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 XR(ω)?(a) \(\sum_{n=0}^∞\)xR (n)cosωn-xI (n)sinωn(b) \(\sum_{n=0}^∞\)xR (n)cosωn+xI (n)sinωn(c) \(\sum_{n=-∞}^∞\)xR (n)cosωn+xI (n)sinωn(d) \(\sum_{n=-∞}^∞\)xR (n)cosωn-xI (n)sinωnThis question was addressed to me in an international level competition.I would like to ask this question from Properties of Fourier Transformfor Discrete Time Signals in chapter 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 XR(ω)?(a) \(\sum_{n=0}^∞\)xR (n)cosωn-xI (n)sinωn(b) \(\sum_{n=0}^∞\)xR (n)cosωn+xI (n)sinωn(c) \(\sum_{n=-∞}^∞\)xR (n)cosωn+xI (n)sinωn(d) \(\sum_{n=-∞}^∞\)xR (n)cosωn-xI (n)sinωnThis question was addressed to me in an international level competition.I would like to ask this question from Properties of Fourier Transformfor Discrete Time Signals in chapter Frequency Analysis of Signals and Systems of Digital Signal Processing
Answer» CORRECT CHOICE is (c) \(\sum_{n=-∞}^∞\)xR (n)cosωn+xI (n)sinωn Explanation: We know that X(ω)=\(\sum_{n=-∞}^∞\) x(n)e^-jωn By substituting e^-jω = cosω – jsinω in the above equation and separating the real and IMAGINARY parts we get XR(ω)=\(\sum_{n=-∞}^∞\)xR (n)cosωn+xI (n)sinωn

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