Modulation is a process of varying one of the characteristics of high frequency sinusoidal (the carrier) in accordance with the instantaneous values of the modulating (the information) signal. The high frequency carrier signal is mathematically represented by the equation 2.1.
Any of the last three characteristics or parameters of the carrier can be varied by the modulating (message) signal, giving rise to amplitude, frequency or phase modulation respectively.
Need for modulation:
1. Practicability of antenna
In the audio frequency range, for efficient radiation and reception, the transmitting and receiving antennas must have sizes comparable to the wavelength of the
frequency of the signal used. It is calculated using the relation fλ = c . The wavelength is
75 meters at 1MHz in the broadcast band, but at 1 KHz, the wavelength turns out to be 300 Kilometers. A practical antenna for this value of wavelength is unimaginable and impossible.
2. Modulation for ease of radiation
For efficient radiation of electromagnetic waves, the antenna dimension required is of the order of λ /4 to λ /2 . It is possible to construct practical antennas only by increasing the frequency of the base band signal.
3. Modulation for multiplexing
The process of combining several signals for simultaneous transmission on a single channel is called multiplexing. In order to use a channel to transmit the different
base band signals (information) at the same time, it becomes necessary to translate different signals so as to make them occupy different frequency slots or bands so that they do not interfere. This is a accomplished by using carrier of different frequencies.
4. Narrow banding:
Suppose that we want to transmit audio signal ranging from 50 - 104 Hz using suitable antenna. The ratio of highest to lowest frequency is 200. Therefore an antenna suitable for use at one end of the frequency range would be entirely too short or too long for the other end. Suppose that the audio spectrum is translated so that it occupies the range from 50+106 to 104+106 Hz. Then the ratio of highest to lowest frequency becomes
1.01. Thus the process of frequency translation is useful to change wideband signals to
narrow band signals.
At lower frequencies, the effects of flicker noise and burst noise are severe.
2.1 Amplitude modulation
In amplitude modulation, the amplitude of the carrier signal is varied by the
modulating/message/information/base-band signal, in accordance with the instantaneous values of the message signal. That is amplitude of the carrier is made proportional to the instantaneous values (amplitude) of the modulating signal.
If m(t) is the information signal and c(t) = Ac cos(2πfc t + φ) is the carrier, the amplitude of
the carrier signal is varied proportional to the m(t ) .
The peak amplitude of carrier after modulation at any instant is given by [ Ac + m(t ) ]. The
carrier signal after modulation or the modulated signal is represented by the equation 2.2.
The modulation index m of AM system is defined as the ratio of peak amplitude of message signal to peak amplitude of carrier signal.
The following figure 2.1 shows the message, carrier and amplitude modulated waveforms.
Note:
(1) m is also called depth of modulation.
(2) m specifies the system clarity. As m increases, the system clarity also increases.
Consider the Amplitude Modulated waveform shown in figure 2.2.
Figure 2.2: Message and amplitude modulated signal
We have the modulation index given by
Dividing the equation (2.7) by (2.9), we get
Here
Amax is the maximum amplitude and Amin is minimum amplitude of the modulated signal.
Modulation index m has to be governed such that it is always less than unity;
otherwise it results in a situation known as ‘over-modulation’ ( m >1). The over- modulation occurs, whenever the magnitude of the peak amplitude of the modulating signal exceeds the magnitude of the peak amplitude of the carrier signal. The signal gets distorted due to over modulation. Because of this limitation on‘ m ’, the system clarity is also limited. The AM waveforms for different values of modulation index m are as shown in figure 2.3.
Note: If the modulation index exceeds unity the negative peak of the modulating waveform is clipped and [Ac + m(t )] goes negative, which mathematically appears as a phase reversal rather than a clamped level.
Example 1.1
A modulating signal consists of a symmetrical triangular wave, which has zero dc component and peak-to-peak voltage 11v. It is used to amplitude modulate a carrier of peak voltage 10v. Calculate the modulation index?
The amplitude of the modulating signal is 11/2 = 5.5volts
The modulation index is m = Am/Ac =5.5/10=0.55
1.3 Single tone Amplitude Modulation/ Sinusoidal AM
Consider a modulating wave frequency component given by that consists of a single tone or single
From the equation (2.15), the spectrum of AM wave obtained is as shown in figure 2.5, for fc > W . This spectrum consists of two delta functions weighted by the factor c , and occurring at 2 ± fc , and two versions of the base band spectrum translated in frequency by ± fc . From the spectrum, the following points are noted.
(i) For positive frequencies, the highest frequency component of the AM wave is c + W , and the lowest frequency component is fc − W . The difference between these two frequencies defines the transmission bandwidth BT which is exactly twice the message bandwidth W.
∴BT = 2W
(ii) For positive frequencies, the portion of the spectrum of an AM wave lying above the carrier frequency fc , is referred to as the Upper Side Band (USB), where as the symmetric portion below fc , is called the Lower Side Band (LSB). For negative frequencies, the USB is the portion of the spectrum below − fc and the LSB is the portion above overlap.
The AM wave s(t ) is a voltage or current wave. In either case, the average power delivered to 1Ω resistor by s(t ) is comprised of three components.
Example 1.2 A carrier wave of frequency 10 MHz and peak value 10V is amplitude modulated by a 5 KHz sine wave of amplitude 6V. Determine the modulation index and amplitude of the side frequencies.
The side frequencies are 10.005 MHz and 9.995 MHz.
The amplitude of side frequencies is given by
Consider the expression for single tone/sinusoidal AM wave
This expression contains three components. They are carrier component, upper side band and lower side band. Therefore Average power of the AM wave is sum of these three components.
Therefore the total power in the amplitude modulated wave is given by
Where all the voltages are rms values and R is the resistance, in which the power is dissipated.
The ratio of total side band power to the total power in the modulated wave is given by
The ratio is called the efficiency of AM system and it takes maximum value of 33% at m=1.
Example 1.3 A broadcast radio transmitter radiates 10KW, when the modulation percentage is 60. How much of this is carrier power.
Example 1.4 A radio transmitter radiates 10 KW and carrier power is 8.5 KW. Calculate modulation index.
1.6 Effective voltage and current for sinusoidal AM
In AM systems, the modulated and unmodulated currents are necessary to calculate the modulation index from them.
The effective or rms value of voltage Et of the modulated wave is defined by the equation
Similarly the effective or root mean square voltage Ec of carrier component is
Where It is the rms current of modulated wave and Ic is the rms current of unmodulated carrier.
Note: The maximum power in the AM wave is Pt = 1.5Pc, when m=1. This is important, because it is the maximum power that relevant amplifiers must be capable of handling without distortion.
Example 1.5 A 400 W carrier is modulated to a depth of 7.5 %. Calculate total power in the modulated wave. (Ans: Pt=512.5w)
Example 1.6 The antenna current of an AM transmitter is 8 Amps, when only the carrier is sent, but it increases to 8.93A, when the carrier is modulated by a single sine wave. Find percentage modulation. Determine the antenna current when the percent modulation changes to 0.8. (Ans: m=70.1%, It=9.19A)
1.7 Nonsinusoidal Modulation
When a sinusoidal carrier signal is modulated by a non-sinusoidal modulating signal, the process is called Non-sinusoidal modulation. Consider a high frequency sinusoidal signal c(t ) = Ac cos(2πfct) and the non-sinusoidal message signal m(t) as shown in figure 2.6. The non-sinusoidal modulating signal has a line spectrum that is many frequency components of different amplitudes.
Figure 2.6: Non-sinusoidal message signal and spectrum
The expression for the non-sinusoidal AM is given by
