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 6 Summer 13 Quiz #1 June 1, 13 NAME: (FIRST) (LAST) GT username: (e.g., gtxyz13) Circle your recitation section (otherwise you lose 3 points!): Mon Tue 1 11:4am L (Moore) 1 1:4pm L3 (Moore) 4 :4pm L1 (Barry) Important Notes: DO NOT unstaple the test. One two-sided page (8. 11 ) of hand-written notes permitted. JUSTIFY your reasoning CLEARLY to receive partial credit. You must write your answer in the space provided on the exam paper itself. Only these answers will be graded. Circle your answers, or write them in the boxes provided. If more space is needed for scratch work, use the backs of the previous pages. Problem Value Score Earned No/Wrong Rec 3 Total

2 PROB. Su13-Q1.1. The figure below shows the locations of twelve points in the complex plane, along with the unit circle: Im{ } C B F E A D L Re{ } H G I K Match each point above to one of the functions of an unspecified complex number z shown in the table below, by arranging the letters {A, B,... M} in the second column of the table below: J (1) () (3) (4) () (6) (7) (9) (9) (1) (11) (1) z 1/z z* 1/z* z + z* z z* zz* z/z* z jz z + j z + 1

3 PROB. Su13-Q1.. The signal x( t ) = B + Acos(πf t + θ) achieves a peak value of zero at time t =. and again at time t =.4, as shown below: x( t )..4 t 4 (a) Specify numeric values for the following parameters: B = A = f = Hz θ = ( π, π]. (b) Consider the following MATLAB code: tt = :.1:dur; % dur is the duration in seconds xx = real((1-j)*exp(j*16*pi*tt)) + real((-j)*exp(j*16*pi*tt)); The variable xx represents a sinusoidal signal x( t ) = Acos(ωt + θ) in standard form, where: ω = rad/s A = θ = ( π, π].

4 PROB. Su13-Q1.3. Several signals are plotted below along with their corresponding spectra. However, they are in a scrambled order. For each of the signals below, identify its corresponding spectrum by writing a letter {A, B,... G} into each answer box: A e jπ e jπ e jπ e jπ (1) t (sec) B 1 1 f (Hz) 1.e jπ 1.e jπ ().e jπ.e jπ t (sec) t (sec) t (sec) t (sec) F C D E 1 1 f (Hz).e jπ.e jπ.e jπ.e jπ 1 1 f (Hz) 1..e j.π.e j.π f (Hz) ().e j.π.e j.π f (Hz) e j.π e j.π (3) (4) (6) t (sec) G 1 1 f (Hz) e j.π e j.π (7) t (sec) 1 1 f (Hz)

5 PROB. Su13-Q Consider the periodic signal x( t ) shown below: x( t ) t (a) The fundamental period of x( t ) is T = seconds. (b) The fundamental frequency is f = Hz. P (c) In the Fourier series representation x( t )= k = a k e jkπt/t, the DC component is a =. (d) The spectrum for x( t ) will have a line at 9 Hz. Explain. (Check one.) YES NO NOT ENOUGH INFORMATION

6 PROB. Su13-Q1.. Suppose a sinusoidal signal x( t ) = cos(πf t ) with frequency f is sampled with a sampling rate of f s =3 samples/sec, and suppose that the resulting discrete-time sequence is as sketched below: x[ n ] 1 x( t ) = cos(πf t ) IDEAL C-to-D CONVERTER x[ n ] 1 1 n f s = 3 Hz -1 There are many possible values for the frequency f. Name any two. f = Hz, or f = Hz. The remainder of this problem concerns the following piece of MATLAB code: fsamp = 8; dt = 1/fsamp; dur =.; A3 = phi3 = tt = : dt : dur; x1 = cos(*pi*tt ); x = cos(*pi*tt +.*pi); x3 = A3*cos(*pi*tt + phi3 ); x = x1 + x + x3; (a) The length of the vector x1 is length(x1) =. (b) The sum x = x1 + x + x3 represents the sum of three sinusoids. In order for this sum to zero, the variables A3 and phi3 need to be: A3 =, phi3 = ( π, π].

7 GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING ECE 6 Summer 13 Quiz #1 June 1, 13 NAME: VERSION A (FIRST) (LAST) GT username: (e.g., gtxyz13) Circle your recitation section (otherwise you lose 3 points!): Mon Tue 1 11:4am L (Moore) 1 1:4pm L3 (Moore) 4 :4pm L1 (Barry) Important Notes: DO NOT unstaple the test. One two-sided page (8. 11 ) of hand-written notes permitted. JUSTIFY your reasoning CLEARLY to receive partial credit. You must write your answer in the space provided on the exam paper itself. Only these answers will be graded. Circle your answers, or write them in the boxes provided. If more space is needed for scratch work, use the backs of the previous pages. Problem Value Score Earned No/Wrong Rec 3 Total

8 PROB. Su13-Q1.1. The figure below shows the locations of twelve points in the complex plane, along with the unit circle: Im{ } C B F E A D L Re{ } H G I K Match each point above to one of the functions of an unspecified complex number z shown in the table below, by arranging letters {A, B,... M} in the second column of the table below: J (1) () (3) (4) () (6) (7) (9) (9) (1) (11) (1) z 1/z z* 1/z* z + z* z z* zz* z/z* z jz z + j z + 1 G E B H A J L F C K D I

9 PROB. Su13-Q1.. The signal x( t ) = B + Acos(πf t + θ) achieves a peak value of zero at time t =. and again at time t =.4, as shown below: x( t )..4 t 4 (a) Specify numeric values for the following parameters: B = A = f = Hz θ = ( π, π]. (b) Consider the following MATLAB code: tt = :.1:dur; % dur is the duration in seconds xx = real((1-j)*exp(j*16*pi*tt)) + real((-j)*exp(j*16*pi*tt)); The variable xx represents a sinusoidal signal x( t ) = Acos(ωt + θ) in standard form, where: ω = 16π rad/s (1 j) + ( j) = 3 j 3.6e j.19π A = θ =.19π ( π, π].

10 PROB. Su13-Q1.3. Several signals are plotted below along with their corresponding spectra. However, they are in a scrambled order. For each of the signals below, identify its corresponding spectrum by writing {A, B,... G} into each answer box: A e jπ e jπ e jπ e jπ (1) C t (sec) t (sec) t (sec) t (sec) t (sec) t (sec) F B C D E G t (sec) 1 1 f (Hz) 1.e jπ 1.e jπ.e jπ.e jπ 1 1 f (Hz).e jπ.e jπ.e jπ.e jπ 1 1 f (Hz) 1..e j.π.e j.π f (Hz).e j.π.e j.π f (Hz) e j.π e j.π 1 1 f (Hz) e j.π e j.π (7) 1 1 f (Hz) () B (3) E (4) A () F (6) D G

11 PROB. Su13-Q Consider the periodic signal x( t ) shown below: x( t ) t (a) The fundamental period of x( t ) is T =.6 seconds. (b) The fundamental frequency is f = Hz. f = 1/T P (c) In the Fourier series representation x( t )= k = a k e jkπt/t, the DC component is a = 3. By inspection, or by integration: a = (1/T )(area under one period) = (1/.6)(.18) = 3 (d) The spectrum for x( t ) will have a line at 9 Hz. Explain. (Check one.) YES NO NOT ENOUGH INFORMATION 9 Hz is not an integer multiple of f.

12 PROB. Su13-Q1.. Suppose a sinusoidal signal x( t ) = cos(πf t ) with frequency f is sampled with a sampling rate of f s =3 samples/sec, and suppose that the resulting discrete-time sequence is as sketched below: x[ n ] = cos(πn/16) 1 x( t ) = cos(πf t ) IDEAL C-to-D CONVERTER x[ n ] 1 1 n f s = 3 Hz -1 There are many possible values for the frequency f. Name any two. f = Hz, or f = + 3 Hz. for any integer From the stem plot we can write x[ n ] = cos(πn/16). Since this is a sinusoid whose digital frequency is less than π, we can convert to continuous time via the substitution n = tf s x( t ) = cos(π(f s /16)t) f = f s /16 = Hz. Other solutions can be found by adding f s for any integer. The remainder of this problem concerns the following piece of MATLAB code: fsamp = 8; dt = 1/fsamp; dur =.; A3 = phi3 = tt = : dt : dur; x1 = cos(*pi*tt ); x = cos(*pi*tt +.*pi); x3 = A3*cos(*pi*tt + phi3 ); x = x1 + x + x3; (a) The length of the vector x1 is length(x1) = 41. length = dur/dt + 1 = (.)(8) + 1 = 41 (b) The sum x = x1 + x + x3 represents the sum of three sinusoids. In order for this sum to zero, the variables A3 and phi3 need to be: A3 = 1.848, phi3 =.87π ( π, π]. 1 + e j.π + A 3 e jϕ 3 = A 3 e jϕ 3 = (1 + e j.π ) 1.848e j.87π

GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING

GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING ECE 26 Summer 14 Quiz #1 June 11, 14 NAME: (FIRST) (LAST) GT username: (e.g., gtxyz123) Circle your recitation section (otherwise