Consider a complex sequence h[n] = hr [n] + jhi[n], where hr[n] and hi[n] are both real...

Consider a complex sequence h[n] = h_r [n] + jhi[n], where h_r[n] and h_i[n] are both real sequences, and let H(e^jω) = H_R(e^jω) + j_HI (e^jω) denote the Fourier transform of h[n], where H_R(e^jω) and H_I (e^jω) are the real and imaginary parts, respectively, of H(e^jω).

Let H_ER(e^jω) and H_OR(e^jω) denote the even and odd parts, respectively, of H_R(e^jω), and let H_EI(e^jω), and H_OI(e^jω) denote the even and odd parts, respectively, of H_I(e^jω). Furthermore, let H_A(e^jω) and H_B(e^jω) denote the real and imaginary parts of the Fourier transform of h_r[n], and let H_C(e^jω) and H_D(e^jω) denote the real and imaginary parts of the Fourier transform of h_i[n]. Express H_A(e^jω), H_B(e^jω), H_C(e^jω), and H_D(e^jω) in terms of H_ER(e^jω), H_OR(e^jω), H_EI(e^jω), and H_OI(e^jω).