Question

If x(n) is a complex valued sequence given by x(n)=xR(n)+jxI(n), then what is the DFT of xR(n)?

(a) \(\sum_{n=0}^N x_R (n) cos⁡\frac{2πkn}{N}+x_I (n) sin⁡\frac{2πkn}{N}\)

(b) \(\sum_{n=0}^N x_R (n) cos⁡\frac{2πkn}{N}-x_I (n) sin⁡\frac{2πkn}{N}\)

(c) \(\sum_{n=0}^{N-1} x_R (n) cos⁡\frac{2πkn}{N}-x_I (n) sin⁡\frac{2πkn}{N}\)

(d) \(\sum_{n=0}^{N-1} x_R (n) cos⁡\frac{2πkn}{N}+x_I (n) sin⁡\frac{2πkn}{N}\)

This question was addressed to me in an online quiz.

My doubt stems from Properties of DFT topic in portion Discrete Fourier Transform – Properties and Applications of Digital Signal Processing

Answer 1

Right answer is (d) \(\sum_{n=0}^{N-1} x_R (n) cos⁡\frac{2πkn}{N}+x_I (n) sin⁡\frac{2πkn}{N}\)

Explanation: Given x(n)=xR(n)+jxI(n)=>xR(n)=1/2(x(n)+x*(n))

Substitute the above equation in the DFT expression

Thus we get, XR(k)=\(\sum_{n=0}^{N-1} x_R (n) cos⁡\frac{2πkn}{N}+x_I (n) sin⁡\frac{2πkn}{N}\)