If x(n)=xR(n)+jxI(n) is a complex sequence whose Fourier transform is given as X(ω)=XR(ω)+jXI(ω), then what is the value of xI(n)?(a) \(\frac{1}{2π} \int_0^{2π}\)[XR(ω) sinωn+ XI(ω) cosωn] dω(b) \(\int_0^{2π}\)[XR(ω) sinωn+ XI(ω) cosωn] dω(c) \(\frac{1}{2π} \int_0^{2π}\)[XR(ω) sinωn – XI(ω) cosωn] dω(d) None of the mentionedThis question was addressed to me at a job interview.My question comes from Properties of Fourier Transformfor Discrete Time Signals topic in division Frequency Analysis of Signals and Systems of Digital Signal Processing

If x(n)=xR(n)+jxI(n) is a complex sequence whose Fourier transform is

1.	If x(n)=xR(n)+jxI(n) is a complex sequence whose Fourier transform is given as X(ω)=XR(ω)+jXI(ω), then what is the value of xI(n)?(a) \(\frac{1}{2π} \int_0^{2π}\)[XR(ω) sinωn+ XI(ω) cosωn] dω(b) \(\int_0^{2π}\)[XR(ω) sinωn+ XI(ω) cosωn] dω(c) \(\frac{1}{2π} \int_0^{2π}\)[XR(ω) sinωn – XI(ω) cosωn] dω(d) None of the mentionedThis question was addressed to me at a job interview.My question comes from Properties of Fourier Transformfor Discrete Time Signals topic in division Frequency Analysis of Signals and Systems of Digital Signal Processing
Answer» RIGHT option is (a) \(\frac{1}{2π} \int_0^{2π}\)[XR(ω) sinωn+ XI(ω) cosωn] dω Explanation: We KNOW that the INVERSE transform or the synthesis equation of a signal x(n) is given as x(n)=\(\frac{1}{2π} \int_0^{2π}\) X(ω)e^jωn dω By substituting e^jω = cosω + jsinω in the above equation and separating the real and imaginary parts we get xI(n)=\(\frac{1}{2π} \int_0^{2π}\)[XR(ω) sinωn+ XI(ω) cosωn] dω

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