Mathematics MODULATION AND AMPLITUDE MODULATION

### Topic Covered

color{blue}{star} MODULATION AND ITS NECESSITY
color{blue}{star} AMPLITUDE MODULATION

### MODULATION AND ITS NECESSITY

color{blue} ✍️As already mentioned, the purpose of a communication system is to transmit information or message signals.
Message signals are also called "baseband signals," which essentially designate the band of frequencies representing the original signal, as delivered by the source of information.

color{blue} ✍️No signal, in general, is a single frequency sinusoid, but it spreads over a range of frequencies called the signal bandwidth.

color{blue} ✍️Suppose we wish to transmit an electronic signal in the audio frequency (AF) range (baseband signal frequency less than 20 kHz) over a long distance directly. Let us find what factors prevent us from doing so and how we overcome these factors.

color{brown}bbul("Size of the antenna or aerial")
color{blue} ✍️For transmitting a signal, we need an antenna or an aerial. This antenna should have a size comparable to the wavelength of the signal (at least lamda//4 in dimension) so that the antenna properly senses the time variation of the signal.

color{blue} ✍️For an electromagnetic wave of frequency 20 kHz, the wavelength lamda is 15 km. Obviously, such a long antenna is not possible to construct and operate. Hence direct transmission of such baseband signals is not practical.

color{blue} ✍️We can obtain transmission with reasonable antenna lengths if transmission frequency is high (for example, if n is 1 MHz, then lamda is 300 m).

color{blue} ✍️Therefore, there is a need of translating the information contained in our original low frequency baseband signal into high or radio frequencies before transmission.

color{brown}bbul("Effective power radiated by an antenna")
color{blue} ✍️A theoretical study of radiation from a linear antenna (length l) shows that the power radiated is proportional to (lamda//l)^2 .

color{blue} ✍️This implies that for the same antenna length, the power radiated increases with decreasing l, i.e., increasing frequency.

color{blue} ✍️Hence, the effective power radiated by a long wavelength baseband signal would be small. For a good transmission, we need high powers and hence this also points out to the need of using high frequency transmission.

color{brown}bbul("Mixing up of signals from different transmitters")
color{blue} ✍️Another important argument against transmitting baseband signals directly is more practical in nature.
Suppose many people are talking at the same time or many transmitters are transmitting baseband information signals simultaneously. All these signals will get mixed up and there is no simple way to distinguish between them.

color{blue} ✍️This points out towards a possible solution by using communication at high frequencies and allotting a band of frequencies to each message signal for its transmission.

color{blue} ✍️The above arguments suggest that there is a need for translating the original low frequency baseband message or information signal into high frequency wave before transmission such that the translated signal continues to possess the information contained in the original signal.

color{blue} ✍️ In doing so, we take the help of a high frequency signal, known as the carrier wave, and a process known as modulation which attaches information to it. The carrier wave may be continuous (sinusoidal) or in the form of pulses as shown in Fig. 15.7.

color{blue} ✍️A sinusoidal carrier wave can be represented as

color{blue}(c(t)=A_csin(omega_ct+phi))

............(15.2)

color{blue} ✍️where c(t) is the signal strength (voltage or current), A_c is the amplitude, omega_c ( = 2pnc) is the angular frequency and f is the initial phase of the carrier wave.

color{blue} ✍️During the process of modulation, any of the three parameters, viz A_c, omega_c and phi, of the carrier wave can be controlled by the message or information signal.

color{blue} ✍️This results in three types of modulation: (i) Amplitude modulation (AM), (ii) Frequency modulation (FM) and (iii) Phase modulation (PM), as shown in Fig. 15.8.

color{blue} ✍️Similarly, the significant characteristics of a pulse are: pulse amplitude, pulse duration or pulse Width, and pulse position (denoting the time of rise or fall of the pulse amplitude) as shown in Fig. 15.7(b).

color{blue} ✍️Hence, different types of pulse modulation are: (a) pulse amplitude modulation (PAM), (b) pulse duration modulation (PDM) or pulse width modulation (PWM), and (c) pulse position modulation (PPM). In this chapter, we shall confine to amplitude modulation on ly.

### AMPLITUDE MODULATION

color{blue} ✍️In amplitude modulation the amplitude of the carrier is varied in accordance with the information signal.
Here we explain amplitude, let's color{purple}(c(t) = A_c sin omega_ct) represent carrier wave and color{purple}(m(t) = A_m sin omega_mt) represent the message or the modulating signal where omega_m = 2pif_m is the angular frequency of the message signal.

color{blue} ✍️The modulated signal c_m (t ) can be written as

color{purple}(c_m (t) = (A_c+A_m sin omega_mt) sinomega_ct)

color{blue}(=A_c1+(A_m)/(A_c) sinomega_mt sinomega_ct)

..............(15.3)

color{brown} {"Note"} that the modulated signal now contains the message signal. This can also be seen from Fig. 15.8(c). From Eq. (15.3), we can write,

color{blue}(C_m (t)=A_Csinomega_ct muA_c sinomega_mt sinomega_ct)

.............(15.4)

color{blue} ✍️Here μ = A_m//A_c is the modulation index; in practice,μ is kept <=1 to avoid distortion.
Using the trignomatric relation color{purple}("sinA sinB = ½ (cos(A – B) – cos (A + B),") we can write c_m (t) of Eq............ (15.4)

as

color{blue}(C_m(t)=A_csinomega_ct+(muA_c)/2 cos(omega_c-omega_m)t-(muA_c)/2 cos(omega_c+omega_m)t)

.......(15.5)

color{blue} ✍️Here omega_c – omega_m and omega_c + omega_m are respectively called the lower side and upper side frequencies.

color{blue} ✍️The modulated signal now consists of the carrier wave of frequency wc plus two sinusoidal waves each with a frequency slightly
different from, known as side bands. The frequency spectrum of the amplitude modulated signal is shown in Fig. 15.9.

color{blue} ✍️As long as the broadcast frequencies (carrier waves) are sufficiently spaced out so that sidebands do not overlap, different stations can operate without interfering with each other.
Q 3189880717

A message signal of frequency 10 kHz and peak voltage of 10 volts is used to modulate a carrier of frequency 1 MHz and peak voltage of 20 volts. Determine (a) modulation index, (b) the side bands produced.
Class 12 Chapter 15 Example 2
Solution:

Modulation index =10//20 = 0.5
(b) The side bands are at (1000+10 kHz)=1010 kHz and (1000 –10 kHz) = 990 kHz.

### PRODUCTION OF AMPLITUDE MODULATED WAVE

color{blue} ✍️Amplitude modulation can be produced by a variety of methods. A conceptually simple method is shown in the block diagram of Fig. 15.10.

color{blue} ✍️Here the modulating signal A_m sin omega_mt is added to the carrier signal A_c sin omega_ct to produce the signal x (t).

color{blue} ✍️This signal color{purple}(x (t) = A_m sinomega_mt) + A_c sin omega_ct is passed through a square law device which is a non-linear device which produces an output color{purple}(y (t ) = B x (t ) + Cx^2 (t ))...(15.6) where B and C are constants. Thus,

color{purple}(y (t )= BA_m sin omega_mt + BA_c sin omega_ct)

color{blue}(+CA_(m)^(2) sin^2 omega_mt + A_(c)^(2) sin^2 omega_ct+2A_mA_c sin omega_m t sin omega_ct)

........(15.7)

color{purple}(= BA_m sin omega_mt + BA_c sin omega_ct)

color{purple}(+ (CA_(m)^(2))/2 +A_(c)^(2) - (CA_(m)^(2))/2 cos2 omega_mt - (CA_(C)^(2))/2 cos 2omega_ct)

color{blue}(+CA_mA_c cos(omega_c-omega_m) t - CA_mA_Ccos(omega_c+omega_m)t)

.....(15.8)

color{blue} ✍️where the trigonometric relations color{purple}(sin^2A = (1 – cos2A)//2) and the relation for sinA sin B mentioned earlier are used.

color{blue} ✍️In Eq. (15.8), there is a dc term color{purple}(C//2 (A_(m)^(2)+A_(c)^(2)) and sinusoids of frequencies color{purple}(omega_m, 2omega_m, omega_c, 2omega_c, omega_c – omega_m) and color{purple}(omega_c + omega_m.)

color{blue} ✍️As shown in Fig. 15.10 this signal is passed through a band pass filter* which rejects dc and the sinusoids of frequencies omega_m , 2omega_m and 2 omega_c and retains the frequencies color{navy}(omega_c, omega_c – omega_m) and omega_c + omega_m.

color{blue} ✍️The output of the band pass filter therefore is of the same form as Eq. (15.5) and is therefore an AM wave.It is to be mentioned that the modulated signal cannot be transmitted as such.

color{blue} ✍️The modulator is to be followed by a power amplifier which provides the necessary power and then the modulated signal is fed to an antenna of appropriate size for radiation as shown in Fig. 15.11.

### DETECTION OF AMPLITUDE MODULATED WAVE

The transmitted message gets attenuated in propagating through the channel. The receiving antenna is therefore to be followed by an amplifier and a detector.

In addition, to facilitate further processing, the carrier frequency is usually changed to a lower frequency by what is called an intermediate frequency (IF) stage preceding the detection.

The detected signal may not be strong enough to be made use of and hence is required to be amplified. A block diagram of a typical receiver is shown in Fig. 15.12

Detection is the process of recovering the modulating signal from the modulated carrier wave. We just saw that the modulated carrier wave contains the frequencies omega_c and omega_c ± omega_m.

In order to obtain the original message signal m(t) of angular frequency omega_m, a simple method is shown in the form of a block
diagram in Fig. 15.13.

The modulated signal of the form given in (a) of fig. 15.13 is passed through a rectifier to produce the output shown in (b). This envelope of signal (b) is the message signal. In order to retrieve m(t ), the signal is passed through an envelope detector (which may consist of a simple RC circuit).

In the present chapter we have discussed some basic concepts of communication and communication systems. We have also discussed one specific type of analog modulation namely Amplitude Modulation (AM).

Other forms of modulation and digital communication systems play an important role in modern communication. These and other exciting developments are taking place everyday.

So far we have restricted our discussion to some basic communication systems.
Before we conclude this chapter, it is worth taking a glance at some of the communication systems (see the box) that in recent times have brought major changes in the way we exchange information even in our day-to-day life: