Wave Modulation

Introduction

•Modulation of wave is a process of modifying a certain characteristic of a wave (carrier wave) according to other wave (modulator wave) oscillating pattern.

•Using this method, the second wave characteristic is carried by the first wave, which will be substracted in later process.

•The carrier wave has much higher frequency and power so it can be propagate a long distance.

Advantages of Modulation

In communication system, the modulation method offer several advantages:

•Enable a weak signal to be transmitter by strong power carrier wave.

•Reduction in antenna size, since the carrier wave has much higher frequency (shorter wavelength).

•Enable simultaneous use of the same medium to transmit different carrier waves.

•Enable shifting in signal frequency such that it can be designed in frequency area suitable to the processing equipment.

Advantages of Modulation

• In scientific measuerements, the modulation method offer several advantages:

• Increasing the signal to noise ratio (S/N)

•Enable to convert domain of the data, e.g from voltage to frequency or analog signal to digital signal.

Category

•There are two groups of modulations :

1. Analog signal : the carrier wave is continuous or pure harmonic wave:

Some of the modulation methods are:

a. Linear modulation : DSB, AM, etc

b. Non linear modulation : FM, PM

2. Digital signal: the carrier wave is a train of pulses.

a. Pulse Amplitude modulation (PAM)

b. Pulse Width Modulation (PWM)

c. Pulse Position Modultaion (PMM)

Double Side Band Modulation (DSB)

For the analysis we only look at the time variation properties (oscillation).

The carrier wave is a pure sinusoidal wave:𝜓𝐶 𝑡 = 𝜓𝐶 cos(𝜔𝐶𝑡)

Suppose the modulator wave is also a pure sinusoidal:𝜓𝑚 𝑡 = 𝜓𝑚 cos(𝜔𝑚𝑡)

The result of DSB modulation is :𝜓𝐷𝑆𝐵 𝑡 = 𝜓𝑐𝜓𝑚 𝑡 cos(𝜔𝑐𝑡)

𝜓𝐷𝑆𝐵 𝑡 = 𝜓𝑐𝜓𝑚 cos(𝜔𝑐𝑡) cos(𝜔𝑚𝑡)

𝜓𝐷𝑆𝐵 𝑡 =1

2𝜓𝑐𝜓𝑚 cos (𝜔𝑐 − 𝜔𝑚 𝑡 + cos( 𝜔𝑐 +𝜔𝑚 𝑡)}

DSB Spectrum for pure sinus modulation

This “multiplication” operation is called : “mixing” or “heterodyning”.

In frequency domain, the result will be two components side bands, each having frequencies of : 𝜔𝐶 ± 𝜔𝑚

To get a picture of frequency spectrum, we do the Fourier Transforms for each waves and the DSB wave:

For the carrier wave:

𝑔𝑐 𝜔 = 𝐹𝑇 𝜓𝑐 𝑡 = 𝐹𝑇(𝜓𝑐 cos𝜔𝑐𝑡)

𝑔𝑐 𝜔 = 𝜋𝜓𝑐 𝛿 𝜔 − 𝜔𝑐 + 𝛿 𝜔 + 𝜔𝑐

𝑔𝑐 𝜔 =1

2𝜓𝑐 𝛿 𝜈 − 𝜈𝑐 + 𝛿 𝜈 + 𝜈𝐶

DSB Spectrum for pure sinus modulation

• For the modulator wave:

𝑔𝑚 𝜔 = 𝐹𝑇 𝜓𝑚 𝑡 = 𝐹𝑇(𝜓𝑚 cos𝜔𝑚𝑡)

𝑔𝑚 𝜔 = 𝜋𝜓𝑚 𝛿 𝜔 − 𝜔𝑚 + 𝛿 𝜔 + 𝜔𝑚

𝑔𝑚 𝜔 =1

2𝜓𝑚 𝛿 𝜈 − 𝜈𝑚 + 𝛿 𝜈 + 𝜈𝑚

• For the DSB wave:

𝑔𝐷𝑆𝐵 𝜔 = 𝐹𝑇 𝜓𝐷𝑆𝐵 𝑡 =

= 𝐹𝑇(1

2𝜓𝑚𝜓𝐶 cos (𝜔𝑐 − 𝜔𝑚 𝑡 + cos( 𝜔𝑐 +𝜔𝑚 𝑡)})

𝑔𝐷𝑆𝐵 𝜔

=𝜋

2𝜓𝑚𝜓𝐶

𝛿 𝜔 − 𝜔𝐶 +𝜔𝑚 + 𝛿 𝜔 + 𝜔𝑐 − 𝜔𝑚 +

𝛿 𝜔 − 𝜔𝐶 − 𝜔𝑚 + 𝛿 𝜔 + 𝜔𝑐 + 𝜔𝑚

Efect of DSB modulation to the spectrum

• It clear from the Fourier Transforms in the frequency domain, the effect of DSB modulation is translation in frequency domain of the modulator wave spectrum ±𝜔𝑚 by the amount of ± 𝜔𝑐thus it becomes ±𝜔𝐶 ±𝜔𝑚

DSB modulation : General Modulator

• In general the modulator wave is not sinusoidal : 𝜓𝑚(𝑡)

• The DSB wave is then :

𝜓𝐷𝑆𝐵 𝑡 = 𝜓𝐶𝜓𝑚 𝑡 cos(𝜔𝑐𝑡)

• To get the spectrum, we do the inverse FT for the modulatorwave:

𝜓𝑚 𝑡 =1

2𝜋∫ 𝑔𝑚 𝜔

′ 𝑒𝑖𝜔′𝑡𝑑𝜔′

So the DSB wave:

𝜓𝐷𝑆𝐵 𝑡 =1

2𝜋

𝜓𝐶

2𝑒𝑖𝜔𝑐𝑡 + 𝑒−𝑖𝜔𝑐𝑡 ∫ 𝑔𝑚 𝜔

′ 𝑒𝑖𝜔′𝑡𝑑𝜔′

DSB modulation : General Modulator

The FT of 𝜓𝐷𝑆𝐵 𝑡 :

𝑔𝐷𝑆𝐵 𝜔 =1

2𝜋

𝜓𝐶

2∫ 𝑒𝑖𝜔𝑐𝑡 + 𝑒−𝑖𝜔𝑐𝑡 𝑒−𝑖𝜔𝑡 ∫ 𝑔𝑚 𝜔

′ 𝑒𝑖𝜔′𝑡𝑑𝜔′𝑑𝑡

𝑔𝐷𝑆𝐵 𝜔 =𝜓𝐶2∫ 𝑔𝑚 𝜔

′1

2𝜋∫ 𝑒𝑖(𝜔𝑐−𝜔+𝜔

′)𝑡 + 𝑒−𝑖(𝜔𝑐+𝜔−𝜔′)𝑡 𝑑𝑡 𝑑𝜔′

𝑔𝐷𝑆𝐵 𝜔 =𝜓𝐶2∫ 𝑔𝑚 𝜔

′ 𝛿(𝜔𝑐−𝜔 + 𝜔′ + 𝛿(𝜔𝑐+𝜔 − 𝜔

′)] 𝑑𝜔′

𝑔𝐷𝑆𝐵 𝜔 =𝜓𝐶2[𝑔𝑚 𝜔 −𝜔𝑐 + 𝑔𝑚 𝜔 +𝜔𝑐 ]

DSB modulation : Interpretation

From the last result:

𝑔𝐷𝑆𝐵 𝜔 =𝜓𝐶2[𝑔𝑚 𝜔 −𝜔𝑐 + 𝑔𝑚 𝜔 +𝜔𝑐 ]

It clear that the effect to general modulator signal is thetranslation of its spectrum by ±𝜔𝑐

Suppose the spectrum of the modulator wave is :

−𝜔𝑚 0 𝜔𝑚 𝜔

𝑔𝑚(𝜔)

DSB modulation : Interpretation

Because of the DSB modulation, the resulting spectrumbecomes:

𝑔𝐷𝑆𝐵 𝜔 =𝜓𝐶2[𝑔𝑚 𝜔 −𝜔𝑐 + 𝑔𝑚 𝜔 +𝜔𝑐 ]

−𝜔𝑐 − 𝜔𝑚 −𝜔𝑐 −𝜔𝑐+ 𝜔𝑚 𝜔𝑐−𝜔𝑚 𝜔𝑐 𝜔𝑐+ 𝜔𝑚

1

2𝜓𝐶𝑔𝑚(𝜔)

𝑈𝑆𝐵L𝑆𝐵

Band Width

• The effect of DSB modulation, the bandwidth is widenedtwice band the amplitude is reduced by half.

• The frequency above 𝜔𝐶 is called USB (Upper side band) andthe one below 𝜔𝑐 is called LSB (lower side band).

• Bandwidth (B) of the DSB wave is 2 x bandwidth of themodulator wave (see the graph):

𝐵 = 2𝜔𝑚 and 𝐵 = 2𝜈𝑚 where m: maximum!

Average Power

• The average power is proportional to 𝜓 2, it is defined:

𝑁 = lim𝑇→∞

1

𝑇

−𝑇2

𝑇2

𝜓 𝑡 2𝑑𝑡

Substitute the 𝜓𝐷𝑆𝐵(𝑡) :

𝑁 = lim𝑇→∞

1

𝑇

−𝑇2

𝑇2

{𝜓𝐶𝜓𝑚 𝑡 cos(𝜔𝑐𝑡)}2𝑑𝑡

Average Power

𝑁 = lim𝑇→∞

1

𝑇

𝜓𝑐2

2

−𝑇2

𝑇2

𝜓𝑚2 (𝑡)𝑑𝑡 +

−𝑇2

𝑇2

𝜓𝑚2 (𝑡) cos(2𝜔𝑐𝑡) 𝑑𝑡

Since usually 𝜔𝑐 ≫ 𝜔𝑚, then approximately:

lim𝑇→∞

1

𝑇

−𝑇2

𝑇2

𝜓𝑚2 (𝑡) cos(2𝜔𝑐𝑡) 𝑑𝑡 ≈ 0

Which result in 𝑁 = 𝑁𝐶𝑁𝑚

Average Power

𝑁𝐶 = lim𝑇→∞

1

𝑇

−𝑇2

𝑇2

𝜓𝑐2 cos2(𝜔𝑐𝑡) 𝑑𝑡 ≈

1

2𝜓𝑐2

𝑁𝑚 = lim𝑇→∞

1

𝑇

−𝑇2

𝑇2

𝜓𝑚2 𝑑𝑡

Demodulation of DSB

Demodulation is a process to get the modulator 𝜓𝑚(𝑡) signalback from 𝜓𝐷𝑆𝐵(𝑡).

Steps in demodulation of DSB:

a. Multiplication with synchronous local oscillator

b. Filtering

Multiplication

The 𝜓𝐷𝑆𝐵(t) is multiplied with 2 cos𝜔𝑐𝑡:𝜓 𝑡 = 𝜓𝐷𝑆𝐵 𝑡 2 cos(𝜔𝐶𝑡)𝜓 𝑡 = 𝜓𝐶𝜓𝑚 𝑡 2 cos

2(𝜔𝐶𝑡)

𝜓 𝑡 = 𝜓𝐶𝜓𝑚 𝑡 + 𝜓𝐶𝜓𝑚 𝑡 cos (2𝜔𝐶𝑡)

Demodulation : Spectrum

If the FT of 𝜓𝑚(𝑡) is 𝑔𝑚 𝜔 , then𝑔 𝜔 = 𝐹𝑇(𝜓(𝑡))

𝑔(𝜔) = 𝜓𝐶𝐹𝑇{𝜓𝑚 𝑡 } + 𝜓𝐶 𝐹𝑇{𝜓𝑚 𝑡 𝑐𝑜𝑠(2𝜔𝐶𝑡)}

𝑔(𝜔) = 𝜓𝐶𝑔𝑚(𝜔) +𝜓𝐶2𝑔𝑚 𝜔 − 2𝜔𝐶 + 𝑔𝑚 𝜔 + 2𝜔𝐶

−2𝜔𝑐 −𝜔𝑚 −2𝜔𝑐 −2𝜔𝑐+ 𝜔𝑚 2𝜔𝑐−𝜔𝑚 2𝜔𝑐 2𝜔𝑐+ 𝜔𝑚−𝜔𝑚 𝜔𝑚

1

2𝜓𝐶𝑔𝑚(𝜔)

1

2𝜓𝐶𝑔𝑚(𝜔)

𝜓𝐶𝑔𝑚(𝜔)

Demodulation : Filtering

The modulator wave is separated from others, since 𝜔𝑚 ≪ 𝜔𝑐then this signal can be obtained using LPF (low pass filter).

As long as the bandwidth B satisfies :𝜔𝑚 < 𝐵 < 2𝜔𝑐 − 𝜔𝑚

The problem with DSB modulation technique is the need forsynchronous local oscillator, which in reality is quite difficult.