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**Angle Modulation Objectives**

To introduce phase modulation (PM) and frequency modulation (FM) To define frequency deviation To describe the relationship between PM and FM To determine the bandwidth of FM signal To explain the methods for generating wideband FM signals To explain wideband FM demodulation method To illustrate frequency division multiplexing (FDM) method To compare AM and FM

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**Angle Modulation Angle modulation**

Angle modulation is the process of varying the total phase angle of a carrier wave in accordance with the instantaneous value of the message signal, while keeping the amplitude of the carrier constant. Because the amplitude of the carrier does not change, angle modulation has the advantage of using the peak power of a transmitter (P = A2 /2). However, angle modulation requires more complicated transmitters and receivers as compared to amplitude modulation.

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Angle Modulation A generalized sinusoidal signal of constant peak amplitude A can be expressed as s(t) = A cos(t) where (t) is the total phase angle (instantaneous or generalized angle). Types of angle modulation There are two types of angle modulation, according to two ways of varying the phase angle (t).

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**Angle Modulation Phase modulation (PM)**

The phase angle is varied linearly with a modulating signal m(t) PM(t) = ct + 0 + kPMm(t) where kPM is known as phase sensitivity of the modulator. The PM waveform can then be expressed as sPM(t) = A cos[ct + 0 + kPMm(t)]

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**Angle Modulation Frequency modulation (FM)**

The instantaneous frequency (t) is varied linearly with a modulating signal m(t) about an unmodulated frequency c, i.e. FM(t) = c + kFMm(t) where kFM represents the frequency sensitivity of the modulator. The total phase angle can be obtained by integrating the instantaneous frequency FM(t) The corresponding FM wave is given by

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**Angle Modulation Frequency deviation**

The instantaneous frequency of FM signal, FM(t) = c + kFMm(t), varies with time. Frequency deviation is the maximum frequency change from the average c, i.e. = kFMm(t)max The frequency deviation is a useful parameter for determining the bandwidth of the FM signals.

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**Angle Modulation Relationship between PM and FM**

Both the net effect of PM and FM is variation in total phase angle. In PM, phase angle varies linearly with m(t) while in FM, phase angle varies linearly with the integral of m(t). FM signal can be obtained by using phase modulator and PM wave can be obtained by using frequency modulator.

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**Angle Modulation Example: Tone modulation The message signal**

m(t) = Am cosmt PM (Tone modulation) sPM(t) = A cos[ct + 0 + kPMm(t)] = A cos[ct + 0 + kPM Amcosmt] = A cos[ct + 0 + mp cosmt] where mp = kPMAm is called the phase modulation index, representing the maximum phase deviation .

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**Angle Modulation FM (Tone modulation)**

where mf = kFMAm / m = / m, i.e. the ratio of frequency deviation to the modulating frequency, is called the frequency modulation index. mf = / m The frequency deviation in FM represents the maximum phase deviation = mf = / m

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Angle Modulation In the case of tone modulation, it is difficult to distinguish PM and FM waves.

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**Angle Modulation We therefore focus our discussion on FM.**

The bandwidth of an FM signal depends on the frequency deviation. When the frequency deviation (and hence mf) is high, the bandwidth will be large. Types of frequency modulation If mf < 1, this is known as narrowband FM. If mf > 1 it is wideband FM ( also known as broadcast FM). What is the significance of bandwidth for FM signal?

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Angle Modulation Since the amplitude of an FM signal is kept constant, the power of an FM signal is the same as that of an unmodulated carrier, i.e., PFM = A2/2. In other words, the power is independent of the frequency modulation process since the power is not dependent on the signal’s frequency. Most of the FM signal power is contained in its bandwidth.

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**Bandwidth of a sinusoidally modulated FM signal **

Angle Modulation It can be shown that 98% of the total power of an FM signal is contained in the bandwidth. ( Then, how to calculate the bandwidth?) Bandwidth of a sinusoidally modulated FM signal WFM = 2(mf + 1)m = 2( + m) or BFM = 2(mf + 1)fm = 2(f + fm) The bandwidth is twice the sum of the maximum frequency deviation and the modulating frequency ---- Carson’s rule.

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Angle Modulation Bandwidth of FM signal with arbitrary modulating signals: WFM = 2( + m) or BFM = 2(f + fm)

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**Angle Modulation Example**

A 10 MHz carrier is frequency modulated by a sinusoidal signal such that the peak frequency deviation is 50 kHz. Determine the bandwidth of the FM signal if the frequency of the modulating sinusoidal is (a) 500 Hz; (b) 10 kHz. Solution mf = f / fm = 50000/500 = 100 B = 2(mf + 1)fm = 2(101)500 = 101 kHz (b) mf = f / fm = 50/10 = 5 B = 2(mf + 1)fm = 2(6)10 = 120 kHz

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Angle Modulation Exercise If f = 75 kHz and fm = 15 kHz, determine the FM bandwidth and compare this with the AM bandwidth. What is your finding?

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**Angle Modulation Wideband modulation methods**

Direct method, voltage-controlled oscillator (VCO) The direct method depends on varying the frequency of an oscillator linearly with m(t) for FM.

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Angle Modulation In the VCO, the modulating signal varies the voltage across the capacitor, as a consequence, the capacitance changes and causes a corresponding change in the oscillator frequency, i.e. C = C0 + C = C0 + k0m(t), C = C0 (1 + C/C0) where k0 is a constant. Assume that where k is also a constant and (1 - x)-1/2 1 – x/2 is used.

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**Angle Modulation Wideband FM demodulation method**

Apply sFM(t) to a differentiator, the output is The response of an envelope detector becomes A[c + kFMm(t)] Blocking the dc term Ac, the output is so(t) = AkFMm(t)

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**Angle Modulation Multiplexing**

Multiplexing is combining several messages into a composite signal for transmission over a single communication channel. When the multiplexing is carried out in the frequency domain, it is called Frequency division multiplexing (FDM). How to separate different messages at the receiver side? Provided that there is no spectral overlap of the sidebands of one signal with those of the adjacent signal, the messages can be separated at the receiver by bandpass filtering. In order to make efficient use of transmitter power and of bandwidth, SSB modulation would normally be used with FDM.

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Angle Modulation Frequency division multiplexing (FDM)

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Angle Modulation FDM can be used in long distance telephone, radio/TV broadcasts, etc. But adjacent FDM signals are vulnerable to crosstalk and the requirement for bandpass filters increase the cost and complexity of the FDM system. Main advantages and disadvantages of FDM Ability to convey many simultaneous messages Expense and size of the band-pass filters Vulnerability to crosstalk

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**Angle Modulation Comparison between AM and FM Noise performance**

Wideband FM has better noise performance than AM. The greater the bandwidth, the better is the noise performance. Narrowband FM has a noise performance equivalent to AM. Channel bandwidth The wideband FM has a larger bandwidth as compared to AM. Example In a typical broadcast system, each channel bandwidth in AM is 15kHz, whereas, in FM, it is 150kHz.

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**Questions (Angle Modulation)**

1. What is FM ? Write the waveform expression for FM. 2. What is PM? Write the waveform expression for PM. 3. How to generate FM signal by using phase modulator? 4. How to generate PM signal by using frequency modulator? 5. What is frequency modulation index? 6. How to determine the bandwidth of FM signal? 7. How to generate wideband FM signal in a direct method? 8. How to demodulate wideband FM signal? 9. What is FDM and what is its main advantage?

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**Exercise Problems (Angle Modulation)**

1. An angle modulated signal is s(t) = Acos(ct + 100cosft). (a) For a PM, what is m(t) when kPM = 2? (b) What is the peak frequency change caused by m(t) in (a)? 2. Work problem 1 assuming modulation is FM and kFM = 2. 3. A 100 MHz carrier is frequency modulated by a sinusoidal signal of frequency 5kHz. If the peak frequency deviation is 50kHz, what is the approximate band of frequencies occupied by the FM waveform.

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**Exercise Problems (Angle Modulation)**

4. Twelve speech signals are to be combined using FDM for transmission over a wideband radio link. Each signal is concentrated in 300Hz < f < 3.4kHz and is allocated a nominal low pass channel bandwidth of 4kHz to allow for the use of practical filters with finite transition regions. Estimate the radio bandwidth required if the signals are combined using SSB-SC frequency division multiplexing and the radio receiver employs conventional amplitude modulation.

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**Exercise Problems (Angle Modulation)**

5. For a carrier wave with frequency of 1 MHz and amplitude of 3V, if the message signal is a cosine function, has unit amplitude and its frequency is 2 kHz, and the maximum frequency deviation is 4 kHz, determine the time domain expression of the FM wave. 6. A carrier wave has fc = 50 MHz, is frequency modulated by a sinusoidal wave of frequency fm = 20 kHz, (a) if the frequency deviation is 20Hz, determine the minimum bandwidth; (b) if the frequency deviation is increased to 1 MHz, what is the corresponding bandwidth?

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**Exercise Problems (Angle Modulation)**

7. The carrier wave of frequency 100MHz, is frequency modulated by a sinusoidal message signal of frequency 10kHz, the maximum frequency deviation is 1MHz, (a) determine the approximate bandwidth of the FM wave; (b) if the amplitude of modulating signal is doubled, determine the bandwidth of the FM wave; (c) if the frequency of the modulating wave is also doubled, determine the bandwidth of the FM wave. 8. A sinusoidal signal of 10kHz is used to modulate a carrier wave of 100MHz, assuming that the maximum frequency deviation is 50kHz, determine the bandwidth for (a) AM wave; (b) SSB wave; (c) FM wave.

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