If x(n) is a real sequence, then what is the value of XI(ω)?(a) \(\sum_{n=-∞}^∞ x(n)sin⁡(ωn)\)(b) –\(\sum_{n=-∞}^∞ x(n)sin⁡(ωn)\)(c) \(\sum_{n=-∞}^∞ x(n)cos⁡(ωn)\)(d) –\(\sum_{n=-∞}^∞ x(n)cos⁡(ωn)\)The question was asked by my college director while I was bunking the class.This interesting question is from Properties of Fourier Transformfor Discrete Time Signals in portion Frequency Analysis of Signals and Systems of Digital Signal Processing

If x(n) is a real sequence, then what is the value of XI(ω)?(a) \(\su

1.	If x(n) is a real sequence, then what is the value of XI(ω)?(a) \(\sum_{n=-∞}^∞ x(n)sin⁡(ωn)\)(b) –\(\sum_{n=-∞}^∞ x(n)sin⁡(ωn)\)(c) \(\sum_{n=-∞}^∞ x(n)cos⁡(ωn)\)(d) –\(\sum_{n=-∞}^∞ x(n)cos⁡(ωn)\)The question was asked by my college director while I was bunking the class.This interesting question is from Properties of Fourier Transformfor Discrete Time Signals in portion Frequency Analysis of Signals and Systems of Digital Signal Processing
Answer» Right choice is (b) –\(\sum_{n=-∞}^∞ X(n)sin⁡(ωn)\) The best EXPLANATION: If the signal x(n) is real, then xI(n)=0 We KNOW that, XI(ω)=\(\sum_{n=-∞}^∞ x_R (n)sinωn-x_I (n)cosωn\) Now substitute xI(n)=0 in the above equation=>xR(n)=x(n) => XI(ω)=-\(\sum_{n=-∞}^∞ x(n)sin⁡(ωn)\).

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