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ANGLE MODULATION 1

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Introduction 2 Another class of modulation methods are frequency and phase modulation which referred to as angle- modulation methods. In frequency-modulation (FM), the frequency of a carrier wave is changed by the message signal. In phase-modulation (PM), the phase of the carrier is changed according to the variations in the message signal.

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Introduction 3 For all AM modulation schemes Modulated spectrum is the frequency translated message spectrum Transmission BW never exceeds twice the message BW In Angle modulation Modulated spectrum is not a translated copy of the message spectrum. Transmission BW is usually much greater than twice the message BW. The major benefit of the FM and PM modulation is their high degree of noise immunity.

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Introduction 4 Consider a sinusoid, A c cos ( ω c t+ φ ) where A c is the (constant) amplitude, ω c is the (constant) frequency and φ is the initial phase. In the AM modulation, the condition that A c be a constant is relaxing and the amplitude become a function of the message signal m(t). the frequency and the phase remain constant and don’t change or effect by the m(t).

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Introduction 5 In the FM and PM modulation, A c is a constant but ω c t+ φ, instead of being constants it will be a function of m(t). We must extend the concept of a sinusoid to a generalized function whose frequency vary with time

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Generalized Sinusoidal Signal 6 Let us consider a generalized sinusoidal signal given as A c cos ( θ (t) ) where the θ (t) is the instantaneous angle and is a function of t. The generalized angle for the conventional sinusoid A c cos ( ω c t+ φ ) is θ (t) = ω c t+ φ

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Angle Modulated signal 7 So for angle modulation, the modulated carrier represented by S angle_mod (t) = A c cos ( θ (t) ) where A c is a constant amplitude and θ (t) is a function of the message signal m(t). We define the instantaneous radian frequency of the angle modulated wave ω i (t) as:

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Phase Modulation (PM) 9 In phase modulation the angle is varied linearly with the message signal m(t) as θ (t) = ω c t + k p m(t) where k p is the phase deviation or sensitivity constant. Thus the phase modulated signal is defined as: S PM (t) = A c cos ( ω c t + k p m(t) ) The instantaneous radian frequency of S PM (t) is

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Example 10 If the message signal m(t) = a cos ( ω m t) is used to phase modulate the carrier A c cos ( ω c t) Find the PM modulated signal S PM (t) = A c cos ( ω c t + a k p cos ( ω m t) )

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Frequency Modulation (FM) 11 In frequency modulation the angle is varied linearly with the integral of message signal m(t) as where k f is the frequency deviation or sensitivity constant. Thus the frequency modulated signal is defined as: The instantaneous radian frequency of S FM (t) is

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Example 12 If the message signal m(t) = a cos ( ω m t) is used to phase modulate the carrier A c cos ( ω c t) Find the FM modulated signal

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Example 13 In a frequency modulation process m(t) be a periodic triangular wave with m max (t) =1 and m min (t)=-1 the carrier frequency is 100 kHz k f = 10 4 Hz/volt find the maximum and minimum values of the instantaneous frequency f i_max (t) = 100 *10 3 + 10 4 * 1= 100 *10 3 + 10 4 =110 kHz f i_min (t) = 100 *10 3 + 10 4 * -1 = 100 *10 3 - 10 4 =90 kHz

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The Relationship Between FM and PM 14 There is a close relation between FM and PM modulations. An FM modulated wave can be generated by first integrating the message signal m(t) with respect to time t and thus using the resulting signal as the input to a phase modulation. A PM modulated wave can be generated by first differentiating m(t) with respect to time t and then using the resulting signal as the input to a frequency modulator

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Bandwidth 16 The FM modulated wave is not band-limited. It has an infinite bandwidth and is not related to the modulating signal spectrum in any simple way, as was the case in AM modulation. Although the theoretical bandwidth of an FM wave is infinite, the most of the modulated signal power resides in a finite bandwidth. There are two distinct possibilities in terms of bandwidth: narrow-band FM and wide-band FM.

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Single-Tone Frequency Modulation 17 Consider a sinusoidal modulating signal defined as m(t) = A m cos( 2 π f m t) So, the instantaneous frequency (in Hertz) of the FM signal is f i (t) =f c + k f A m cos( 2 π f m t) = f c + ∆f cos( 2 π f m t) where ∆f is called the frequency deviation given by ∆f =k f A m The resultant FM signal is is the modulation index

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Single-Tone Frequency Modulation 18 The frequency deviation factor indicates the amount of frequency change in the FM signal from the carrier frequency f c on either side of it. Thus FM signal will have the frequency components between (fc - ∆f ) to (fc +∆f ). The modulation index, β represents the phase deviation of the FM signal and is measured in radians. Depending on the value of β, FM signal can be classified into two types: 1. Narrow band FM ( β << 1) and 2. Wide band FM ( β >> 1).

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