GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING

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1 GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING ECE 226 Summer 28 Quiz #2 July 6, 28 NAME: (FIRST) (LAST) GT username: (e.g., gtxyz23) To avoid losing 3 oints, circle your recitation section: :5 :55am L (Beck) 2:3 2:2m L2 (Beck) 2:35 4:25m L3 (Bhattacharjea) Imortant Notes: Do not unstale the test. Two double-sided age (8.5 ) of hand-written notes ermitted. Calculators are allowed, but no smarthones/wifi/etc. JUSTIFY your reasoning CLEARLY to receive artial credit. Exress all angles as a fraction of. For examle, write. as oosed to 8 or.342 radians. You must write your answer in the sace rovided on the exam aer itself. Only these answers will be graded. Write your answers in the rovided answer boxes. If more sace is needed for scratch work, use the backs of the revious ages. Problem Value Score Earned Total

2 PROB. Su8-Q2.. (a) If x[ n ] = 3d[ n ] + 2d[ n 2 ], and if y[ n ] = x[ n ] * x[ n ] is the convolution of x[ n ] with itself, then find numerical values for the following: y[ ] y[ ] y[ 2 ] y[ 3 ] y[ 4 ] y[ 5 ] (b) The inverse DTFT of X(e j e ) = jŵ e jŵ is x[ n ] =. (Give an equation that is valid for all n.)

3 PROB. Su8-Q2.2. Consider an LTI system defined by the difference equation y[ n ] = x[ n ] + Bx[n ] x[n 2], where B is an unsecified constant that may be different for each art below. (a) If the outut in resonse to the constant signal x[ n ] = -- (for all n) is y[ n ] = (for all n), then 3 B =. (b) If the outut in resonse to the signal x[ n ] = 4cos(n) is y[ n ] = 2cos(n), then B =. (c) If B =.756, and if the outut in resonse to the constant-lus-sinusoidal signal x[ n ] = + cos( n) is the constant signal y[ n ] = C, then it must be that: C =, =

4 PROB. Su8-Q2.3. Consider three LTI filters whose imulse resonses are as follows: sin(.6n) h [ n ] = , n sin(.2n) h 2 [ n ] = d[ n ] , n sin(.4n) h 3 [ n ] = 2cos(.5n) n (a) Indicate what kind of filter each is by writing LPF, BPF, or HPF into each answer box above. (b) Listed below are six different imulse resonses h[ n ], where the symbol * denotes convolution. Match each to its corresonding magnitude resonse jh(e j ) j shown to the right (A to F). Indiciate your answers by writing a letter from fa,... Fg into each answer box. jh(e j ) j A (i) h [ n ] B (ii) h 2 [ n ] C (iii) h 3 [ n ] D (iv) (h [ n ] h 2 [ n ]) * h 3 [ n ] (v) (h [ n ] h 3 [ n ]) * h 2 [ n ] E (vi) (h 2 [ n ] h 3 [ n ]) * h [ n ] F

5 PROB. Su8-Q2.4. Below are a list of LTI filters (labeled A to H) secified in the time domain: either by their imulse resonse h[ n ], their difference equation, or their MATLAB imlementation. Match each filter to its corresonding magnitude resonse shown on the right. Indicate your answers by writing a letter from fa, B,... Hg in each answer box. (A) y[ n ] =.5x[ n ].5x[ n ] (B) y = conv(x,[2/3, /3]); (C) h[ n ] =.5x[ n 5 ] +.5x[ n 6 ] P 99 (D) y[ n ] =. k = (.9)k x[ n k] P 99 (E) y[ n ] =. k = (.9)k x[ n k] P 3 (F) y[ n ] =.25 x[ n k] k= (G) h[ n ] = -- 6 P 5 k= d[ n k] (H) y = conv(x,cos(i*(:5))/6);

6 PROB. Su8-Q2.5. Consider an LTI filter whose real-valued frequency resonse satisfies H(e j ) = j j, for j ^w j <, as illustrated below: H(e j) (a) The DC gain of the filter is. (b) If the inut is the sinusoid x[ n ] = cos(.5n), then the outut at n = 8 will be y[8] =. (c) If the outut of this system in resonse to the inut x[ n ] = cos( n) + cos( (n )) is y[ n ] = cos( n + j), then = j =

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8 GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING ECE 226 Summer 28 Quiz #2 July 6, 28 NAME: KEY (FIRST) (LAST) GT username: (e.g., gtxyz23) To avoid losing 3 oints, circle your recitation section: :5 :55am L (Beck) 2:3 2:2m L2 (Beck) 2:35 4:25m L3 (Bhattacharjea) Imortant Notes: Do not unstale the test. Two double-sided age (8.5 ) of hand-written notes ermitted. Calculators are allowed, but no smarthones/wifi/etc. JUSTIFY your reasoning CLEARLY to receive artial credit. Exress all angles as a fraction of. For examle, write. as oosed to 8 or.342 radians. You must write your answer in the sace rovided on the exam aer itself. Only these answers will be graded. Write your answers in the rovided answer boxes. If more sace is needed for scratch work, use the backs of the revious ages. Problem Value Score Earned Total

9 PROB. Su8-Q2.. (a) If x[ n ] = 3d[ n ] + 2d[ n 2 ], and if y[ n ] = x[ n ] * x[ n ] is the convolution of x[ n ] with itself, then find numerical values for the following: y[ ] y[ ] y[ 2 ] y[ 3 ] y[ 4 ] y[ 5 ] x[n] x[]x[n] x[]x[n-] x[2]x[n-2] (b) The inverse DTFT of X(e j e ) = 8jŵ is x[ n ] =. e jŵ (Give an equation that is valid for all n.) (.) n 8 u[ n 8] Simlifies to X(e j ) = e jŵ = e 8j.e jŵ e jŵ From Table, the second factor is the DTFT of (.) n u[ n ]. From Table 2, multilying the DTFT by e 8j delays by 8.

10 PROB. Su8-Q2.2. Consider an LTI system defined by the difference equation y[ n ] = x[ n ] + Bx[n ] x[n 2], where B is an unsecified constant that may be different for each art below. (a) If the outut in resonse to the constant signal x[ n ] = -- (for all n) is y[ n ] = (for all n), then 3 The DC gain is zero ) + B = ) B = 2 2 B =. (b) If the outut in resonse to the signal x[ n ] = 4cos(n) is y[ n ] = 2cos(n), then The gain at is five ) B = 5 ) B = 7 7 B =. (c) If B =.756, and if the outut in resonse to the constant-lus-sinusoidal signal x[ n ] = + cos( n) is the constant signal y[ n ] = C, then it must be that: The frequency resonse of this [, B, ] filter is H(e j ) = + B e j e 2j = e j ( e j + B e j ) = e j (B 2cos( )) The nulled frequency must therefore satisfy 2cos( ) = B ) = cos (B/2) = cos (.756/2) =.7 C =, = Equating the DC gain B 2 to the ratio C/ yields: C/ = B 2 = ) C = 37.56

11 PROB. Su8-Q2.3. Consider three LTI filters whose imulse resonses are as follows: LPF HPF BPF sin(.6n) h [ n ] = , n sin(.2n) h 2 [ n ] = d[ n ] , n sin(.4n) h 3 [ n ] = 2cos(.5n) n (a) Indicate what kind of filter each is by writing LPF, BPF, or HPF into each answer box above. (b) Listed below are six different imulse resonses h[ n ], where the symbol * denotes convolution. Match each to its corresonding magnitude resonse jh(e j ) j shown to the right (A to F). Indiciate your answers by writing a letter from fa,... Fg into each answer box. jh(e j ) j A C (i) h [ n ] B A (ii) h 2 [ n ] C E (iii) h 3 [ n ] D (iv) (h [ n ] h 2 [ n ]) * h 3 [ n ] D F (v) (h [ n ] h 3 [ n ]) * h 2 [ n ] E B (vi) (h 2 [ n ] h 3 [ n ]) * h [ n ] F

12 PROB. Su8-Q2.4. Below are a list of LTI filters (labeled A to H) secified in the time domain: either by their imulse resonse h[ n ], their difference equation, or their MATLAB imlementation. Match each filter to its corresonding magnitude resonse shown on the right. Indicate your answers by writing a letter from fa, B,... Hg in each answer box. (A) y[ n ] =.5x[ n ].5x[ n ]. (B) y = conv(x,[2/3, /3]); (C) h[ n ] =.5x[ n 5 ] +.5x[ n 6 ] P 99 (D) y[ n ] =. k = (.9)k x[ n k] P 99 (E) y[ n ] =. k = (.9)k x[ n k] P 3 (F) y[ n ] =.25 x[ n k] k= (G) h[ n ] = -- 6 P 5 k= d[ n k] D G B C F H A (H) y = conv(x,cos(i*(:5))/6); E

13 PROB. Su8-Q2.5. Consider an LTI filter whose real-valued frequency resonse satisfies H(e j ) = j j, for j ^w j <, as illustrated below: H(e j) (a) The DC gain of the filter is..5 (b) If the inut is the sinusoid x[ n ] = cos(.5n), then the outut at n = 8 will be y[8] =. The freq resonse at the sinusoid frequency H(e j.5 ) =.5, ) y[n] = rcos(.5n + q) =.5cos(.5n) ) y[8] =.5 olar form re jq where r =.5, q = (c) If the outut of this system in resonse to the inut x[ n ] = cos( n) + cos( (n )) is y[ n ] = cos( n + j), then From hasor addition, inut simlifies to Ae jq = + e j ) A = 2cos( /2 ), q = /2 x[ n ] = 2cos( /2)cos( n /2 ). Filter scales by, no hase change ) inut amlitude must be = j = ) 2cos( /2) = ) = 2/3 alternative solution: = j = anything