t) \,cos⁡2π \,F_c \,t-u_c (t) \,sin⁡2π \,F_c \,t\)

The question was asked during a job interview.

This is a very interesting question from The Representation of Bandpass Signals topic in portion Sampling and Reconstruction of Signals of Digital Signal Processing

Answer 1

Correct answer is (b) x(t)=\(u_c (t) \,cos⁡2π \,F_c \,t-u_s (t) \,sin⁡2π \,F_c \,t\); ẋ(t)=\(u_s (t) \,cos⁡2π \,F_c t+u_c (t) \,sin⁡2π \,F_c \,t\)

Best explanation: If we substitute the given equation in other, then we get the required resu