Solution of ECE 342 Test 3 S12
There is a table of Bessel function values on the last two pages of the test.
1.
A signal is described by x c ( t ) = 30 cos ( 2000π t + 4sin (10π t )) .
(a)
What is its numerical total phase ( θ c ) at time t = 0.73 s (in radians)?
θ c ( t ) = 2000π t + 4sin (10π t ) ⇒ θ c ( 0.73) = 2000π ( 0.73) + 4sin (10π ( 0.73)) = 4583.5 radians
(b)
What is its numerical instantaneous cyclic frequency ( f ( t ) ) at time t = 1.27 s ?
Instantaneous cyclic frequency is the first time derivative of total phase divided by 2π .
f (t ) =
2000π + 40π cos (10π t )
1 d
⇒ f (1.27 ) = 1000 + 20 cos (10π (1.27 )) = 988.2443
(θ c (t )) =
2π dt
2π
2.
A cosine carrier of frequency 1 MHz and unit amplitude ( Ac = 1 ) is FM tone-modulated by a cosine modulation
of unit amplitude ( Am = 1 ), modulation frequency 1 kHz ( fm = 1000 ) with a frequency deviation fΔ = 3000 . Fill in the
table below with the numerical frequency and numerical average signal power (not amplitude) of the six strongest
sinusoidal components in the frequency spectrum of the modulated carrier. (The average signal power of any sinusoid is
one-half of the square of its amplitude.)
f = ____________ MHz Signal Power = ____________
f = ____________ MHz Signal Power = ____________
f = ____________ MHz Signal Power = ____________
f = ____________ MHz Signal Power = ____________
f = ____________ MHz Signal Power = ____________
f = ____________ MHz Signal Power = ____________
⎡ Ac J n ( β ) ⎤⎦
x c ( t ) = Ac ∑ J n ( β ) cos ((ω c + nω m ) t ) ⇒ Pn = ⎣
2
n = −∞
β = ( Am / fm ) fΔ = 3
∞
−7
−0.003
n
J n ( 3)
⎡⎣ Ac J n ( 3) ⎤⎦
2
−6
0.011
−5
−0.043
−4
0.132
−3
−0.309
−2
0.486
−1
−0.339
0
−0.26
1
0.339
2
0.486
2
3
0.309
4
0.132
5
0.043
6
0.011
7
0.003
2
0
0.0001 0.0009 0.0087 0.0478 0.1181 0.0575 0.0338 0.0575 0.1181 0.0478 0.0087 0.0009 0.0001
f = 0.997MHz Signal Power = 0.0478
f = 0.998MHz Signal Power = 0.1181
f = 0.999MHz Signal Power = 0.0575
f = 1.001MHz
Signal Power = 0.0575
f = 1.002MHz
Signal Power = 0.1181
f = 1.003MHz
Signal Power = 0.0478
0
3.
In the circuit below, let C1 = 50pF , L = 10 µΗ , VB = 4 V and Cv ( t ) =
10
pF . Assume the
VB + 0.1x ( t )
capacitance of the DC blocking capacitor is large enough to consider it to have zero impedance at the tuned circuit's
resonant frequency. Also assume that the inductance of the RFC is large enough to consider its impedance at the tuned
1 ⎞
⎛
circuit's resonant frequency to be infinite. ⎜ Resonant radian frequency of a parallel LC circuit is ω 0 =
⎟
⎝
LC ⎠
(a)
What is the numerical carrier frequency of this FM modulator in MHz?
The carrier frequency is the resonant frequency with x ( t ) = 0 . Cv ( t ) =
10
pF = 5pF
4
The total capacitance of the LC resonator is 55 pF.
ω0 =
(b)
1
1
4.264 × 10 7
=
= 4.264 × 10 7 rad/s ⇒ f0 =
= 6.7864 MHz
2π
LC
10 −5 × 55 × 10 −12
What are the numerical maximum and minumum instantaneous frequencies produced by a modulation
x ( t ) = 3sin ( 30000π t ) ?
Maximum modulation swing is ±3 .
10
1
pF = 4.8224 pF ⇒ f0 =
= 6.7974 MHz
At x ( t ) = 3, Cv ( t ) =
4 + 0.3
2π LC
10
1
pF = 5.1988 pF ⇒ f0 =
= 6.7742 MHz
At x ( t ) = −3, Cv ( t ) =
4 − 0.3
2π LC
(c)
What is the effective numerical frequency deviation fΔ for this FM modulator?
6.7974 × 10 6 − 6.7864 × 10 6
= 3700
3
6
6
6.7864 × 10 − 6.7742 × 10
= 4100
Using the negative deviation, fΔ =
3
Using the positive deviation, fΔ =
This difference is due to the nonlinearity of the varactor diode's capacitance with voltage.
The average of the two is 3900.
4.
An FM modulated signal of unit amplitude ( Ac = 1 ) , carrier frequency 10 MHz and maximum instantaneous
frequency deviation of 40 kHz, is the input signal to a slope detector. In the frequency range near 10 MHz the slope
f − 10 7
detector's frequency response magnitude can be approximated by H ( f ) = 5 +
. What will be the numerical
10 5
maximum value of the envelope of the output signal from the slope detector?
Maximum amplitude occurs when H ( f ) = 5 +
That occurs at H (10.04 MHz ) = 5 +
f − 10 7
is a maximum.
10 5
1.004 × 10 7 − 10 7
= 5.4
10 5
Below is a table of Bessel function values for J n ( x ) . Also J − n ( x ) = ( −1) J n ( x ) .
n
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.0
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5.0
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
6.0
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
7.0
7.1
7.2
7.3
7.4
n = 0
1.000
0.998
0.990
0.978
0.960
0.938
0.912
0.881
0.846
0.808
0.765
0.720
0.671
0.620
0.567
0.512
0.455
0.398
0.340
0.282
0.224
0.167
0.110
0.056
0.003
-0.048
-0.097
-0.142
-0.185
-0.224
-0.260
-0.292
-0.320
-0.344
-0.364
-0.380
-0.392
-0.399
-0.403
-0.402
-0.397
-0.389
-0.377
-0.361
-0.342
-0.321
-0.296
-0.269
-0.240
-0.210
-0.178
-0.144
-0.110
-0.076
-0.041
-0.007
0.027
0.060
0.092
0.122
0.151
0.177
0.202
0.224
0.243
0.260
0.274
0.285
0.293
0.298
0.300
0.299
0.295
0.288
0.279
n = 1
0.000
0.050
0.100
0.148
0.196
0.242
0.287
0.329
0.369
0.406
0.440
0.471
0.498
0.522
0.542
0.558
0.570
0.578
0.582
0.581
0.577
0.568
0.556
0.540
0.520
0.497
0.471
0.442
0.410
0.375
0.339
0.301
0.261
0.221
0.179
0.137
0.095
0.054
0.013
-0.027
-0.066
-0.103
-0.139
-0.172
-0.203
-0.231
-0.257
-0.279
-0.298
-0.315
-0.328
-0.337
-0.343
-0.346
-0.345
-0.341
-0.334
-0.324
-0.311
-0.295
-0.277
-0.256
-0.233
-0.208
-0.182
-0.154
-0.125
-0.095
-0.065
-0.035
-0.005
0.025
0.054
0.083
0.110
n = 2
0.000
0.001
0.005
0.011
0.020
0.031
0.044
0.059
0.076
0.095
0.115
0.137
0.159
0.183
0.207
0.232
0.257
0.282
0.306
0.330
0.353
0.375
0.395
0.414
0.431
0.446
0.459
0.470
0.478
0.483
0.486
0.486
0.484
0.478
0.470
0.459
0.445
0.428
0.409
0.388
0.364
0.338
0.311
0.281
0.250
0.218
0.185
0.151
0.116
0.081
0.047
0.012
-0.022
-0.055
-0.087
-0.117
-0.146
-0.174
-0.199
-0.222
-0.243
-0.261
-0.277
-0.290
-0.300
-0.307
-0.312
-0.314
-0.312
-0.308
-0.301
-0.292
-0.280
-0.266
-0.249
n = 3
0.000
0.000
0.000
0.001
0.001
0.003
0.004
0.007
0.010
0.014
0.020
0.026
0.033
0.041
0.050
0.061
0.073
0.085
0.099
0.113
0.129
0.145
0.162
0.180
0.198
0.217
0.235
0.254
0.273
0.291
0.309
0.326
0.343
0.359
0.373
0.387
0.399
0.409
0.418
0.425
0.430
0.433
0.434
0.433
0.430
0.425
0.417
0.407
0.395
0.381
0.365
0.347
0.327
0.305
0.281
0.256
0.230
0.202
0.174
0.145
0.115
0.085
0.054
0.024
-0.006
-0.035
-0.064
-0.092
-0.118
-0.144
-0.168
-0.190
-0.210
-0.228
-0.244
n = 4
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.001
0.001
0.002
0.002
0.004
0.005
0.007
0.009
0.012
0.015
0.019
0.023
0.028
0.034
0.040
0.048
0.056
0.064
0.074
0.084
0.095
0.107
0.119
0.132
0.146
0.160
0.174
0.189
0.204
0.220
0.235
0.251
0.266
0.281
0.296
0.310
0.324
0.336
0.348
0.359
0.369
0.378
0.385
0.391
0.396
0.398
0.400
0.399
0.397
0.393
0.387
0.379
0.369
0.358
0.344
0.329
0.313
0.295
0.275
0.254
0.231
0.208
0.183
0.158
0.132
0.105
0.078
0.051
n = 5
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.001
0.001
0.001
0.002
0.002
0.003
0.004
0.006
0.007
0.009
0.011
0.013
0.016
0.020
0.023
0.027
0.032
0.037
0.043
0.049
0.056
0.064
0.072
0.080
0.090
0.099
0.110
0.121
0.132
0.144
0.156
0.169
0.182
0.195
0.208
0.221
0.235
0.248
0.261
0.274
0.287
0.299
0.310
0.321
0.331
0.340
0.349
0.356
0.362
0.367
0.371
0.373
0.374
0.374
0.372
0.368
0.363
0.356
0.348
0.338
0.327
0.314
0.299
n = 6
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.001
0.001
0.001
0.002
0.002
0.003
0.003
0.004
0.005
0.006
0.008
0.010
0.011
0.014
0.016
0.019
0.022
0.025
0.029
0.034
0.038
0.043
0.049
0.055
0.062
0.069
0.076
0.084
0.093
0.102
0.111
0.121
0.131
0.142
0.153
0.164
0.175
0.187
0.199
0.210
0.222
0.234
0.246
0.257
0.269
0.279
0.290
0.300
0.309
0.318
0.326
0.333
0.339
0.344
0.349
0.352
0.353
n = 7
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.001
0.001
0.001
0.001
0.002
0.002
0.003
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.011
0.013
0.015
0.018
0.020
0.023
0.026
0.030
0.034
0.038
0.043
0.048
0.053
0.059
0.065
0.072
0.079
0.087
0.094
0.103
0.111
0.120
0.130
0.139
0.149
0.159
0.170
0.180
0.191
0.201
0.212
0.223
0.234
0.244
0.254
0.264
0.274
n = 8
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.001
0.001
0.001
0.001
0.002
0.002
0.002
0.003
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.011
0.012
0.014
0.016
0.018
0.021
0.024
0.027
0.030
0.034
0.038
0.042
0.046
0.051
0.057
0.062
0.068
0.074
0.081
0.088
0.095
0.103
0.111
0.119
0.128
0.137
0.146
0.155
0.165
n = 9
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.001
0.001
0.001
0.001
0.001
0.002
0.002
0.002
0.003
0.003
0.004
0.005
0.006
0.006
0.007
0.009
0.010
0.011
0.013
0.015
0.017
0.019
0.021
0.024
0.027
0.030
0.033
0.037
0.040
0.045
0.049
0.054
0.059
0.064
0.070
0.076
0.082
n = 10
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.001
0.001
0.001
0.001
0.001
0.001
0.002
0.002
0.002
0.003
0.003
0.004
0.005
0.005
0.006
0.007
0.008
0.009
0.010
0.012
0.013
0.015
0.017
0.019
0.021
0.024
0.026
0.029
0.032
0.035
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
= 7.5
= 7.6
= 7.7
= 7.8
= 7.9
= 8.0
= 8.1
= 8.2
= 8.3
= 8.4
= 8.5
= 8.6
= 8.7
= 8.8
= 8.9
= 9.0
= 9.1
= 9.2
= 9.3
= 9.4
= 9.5
= 9.6
= 9.7
= 9.8
= 9.9
= 10.0
0.266
0.252
0.235
0.215
0.194
0.172
0.148
0.122
0.096
0.069
0.042
0.015
-0.013
-0.039
-0.065
-0.090
-0.114
-0.137
-0.158
-0.177
-0.194
-0.209
-0.222
-0.232
-0.240
-0.246
0.135
0.159
0.181
0.201
0.219
0.235
0.248
0.258
0.266
0.271
0.273
0.273
0.270
0.264
0.256
0.245
0.232
0.217
0.200
0.182
0.161
0.140
0.117
0.093
0.068
0.043
-0.230
-0.210
-0.187
-0.164
-0.139
-0.113
-0.086
-0.059
-0.032
-0.005
0.022
0.049
0.075
0.099
0.123
0.145
0.165
0.184
0.201
0.215
0.228
0.238
0.246
0.251
0.254
0.255
-0.258
-0.270
-0.279
-0.285
-0.289
-0.291
-0.290
-0.287
-0.281
-0.273
-0.263
-0.250
-0.235
-0.219
-0.201
-0.181
-0.160
-0.137
-0.114
-0.090
-0.065
-0.040
-0.015
0.010
0.034
0.058
0.024
-0.003
-0.030
-0.056
-0.081
-0.105
-0.129
-0.151
-0.171
-0.190
-0.208
-0.223
-0.237
-0.249
-0.258
-0.265
-0.271
-0.274
-0.274
-0.273
-0.269
-0.263
-0.255
-0.245
-0.233
-0.220
0.283
0.354
0.266
0.354
0.248
0.352
0.228
0.348
0.207
0.344
0.186
0.338
0.163
0.330
0.140
0.321
0.116
0.311
0.092
0.300
0.067
0.287
0.042
0.273
0.018
0.257
-0.007
0.241
-0.031
0.223
-0.055
0.204
-0.078
0.185
-0.101
0.164
-0.122
0.143
-0.142
0.122
-0.161
0.099
-0.179
0.077
-0.195
0.054
-0.210
0.031
-0.223
0.008
-0.234 -0.014
0.283
0.292
0.300
0.308
0.314
0.321
0.326
0.330
0.334
0.336
0.338
0.338
0.337
0.335
0.332
0.327
0.322
0.315
0.307
0.297
0.287
0.275
0.262
0.248
0.233
0.217
0.174
0.184
0.194
0.204
0.214
0.223
0.233
0.243
0.252
0.261
0.269
0.278
0.285
0.292
0.299
0.305
0.310
0.315
0.319
0.321
0.323
0.324
0.324
0.323
0.321
0.318
0.089
0.096
0.103
0.111
0.118
0.126
0.135
0.143
0.152
0.160
0.169
0.178
0.188
0.197
0.206
0.215
0.224
0.233
0.241
0.250
0.258
0.265
0.273
0.280
0.286
0.292
0.039
0.043
0.047
0.051
0.056
0.061
0.066
0.071
0.077
0.083
0.089
0.096
0.103
0.110
0.117
0.125
0.132
0.140
0.148
0.157
0.165
0.173
0.182
0.190
0.199
0.207