Principle of Communication Chapter 3 Solution

QUESTIONS

3-1.            Define amplitude modulation.

·           Amplitude modulation (AM) is the process of changing the amplitude of a relatively high frequency carrier signal in proportion with the instantaneous value of the modulating signal (information).

3-2.            Describe the basic operation of an AM modulator.

·           In the modulator, the information acts on or modulates the RF carrier producing a modulated waveform. The information signal may be a single frequency or more likely consists of a range of frequencies. For example, typical voice-grade communications systems utilize a range of information frequency between 300Hz and 3000 Hz.

3-3.            What is meant by the term RF?

·           RF or radio frequencies are frequencies that are high enough to be efficiently radiated by an antenna and propagated through free space.

3-4.            How many inputs are there to an amplitude modulator? What are they?

·           Amplitude modulator has two inputs. One input is a single, high frequency carrier signal of constant amplitude and the second input is comprised of relatively low-frequency information signals, which may be a single frequency, or a complex waveform made up of many frequencies.

3-5.            In an AM communications system, what is meant by the terms modulating signal, carrier, modulated wave, and AM envelope?

·           The term modulating signal is the information signal, its shape depends on the type of information, such as voice, video or binary data. Carrier is a high frequency analog signal that carries the information through the system. A carrier that has been acted upon by an information signal is called a modulated wave or modulated signal. The term AM envelope is the shape of the modulated wave.  

3-6.            What is meant by the repetition rate of the AM envelope?

·           The repetition rate of the envelope is equal to the frequency of the modulating signal, and the shape of the envelope is identical to the shape of the modulating signal.

3-7.            Describe upper and lower sidebands and the upper and lower side frequencies.

·           The band of frequencies between the carrier frequency f_c and f_c + f_m(max) is called the upper sideband (USB). The lower sideband (LSB) is the band of frequencies between the f_c – f_m(max) and the carrier frequency f_c.

·           Upper side frequencies (USF) are frequencies within the upper sideband. The lower side frequencies (LSF) are frequencies within the lower sideband.

3-8.            What is the relationship between the modulating signal frequency and the bandwidth in a conventional AM system?

·           Two times the highest modulating signal frequency is the bandwidth (B) of a conventional AM or AM DSBFC wave (B = 2f_m(max)).

3-9.            Define modulation coefficient and percent modulation.

·           Modulation coefficient is a term used to describe the amount of amplitude change (modulation) present in an AM waveform.

·           Percent modulation is simply the coefficient of modulation stated as a percentage. It gives the percentage change in the amplitude of the output wave when the carrier is acted on by a modulating signal.

3-10.        What is the highest modulation coefficient and percent modulation possible with a conventional AM system without causing excessive distortion?

·           The maximum percent modulation that can be imposed without causing excessive distortion is 100%.

3-11.        For 100% modulation, what is the relationship between the voltage amplitudes of the side frequencies and the carrier?

·           For 100% modulation, m = 1 and the amplitude of the upper and lower side frequencies are each equal to one-half the amplitude of the carrier (E_C/2).

3-12.        Describe the meaning of the following expression:

            v_am (t) = E_csin(2πf_ct) – (mE_c/2)cos[2π(f_c  + f_m)t] + (mE_c/2)cos[2π(f_c  + f_m)t]

where   E_csin (2pf_ct) = carrier signal (volts)

- (mE_c/2)cos[2p(f_c + f_m)t] = upper side frequency signal (volts)

+ (mE_c/2)cos[2p(f_c - f_m)t] = lower side frequency signal (volts)

·           First, the amplitude of the carrier after the modulation is the same as it was before modulation (E_c). Therefore, the amplitude of the carrier is unaffected by the modulation process. Second, the amplitude of the upper and lower side frequency depends on both the carrier amplitude and the coefficient of modulation.

3-13.        Describe the meaning of each term in the following expression:

             v_am (t) = 10 sin(2π500kt) – 5 cos(2π515kt) + 5 cos(2π485kt)

·           The peak amplitude of the modulated carrier is 10 V_p and the upper and lower side frequency voltage is 5 V_p. The upper side frequency is 515 kHz and the lower side frequency is 485 kHz.

3-14.        What effects does modulation have on the amplitude of the carrier component of the modulated signal spectrum?

·           The amplitude of the carrier after the modulation is the same as it was before modulation. Therefore, the amplitude of the carrier is unaffected by the modulation process.

3-15.    Describe the significance of the following formula:

                                               

P_t = P_c (1 +  (m­²/2))

·           The carrier power in the modulated wave is the same as the carrier power in the unmodulated wave. Thus, it is evident that the power of the carrier is unaffected by the modulation process. Also, because the total power in an AM wave is the sum of the carrier and sideband powers, the total power in an AM envelope increases with modulation.

3-16.        What does AM DSBFC stand for?

·           AM DSBFC stands for amplitude modulation double-sideband full carrier.

3-17.        Describe the relationship between the carrier and sideband powers in an AM DSBFC wave.

·           The total power in an AM DSBFC wave is the sum of the carrier and sideband powers, the power in an AM envelope increases with modulation. For 100% modulation, the maximum power in the upper or lower sideband is equal to only one-fourth the power in the carrier. Thus, the maximum total sideband power is equal to one-half the carrier power.

3-18.        What is the predominant disadvantage of AM DSBFC?

·           The most significant disadvantage of AM DSBFC transmission is the fact that the information is contained in the sidebands although most of the power is wasted in the carrier.

3-19.        What is the predominant advantage of AM DSBFC?

·           The power in the carrier is not totally wasted because it allows for the use of relatively simple, inexpensive demodulator circuits in the receiver.

3-20.        What is the primary disadvantage of low-level AM?

·           An obvious disadvantage of low-level modulation is in high-power applications when all the amplifiers that follow the modulator stage must be linear amplifiers, which is extremely inefficient.

3-21.        Why do any amplifiers that follow the modulator circuit in an AM DSBFC transmitter have to be linear?

·           They have to be linear so that the output is simply the carrier amplified by the quiescent voltage gain.

3-22.        Describe the differences between low- and high-level modulators.

·           With low-level modulation, the modulation takes place prior to the output element of the final stage of the transmitter, in other words, prior to the collector of the output transistor in a transistorized transmitter, prior to the drain of the output FET in a FET transistor, or prior to the plate of the output tube in a vacuum-tube transmitter.

·           In high-level modulators, the modulation takes place in the final element of the final stage where the carrier signal is at its maximum amplitude, and, thus, requires a much higher amplitude-modulating signal to achieve a reasonable percent modulation.

 

3-23.        List the advantages of low-level modulation; high-level modulation.

·           An advantage of low-level modulation is that less modulating signal power is required to achieve a high percentage of modulation.

·           With high-level modulation, the final modulating signal amplifiers must supply all the sideband power, which could be as much as 33% of the total transmit power.

3-24.        What are the advantages of using linear-integrated circuit modulators for AM?

·           It uses a unique arrangement of transistors and FETs to perform signal multiplication, which is a characteristic that makes them ideally suited for generating AM waveforms. It can precisely match current flow, amplifier voltage gain, and temperature variations. Offer excellent frequency stability, symmetrical modulation characteristics, circuit miniaturization, fewer components, temperature immunity, and simplicity of design and troubleshooting.

3-25.        What is the advantage of using a trapezoidal pattern to envelope an AM envelope?

·           The advantage of trapezoidal pattern is that it is more easily and accurately interpreted.

 

 

PROBLEMS

3-1.      For an AM DSBFC modulator with a carrier frequency f_c = 100 kHz and a maximum modulating signal f_m(max) = 5 kHz, determine

(a)   Frequency limits for the upper and lower sidebands.

(b)   Bandwidth.

(c)    Upper and lower side frequencies produced when the modulating signal is a single frequency 3-kHz tone.

Then

(d)   Sketch the output frequency spectrum.

Solution:

a. f_USB   = f_c + f_{m (max)}                                                                

= 100 kHz + 5 kHz                                   

= 105 kHz

                            f_LSB  = f_c - f_{m (max)}

                                                = 100 kHz + 5 kHz

                                    = 95 kHz

                                                     

b. BW   = f_USB - f_LSB

                                    = 105 fHz – 95 kHz

                        = 10 kHz

c. USF  = 100 kHz + 3 kHz

                        = 103 kHz

    LSF = 100 kHz – 3 kHz

                                    = 97 kHz

3-2.            What is the maximum modulating signal frequency that can be used with an AM DSBFC system with a 20-kHz bandwidth?

Solution:

                        BW       = 2f_m(max)

                        20 kHz             = 2f_m(max)

                        f_m(max)   = 20 kHz / 2

                        f_m(max)   = 10 k Hz

3-3.      If a modulated wave with an average voltage of 20 V_p changes in amplitude + 5V, determine the minimum and maximum envelope amplitudes, the modulation   coefficient, and the percent modulation.

Solution:

            For maximum and minimum amplitudes of the envelope:

                        V_(max) = 20 + 5 = 25 Vp

                        V_(min)  = 20 – 5 = 15 Vp

            For modulation coefficient:

                              m = E_m/E_c

                                  = 5 / 20

                                  = 0.25

            For percent modulation:

                              M = m x 100

                                  = 0.25 x 100

                                  = 25 %

3-4.      Sketch the envelope for Problem 3-3 (label all pertinent voltages).

3-5.            For a 30-V_p carrier amplitude, determine the maximum upper and lower side frequency amplitudes for an AM DSBFC envelope.

Solution:

                               E_usf = E_isl    = mE_c/2

                                                = 1(30 V_p)/2

                                                = 15 V_p

3-6.            For a maximum positive envelope voltage of +12 AM DSBFC and minimum positive envelope amplitude of +4 V, determine the modulation coefficient and percent modulation.

Solution:

                                E_c = ½ (12 + 4) = 8

                                E_m                = ½ (12 – 4) = 4

                                 m            = E_m/E_c

                                    = 4 / 8

                                    = 0.5

                                 M            = m x 100

                                    = 0.5 x 100

                                    = 50%

3-7.            Sketch the envelope for Problem 3-6 (label all pertinent voltages).

3-8.            For an AM DSBFC envelope with a +V_max = 40 V and +V_min = 10 V, determine

(a)   Unmodulated carrier amplitude.

(b)   Peak change in amplitude of the modulated wave.

(c)    Coefficient of modulation and percent modulation.

Solution:

                a. E_c = ½(V_max + V_min)                            c. m = E_m/E_c

                        = ½(40 + 10)                                        = 15/25

                        = 25                                                     = 0.6

               b. E_m             = ½(V_max - V_min)                                            M = m x 100

                        = ½(40 - 10)                                         = 0.6 x 100  

                        = 15                                                     = 60 %

3-9.            For an unmodulaated carrier amplitude of 16 V_p and a modulation coefficient m = 0.4, determine the amplitude of the modulated carrier and side frequencies.

Solution:

                              E_USF = E_LSF            = mE_c

                                                = 0.4 (16)

                                                = 6.4 V_p

3-10.        Sketch the envelope for Problem 3-9 (label all pertinent voltages).

3-11.        For the AM envelope shown below, determine

(a)   Peak amplitude of the upper and lower side frequencies.

(b)   Peak amplitude of the carrier.

(c)    Peak change in the amplitude of the envelope.

(d)   Modulation coefficient.

(e)    Percent modulation.

Solution:

            a. E_usf = E_lsf = 4 V_p                                   d. m             = E_m/E_c

            b. E_c    = ½(V_max + V_min)                                   = 8 / 12

                        = ½ (20 + 4)                                         = 0.667

                        = 12 V_p                                                    e. M             = m x 100

            c. E_c    = ½(V_max - V_min)                                                = 0.667 x 100

                        = ½(20 - 4)                                           = 66.7 %

                        = 8 V_p

3-12.        One input to an AM DSBFC modulator is an 800-kHz carrier with an amplitude input is a 25-kHz modulating signal whose amplitude is sufficient to produce a + 10 V change in the amplitude of the envelope. Determine

(a)   Upper and lower side frequencies.

(b)   Modulation coefficient and percent modulation.

(c)    Maximum and minimum positive peak amplitudes of the envelope.

Then

(d)   Draw the output frequency spectrum.

(e)    Draw the envelope (label all pertinent voltages).

Solution:

a. f_USF   = f_c + f_m                                         f_LSF = f_c - f_m

            = 800 + 25                                           = 800 - 25

            = 825                                                   = 775

b. m     = E_m/E_c                                            M = m x 100

            = 10 / 40                                              = 0.25 x 100

            = 0.25                                                  = 25%

c. V_(max) = E_c + E_m                                    V_(min) = E_c - E_m

            = 40 + 10                                             = 40 - 10

            = 50                                                     = 30

d.

e.

3-13.        For a modulation coefficient m = 0.2 and an unmodulated carrier power P_c = 1000W, determine

(a)   Total sideband power.

(b)    Upper and lower sideband power.

(c)    Modulated carrier power.

(d)   Total transmitted power.

Solution:

                        a. P_sbt   = m²P_c/2= 0.2²(1000)/2 = 20 W

b. P_USB             = P_LSB = m²P_c²/4= 0.2²(1000)/4  = 10 W

c. P_c     = 1000 W

d. P_t     = P_c(1 + m²/ 2)

                        = 1000 (1 + 0.2² / 2)

                        = 1020 W

       

3-14.        Determine the maximum upper, lower, and total sideband power for an unmodulated carrier power P_c = 2000 W.

Solution:

a. P_USB = P_LSB = m²P_c²/4 = 1² (2000) = 500 W

            b. P_sbt    = m²P_c/2 = 1² (2000) = 1000 W

3-15.        Determine the maximum total transmitted power (P_t) for the AM system described in Problem 3-14.

Solution:

        P_t             = P_c (1 + m² / 2)

                        = 2000 (1 + 1² / 2)

                        = 3000 W

3-16.        For an AM DSBFC wave with an unmodulatedd carrier voltage of 25 V_p and a load resistance of 50 Ω, determine

(a)   Power in the unmodulated carier.

(b)   Power of the modulated carier, upper and lower sidebands, and total transmited power for a modulation coefficient m = 0.6.

Solution:

                        a.     P_c             = [( 2 / 2) (25 V_p)]²/50W= 6.25 W

b.        P_c = 6.25 W

    P_USB             = P_LSB = m²P_c²/4= 0.6² (6.25)²/4 = 3.516 W

                    P_t = P_c (1 + m² / 2)

= 6.25 (1 + 0.6² / 2)

                        = 7.375 W

3-17.        For a low-power transistor modulator with a modulation coefficient m = 0.4, a quiescent voltage gain A_q = 80, and an input carrier amplitude of 0.002 V; determine

(a)   Maximum and minimum voltage gains

(b)   Maximum and minimum voltages for v_out. Then

(c)    Sketch the modulated envelope.

Solution:

            a. A_max = 80 (1 + 0.4)                           b. V_out(max)         = 112 (0.002)

                        = 112                                                               = 0.224

                A_min =  80 (1 - 0.4)                              V_out(min)         = 48 (0.002)

                        =   48                                                               = 0.096

            c.

3-18.        For the trapezoidal pattern shown below, determine

(a)   Modulation coefficient.

(b)   Percent modulation

(c)    Carrier amplitude

(d)   Upper and lower side frequency amplitudes.

Solution:

a.        m              = (V_max - V_min)/(V_max + V_min) = (28 – 12)/(28 + 12) = 0.4

b.       M = m x 100 = 0.4 x 100 = 40%                                  

                        c.     E_c             = ½ (V_max + V_min)                                                              

                                    = ½ (28 + 12)                                                           

                                    = 20  

d. E_usf = E_lsf = mE_c/2 = 1(20)/2 = 10 V_p

3-19.        Sketch the approximate trapezoidal pattern for the following percent modulations and modulation conditions:

(a)   100%

(b)    50%

(c)    >100%

(d)   Improper phase relationship.

(e)    Nonsymmetrical AM modulation

Solution:

                        a. for 100%

                            M = [(V_max – 0)/(V_max + 0)] x 100  = 100%

                        b. for 50%

                            M = [(V_max – V_min)/(V_max + V_min)] x 100 = 50 %

                                   

                        c. >100%

                        d. Improper phase relationship

                        e. Nonsymmetrical AM modulation

3-20.        For an AM modulator with a carrier frequency f_c = 200 kHz and a maximum modulating signal frequency f_m(max) = 10 kHz, determine

(a)   Frequency limits for the upper and lower sidebands.

(b)   Upper and lower side frequencies produced when the modulating signal is a single frequency 7-kHz tone.

(c)    Bandwidth necessary to pass the maximum modulating signal frequency. Then

(d)   Draw the output spectrum.

Solution:

            a. Frequency limits for upper sidebands and lower sidebands:

                USB             = f_c to [ f_c + f_m(max)]

                        = 200 kHz to [200 kHz +10 kHz]

                        = 200 kHz to 210 kHz

             

   LSB = [f_c – f_m(max)] to f_c

                        = [200 kHz – 10 kHz] to 200 kHz

                        = 190 kHz to 200 kHz

b. f_USF   = f_c + f_m                                       f_LSF    = f_c - f_m

            = (200 + 7)kHz                                                = (200 – 7)kHz

            = 207 kHz                                            = 775 kHz

c. BW   = 2 f_m(max)

= 2 (7kHz)

                        = 14 kHz

            d.

3-21.        For the unmodulated carrier voltage of 10 V_p and a + 4 V change in amplitude of the envelope, determine

(a)   Modulation coefficient.

(b)   Percent modulation.

Solution:

            a. m = E_m/E_c                 b.    M = m x 100

                    = 4 / 10                            = 0.4 x 100

                    = 0.4                                = 40%

3-22.        For a maximum positive envelope voltage V_max = +20 V and a minimum positive enveope amolitude of +6 V, determine

(a)   Modulation coefficient

(b)   Percent modulation

(c)    Carrier amplitude.

Solution:

            E_c    = ½(V_max + V_min)                                 E_m               = ½(V_max - V_min)

                    = ½ (20 + 6)                                             = ½(20 – 6)

                    = 13                                                         = 7

a. m = E_m/E_c                                         b.  M    = m x 100

                    = 7 / 13                                                    = 0.54 x 100

                    = 0.54                                                      = 54%

            c. E_c = ½ (V_max + V_min)

                     = ½ (20 + 6)

                     = 13

3-23.        For an envelope with +V_max = +30V_p and a + V_min = +10 V_p, determine

(a)   Unmodulated carrier amplitude.

(b)   Modulated carrier amplitude

(c)    Peak change in the amplitude of the envelope.

(d)   Modulation coefficient.

(e)    Percent modulation.

Solution:

            a. E_c = ½(V_max + V_min)                           b. E_c(modulated)     = E_{c(unmodulated)}

                     = ½(30 + 10)                                                       = 20 V_p

                     = 20 V_p

            c. E_m = ½(V_max - V_min)                            d. m     = E_m/E_c

                     = ½(30 – 10)                                           = 10 / 20

                     = 10 V_p                                                   = 0.5

           

e. M  = m x 100

                     = 0.5 x 100

                     = 50%

3-24.        Write the expression for an AM voltage wave with the following values:

Unmodulated carrier = 20 V_p

Modulation coefficient = 0.4

Modulating signal frequency = 5 kHz

Carrier frequency – 200 kHz

3-25.        For an unmodulated carrier amplitude of 12 V_p and a modulation coefficient of 0.5, determine the following:

(a)   Percent modulation

(b)   Peak voltages of the carrier and side frequencies.

(c)    Maximum positive envelope voltage.

(d)   Minimum positive envelope voltage.

Solution:

                        a. M     = m x 100

                                    = 0.5 x 100

                                    = 50%                                                       E_USF = E_LSF             = E_m/ 2

                        b. E_c     = 0.5 (12)                                                                     = 6 /2              

                                    = 6 Vp                                                                         = 3

c. V_max = E_c + E_m                                                        d. V_min    = E_c - E_m

                                    = 12 + 6                                                                       = 12 - 6

                                    = 18 Vp                                                                       = 6 Vp

3-26.        Sketch the envelope for Problem 3-25.

3-27.        For an AM envelope with a maximum peak voltage of 52 V and a minimum peak-to-peak voltage of 24 V, determine the following:

(a)   Percent modulation.

(b)   Peak voltages of the carrier and side frequencies.

(c)    Maximum positive envelope voltage.

(d)   Minimum positive envelope voltage.

Solution:

            a. M     = [(V_max – V_min)/(V_max + V_min) ] x 100                            

                         = [(52 – 24)/(52 + 24)] x 100                                                  

                       = 36.84 %

            b. E_c     = ½(V_max + V_min)

                        = ½(52 +24)

                        = 38 V_p

            c. V_max = 2E_c

                        = 2 (38)

                        = 76

c. V_min   = 0 V

3-28.        One input to an AM DSBFC modulator is a 500-kHz carrier with a peak amplitude of 32 V. The second input is a 12-kHz modulating signal whose amplitude is sufficient to produce a +14 V_p change in the amplitude of the envelope. Determine the following:

(a)   Upper and lower side frequencies.

(b)   Modulation coefficient and percent modulation.

(c)    Maximum and minimum amplitudes of the envelope, Then

(d)   Draw the output envelope.

(e)    Draw the output frequency spectrum.

Solution:

a. f_USF   = f_c + f_m                                        f_LSF  = f_c - f_m

           = (500 + 12) kHz                                 = (500 – 12) kHz

           = 512 kHz                                            = 488 kHz

b. m     = E_m/E_c                                                       M  = m x 100

                        = 14/32                                                            = 0.4375 x 100

                        = 0.4375                                             = 43.75%

            c. V_max = E_c + E_m                                     V_min   = E_c - E_m

                        = 32 + 14                                             = 32 - 14

                        = 46 V_p                                                            = 18 V_p

            d.

            e.

3-29.        For a modulation coefficient of 0.4 and a carrier power of 400 W, determine

(a)   Total sideband power.

(b)   Total transmitted power.

Solution:

a. P_sbt   = m²P_c/2 = 0.4²(400 W)/2 = 32 W

b. P_t     = P_c(1 + m²/ 2)

= 400 (1 + 0.4² / 2)

            = 432 W

3-30.        For an AM DSBFC wave with an unmodulated carrier voltage of 18 V_p and a load resistance of 72 Ω, determine

(a)   Unmodulated carrier power.

(b)   Modulated carrier power.

(c)    Total sideband power.

(d)   Upper and lower sideband powers.

(e)    Total transmitted power.

Solution:

                        a. P_c     = E_c²/2 = 18²/ 2 = 2.25

                        b. P_c     = E_c²/2 = 18² / 2 = 2.25

c. P_sbt   = m²P_c/2= 1²(2.25 W)/2 = 1.125 W

d. P_usb = P_lsb = m²P_c/ 4 = 1²(2.25)/4 = 0. 5625

e. P_t     = P_c(1 + m²/ 2)

                        = 2.25 (1 + 1² / 2)

                        = 3.375

3-31.        For a low-power AM modulator with a modulation coefficient of 0.8, a quiescent gain of 0, and an input carrier amplitude of 10 mV_p, determine

(a)   Maximum and minimum voltage gains.

(b)   Maximum and minimum envelope voltages, then

(c)    Sketch the AM envelope.

Solution:

                        a. A_max = A_q (1 + m)                                         b. V_max = A_maxV_c

                        = 90 (1 + 0.8)                                                   = 162 (0.01)

                        = 162                                                               = 1.62 V

                A_min  = A_q (1 – m)                                             V_min  = A_minV_c

                        = 90 (1 – 0.8)                                                   = 18 (0.01)

            = 18                                                                 = 0.18 V