EE 350 Signals and Systems Spring 2005 Sample Exam #2 - Solutions
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1 EE 35 Signals an Sysms Spring 5 Sampl Exam # - Soluions. For h following signal x( cos( sin(3 - cos(5 - T, /T x( j j 3 j 3 j j 5 j 5 j a -, a a -, a a - ½, a 3 /j-j -j/, a -3 -/jj j/, a 5 -½, a -5 -½, all ohr a k ar. For a prioic funcion wih prio T4, h non-zro Fourir sris cofficins ar a, a a -, a 3 a* -3 j (a Exprss h funcion x( in h form x( Σ A k cos( k φ k /T/4 ½ a 3 j j/, a -3 -j -j/ No: his problm is much asir if you convr all a k o polar form firs. x(a j a j(/ a - -j(/ a 3 3j(/ a -3-3j(/ x( j(/ -j(/ j/ 3j(/ -j/ -3j(/ x( (4 j j ( j( j( 4cos((/ cos((3// (b Graph h magniu of a k vrsus. (c Graph h phas (angl of a k vrsus. a k a k -3/ / 3/ -3/ / / / 3/ ( If x( is pass hrough a ral filr h( wih magniu an phas shown blow Call h cofficins for y(, b k. Thn b k a k * H( H( (muliply magniu, a angls b *H(j*, b * H(j / H(j / (* j( 4 j b - * H(-j / H(-j / (* j( 4 j, b 3 * j/ * H(j3 / H(j3 / (* j(// ( j( b -3 * -j/ * H(-j 3 / H(-j 3/ (* j(-/-/ (* j(- ( Wha is y( for hs Fourir sris cofficins? x( 4cos((/ cos((3// y(*h(j 4* H(j / cos((/ H(j / H(j / cos((3// H(j 3/ H(, H(j/, H(j3 / / y(* 4*cos((/ *cos((3/// y( 8cos((/ cos((3/ 3. ( poins Drmin h (xponnial Fourir sris of x( as shown blow.
2 x( T 6 jk ak T x( T a k a k jk jk jk 4 jk jk jk jk jk jk jk jk ( 4 jk ( jk ( jk ( jk ( 4 a k jk ( k jk ( jk jk jk jk jk a T x( 4 jk jk - 4 jk ( jk - jk - ( jk 4 jk jk jk ( ( jk jk (4 4 jk jk jk jk a k [ ] [ ] ( k 4 ( k j ( k jk / jk / jk / ( j jk a k [ sin( k / ] 4( ( k k k o k vn jk T ( jk 4. a Tabls: X a (j sin(t / sin(/ (from Tabl 4., Eq#8 Ingral: x( j ( j j j j b x b ( x a (x a (- (im shif & linariy X b (j X a (jx a (j -j (sin(/( -j -or- x b ( x a ((-/ X b (j X(j -j -j sin(/ * hs wo forms ar quivaln. Us Eulr s o prov. c x c ( x a ( x a (- (im shif & linariy X c (j X(j-X(j -j (sin(/(- -j x ( x c ( (im scaling X (j ½ X c (j/ ½ (sin(//(/( -j/ x ( x( x(- (im shif & linariy X (j X(j X(j -j (sin(/( -j f x f ( x( jk ( jk j j j j ( sin( j
3 sin X f (j ( ( j cos( ( sin( X j j j g x g ( x( x( x f ( x( X g (j sin( j ( cos( ( sin( 5. Dualiy: if x(g( X(jf(, hn y(f( Y(jg(- from problm (3: < < X a (j sin(/ (from Tabl 4., Eq#8 x( ls so < <, f( sin(/ g( ls f( sin(/, < < g( ls < < < < g( ls ls < < g ( ls so, sin(/ < < ls an by linariy y( sin(/( sin(/( Y(j < < ls which agrs wih Tabl 4. #9 6. x( -u( h( -4 u( a Cas I h(-τ -4(-τ x(τ - < y( Cas II h(-τ -4(-τ x(τ τ - τ > τ 4( τ 4 τ y( ( τ ( τ τ τ y( ½ - ¼ - ¼ -4 So y( (½ - ¼ - ¼ -4 u( b X(j /(j, H(j /(4j Y(j X(jH(j /[(j (4j] -/4 (/(j ½ (/(j ¼ (/(4j So y( ( ¼ - ½ - ¼ -4 u( No: For his problm you o n o valua h ingral. I rcommn having h ingrals for a, a, an a on your no sh. 7. 3(3 j H ( j 3/(/(4j 3/(/(j (4 j( j a h( 3/ -4 u( 3/ - u(
4 b causal: ys-bcaus h( for all < bcaus of u( in formula sabl: ys bcaus ara of magniu 9/8 < infiniy c (bo plo iagram 3(3 ( 3 H ( j (4 ( ( ( Bo Diagram Ma gni u - (B Ph as ( g Frquncy (ra/sc 8. y 3y y x( a x( u( r 3r r-, - y h ( K - K - y p (A y p, y p 3( (A A ½ This is a low pass filr h( / {y s (} ( - - u( b ( Y( 3(Y( Y( X( Y ( H( X ( ( 3( ( ( h( - u( - u( (machs answr from a ( c H (, log ( ½ -6 ( Bo Diagram y( y h (y p ( K - K - ½ y( K K ½ y ( -K K K -, K ½ y s ( (- - ½ - ½ u( - Ma gni u -4 (B Ph as ( g Frquncy (ra/sc Low Pass Filr 9. KVL: V in V r V L V c V ou, i (/RV ou, i C / V c Vou R R a Vin R L ( Vou Vc Vou
5 Vin Vin Vin L Vou ( Vou Vc R L i Vou ( Vou R C L Vou Vou ( Vou R RC Vou Vou Vou Vin x(, vou y( X y L/Ry /(RCy b X( (Y( (L/R( Y( /(RCY( H( Y ( X ( c H( L R ( ( RC h( 5. - u(.5 -. u( 5 H( - ( (. ( 5 Bo Diagram ( ( (. 5.5 log (54 - Ma gni -3 u (B Ph as ( g Frquncy (ra/sc x( 3cos(5g w H(j y( 3( ½ cos(5 g g H( ½ y( 3/ cos(5g 5 j( j 5 j ( j angl(h( g. (awha is h quaion scribing X(j? sin( X ( j F{cos( } F{ } X ( j [ δ ( δ ( ] <, < X ( j ls X(j / (b (c Nyquis: s >( max s >( s > ( Nyquis Frquncy is ( Nyquis Ra is (f p 34.5, p 69 < >
6 X(j - -5 / X(j - -5 / (g X(j -8 - /
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