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Unformatted text preview: ECE 108 Signals and Systems Spring 2006, Solutions to Midterm 2 Instructors: Prof. Bruce Fritchman and Prof. Tiffany Jing Li Given: Fri. 04/07/2006 9:00-10:00am 1. (30 points) True or False: (a) For a discrete-time signal with period N , the Fourier coefficients have at the most N distinctive values ( T ) Comment: For a discrete-time periodic signal, its Fourier coefficients repeat with the period N . (b) Signal sin ( 4 π 5 t ) has period T = 5 ( F ) Comment: period T = 2 π 4 π/ 5 = 5 / 2 (c) Signal cos [ 3 π 5 n ] has period N = 10 ( T ) Comment: cos [ 2 πm 1 m 2 n ] (where m 1 and m 2 are relatively prime) has period N = m 2 . (d) Consider x ( t ) = s ( t ) + g ( t ) , where x ( t ) , s ( t ) and g ( t ) are all real, • | X ( jw ) | = | S ( jw ) | + | G ( jw ) | ( F ) • ∠ X ( jw ) = ∠ S ( jw ) + ∠ G ( jw ) ( F ) Comment: By the linearity property, X ( jw ) = S ( jw ) + G ( jw ) , but | X ( jw ) | 6 = | S ( jw ) | + | G ( jw ) | and ∠ X ( jw ) 6 = ∠ S ( jw ) + ∠ G ( jw ) . That is, neither amplitude nor phase is a linear operation. (e) The Fourier transform of the signal x ( t ) is X ( jw ) = j 2 δ ( w- 2)- j 2 δ ( w + 2) • x ( t ) must be real ( T ) • x ( t ) must be odd ( T ) • x ( t ) must be periodic ( T ) • x ( t ) must be a sinusoidal function ( T ) Comment: If x ( t ) is real, then X ( jw ) is conjugate symmetric. If x ( t ) is real and even, then X ( jw ) is purely real and even. If x ( t ) is real and odd, then X ( jw ) is purely imaginary and odd. The reverse is also true. Further, if X ( jw ) consists of only discrete spectral lines, then x ( t ) must be periodic, and these spectral lines must be located at harmonically related frequencies.must be located at harmonically related frequencies....
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- Spring '08
- Fourier Series, Fourier series coefficients