Question

7.1-12. Determine which of the following functions can and cannot be valid power density spectrums. For those that are not, explain why. (a) 06 +302 +3 (b) exp[-(0-1)?] 04 (0) 360) (1 + 0 + 0* +1 7.1-13. Work Problem 7.1-12 for the following functions: (a) cos(30) (b) — 1 ” (1 + m2)2 1+02 - (d) T (d) TI-30² (c) lol 1+20 +02 N u ore la is a set of real constants and the processes

Answer 1

7.) -12. Solutions w2 a w6+360+ 3 The above function is a valid power density Spectoum, becuase to All the values of w are positive . All w are even functions iv. It is a real femetion. exp [- (W-1)? The above function is not a valid powder density Spectrum because it is not an even function of ww... C W2 -S() 10441 The above function is not a valid power . density Spectoum because not every team is non- negative wy d) 1707 jub The above function has imaginary terms and ol is not real. Therefore, the given en not a valid power density spectrum the given function

7.)-13 a) cos (sw) لایا + SKW) = COS (34-w)) (os 3W Sylw) 1+ (-coj? Trož - Sylw) Six (W) Lo for wat [3no odd The above function es not valid power density Spectrum. b) SzX(60) , 1 SX-w) (1+02)2 I (1 + (-w)2)2 (+60272 S2X (o) 20 S2X26) is a deal function The above function is valid power density Spec-toum. 100 C) 12W +62 kool - szk(-40) S3x6. 0 i+266)+(c) 1-2016 The above function is not valid power density spectrum

✓ $-3 m2 Suw) = 5 1-3 (0)2 5 Sy (40). 51-3602 for all w Su (W) co and not a real value for 3²7 The above function is not valid powder Spectrum

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