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Problem 9.20. Assume that the narrowband noise n(t) is Gaussian and its power spectral density SN(f) is symmetric about the midband frequency fc. Show that the in-phase and quadrature components of n(t) are statistically independent. Solution The narrowband noise n(t) can be expressed as: 2 ( ) ( ) cos(2 ) ( )sin(2 ) Re ( ) c I c Q j f t n t n t f t n t f t z t e \u3c0 c\u3c0 \u3c0= \u2212 \u23a1 \u23a4= \u23a3 \u23a6 , where nI(t) and nQ(t) are in-phase and quadrature components of n(t), respectively. The term z(t) is called the complex envelope of n(t). The noise n(t) has the power spectral density SN(f) that may be represented as shown below We shall denote ( )nnR \u3c4 , ( )I In nR \u3c4 and ( )Q Qn nR \u3c4 as autocorrelation functions of n(t), nI(t) and nQ(t), respectively. Then [ ] { } ( ) ( ) ( ) ( ) cos(2 ) ( )sin(2 ) ( ) cos(2 ( )) ( )sin(2 ( )) 1 1( ) ( ) cos(2 ) ( ) ( ) cos(2 (2 )) 2 2 1 ( ) ( ) 2 I I Q Q I I Q Q Q I I Q nn I c Q c I c Q c n n n n c n n n n c n n n n R n t n t n t f t n t f t n t f t n t f t R R f R R f t R R \u3c4 \u3c4 \u3c0 \u3c0 \u3c4 \u3c0 \u3c4 \u3c4 \u3c0 \u3c4 \u3c4 \u3c0 \u3c4 \u3c4 \u3c4 \u3c0 \u3c4 \u3c4 \u3c4 = + \u23a1 \u23a4 \u23a1= \u2212 \u22c5 + + \u2212 +\u23a3 \u23a6 \u23a3 \u23a1 \u23a4 \u23a1 \u23a4= + + \u2212 +\u23a3 \u23a6 \u23a3 \u23a6 \u23a1 \u23a4\u2212 \u2212\u23a3 \u23a6 E E \u3c4 \u23a4+ \u23a6 1sin(2 ) ( ) ( ) sin(2 (2 )) 2 Q I I Qc n n n n c f R R f t\u3c0 \u3c4 \u3c4 \u3c4 \u3c0 \u3c4\u23a1 \u23a4\u2212 + +\u23a3 \u23a6 Since n(t) is stationary, the right-hand side of the above equation must be independent of t, this implies ( ) ( ) I I Q Qn n n n R R\u3c4 \u3c4= (1) ( ) ( ) I Q Q In n n n R R\u3c4 \u3c4= \u2212 (2) Continued on next slide Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or instructional purposes only to students enrolled in courses for which the textbook has been adopted. Any other reproduction or translation of this work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. Problem 9.20 continued Substituting the above two equations into the expression for Rnn(\u3c4), we have ( ) ( )cos(2 ) ( )sin(2 ) I I Q Inn n n c n n c R R f R f\u3c4 \u3c4 \u3c0 \u3c4 \u3c4 \u3c0= \u2212 \u3c4 (3) The autocorrelation function of the complex envelope ( ) ( ) ( )I Qz t n t jn t= + is *( ) ( ) ( ) 2 ( ) 2 ( I I Q I zz n n n n R E z t z t R j R \u3c4 \u3c4 )\u3c4 \u3c4 \u23a1 \u23a4= +\u23a3 \u23a6 = + (4) From the bandpass to low-pass transformation of Section 3.8, the spectrum of the complex envelope z is given bye ( ) ( ) 0 oth N c Z S f f f f S f \u23a7 + > \u2212\u23aa= \u23a8\u23aa\u23a9 erwise c Since SN(f) is symmetric about fc, SZ(f) is symmetric about f = 0. Consequently, the inverse Fourier transform of SZ(f) = Rzz(\u3c4) must be real. Since ( )zzR \u3c4 is real valued, based on Eq. (4), we have ( ) 0 Q In n R \u3c4 = , which means the in-phase and quadrature components of n(t) are uncorrelated. Since the in-phase and quadrature components are also Gaussian, this implies that they are also statistically independent. Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or instructional purposes only to students enrolled in courses for which the textbook has been adopted. Any other reproduction or translation of this work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful.