EE 2950 Final: Chapter 6 HW
1
CHAPTER 6
Exercises
E6.1 (a) The frequency of
)20002cos(2)(
in
ttv
is 2000 Hz. For this
frequency
.602)(
fH
Thus,
60402602)( in
out VV
fH
and we have
).6020002cos(4)(
out
ttv
(b) The frequency of
)2030002cos()(
in
ttv
is 3000 Hz. For this
frequency
.0)(
fH
Thus,
0020)( in
out
VV
fH
and we have
.0)(
out
tv
E6.2 The input signal
)15002cos(3)205002cos(2)(
tttv
has two
components with frequencies of 500 Hz and 1500 Hz. For the 500-Hz
component we have:
357202155.3)500(in
out,1 VV
H
)355002cos(7)(
out,1
ttv
For the 1500-Hz component:
455.703455.2)1500( in
out,2 VV
H
)4515002cos(5.7)(
out,2
ttv
Thus the output for both components is
)4515002cos(5.7)355002cos(7)(
out
tttv
E6.3 The input signal
)30002cos(3)10002cos(21)(
tttv
has three
components with frequencies of 0, 1000 Hz and 3000 Hz.
For the dc component, we have
414)()0()( 1,out,1
tvHtv in
For the 1000-Hz component, we have:
30602303)1000( in,2out,2 VV
H
)3010002cos(6)(
out,1
ttv
For the 3000-Hz component:
0030)3000(in,3out,3
VV
H
0)(
out,3
tv
Thus, the output for all three components is
)3010002cos(64)(
out
ttv
© 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication
is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to:
Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
© 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication
is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to:
Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
2
E6.4 Using the voltage-division principle, we have:
fLjR
R
2
in
out
VV
Then the transfer function is:
B
fjfRfLjfLjR
R
fH
/1
1
/21
1
2
)(
in
out
ππV
V
E6.5 From Equation 6.9, we have
Hz 200)2/(1
RCfB
, and from Equation
6.9, we have
.
/1
1
)(
in
out
B
fjf
fH
V
V
For the first component of the input, the frequency is 20 Hz,
,71.5995.0)(
fH
010
in V
, and
71.595.9)( in
out VV
fH
Thus the first component of the output is
)71.540cos(95.9)(
out,1
ttv
π
For the second component of the input, the frequency is 500 Hz,
,2.68371.0)(
fH
05
in V
, and
2.6886.1)( in
out VV
fH
Thus the second component of the output is
)2.6840cos(86.1)(
out,2
ttv
π
For the third component of the input, the frequency is 10 kHz,
,9.88020.0)(
fH
05
in V
, and
9.88100.0)( in
out VV
fH
Thus the third component of the output is
)9.88102cos(100.0)( 4
out,2
ttv
π
Finally, the output with for all three components is:
)9.88102cos(100.0
)2.6840cos(86.1)71.540cos(95.9)(
4
out
t
tttv
π
ππ
E6.6
dB 98.33)50log(20)(log20)( dB
fHfH
E6.7 (a)
dB 15)(log20)( dB
fHfH
0.75 /2015)(log
fH
623.510 )( 0.75
fH
© 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication
is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to:
Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
© 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication
is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to:
Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
3
(b)
dB 30)(log20)( dB
fHfH
1.5 /2030)(log
fH
62.3110 )( 1.5
fH
E6.8 (a)
Hz 400021000 2
is two octaves higher than 1000 Hz.
(b)
1252/1000 3
Hz is three octaves lower than 1000 Hz.
(c)
100101000 2
kHz is two decades higher than 1000 Hz.
(d)
10010/1000
Hz is one decade lower than 1000 Hz.
E6.9 (a) To find the frequency halfway between two frequencies on a
logarithmic scale, we take the logarithm of each frequency, average the
logarithms, and then take the antilogarithm. Thus
2.3161010 5.22/)]1000log()100[log(
f
Hz
is half way between 100 Hz and 1000 Hz on a logarithmic scale.
(b) To find the frequency halfway between two frequencies on a linear
scale, we simply average the two frequencies. Thus (100 + 1000)/2 = 550
Hz is halfway between 100 and 1000 Hz on a linear scale.
E6.10 To determine the number of decades between two frequencies we take
the difference between the common (base-ten) logarithms of the two
frequencies. Thus 20 Hz and 15 kHz are
875.2)20log()1015log( 3
decades apart.
Similarly, to determine the number of octaves between two frequencies
we take the difference between the base-two logarithms of the two
frequencies. One formula for the base-two logarithm of
z
is
)log(322.3
)2log(
)log(
)(log2
z
z
z
Thus the number of octaves between 20 Hz and 15 kHz is
551.9
)2log(
)20log(
)2log(
)1015log( 3
E6.11 The transfer function for the circuit shown in Figure 6.17 in the book is
B
in
out
/1
1
21
1
)2/(1
)2/(1
)(
fjfRCfjfCjR
fCj
fH
ππ
π
V
V
© 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication
is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to:
Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
© 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication
is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to:
Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
