If x[n] = x*[n] is a real-valued sequence, then the DTFT X(f) has Hermitian symmetry, X()...
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If x[n] = x*[n] is a real-valued sequence, then the DTFT X(f) has Hermitian symmetry, X(ƒ) = X*(-ƒ). Decompose X(f) into its real and imaginary parts, X(f) = Xr(f) +jXi(f), (a) Prove that the real part is even, Xr(f) = X(-ƒ). X.(f) = jX()+X*()] X; (f) = {x (ƒ) − x* (ƒ)] (b) Prove that the imaginary part is odd, X; (f) = -X¿(−f). (c) Prove that the magnitude is even, |X(ƒ)| = |X(-ƒ)|. (d) Prove that the phase is odd, X(ƒ) = −ZX(−ƒ). (e) For real sequences x[n], explain why it is sufficient to visualize the DTFT X(f) for only positive frequencies, 0≤f≤0.5. (f) Suppose x[n] is real and consider the partial plots of Xr(f), Xi(f), |X(ƒ)\, and ZX(ƒ) for 0 < ƒ < 0.5 shown below. Given the symmetry requirements on X(f), sketch the real part, imaginary, part, magnitude, and phase for -0.5< x < 0. magnitude (linear] real (linear] 0.8 1 0.6 0.4 0.2 0 -0.2 -0.4- -0.6 -0.8 -1 0.8 1 0.6 0.4 ོ 0.2 20 0 -0.2 -0.4 -0.6 -0.8 -0.5 -1 -0.5 -0.4 -0.3 -0.2 -0.1 0 -0.4 -0.3 imag [linear] 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0.1 0.2 0.3 0.4 0.5 -0.5 -0.4 -0.3 -0.2 -0.1 0 frequency [cycles/sample] M phase [degrees] 3 2 -2 -3 0.1 0.2 0.3 0.4 0.5 frequency (cycles/sample] -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 frequency [cycles/sample] frequency (cycles/sample] If x[n] = x*[n] is a real-valued sequence, then the DTFT X(f) has Hermitian symmetry, X(ƒ) = X*(-ƒ). Decompose X(f) into its real and imaginary parts, X(f) = Xr(f) +jXi(f), (a) Prove that the real part is even, Xr(f) = X(-ƒ). X.(f) = jX()+X*()] X; (f) = {x (ƒ) − x* (ƒ)] (b) Prove that the imaginary part is odd, X; (f) = -X¿(−f). (c) Prove that the magnitude is even, |X(ƒ)| = |X(-ƒ)|. (d) Prove that the phase is odd, X(ƒ) = −ZX(−ƒ). (e) For real sequences x[n], explain why it is sufficient to visualize the DTFT X(f) for only positive frequencies, 0≤f≤0.5. (f) Suppose x[n] is real and consider the partial plots of Xr(f), Xi(f), |X(ƒ)\, and ZX(ƒ) for 0 < ƒ < 0.5 shown below. Given the symmetry requirements on X(f), sketch the real part, imaginary, part, magnitude, and phase for -0.5< x < 0. magnitude (linear] real (linear] 0.8 1 0.6 0.4 0.2 0 -0.2 -0.4- -0.6 -0.8 -1 0.8 1 0.6 0.4 ོ 0.2 20 0 -0.2 -0.4 -0.6 -0.8 -0.5 -1 -0.5 -0.4 -0.3 -0.2 -0.1 0 -0.4 -0.3 imag [linear] 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0.1 0.2 0.3 0.4 0.5 -0.5 -0.4 -0.3 -0.2 -0.1 0 frequency [cycles/sample] M phase [degrees] 3 2 -2 -3 0.1 0.2 0.3 0.4 0.5 frequency (cycles/sample] -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 frequency [cycles/sample] frequency (cycles/sample]
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a To prove that the real part of Xf is even we need to show that Xrf Xrf Given that Xf Xrf jXif we can write Xf as Xf Xrf jXif Since Xf has Hermitian ... View the full answer
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