CHAPTER 5 5.1 The ramp function appears as the following:
T = 0.02 sec o = 2f = 2(1/T) = 2(1/0.02) = 100 rad/sec f(t) = 100t 0 t 0.01 f(t) = 100t 2 0.01 t 0.02 From Eq. 5.5, bn
2 T
T
0
f (t ) sin n o tdt
Thus, b1
2 0.02
0.01
0
100t sin(100t )dt
0.02
0.01
100 3.1831 10
3
(100t 2) sin(100t )dt
3.1831 10 3
0.6366
b2
2 0.02
0.01
0
100t sin( 2 100t )dt
100 15916 . 10 0.3183
0.02
0.01
3
(100t 2) sin(2 100t )dt
. 10 15916 3
ao is the average over the period, Eq. 5.3
5.1
ao
1 T
f t dt T
0
Thus, a0
1 0.02
0.01
0
50 5 10
100t cos(0)dt
3
0.02
0.01
5 10
(100t 2) cos(0)dt
3
0
From Eq. 5.4, an
2 T
T
0
a1
f (t ) cos n o tdt
2 0.02
0.01
0.02
0
0.01
100t cos(100t )dt
100 2.023 10 0
a2
2 0.02
3
2.026 10
3
0.01
0
(100t 2) cos(100t )dt
100t cos( 2 100t )dt
0.02
0.01
(100t 2) cos( 2 100t )dt
0
Without actually evaluating the values for a o, a1, and a2, we could have found that they were each equal to zero since f(t) is an odd function. 5.2 The ramp function appears as the following:
T = 0.04 sec o = 2f = 2(1/T) = 2(1/0.04) = 50 rad/sec f(t) = 50t f(t) = 50t 2
0 t 0.02 0.02 t 0.04
From Eq. 5.5, bn
2 T
f (t ) sin n tdt T
0
o
Thus,
5.2
0.02 0.04 2 50t sin(50 t ) dt 0.02 (50t 2) sin(50 t ) dt 0 0.04 50 6.3662 10 3 6.3662 10 3
b1
0.6366
0.02 0.04 2 50t sin( 2 50 t )dt 0.02 (50t 2) sin( 2 50 t ) dt 0 0.04 50 3.1831 10 3 3.1831 10 3
b2
0.3183
ao is the average over the period,Eq. 5.3 ao
1 T
f t dt T
0
Thus,
0.02 0.04 1 50 t cos( 0 ) dt (50t 2) cos(0) dt 0.02 0.04 0 25 0.01 0.01 0
a0
From Eq. 5.4, an
2 T
T
0
f (t ) cos n o tdt
0.02 0.04 2 50 t cos( 50 t ) dt (50t 2) cos(50 t ) dt 0.02 0.04 0 50 4.053 10 3 4.053 10 3
a1
0 a2
2 0.04
0.02
0
50t cos( 2 50 t ) dt 0.02 (50t 2) cos( 2 50 t ) dt 0.04
0
Without actually evaluating the values for a o, a1, and a2, we could have found that they were each equal to zero since f(t) is an odd function.
5.3
25
0 5.3
V(t) = 25t 0 t 1 T = 1sec o = 2f = 2(1/T) = 2(1/1) = 2 rad/sec By Eq. 5.5. bn
2 T
T
0
f (t ) sin n o tdt
Thus, b1
2 1 25t sin( 2t )dt 1 0
= 2(-3.9789) = -7.9578 b2
2 1 25t sin( 2 2t )dt 1 0
= 2(-1.9894) = -3.9788 ao is the average, Eq. 5.3 ao
1 T f t cos n o tdt T 0
Thus, a0
1 1 25t cos(0)dt 1 0
= 12.5 2 T f t cos n o tdt T 0 2 1 a1 25t cos( 2t )dt 1 0
and an
=0
5.4
1
2
a2
2 1 25t cos(2 2t )dt 1 0
=0
5.5
10
0 5.4
0t2 V(t) = 10t T = 2sec o = 2f = 2(1/T) = 2(1/2) = 1 rad/sec
By Eq. 5.5. 2 T
bn
f (t ) sin n tdt T
o
0
Thus, 2 2 10t sin(t )dt 2 0
b1
= -6.3662 2 2 10t sin(2 t )dt 2 0
b2
= -3.1831 ao is the average, Eq. 5.3 ao
1 T
T
0
f t cos notdt
Thus, a0
1 2 10t cos(0)dt 2 0
= 0.5(20) = 10 2 T f t cos n o tdt T 0 2 2 a1 010t cos( t ) dt 2
and an
=0 a2
2 2 10t cos(2 t )dt 2 0
5.6
2
4
=0
5.7
time -0.01 -0.0098 -0.0096 -0.0094 -0.0092 -0.009 -0.0088 -0.0086 -0.0084 -0.0082 -0.008 -0.0078 -0.0076 -0.0074 -0.0072 -0.007 -0.0068 -0.0066 -0.0064 -0.0062 -0.006 -0.0058 -0.0056 -0.0054 -0.0052 -0.005 -0.0048 -0.0046 -0.0044 -0.0042 -0.004 -0.0038 -0.0036 -0.0034 -0.0032 -0.003 -0.0028 -0.0026 -0.0024 -0.0022 -0.002 -0.0018 -0.0016 -0.0014 -0.0012 -0.001 -0.0008 -0.0006 -0.0004 -0.0002 0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002 0.0022 0.0024 0.0026 0.0028 0.003 0.0032 0.0034 0.0036 0.0038 0.004 0.0042 0.0044 0.0046 0.0048 0.005 0.0052 0.0054 0.0056 0.0058 0.006 0.0062 0.0064 0.0066 0.0068 0.007 0.0072 0.0074 0.0076 0.0078 0.008 0.0082 0.0084 0.0086 0.0088 0.009 0.0092 0.0094 0.0096 0.0098 0.01
f(t) n> -1.0000 -0.9800 -0.9600 -0.9400 -0.9200 -0.9000 -0.8800 -0.8600 -0.8400 -0.8200 -0.8000 -0.7800 -0.7600 -0.7400 -0.7200 -0.7000 -0.6800 -0.6600 -0.6400 -0.6200 -0.6000 -0.5800 -0.5600 -0.5400 -0.5200 -0.5000 -0.4800 -0.4600 -0.4400 -0.4200 -0.4000 -0.3800 -0.3600 -0.3400 -0.3200 -0.3000 -0.2800 -0.2600 -0.2400 -0.2200 -0.2000 -0.1800 -0.1600 -0.1400 -0.1200 -0.1000 -0.0800 -0.0600 -0.0400 -0.0200 0.0000 0.0200 0.0400 0.0600 0.0800 0.1000 0.1200 0.1400 0.1600 0.1800 0.2000 0.2200 0.2400 0.2600 0.2800 0.3000 0.3200 0.3400 0.3600 0.3800 0.4000 0.4200 0.4400 0.4600 0.4800 0.5000 0.5200 0.5400 0.5600 0.5800 0.6000 0.6200 0.6400 0.6600 0.6800 0.7000 0.7200 0.7400 0.7600 0.7800 0.8000 0.8200 0.8400 0.8600 0.8800 0.9000 0.9200 0.9400 0.9600 0.9800 1.0000 ao = 0
f(t)sinwt 1 2.65359E-06 0.061537253 0.12032233 0.176140739 0.228796872 0.278117339 0.323951517 0.366171967 0.404674727 0.439379478 0.470229576 0.497191956 0.520256918 0.539437779 0.554770412 0.56631266 0.574143647 0.578362966 0.579089776 0.576461797 0.570634205 0.561778455 0.550081016 0.535742036 0.518973944 0.5 0.479052791 0.456372692 0.432206294 0.406804811 0.380422475 0.353314924 0.325737592 0.297944123 0.270184791 0.242704958 0.215743575 0.18953172 0.164291194 0.140233179 0.117556965 0.096448751 0.077080528 0.059609054 0.044174911 0.030901674 0.019895175 0.011242869 0.005013325 0.001255809 0 0.001255809 0.005013325 0.011242869 0.019895175 0.030901674 0.044174911 0.059609054 0.077080528 0.096448751 0.117556965 0.140233179 0.164291194 0.18953172 0.215743575 0.242704958 0.270184791 0.297944123 0.325737592 0.353314924 0.380422475 0.406804811 0.432206294 0.456372692 0.479052791 0.5 0.518973944 0.535742036 0.550081016 0.561778455 0.570634205 0.576461797 0.579089776 0.578362966 0.574143647 0.56631266 0.554770412 0.539437779 0.520256918 0.497191956 0.470229576 0.439379478 0.404674727 0.366171967 0.323951517 0.278117339 0.228796872 0.176140739 0.12032233 0.061537253 2.65359E-06 b1 = 0.636411395
f(t)sin2wt 2 -5.30718E-06 -0.122831626 -0.238747029 -0.34604144 -0.443217317 -0.529010205 -0.602404449 -0.662643891 -0.709237464 -0.741959702 -0.760846263 -0.766184661 -0.758500506 -0.738539597 -0.707246305 -0.665738758 -0.615281351 -0.557255192 -0.49312709 -0.424417719 -0.352669606 -0.279415566 -0.206148202 -0.13429104 -0.065171858 1.32679E-06 0.060161165 0.114398436 0.161975758 0.202337364 0.235114788 0.260128459 0.277385206 0.287071823 0.289544888 0.285317102 0.275040508 0.259486972 0.239526396 0.216103147 0.190211238 0.162868796 0.135092395 0.107871788 0.082145597 0.058778482 0.038540264 0.022087455 0.009947587 0.002506663 0 0.002506663 0.009947587 0.022087455 0.038540264 0.058778482 0.082145597 0.107871788 0.135092395 0.162868796 0.190211238 0.216103147 0.239526396 0.259486972 0.275040508 0.285317102 0.289544888 0.287071823 0.277385206 0.260128459 0.235114788 0.202337364 0.161975758 0.114398436 0.060161165 1.32679E-06 -0.065171858 -0.13429104 -0.206148202 -0.279415566 -0.352669606 -0.424417719 -0.49312709 -0.557255192 -0.615281351 -0.665738758 -0.707246305 -0.738539597 -0.758500506 -0.766184661 -0.760846263 -0.741959702 -0.709237464 -0.662643891 -0.602404449 -0.529010205 -0.443217317 -0.34604144 -0.238747029 -0.122831626 -5.30718E-06 b2 = -0.317891435
f(t)coswt 1 1 0.978066034 0.952429807 0.923349576 0.89109595 0.8559502 0.818202551 0.778150429 0.736096709 0.692347943 0.647212597 0.6009993 0.554015108 0.506563799 0.458944213 0.411448625 0.364361185 0.317956412 0.272497763 0.228236274 0.185409288 0.14423927 0.104932719 0.067679178 0.032650354 -6.63397E-07 -0.03014006 -0.057653845 -0.082448283 -0.104450206 -0.123607202 -0.139887686 -0.153280856 -0.163796518 -0.171464804 -0.176335769 -0.178478877 -0.177982378 -0.174952575 -0.169512995 -0.161803461 -0.151979073 -0.140209102 -0.126675809 -0.111573192 -0.09510566 -0.077486657 -0.058937237 -0.039684589 -0.019960535 0 0.019960535 0.039684589 0.058937237 0.077486657 0.09510566 0.111573192 0.126675809 0.140209102 0.151979073 0.161803461 0.169512995 0.174952575 0.177982378 0.178478877 0.176335769 0.171464804 0.163796518 0.153280856 0.139887686 0.123607202 0.104450206 0.082448283 0.057653845 0.03014006 6.63397E-07 -0.032650354 -0.067679178 -0.104932719 -0.14423927 -0.185409288 -0.228236274 -0.272497763 -0.317956412 -0.364361185 -0.411448625 -0.458944213 -0.506563799 -0.554015108 -0.6009993 -0.647212597 -0.692347943 -0.736096709 -0.778150429 -0.818202551 -0.8559502 -0.89109595 -0.923349576 -0.952429807 -0.978066034 -1 a1 = 1.39888E-16
f(t)cos2wt 2 -1 -0.972271768 -0.929838618 -0.87398817 -0.806199982 -0.728112768 -0.641489579 -0.548181607 -0.450091346 -0.34913579 -0.247210365 -0.146154254 -0.047717735 0.046467885 0.134917249 0.216314369 0.289532139 0.353647637 0.407953028 0.451961946 0.48541132 0.508258735 0.520675445 0.523035292 0.515899825 0.5 0.476214903 0.445547975 0.409101276 0.368048355 0.323606299 0.277007554 0.229472106 0.182180592 0.136248882 0.092704644 0.052466359 0.016325177 -0.01507003 -0.041224142 -0.061803601 -0.076640428 -0.085732402 -0.089239439 -0.087476288 -0.080901731 -0.070104551 -0.055786596 -0.038743329 -0.019842294 0 0.019842294 0.038743329 0.055786596 0.070104551 0.080901731 0.087476288 0.089239439 0.085732402 0.076640428 0.061803601 0.041224142 0.01507003 -0.016325177 -0.052466359 -0.092704644 -0.136248882 -0.182180592 -0.229472106 -0.277007554 -0.323606299 -0.368048355 -0.409101276 -0.445547975 -0.476214903 -0.5 -0.515899825 -0.523035292 -0.520675445 -0.508258735 -0.48541132 -0.451961946 -0.407953028 -0.353647637 -0.289532139 -0.216314369 -0.134917249 -0.046467885 0.047717735 0.146154254 0.247210365 0.34913579 0.450091346 0.548181607 0.641489579 0.728112768 0.806199982 0.87398817 0.929838618 0.972271768 1 a2 = -2.66454E-17
5.8
5.5 The spreadsheet is shown at the left. The values determined are b1 = 0.6364, b2 = -0.3179, ao = 0, a1 = 0, and a2 =0. These compare to exact respective values of 0.6366, -0.3183, 0, 0 and 0.
note: The integration is from -T/2 to T/2 which is equivalent to 0 to T
time
f(t) 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.4 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.5 0.51 0.52 0.53 0.54 0.55 0.56 0.57 0.58 0.59 0.6 0.61 0.62 0.63 0.64 0.65 0.66 0.67 0.68 0.69 0.7 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.8 0.81 0.82 0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1
f(t)sinwt 1
n> 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4 4.25 4.5 4.75 5 5.25 5.5 5.75 6 6.25 6.5 6.75 7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10 10.25 10.5 10.75 11 11.25 11.5 11.75 12 12.25 12.5 12.75 13 13.25 13.5 13.75 14 14.25 14.5 14.75 15 15.25 15.5 15.75 16 16.25 16.5 16.75 17 17.25 17.5 17.75 18 18.25 18.5 18.75 19 19.25 19.5 19.75 20 20.25 20.5 20.75 21 21.25 21.5 21.75 22 22.25 22.5 22.75 23 23.25 23.5 23.75 24 24.25 24.5 24.75 25
bo =
a1 = 12.5
f(t)sin2wt 2
0 0.015697617 0.062666564 0.140535869 0.248689682 0.386270928 0.552186385 0.745113172 0.963506604 1.205609381 1.469462057 1.752914735 2.053639925 2.369146504 2.696794692 3.033811974 3.377309882 3.724301543 4.071719906 4.416436545 4.755280941 5.085060141 5.402578676 5.704658653 5.988159891 6.25 6.487174298 6.696775446 6.876012704 7.022230693 7.132927562 7.205772459 7.238622205 7.229537071 7.176795585 7.078908254 6.934630146 6.742972238 6.503211473 6.214899449 5.877869697 5.49224348 5.058434088 4.577149582 4.049393962 3.476466739 2.859960895 2.201759236 1.504029131 0.769215658 3.31699E-05 -0.800544682 -1.629296443 -2.482765809 -3.357276003 -4.248945501 -5.153705051 -6.0673159 -6.985389162 -7.90340623 -8.816740142 -9.720677803 -10.61044296 -11.48121983 -12.32817724 -13.14649321 -13.9313798 -14.6781082 -15.38203377 -16.03862113 -16.64346894 -17.19233448 -17.68115762 -18.10608444 -18.46348991 -18.75 -18.96251265 -19.09821786 -19.15461651 -19.12953802 -19.02115657 -18.82800588 -18.54899256 -18.18340764 -17.7309366 -17.19166748 -16.56609727 -15.85513639 -15.06011123 -14.18276475 -13.22525512 -12.19015235 -11.08043291 -9.899472362 -8.651035991 -7.339267499 -5.968675728 -4.544119506 -3.070790643 -1.554195141 -0.000132679 a2 = -7.955142436
f(t)coswt 1
0 0.031333282 0.124344841 0.276093192 0.481753302 0.734731029 1.026819963 1.348397346 1.688654941 2.035859953 2.377640471 2.701289338 2.994079945 3.243587149 3.438006352 3.566463781 3.619311102 3.588397793 3.467315073 3.251605736 2.938934849 2.529217044 2.024696981 1.429980448 0.752014566 1.65849E-05 -0.814648221 -1.678638001 -2.576852526 -3.492694581 -4.408370071 -5.305221481 -6.164088622 -6.965689901 -7.691016885 -8.321734472 -8.840578812 -9.231744957 -9.481256326 -9.577308257 -9.510578283 -9.27449628 -8.865468299 -8.283048635 -7.530055615 -6.612627561 -5.540216457 -4.325517995 -2.984337864 -1.535395321 -6.63397E-05 1.597930252 3.232899034 4.877581018 6.503606793 8.081982278 9.583598821 10.97975875 12.24270709 13.34615969 14.26581822 14.97986207 15.46940789 15.71892759 15.71661638 15.45470303 14.92969558 14.14255658 13.09880334 11.80852967 10.28634711 8.551244936 6.62636985 4.538727408 2.318808986 0.000149264 -2.381179375 -4.787127939 -7.17827867 -9.514489938 -11.75556765 -13.86195198 -15.79540774 -17.5197065 -19.00128838 -20.20989173 -21.11913911 -21.70706873 -21.95660093 -21.85593071 -21.39883804 -20.584909 -19.41966263 -17.91457919 -16.08702791 -13.96009349 -11.56230248 -8.927252545 -6.093149079 -3.102255558 -0.000265359 b1 = -3.973642941
f(t)cos2wt 2
0 0.249506683 0.496057357 0.73671546 0.968583214 1.188820748 1.394664905 1.583447619 1.752613769 1.899738408 2.022543266 2.118912441 2.18690719 2.224779729 2.230985968 2.204197111 2.143310048 2.047456475 1.916010702 1.748596079 1.545090019 1.305627575 1.030603538 0.720673056 0.376750744 8.29247E-06 -0.408129426 -0.84598973 -1.311658984 -1.802990874 -2.317616101 -2.852953428 -3.406222039 -3.97445515 -4.554514807 -5.143107808 -5.736802659 -6.332047489 -6.925188845 -7.512491253 -8.090157466 -8.654349286 -9.201208865 -9.726880369 -10.22753189 -10.69937751 -11.13869937 -11.5418697 -11.90537258 -12.22582542 -12.5 -12.72484295 -12.89749561 -13.01531306 -13.0758823 -13.0770395 -13.01688612 -12.89380385 -12.70646836 -12.45386165 -12.13528299 -11.75035842 -11.29904864 -10.78165531 -10.19882571 -9.551555701 -8.841190915 -8.069426234 -7.238303469 -6.350207267 -5.407859233 -4.41431028 -3.372931225 -2.28740166 -1.161697123 -7.46322E-05 1.192943387 2.412586701 3.653856342 4.911545068 6.180259129 7.454441254 8.728394754 9.996308641 11.25228365 12.49035906 13.70454017 14.88882635 16.03723947 17.14385266 18.2028192 19.20840143 20.15499954 21.03718011 21.84970426 22.58755526 23.24596546 23.82044244 24.30679421 24.70115336 25
0 0.248028679 0.484291607 0.697332452 0.876306885 1.011271633 1.093453595 1.115492984 1.071655024 0.958005351 0.77254501 0.515301769 0.188375372 -0.204064713 -0.655829492 -1.158808051 -1.70311102 -2.277257404 -2.868401329 -3.462594423 -4.045078733 -4.600604433 -5.113765944 -5.569349685 -5.952686292 -6.25 -6.448747807 -6.537941148 -6.50844306 -6.353234181 -6.067641495 -5.649524319 -5.099412855 -4.420595458 -3.619151734 -2.703929616 -1.686465613 -0.580848562 0.596471693 1.826928171 3.090129564 4.364197377 5.626141825 6.852270085 8.018619734 9.101409602 10.07749977 10.92485213 11.62298273 12.15339711 12.5 12.64947109 12.59159894 12.31956588 11.83017746 11.12403085 10.20561775 9.083358284 7.769563897 6.280328131 4.635345769 2.857662038 0.973354855 -0.988845569 -2.997994267 -5.021419532 -7.025253721 -8.97499824 -10.83611328 -12.57462234 -14.15772097 -15.55437913 -16.73592611 -17.67660752 -18.35410376 -18.75 -18.85019853 -18.64526498 -18.13070091 -17.30713671 -16.18043971 -14.76173389 -13.06732927 -11.11856034 -8.94153454 -6.566793469 -4.028891047 -1.365894254 1.381186341 4.169027782 6.952677952 9.686280053 12.32382665 14.81993092 17.13060224 19.21401285 21.0312425 22.54698787 23.73022407 24.55480626 25 b2 =
-4.22194E-05
-4.21777E-05
5.9
5.6
The spreadsheet is shown at the left. The values determined are b1 = -7.955, b2 = -3.974, a0 = 12.5, a1 = 0, and a2 =0. These compare to exact respective values of -7.9578, -3.9788, 12.5, 0 and 0.
5.7 If the function is made into an odd function, i.e. v(t) = -v(-t), then it can be represented with sine functions only. This can be accomplished by adding the function from t = -0.1 to t = 0 having the form 20t + 200t 2. The original function plus this new function form one cycle of a function on which a Fourier analysis can be performed.
Voltage 0.6
0.4
2
v(t) = 20t - 200t
0.2 0 -0.2
2
v(t) = 20t + 200t
-0.4 -0.6 -0.1
-0.05
0
0.05
0.1 Time - sec
The resulting Fourier series will properly represent the original function in the 0 0.1 second interval but will be incorrect outside that interval. 5.8 If the function is made into an odd function, i.e. v(t) = -v(-t), then it can be represented with sine functions only. This can be accomplished by adding the function from t = -0.2 to t = 0 having the form 20t + 100t 2. The original function plus this new function form one cycle of a function on which a Fourier analysis can be performed.
The resulting Fourier series will properly represent the original function in the 0 0.2 second interval but will be incorrect outside that interval. 5.10
5.9 The function can be represented with only cosine terms if it is an even function, f(t) = f(-t). One such function is shown below. v(t)
4 3 2 1 0 -1 -2 -3 -4 -6
-4
-2
0
2
4
6
time - seconds
The resulting Fourier will represent the original function correctly in the 0 - 1.5 second interval but will be incorrect outside that interval. 5.10 The function can be represented with only cosine terms if it is an even function, f(t) = f(-t). One such function is shown below.
The resulting Fourier will represent the original function correctly in the 0 - 2 second interval but will be incorrect outside that interval.
5.11
5.11 f(t) 1.2
1
first 20 terms
0.8
b0 , b1 , b2 , b3 terms
0.6 0.4 0.2 0 -0.2 0
0.1
0.2
0.3
0.4
0.5 0.6 time - sec
The first four terms do not do a very good job of representing this square waves. This is why square waves are a sensitive test of the frequency response of instruments. For reference, the sum of the first 20 terms is also shown.
5.12
f(t) 3 2.871651566 1.183250021 0.424079962 2.12396463 5.422294952 7.803153495 7.336129722 4.403878647 1.359405907 0.421978514 1.707936728 3.17016518 2.498103016 -0.738323335 -4.637804818 -6.620225193 -5.696273486 -3.310949009 -2.003290419 -3.093180742 -5.482581588 -6.652700577 -4.957472928 -1.143869125 2.274093196 3.226047986 1.890628905 0.471250995 1.148357663 4.058389982 7.125272795 7.899143154 5.746285898 2.416680299 0.477612381 1.025127872 2.735401501 3.109028387 0.783125743 -3.177231726 -6.176592366 -6.370160297 -4.274984328 -2.261606546 -2.387137882 -4.525560184 -6.482143137 -5.988753492 -2.77980659 1.124988499 3.188687984 2.58487515 0.879896973 0.558205579 2.72435974 6.057407815 7.96355213 6.890027853 3.715191016 0.961924643 0.543338635 2.072309777 3.257116489 2.023572337 -1.553351437
freq (Hz) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
F(f) 2.99963853703405 3.03107808323142-0.953489727174751i 3.13972600731118-2.19264191621691i 3.39446291040749-4.35277576822253i 4.1506503807292-10.3318395080083i 42.0778018981402-315.139900191749i 1.15673145040832+14.5606087209i 2.0687931824973+7.94063600672844i 2.45177094174856+5.81805890632637i 2.73216871318734+4.80571676035284i 2.99947307192228+4.25004076361989i 3.29223988807091+3.94021835272683i 3.63963995929184+3.79303154623195i 4.07528245235743+3.77711044930519i 4.64861413905354+3.89155708187997i 5.44394983322112+4.16524610793015i 6.62445094289296+4.67557346031329i 8.55830664201133+5.61403491112103i 12.2949154415459+7.54174889789134i 22.5086297516064+12.9766095685749i 162.608034374744+88.4852822222177i -29.8880606599922-15.4066520489326i -13.4508975188424-6.58827818481786i -8.58753125233437-4.00709570329827i -6.2617659309384-2.78971922963066i -4.90140821032262-2.08882384099306i -4.01063677914423-1.63756127310175i -3.38347692063223-1.32532271690295i -2.91901176398376-1.09808332141237i -2.56197898616181-0.926376517565872i -2.2795792213038-0.792797515774453i -2.05111784768694-0.686426317762479i -1.86289456542791-0.600079203753877i -1.70548044113495-0.528846341613602i -1.57217514308172-0.469264699141709i -1.45808779197661-0.418826334322632i -1.35956589935354-0.375673663607068i -1.2738277338382-0.338403845548055i -1.1987183134243-0.305939362053351i -1.13254310168028-0.277440124902437i -1.07395199924385-0.252242399286549i -1.02185674147784-0.229815493691055i -0.975370998471055-0.209730489103945i -0.933766222021456-0.191637292692324i -0.896438617865371-0.175247552311383i -0.862884109708393-0.16032176510352i -0.83267913164779-0.146659432058179i -0.805465730442839-0.134091454589227i -0.780939895538502-0.122474201673354i -0.758842335060168-0.111684835739446i -0.738951125784759-0.101617596738633i -0.721075813698939-9.21808224173024E-002i -0.705052648416223-8.32945390417479E-002i -0.69074071219551-7.48884975129065E-002i -0.67801876123611-6.69005595738468E-002i -0.666782639208102-5.92753608008256E-002i -0.656943154719748-5.19631934414062E-002i -0.648424338491339-4.49190644708364E-002i -0.641162014451039-3.81018935315351E-002i -0.635102633248227-3.14738224842301E-002i -0.630202327871022-2.49996136833994E-002i -0.626426159944596-1.86461181921036E-002i -0.62374753243364-1.23817982669434E-002i -0.622147750372017-6.17629078533588E-003i -0.621615716175162 -0.622147750372011+6.17629078533999E-003i
|F(f)| 2.999638537 3.177511134 3.829563679 5.519876388 11.13439743 317.9366259 14.60648329 8.205705663 6.313556065 5.528079183 5.201892463 5.134604576 5.256811519 5.556481838 6.062493706 6.854623617 8.108288197 10.23533099 14.42369308 25.98135503 185.1243312 33.62530442 14.97771857 9.476418574 6.855089055 5.32794402 4.332068132 3.633785406 3.118720356 2.724318223 2.41350561 2.162929845 1.957158965 1.785592895 1.64071449 1.517048288 1.410514139 1.318011479 1.237143681 1.166030403 1.103176833 1.047380715 0.997665005 0.953228309 0.91340785 0.877651443 0.845496023 0.816551016 0.790485326 0.767017074 0.745905424 0.726944037 0.709955785 0.69478847 0.681311328 0.669412172 0.658995055 0.649978342 0.642293144 0.63588203 0.63069799 0.626703607 0.623870414 0.622178407 0.621615716 0.622178407
5.12 400
Amplitude
time 0 0.007874016 0.015748031 0.023622047 0.031496063 0.039370079 0.047244094 0.05511811 0.062992126 0.070866142 0.078740157 0.086614173 0.094488189 0.102362205 0.11023622 0.118110236 0.125984252 0.133858268 0.141732283 0.149606299 0.157480315 0.165354331 0.173228346 0.181102362 0.188976378 0.196850394 0.204724409 0.212598425 0.220472441 0.228346457 0.236220472 0.244094488 0.251968504 0.25984252 0.267716535 0.275590551 0.283464567 0.291338583 0.299212598 0.307086614 0.31496063 0.322834646 0.330708661 0.338582677 0.346456693 0.354330709 0.362204724 0.37007874 0.377952756 0.385826772 0.393700787 0.401574803 0.409448819 0.417322835 0.42519685 0.433070866 0.440944882 0.448818898 0.456692913 0.464566929 0.472440945 0.480314961 0.488188976 0.496062992 0.503937008 0.511811024
300 200 100 0 0 3 6 9 12 15 18 21 24 27 30 33 36
Frequency - Hz
Note: peaks occur at the frequencies in the generating function
5.13
0.519685039 0.527559055 0.535433071 0.543307087 0.551181102 0.559055118 0.566929134 0.57480315 0.582677165 0.590551181 0.598425197 0.606299213 0.614173228 0.622047244 0.62992126 0.637795276 0.645669291 0.653543307 0.661417323 0.669291339 0.677165354 0.68503937 0.692913386 0.700787402 0.708661417 0.716535433 0.724409449 0.732283465 0.74015748 0.748031496 0.755905512 0.763779528 0.771653543 0.779527559 0.787401575 0.795275591 0.803149606 0.811023622 0.818897638 0.826771654 0.834645669 0.842519685 0.850393701 0.858267717 0.866141732 0.874015748 0.881889764 0.88976378 0.897637795 0.905511811 0.913385827 0.921259843 0.929133858 0.937007874 0.94488189 0.952755906 0.960629921 0.968503937 0.976377953 0.984251969 0.992125984 1
-5.253975531 -6.654266847 -5.256118151 -2.889965405 -2.029518013 -3.540399016 -5.894110744 -6.556461971 -4.297832112 -0.340474383 2.69406209 3.091251797 1.527448962 0.397006012 1.593257113 4.748004319 7.520282281 7.676642507 5.088936023 1.849089024 0.397420617 1.35138986 2.990907747 2.861768829 0.048127139 -3.938796782 -6.460992062 -6.074200387 -3.779732533 -2.081814452 -2.70291604 -5.01886925 -6.625611204 -5.526256294 -1.965143306 1.747018651 3.262000087 2.250726831 0.63688536 0.800648776 3.376056779 6.63168196 7.995961579 6.353922911 3.045588561 0.66618074 0.74951434 2.422132071 3.239542613 1.447528332 -2.374073117 -5.771156984 -6.567937388 -4.774921543 -2.535053219 -2.159292329 -4.024989253 -6.2332661 -6.334502788 -3.565174975 0.422714993 2.999867304
-0.623747532433636+1.23817982669474E-002i -0.626426159944614+1.86461181921076E-002i -0.630202327871017+2.49996136834509E-002i -0.635102633247936+3.14738224842586E-002i -0.641162014451052+3.81018935315351E-002i -0.648424338491355+4.49190644708368E-002i -0.656943154719751+5.19631934414049E-002i -0.666782639208101+5.92753608008274E-002i -0.678018761236112+6.69005595738419E-002i -0.690740712195535+7.4888497512885E-002i -0.705052648416258+8.32945390418962E-002i -0.721075813698932+9.21808224173266E-002i -0.738951125784765+0.101617596738646i -0.758842335060181+0.111684835739462i -0.780939895538509+0.122474201673366i -0.805465730442842+0.134091454589245i -0.832679131647796+0.146659432058196i -0.862884109708437+0.160321765103543i -0.896438617865513+0.175247552311575i -0.933766222021401+0.19163729269233i -0.975370998471051+0.209730489103939i -1.02185674147784+0.229815493691042i -1.07395199924384+0.252242399286547i -1.13254310168027+0.277440124902439i -1.19871831342429+0.305939362053357i -1.27382773383837+0.338403845547929i -1.35956589935359+0.375673663607144i -1.4580877919766+0.418826334322665i -1.57217514308172+0.469264699141715i -1.70548044113496+0.528846341613596i -1.86289456542791+0.600079203753877i -2.05111784768693+0.686426317762493i -2.2795792213038+0.792797515774465i -2.56197898616183+0.926376517565844i -2.91901176398385+1.09808332141246i -3.3834769206321+1.32532271690313i -4.01063677914423+1.63756127310176i -4.90140821032263+2.08882384099306i -6.2617659309384+2.78971922963067i -8.58753125233436+4.00709570329832i -13.4508975188424+6.58827818481792i -29.8880606599922+15.4066520489327i 162.608034374744-88.4852822222182i 22.5086297516063-12.9766095685749i 12.2949154415458-7.54174889789138i 8.55830664201125-5.6140349111211i 6.62445094289294-4.6755734603133i 5.44394983322108-4.16524610793018i 4.64861413905351-3.89155708187998i 4.07528245235733-3.77711044930534i 3.63963995929178-3.79303154623196i 3.29223988807091-3.94021835272679i 2.99947307192227-4.25004076361991i 2.73216871318728-4.80571676035299i 2.45177094174854-5.81805890632638i 2.06879318249724-7.94063600672859i 1.15673145040826-14.5606087209i 42.0778018981418+315.139900191749i 4.15065038072927+10.3318395080083i 3.39446291040754+4.35277576822282i 3.13972600731119+2.19264191621688i 3.03107808323142+0.953489727174949i -1.28-52.1414i
5.14
0.623870414 0.626703607 0.63069799 0.63588203 0.642293144 0.649978342 0.658995055 0.669412172 0.681311328 0.69478847 0.709955785 0.726944037 0.745905424 0.767017074 0.790485326 0.816551016 0.845496023 0.877651443 0.91340785 0.953228309 0.997665005 1.047380715 1.103176833 1.166030403 1.237143681 1.318011479 1.410514139 1.517048288 1.64071449 1.785592895 1.957158965 2.162929845 2.41350561 2.724318223 3.118720356 3.633785406 4.332068132 5.32794402 6.855089055 9.476418574 14.97771857 33.62530442 185.1243312 25.98135503 14.42369308 10.23533099 8.108288197 6.854623617 6.062493706 5.556481838 5.256811519 5.134604576 5.201892463 5.528079183 6.313556065 8.205705663 14.60648329 317.9366259 11.13439743 5.519876388 3.829563679 3.177511134 52.15710876
f(t) 0 0.000787402 0.001574803 0.002362205 0.003149606 0.003937008 0.004724409 0.005511811 0.006299213 0.007086614 0.007874016 0.008661417 0.009448819 0.01023622 0.011023622 0.011811024 0.012598425 0.013385827 0.014173228 0.01496063 0.015748031 0.016535433 0.017322835 0.018110236 0.018897638 0.019685039 0.020472441 0.021259843 0.022047244 0.022834646 0.023622047 0.024409449 0.02519685 0.025984252 0.026771654 0.027559055 0.028346457 0.029133858 0.02992126 0.030708661 0.031496063 0.032283465 0.033070866 0.033858268 0.034645669 0.035433071 0.036220472 0.037007874 0.037795276 0.038582677 0.039370079 0.04015748 0.040944882 0.041732283 0.042519685 0.043307087 0.044094488 0.04488189 0.045669291 0.046456693 0.047244094 0.048031496 0.048818898 0.049606299 0.050393701 0.051181102
freq (Hz) 8 -1.001264893 -5.77992304 2.175730456 0.389538306 -1.472531294 5.106322592 -0.756970292 -7.636525619 2.942428177 5.890432359 -3.355176234 -1.021855252 1.348628445 -4.101026101 2.068165004 6.617226701 -4.756626851 -5.318021241 4.790059281 1.246670585 -1.767720112 3.055511354 -2.769472158 -5.140169668 6.153693982 4.081876412 -6.189160753 -0.894451937 2.562990265 -2.257007351 2.83276874 3.487761877 -6.9245361 -2.337642887 7.253700679 -0.074514159 -3.474443875 1.925282318 -2.368860985 -1.966088397 6.985042837 0.345388418 -7.74206003 1.559736732 4.206625937 -2.16427335 1.601552938 0.839333256 -6.393058266 1.584988948 7.522342121 -3.340637899 -4.494187063 2.939058053 -0.817230892 -0.273494243 5.334752963 -3.159098978 -6.601372449 5.122836396 4.161054415 -4.083169508 0.300925461 0.301581782 -4.083368667
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350 360 370 380 390 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640
F(f) 7.99937 8.00425+0.196478i 8.01896+0.393915i 8.04374+0.593295i 8.079+0.795648i 8.12536+1.00209i 8.18363+1.21383i 8.25491+1.43224i 8.34059+1.65891i 8.4425+1.89567i 8.56297+2.14473i 8.70502+2.4088i 8.8726+2.69124i 9.07099+2.99638i 9.30727+3.3299i 9.59131+3.69944i 9.93715+4.11572i 10.3656+4.5943i 10.9086+5.15889i 11.6189+5.84784i 12.5885+6.728i 13.9972+7.92843i 16.2474+9.73722i 20.4723+12.9624i 31.5848+21.1016i 154.285+108.645i -28.1429-20.869i -7.99645-6.23935i -2.03189-1.66704i 0.979711+0.845228i 2.92885+2.65441i 4.41931+4.20714i 5.72223+5.72158i 7.00261+7.35412i 8.40265+9.26959i 10.0975+11.704i 12.3757+15.0773i 15.8372+20.2899i 22.0866+29.7746i 37.574+53.3399i 150.575+225.3i -66.0264-104.242i -25.4567-42.4607i -15.0649-26.5856i -10.2921-19.2491i -7.54606-14.9862i -5.7612-12.1763i -4.50915-10.1681i -3.5842-8.64881i -2.87522-7.4496i -2.31683-6.47102i -1.8681-5.65069i -1.50207-4.94742i -1.20027-4.33284i -0.949682-3.78672i -0.740847-3.29419i -0.566788-2.84402i -0.422255-2.42751i -0.303287-2.03776i -0.206866-1.66916i -0.130696-1.31704i -0.073063-0.977412i -0.0327292-0.646777i -0.00885656-0.321957i -0.000945956 -0.00885656+0.321957i
|F(f)| 7.99937 8.006661081 8.028629304 8.065590626 8.118084549 8.186920025 8.273160415 8.378236719 8.503965189 8.652708882 8.827475403 9.032147621 9.27177456 9.553070328 9.885014358 10.28003326 10.75574736 11.33813282 12.06696731 13.00753896 14.27362309 16.08669046 18.94179136 24.23094883 37.98522231 188.6997542 35.03623812 10.14261806 2.628231218 1.293925815 3.95272751 6.10166599 8.091995667 10.15478349 12.51118802 15.45778517 19.50597154 25.73901602 37.07215538 65.24530947 270.985093 123.3931929 49.50711686 30.55724692 21.82785315 16.77883226 13.47047539 11.12307023 9.362072637 7.985200699 6.873267134 5.951478395 5.170413808 4.496014962 3.903990812 3.376468871 2.899947999 2.463961056 2.060206014 1.681930036 1.323508899 0.98013898 0.647604577 0.322078792 0.000945956
5.15
5.13 Amplitude
time
300 250 200 150 100 50 0 0
70 140 210 280 350 420 490 560 630
Frequency - Hz
Spikes in FFT occur at the frequencies of the generating function. The spread at the peak frequencies is due to insufficient resolution of signal (sampling rate only about three times maximum frequency.)
0.051968504 0.052755906 0.053543307 0.054330709 0.05511811 0.055905512 0.056692913 0.057480315 0.058267717 0.059055118 0.05984252 0.060629921 0.061417323 0.062204724 0.062992126 0.063779528 0.064566929 0.065354331 0.066141732 0.066929134 0.067716535 0.068503937 0.069291339 0.07007874 0.070866142 0.071653543 0.072440945 0.073228346 0.074015748 0.07480315 0.075590551 0.076377953 0.077165354 0.077952756 0.078740157 0.079527559 0.080314961 0.081102362 0.081889764 0.082677165 0.083464567 0.084251969 0.08503937 0.085826772 0.086614173 0.087401575 0.088188976 0.088976378 0.08976378 0.090551181 0.091338583 0.092125984 0.092913386 0.093700787 0.094488189 0.095275591 0.096062992 0.096850394 0.097637795 0.098425197 0.099212598 0.1
4.160002467 5.123842099 -6.600583844 -3.160351437 5.334672217 -0.272744699 -0.817830481 2.939295569 -4.493355663 -3.341770928 7.52183541 1.586382956 -6.393176027 0.838490071 1.602157717 -2.164487132 4.20599503 1.560899841 -7.741862601 0.343941211 6.985420913 -1.96518981 -2.36954195 1.925441863 -3.473960722 -0.075621455 7.253797831 -2.33624433 -6.925203276 3.486878699 2.833587469 -2.25712012 2.562583776 -0.893458914 -6.189499769 4.080622999 6.154638085 -5.139391821 -2.770464269 3.055620507 -1.767318949 1.245812955 4.790561336 -5.316986349 -4.757795366 6.61664527 2.06932917 -4.101200109 1.34817893 -1.021115184 -3.355753576 5.889653364 2.943737309 -7.636214285 -0.758265265 5.106638981 -1.472011193 0.388865998 2.176305109 -5.779396071 -1.002615148 7.999999861
-0.0327292+0.646777i -0.073063+0.977412i -0.130696+1.31704i -0.206866+1.66916i -0.303287+2.03776i -0.422255+2.42751i -0.566788+2.84402i -0.740847+3.29419i -0.949682+3.78672i -1.20027+4.33284i -1.50207+4.94742i -1.8681+5.65069i -2.31683+6.47102i -2.87522+7.4496i -3.5842+8.64881i -4.50915+10.1681i -5.7612+12.1763i -7.54606+14.9862i -10.2921+19.2491i -15.0649+26.5856i -25.4567+42.4607i -66.0264+104.242i 150.575-225.3i 37.574-53.3399i 22.0866-29.7746i 15.8372-20.2899i 12.3757-15.0773i 10.0975-11.704i 8.40265-9.26959i 7.00261-7.35412i 5.72223-5.72158i 4.41931-4.20714i 2.92885-2.65441i 0.979711-0.845228i -2.03189+1.66704i -7.99645+6.23935i -28.1429+20.869i 154.285-108.645i 31.5848-21.1016i 20.4723-12.9624i 16.2474-9.73722i 13.9972-7.92843i 12.5885-6.728i 11.6189-5.84784i 10.9086-5.15889i 10.3656-4.5943i 9.93715-4.11572i 9.59131-3.69944i 9.30727-3.3299i 9.07099-2.99638i 8.8726-2.69124i 8.70502-2.4088i 8.56297-2.14473i 8.4425-1.89567i 8.34059-1.65891i 8.25491-1.43224i 8.18363-1.21383i 8.12536-1.00209i 8.079-0.795648i 8.04374-0.593295i 8.01896-0.393915i 8.00425-0.196478i
5.16
f(t) 0 0.015748031 0.031496063 0.047244094 0.062992126 0.078740157 0.094488189 0.11023622 0.125984252 0.141732283 0.157480315 0.173228346 0.188976378 0.204724409 0.220472441 0.236220472 0.251968504 0.267716535 0.283464567 0.299212598 0.31496063 0.330708661 0.346456693 0.362204724 0.377952756 0.393700787 0.409448819 0.42519685 0.440944882 0.456692913 0.472440945 0.488188976 0.503937008 0.519685039 0.535433071 0.551181102 0.566929134 0.582677165 0.598425197 0.614173228 0.62992126 0.645669291 0.661417323 0.677165354 0.692913386 0.708661417 0.724409449 0.74015748 0.755905512 0.771653543 0.787401575 0.803149606 0.818897638 0.834645669 0.850393701 0.866141732 0.881889764 0.897637795 0.913385827 0.929133858 0.94488189 0.960629921 0.976377953 0.992125984 1.007874016 1.023622047
freq (Hz) 0 1.690774167 5.238447449 7.443918867 6.318412902 2.876850305 0.222647147 0.749221098 3.981736088 7.005283935 7.093391792 4.166608731 0.864150321 0.163995956 2.697507191 6.18009984 7.471359139 5.406896893 1.848334133 0.004591385 1.538256705 5.066353187 7.407433155 6.450436593 3.058331535 0.289935903 0.641639999 3.796295983 6.90920474 7.173312557 4.350461209 0.986146234 0.114125951 2.520741356 6.035836099 7.489686775 5.57128903 2.010550782 0.018354295 1.391155225 4.891035519 7.361991345 6.57584762 3.24150648 0.36569745 0.541670459 3.610742511 6.805389478 7.244850525 4.532843315 1.114910103 0.073159251 2.346985654 5.885974943 7.498856897 5.731221306 2.177026887 0.04125503 1.249829939 4.712923755 7.307704713 6.694338886 3.425926592 0.44974627 0.44955728 3.425530045
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 15.5 16 16.5 17 17.5 18 18.5 19 19.5 20 20.5 21 21.5 22 22.5 23 23.5 24 24.5 25 25.5 26 26.5 27 27.5 28 28.5 29 29.5 30 30.5 31 31.5 32
F(f) 476.25 -3.75741-0.092231i -3.78098-0.185729i -3.82107-0.281831i -3.87892-0.381998i -3.95643-0.487931i -4.0563-0.601634i -4.18228-0.725621i -4.33958-0.86311i -4.53544-1.01836i -4.78015-1.19724i -5.08862-1.40806i -5.48327-1.66315i -5.99918-1.98164i -6.69433-2.395i -7.67189-2.95904i -9.13438-3.78315i -11.5411-5.11522i -16.2032-7.66263i -28.9632-14.577i -204.081-109.07i 36.5439+20.6991i 16.0018+9.58982i 9.92636+6.28493i 7.02244+4.69157i 5.32486+3.74964i 4.21392+3.12478i 3.43221+2.6781i 2.85361+2.34151i 2.40908+2.07773i 2.05767+1.86463i 1.77351+1.6882i 1.53951+1.53921i 1.3439+1.41125i 1.17832+1.29979i 1.03666+1.2015i 0.914375+1.11389i 0.80799+1.03507i 0.714826+0.963563i 0.632766+0.898192i 0.56013+0.838029i 0.495564+0.782312i 0.437965+0.730424i 0.386425+0.681855i 0.340196+0.636169i 0.29864+0.593005i 0.261244+0.552052i 0.227553+0.513046i 0.197192+0.475753i 0.169851+0.439984i 0.145236+0.405558i 0.123121+0.372317i 0.1033+0.340127i 0.0855925+0.308866i 0.0698582+0.278423i 0.0559598+0.248692i 0.0437916+0.219586i 0.0332556+0.191019i 0.0242768+0.162911i 0.016783+0.135188i 0.0107256+0.107779i 0.00605607+0.0806229i 0.00274071+0.0536478i 0.000764668+0.0267928i 0.000116199 0.000764668-0.0267928i
|F(f)| 476.25 3.7585418 3.785538934 3.83144942 3.897684292 3.986403769 4.10067472 4.244760516 4.424580596 4.648362409 4.927800487 5.279837731 5.729949198 6.317994758 7.109857885 8.222761938 9.886815559 12.6238847 17.92371582 32.42461847 231.3986159 41.99892103 18.65535448 11.74874325 8.445442121 6.512598111 5.246081568 4.353422228 3.691308591 3.18129351 2.776841887 2.448541803 2.1769838 1.948767244 1.754392221 1.586904599 1.441121984 1.31309472 1.19976242 1.098699995 1.007987213 0.926064654 0.851664582 0.783741361 0.721418266 0.663958417 0.610745314 0.561245549 0.515000584 0.471630451 0.430779279 0.392146311 0.355467672 0.32050629 0.287053192 0.2549102 0.223910061 0.193892221 0.164709918 0.136225785 0.108311363 0.080850034 0.053717762 0.02680371 0.000116199
5.17
5.14 400 300
Amplitude
time
200 100 0 0
5
10
15
20
25
30
Frequency - Hz
10 Hz is double function frequency since squaring makes negative portion of sine wave positive leading to an apparent doubling of the frequency. Spike at f = 0 is due to the fact that the average value of the funcion is not zero (there is a DC component of the signal.
1.039370079 1.05511811 1.070866142 1.086614173 1.102362205 1.118110236 1.133858268 1.149606299 1.165354331 1.181102362 1.196850394 1.212598425 1.228346457 1.244094488 1.25984252 1.275590551 1.291338583 1.307086614 1.322834646 1.338582677 1.354330709 1.37007874 1.385826772 1.401574803 1.417322835 1.433070866 1.448818898 1.464566929 1.480314961 1.496062992 1.511811024 1.527559055 1.543307087 1.559055118 1.57480315 1.590551181 1.606299213 1.622047244 1.637795276 1.653543307 1.669291339 1.68503937 1.700787402 1.716535433 1.732283465 1.748031496 1.763779528 1.779527559 1.795275591 1.811023622 1.826771654 1.842519685 1.858267717 1.874015748 1.88976378 1.905511811 1.921259843 1.937007874 1.952755906 1.968503937 1.984251969 2
6.694092366 7.307830518 4.713308442 1.250126617 0.041196172 2.176665567 5.730883344 7.498847049 5.88630209 2.347354792 0.073237512 1.114626915 4.532454042 7.244706192 6.805620237 3.611140276 0.541876547 0.365526022 3.241112121 6.575585942 7.362098315 4.891414678 1.391464667 0.018314983 2.010198165 5.570941079 7.489657254 6.036151628 2.521117408 0.114223423 0.98587723 4.350068303 7.173150051 6.909419172 3.796693991 0.64186268 0.289782455 3.05794033 6.450160397 7.407521027 5.066725888 1.538578154 0.004571715 1.847991082 5.406539806 7.471310017 6.18040298 2.697889236 0.1641124 0.86389616 4.166213154 7.09321151 7.005481516 3.982133364 0.749459828 0.222512056 2.876463212 6.318122866 7.443987427 5.238812782 1.691106834 2.11246E-08
0.00274071-0.0536478i 0.00605607-0.0806229i 0.0107256-0.107779i 0.016783-0.135188i 0.0242768-0.162911i 0.0332556-0.191019i 0.0437916-0.219586i 0.0559598-0.248692i 0.0698582-0.278423i 0.0855925-0.308866i 0.1033-0.340127i 0.123121-0.372317i 0.145236-0.405558i 0.169851-0.439984i 0.197192-0.475753i 0.227553-0.513046i 0.261244-0.552052i 0.29864-0.593005i 0.340196-0.636169i 0.386425-0.681855i 0.437965-0.730424i 0.495564-0.782312i 0.56013-0.838029i 0.632766-0.898192i 0.714826-0.963563i 0.80799-1.03507i 0.914375-1.11389i 1.03666-1.2015i 1.17832-1.29979i 1.3439-1.41125i 1.53951-1.53921i 1.77351-1.6882i 2.05767-1.86463i 2.40908-2.07773i 2.85361-2.34151i 3.43221-2.6781i 4.21392-3.12478i 5.32486-3.74964i 7.02244-4.69157i 9.92636-6.28493i 16.0018-9.58982i 36.5439-20.6991i -204.081+109.07i -28.9632+14.577i -16.2032+7.66263i -11.5411+5.11522i -9.13438+3.78315i -7.67189+2.95904i -6.69433+2.395i -5.99918+1.98164i -5.48327+1.66315i -5.08862+1.40806i -4.78015+1.19724i -4.53544+1.01836i -4.33958+0.86311i -4.18228+0.725621i -4.0563+0.601634i -3.95643+0.487931i -3.87892+0.381998i -3.82107+0.281831i -3.78098+0.185729i -3.75741+0.092231i
5.18
5.15 f(t)
w(n)
0 0.003922 0.007843 0.011765 0.015686 0.019608 0.023529 0.027451 0.031373 0.035294 0.039216 0.043137 0.047059 0.05098 0.054902 0.058824 0.062745 0.066667 0.070588 0.07451 0.078431 400 0.082353 0.086275 350 0.090196 300 0.094118 0.098039 250 0.101961 200 0.105882 0.109804 150 0.113725 0.117647 100 0.121569 50 0.12549 0.129412 0 0.133333 1 0.137255 0.141176 0.145098 0.14902 0.152941 0.156863 0.160784 0.164706 0.168627 0.172549 0.176471 0.180392 0.184314 0.188235 0.192157 0.196078 0.2 0.203922 0.207843 0.211765 0.215686 0.219608 0.223529 0.227451 0.231373 0.235294 0.239216 0.243137 0.247059 0.25098 0.254902 0.258824 0.262745
2 2.560006 2.049354 1.267592 1.01408 1.52654 2.316328 2.507956 1.46111 -0.733176 -3.214141 -4.811389 -4.724188 -2.996915 -0.492563 1.619046 2.537596 2.252438 1.453049 1.00319 1.32937 2.124775 2.561975 1.875572 -0.066945 -2.584947 -4.526322 -4.928061 -3.585873 -1.176101 1.146607 2.423491 2.417133 1.666602 1.053186 4 7 10 1.169499 1.906853 2.530531 2.193828 0.55459 -1.912745 -4.124828 -4.999925 -4.094401 -1.867182 0.593791 2.211444 2.525707 1.89191 1.160509 1.058599 1.68139 2.426268 2.412491 1.112244 -1.22202 -3.622363 -4.936983 -4.503123 -2.541379 -0.024061 1.899749 2.562415 2.110914 1.317452 1.004639 1.466526 2.264766
0 0.000152 0.000607 0.001366 0.002427 0.003791 0.005455 0.007421 0.009685 0.012247 0.015105 0.018258 0.021703 0.025438 0.029462 0.033771 0.038364 0.043237 0.048387 0.053812 0.059507 0.06547 0.071697 0.078184 0.084927 0.091922 0.099165 0.106651 0.114376 0.122335 0.130524 0.138937 0.147569 0.156415 0.16547 13 16 19 0.174728 0.184184 0.193831 0.203664 0.213677 0.223864 0.234219 0.244735 0.255406 0.266226 0.277187 0.288284 0.29951 0.310857 0.322319 0.333889 0.345559 0.357324 0.369175 0.381106 0.393109 0.405177 0.417302 0.429477 0.441696 0.45395 0.466231 0.478534 0.490849 0.50317 0.515489 0.527798 0.540091
fxw
f F(f) [w/o Hann] |F| w/o Hann |F| with Hann 0 0 9.52089907615234 9.52089907615234 9.520899 -1.83056267609769E-002-2.55671310739411E-003i 0.018483 0.000389 1 9.54398225491964+0.592611312520987i 9.54398225491964+0.592611312520987i 9.562363 -1.92929477549381E-002-5.29251415784474E-003i 0.020006 0.001244 2 9.61424520583671+1.19539882911063i 9.61424520583671+1.19539882911063i 9.688276 -2.10542353890507E-002-8.4144408297074E-003i 0.022673 0.001731 3 9.73484028645024+1.8192439357619i 9.73484028645024+1.8192439357619i 9.903371 -2.37887838246068E-002-1.2193253877418E-002i 0.026732 0.002461 4 9.91141884239111+2.47654844894736i 9.91141884239111+2.47654844894736i 10.21614 -2.78364257427318E-002-1.70183637239435E-002i 0.032627 0.005787 5 10.1528186286056+3.18230083105936i 10.1528186286056+3.18230083105936i 10.63987 -3.37659173848057E-002-2.34926070300524E-002i 0.041134 0.012637 6 10.4722432772704+3.95560976741923i 10.4722432772704+3.95560976741923i 11.19441 -4.25464464212656E-002-3.26097854105798E-002i 0.053606 0.01861 7 10.8892513310703+4.8220786016053i 10.8892513310703+4.8220786016053i 11.90917 -5.58960315401776E-002-4.61122117038859E-002i 0.072462 0.014151 8 11.4331748967985+5.8177204760782i 11.4331748967985+5.8177204760782i 12.82823 -7.70384765327427E-002-6.7268570002083E-002i 0.102274 -0.008979 9 12.1492361449759+6.99582401113046i 12.1492361449759+6.99582401113046i 14.01947 -0.112499212438008-0.10272860632803i 0.152346 -0.048549 10 13.1101367239048+8.43983757081067i 13.1101367239048+8.43983757081067i 15.59187 -0.176882453591918-0.167488142711216i 0.243597 -0.087844 11 14.4397515416454+10.2895894577062i 14.4397515416454+10.2895894577062i 17.73082 -0.307691721662172-0.300397592619051i 0.430015 -0.102528 12 16.3666958002919+12.8004374292734i 16.3666958002919+12.8004374292734i 20.77787 -0.622619657668311-0.62453151177262i 0.88187 -0.076236 13 19.3632899198369+16.496558969942i 19.3632899198369+16.496558969942i 25.43764 -1.63437812057132-1.68129435121605i 2.344769 -0.014512 14 24.5860741810464+22.6566529448973i 24.5860741810464+22.6566529448973i 33.4335 -7.84124173583645-8.26794992168868i 11.39491 0.054678 15 35.8134923608497+35.449985230018i 35.8134923608497+35.449985230018i 50.39154 87.90976257063+95.0359566135189i 129.4603 0.097352 16 76.5718111545271+80.8143565447084i 76.5718111545271+80.8143565447084i 111.3293 -123.626530071246-137.091701774767i 184.6013 0.097389 17 -230.884626101933-257.630695046675i -230.884626101933-257.630695046675i 345.9498 42.1659316649976+47.9534504903429i 63.8553 0.070309 -40.3644971813686-47.127448227184i 62.0507 2.92746337045364+3.40598118951088i 4.491186 W/O Hann 18 -40.3644971813686-47.127448227184i With Hann 0.053983 19 -20.4370925215447-24.6262378123621i -20.4370925215447-24.6262378123621i 32.00197 0.825664436417024+0.975311646376146i 1.277871 0.079107 20 -12.8814983153593-15.7101270169791i -12.8814983153593-15.7101270169791i 20.31603 0.340305830822219+0.400551621763411i 0.525595 200 0.139109 21 -8.93619857309896-10.7068633801254i -8.93619857309896-10.7068633801254i 13.94606 0.169916497289174+0.190699988080646i 0.255418 180 9.814375 0.183686 22 -6.5363485821225-7.32107205658236i -6.5363485821225-7.32107205658236i 9.53740986802364E-002+9.129516981575E-002i 0.132027 0.146639 23 -4.94367304927676-4.71251650027881i -4.94367304927676-4.71251650027881i 5.8003808947335E-002+3.2183343037416E-002i 0.066334 160 6.829913 -0.005685 24 -3.83039184025816-2.48254923788094i -3.83039184025816-2.48254923788094i 3.76062146503254E-002-1.32939552229795E-002i 0.039887 140 4.564532 -0.237613 25 -3.03110938233169-0.3942088984944i -3.03110938233169-0.3942088984944i 2.61373517575831E-002-6.01240390550676E-002i 0.06556 120 3.056636 -0.448851 26 -2.45672767371197+1.73419015955065i -2.45672767371197+1.73419015955065i 3.007146 2.03740186151045E-002-0.123478638584714i 0.125148 100 -0.525582 27 -2.06035469896185+4.08873113591887i -2.06035469896185+4.08873113591887i 4.578513 1.99086055927278E-002-0.230227913098095i 0.231087 80 -0.410138 28 -1.82457044481194+6.91998992895116i -1.82457044481194+6.91998992895116i 7.156488 2.81278831041657E-002-0.447811711847271i 0.448694 60 -0.143879 29 -1.7622992003391+10.6491200283317i -1.7622992003391+10.6491200283317i 10.79395 6.04791096851053E-002-0.999899356943792i 1.001727 40 0.14966 30 -1.9375816315704+16.1338971441226i -1.9375816315704+16.1338971441226i 16.24983 0.201099756437104-2.98393755897354i 2.990706 20 25.67786 0.336712 31 -2.54864816647615+25.5510633512045i -2.54864816647615+25.5510633512045i 1.56257067251059-19.9557484735226i 20.01683 0.356694 32 -4.34823496977889+46.7238243570184i -4.34823496977889+46.7238243570184i -9.58854860337493+105.66285785475i 106.097 0 46.92572 0.260682 33 -13.8327950067868+146.330925982862i -13.8327950067868+146.330925982862i 146.9833 11.3840548649404-109.93060570717i 110.5185 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 0.174271 34 17.9608619432907-177.513648754344i 17.9608619432907-177.513648754344i 178.42 -2.78806322628668+23.9918851781958i 24.15334 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 f- Hz 0.204344 35 6.68992755636645-60.1193106141376i 6.68992755636645-60.1193106141376i 60.49038 -0.413763790828728+3.21955914445288i 3.246038 0.351211 36 4.65774442124189-37.6373203945531i 4.65774442124189-37.6373203945531i 37.92443 -0.148295585182181+1.05574525769422i 1.06611 0.490495 37 3.86224455141116-28.0038837284794i 3.86224455141116-28.0038837284794i 28.26897 -7.2408954752263E-002+0.475968363231916i 0.481445 0.446804 38 3.46545081260024-22.603845539905i 3.46545081260024-22.603845539905i 22.86795 -4.17275908629139E-002+0.255032170127355i 0.258423 0.118503 39 3.24345887698067-19.1216980335112i 3.24345887698067-19.1216980335112i 19.39483 -2.66982980903635E-002+0.152523557935908i 0.154843 -0.428195 40 3.11159494425029-16.6727583502849i 3.11159494425029-16.6727583502849i 16.96063 -1.83655556549138E-002+9.84632358361419E-002i 0.100161 -0.966113 41 3.03106859208115-14.8454930337545i 3.03106859208115-14.8454930337545i 15.15177 -1.33249828364367E-002+6.72463349626715E-002i 0.068554 -1.223657 42 2.98178853615779-13.4222859699772i 2.98178853615779-13.4222859699772i 13.7495 -1.00713535814055E-002+4.79543194593184E-002i 0.049 -1.045735 43 2.9524231503951-12.2771179547158i 2.9524231503951-12.2771179547158i 12.62713 -7.86257082364794E-003+3.5385368836958E-002i 0.036248 -0.497092 44 2.93617066994831-11.3318731840992i 2.93617066994831-11.3318731840992i 11.70609 -6.30127032602193E-003+2.6842739725767E-002i 0.027572 0.164591 45 2.92875124347351-10.5354937318627i 2.92875124347351-10.5354937318627i 10.935 -5.16040054871906E-003+2.08318603641251E-002i 0.021462 0.637524 46 2.92737151068435-9.8531576022635i 2.92737151068435-9.8531576022635i 10.27882 -4.30323252236245E-003+1.64782843206885E-002i 0.017031 0.756474 47 2.93015491346819-9.26027251974002i 2.93015491346819-9.26027251974002i 9.712799 -3.64377954386447E-003+1.32468666149042E-002i 0.013739 0.588113 48 2.93581148129834-8.73895669332876i 2.93581148129834-8.73895669332876i 9.218913 -3.12595756926322E-003+1.07974504814177E-002i 0.011241 0.374054 49 2.94343806341119-8.27588189764741i 2.94343806341119-8.27588189764741i 8.783738 -2.71202994918288E-003+8.90668982668203E-003i 0.00931 0.353454 50 2.95239309139856-7.86090035946412i 2.95239309139856-7.86090035946412i 8.397046 -2.3759130018119E-003+7.42375796275876E-003i 0.007795 0.58102 51 2.96221566708471-7.48614175388276i 2.96221566708471-7.48614175388276i 8.050903 -2.09914886887435E-003+6.2442674219692E-003i 0.006588 0.866964 52 2.97257192746298-7.14540241226367i 2.97257192746298-7.14540241226367i 7.739054 -1.86840242115474E-003+5.29438835853882E-003i 0.005614 0.890632 53 2.98321869596771-6.83372188556372i 2.98321869596771-6.83372188556372i 7.456497 -1.67386020180755E-003+4.52088748614375E-003i 0.004821 0.423883 54 2.99397837173733-6.54708294618558i 2.99397837173733-6.54708294618558i 7.199181 -1.50818005108094E-003+3.88471468076608E-003i 0.004167 -0.480387 55 3.0047212897412-6.28219489807689i 3.0047212897412-6.28219489807689i 6.963787 -1.36578626395333E-003+3.3567760594796E-003i 0.003624 -1.467696 56 3.01535314609505-6.03633433041855i 3.01535314609505-6.03633433041855i 6.747569 -1.24238693268426E-003+2.91508897056632E-003i 0.003169 -2.060212 57 3.02580591790716-5.8072262500206i 3.02580591790716-5.8072262500206i 6.548235 -1.13463733893784E-003+2.54283040222814E-003i 0.002784 -1.933989 58 3.03603123173695-5.59295409417398i 3.03603123173695-5.59295409417398i 6.363853 -1.03990127150382E-003+2.22697496402402E-003i 0.002458 -1.122516 59 3.04599547175924-5.39189072782317i 3.04599547175924-5.39189072782317i 6.192784 -9.5607920606733E-004+1.95732925840776E-003i 0.002178 -0.010922 60 3.05567613943102-5.20264490815452i 3.05567613943102-5.20264490815452i 6.033628 -8.814828854118E-004+1.72583733657927E-003i 0.001938 0.885722 61 3.06505912359628-5.02401930081775i 3.06505912359628-5.02401930081775i 5.885181 -8.14742602125932E-004+1.52607447223154E-003i 1.226202 62 3.07413663960986-4.85497722800901i 3.07413663960986-4.85497722800901i 5.746401 -7.54737839044773E-004+1.35287364765499E-003i 1.03614 63 3.08290566458054-4.69461609075404i 3.08290566458054-4.69461609075404i 5.61638 -7.00544819071779E-004+1.2020468243561E-003i 0.662902 3.0913667435362-4.54214594532929i 3.0913667435362-4.54214594532929i -6.51396436445562E-004+1.07017474256731E-003i 0.51788 3.09952307497051-4.39687209809818i 3.09952307497051-4.39687209809818i -6.06651355273534E-004+9.54446846780328E-004i Remaining rows not shown for 0.77403 3.107379808229-4.25818086119435i 3.107379808229-4.25818086119435i -5.65769967673952E-004+8.52538259817063E-004i presentation purposes 1.223179 3.11494350247595-4.12552781514903i 3.11494350247595-4.12552781514903i -5.28295534871413E-004+7.6251442421007E-004i |F|
t
5.19
5.16 The function has frequencies of 5 and 20 Hz. The minimum sampling rate would then be 40 Hz to avoid aliasing. Sampling at 30 Hz would not produce aliases for the 5 Hz signal but would for the 20 Hz signal. The alias frequency can be evaluated using the folding diagram in Section 5.1. fN = 30/2 = 15 Hz. f/fN = 20/15 = 1.3333. From the folding diagram, fa/fN = 0.666. Thus fa = 0.66615 = 10 Hz. This is the difference between the sampling rate and the signal frequency. 5.17 The function has frequencies of 10 and 15 Hz. The minimum sampling rate would then be 30 Hz to avoid aliasing. Sampling at 50 Hz would not produce false aliases for either signals. The alias frequency can be evaluated using the folding diagram in Section 5.1.
5.18 The signal in problem 5.11 has frequencies of 5 and 20 Hz. The minimum sampling rate to avoid aliasing would then be twice the maximum frequency, or 40 Hz in this case. We will evaluate the alias frequencies using the folding diagram in Section 5.1. For a sampling rate of 5 Hz, fN = 2.5 Hz. For the 5 Hz signal, f/fN = 5/2.5 = 2. Reading from the folding diagram, fa/fN = 0. So the alias frequency will be 0 Hz (i.e. DC). For the 20 Hz signal, f/fN = 20/2.5 = 8. This is over the range of Figure A.1, but if the construction of the diagram is examined, it will be noticed that f a/fn = 0 again. So again, fa = 0. Both of these signals are integer multiples of the sampling rate, so the dc alias is to be expected. 5.19 The maximum frequency in the signal of problem 5.11 is 20 Hz. The sampling rate should exceed twice this value, or 40 Hz.
5.20
5.20 The signal has frequencies of 250 and 400 Hz. The minimum sampling rate to avoid aliasing would then be twice the maximum frequency, or 800 Hz in this case. We will evaluate the alias frequencies using the folding diagram in Section 5.1. For a sampling rate of 400 Hz, fN = 200 Hz. For the 250 Hz signal, f/fN = 250/200 = 1.25. Reading from the folding diagram, fa/fN = 0.75 So the alias frequency will be 0.75200 = 150 Hz. For the 400 Hz signal, f/fN = 400/200 = 2. The value of fa/fN = 0. So fa = 0, or dc. 5.21 The signal has frequencies of 50 and 250 Hz. The minimum sampling rate to avoid aliasing would then be twice the maximum frequency, or 500 Hz in this case. We will evaluate the alias frequencies using the folding diagram in Section 5.1. For a sampling rate of 200 Hz, fN = 100 Hz. This is adequate to avoid aliasing for the 50 Hz signal. For the 250 Hz signal, f/f N = 250/100 = 2.5. Reading from the folding diagram, fa/fN = 0.5 So the alias frequency will be 0.5100 = 50 Hz. 5.22 The maximum frequency in the signal of problem 5.11 is 400 Hz. The sampling rate should exceed twice this value, or 800 Hz.
5.21
time 0.00025 0.00275 0.00525 0.00775 0.01025 0.01275 0.01525 0.01775 0.02025 0.02275 0.02525 0.02775 0.03025 0.03275 0.03525 0.03775 0.04025 0.04275 0.04525 0.04775 0.05025 0.05275 0.05525 0.05775 0.06025 0.06275 0.06525 0.06775 0.07025 0.07275 0.07525 0.07775 0.08025 0.08275 0.08525 0.08775 0.09025 0.09275 0.09525 0.09775 0.10025
f(t) 6.816737266 2.897171027 2.897333587 6.817124503 1.273976625 5.193861135 5.193891276 1.2744186 6.817832692 2.898162376 2.898471966 6.818158801 1.27501086 5.194999313 5.194882292 1.275513565 6.818927586 2.899153195 2.899609814 6.819192565 1.276044565 5.196136959 5.195872775 1.276608001 6.820021945 2.900143483 2.900747132 6.820225795 1.277077741 5.197274072 5.196862726 1.277701907 6.821115771 2.901133239 2.901883918 6.821258492 1.278110386 5.198410652 5.197852144 1.278795283 6.822209063
term 1 2.771638978 -1.14806041 -1.148030991 2.771626792 -2.77165421 1.148097184 1.147994217 -2.771611559 2.771669442 -1.148133957 -1.147957442 2.771596326 -2.771684673 1.148170731 1.147920668 -2.771581092 2.771699903 -1.148207504 -1.147883893 2.771565858 -2.771715134 1.148244277 1.147847118 -2.771550623 2.771730363 -1.14828105 -1.147810343 2.771535388 -2.771745592 1.148317822 1.147773567 -2.771520152 2.771760821 -1.148354595 -1.147736792 2.771504916 -2.771776049 1.148391367 1.147700016 -2.771489679 2.771791277
term 2 4.045098287 4.045231437 4.045364578 4.045497711 4.045630835 4.045763952 4.04589706 4.046030159 4.046163251 4.046296334 4.046429408 4.046562475 4.046695533 4.046828583 4.046961624 4.047094657 4.047227682 4.047360699 4.047493707 4.047626707 4.047759699 4.047892682 4.048025657 4.048158624 4.048291582 4.048424532 4.048557474 4.048690408 4.048823333 4.04895625 4.049089158 4.049222059 4.049354951 4.049487834 4.049620709 4.049753576 4.049886435 4.050019285 4.050152128 4.050284961 4.050417787
amplitude
5.23
8 7 6 5 4 3 2 1 0 0
0.02
0.04
0.06
0.08
0.1
0.12
time - seconds
This signal when sampled at the sampling rate of 400 Hz will produce a DC alias of the second term (with amplitude 4.05) and a alias of the first term with frequency 150 Hz.
5.22
5.24 Use Appendix A-3. fN = 2000/2 = 1000 Hz. f/fN = 3500/1000 = 3.5. From Figure A.1, fa/fN = 0.5. Thus fa = 0.5*1000 = 500 Hz. 5.25 Use Appendix A-3. fN = 3000/2 = 1500 Hz. f/fN = 5000/1500 = 3.333. From Figure A.1, fa/fN = 0.666. Thus fa = 0.666*1500 = 9990 Hz. 5.26 Use Appendix A-3. fN = 1500/2 = 750 Hz. f/fN = 1000/750 = 1.333. From Figure A.1, fa/fN = 0.666. Thus fa = 0.666*750 = 500 Hz. 5.27 Use Appendix A-3. fN = 4000/2 = 2000 Hz. f/fN = 3000/2000 = 1.5. From Figure A.1, fa/fN = 0.5. Thus fa = 0.5*2000 = 1000 Hz. 5.28 The dynamic range of an A/D converter can be determined from Eq. 5.18. Since the 16 bit converter is bipolar, one bit is used for the sign and the dynamic range is determined from 15 bits. The dynamic range is then: dynamic range = 20 log10(215) = 90.3 dB 5.29 The dynamic range can be determined from Eq. 5.18. For the 14 bit unipolar converter, the dynamic range is: dynamic range = 20 log10(214) = 84.3 dB
5.23
5.30 The required attenuation is the dynamic range of the A/D converter. For the 12 bit bipolar converter, 1 bit is used for the sign and the dynamic range is: dynamic range = 20 log10(211) = 66.2 dB A second order Butterworth filter will attenuate the signal at a rate of 12 dB octave. Hence, 66.2/12 = 5.5 octaves will be required. The maximum frequency will then be given by Eq. 5.20: fm = 1000025.5 = 452 kHz. This is a rather high frequency and will require a sampling rate greater than 900 kHz. It will probably be better to use a higher order filter. For a fourth order filter, f m will be only 68 kHz. 5.31 From Eq. 5.18, dynamic range = 20 log10(2N) dB = 20 log10(211) dB = 66.2 dB Since 1 bit for sign with bipolar.Next choose a fourth order Butterworth filter which attenuates the signal 46 = 24 dB/octave. From Eq. 5.19, we can find the number of octaves required for attenuation: Dynamic Range Filter Attenuation Rate 66.2 dB 24 dB / oct 2.75 octaves
Noct
Choose fc = 500 Hz and we can find fm from Eq. 5.20:
fm fc 2 N
oct
= 50022.75 = 3363 Hz From Eq. 5.16, we find the actual minimum sampling frequency: fs = 2(3363 Hz) = 6274 Hz, which is less than our maximum sampling rate of 10,000 samples per second. The corner frequency should also be chosen to be equal to f c and thus 500 Hz.
5.32 5.24
From Eq. 5.18, dynamic range = 20 log10(2N) dB = 20 log10(2(81)) dB = 42.14 dB A first order Butterworth filter has a filter attenuation rate of 16 = 6 dB/octave Thus, from Eq. 5.19, Dynamic Range Filter Attenuation Rate 42.14 dB 6 dB / oct 7.02 octaves
Noct
If we choose fc = 100 Hz, from Eq. 5.20:
fm fc 2 N
oct
= 10027.02 = 13015.01 Hz From Eq. 5.16, the minimum sampling rate is fs = 26031 Hz.
5.25
