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**S-72.1140 Transmission Methods in Telecommunication Systems (5 cr)**

Exponential Carrier Wave Modulation

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**Exponential modulation: Frequency (FM) and phase (PM) modulation**

FM and PM waveforms Instantaneous frequency and phase Spectral properties narrow band arbitrary modulating waveform tone modulation - phasor diagram wideband tone modulation Transmission BW Generating FM - signals de-tuned tank circuit narrow band mixer modulator indirect modulators

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**Contents (cont.) Detecting FM/PM**

FM-AM conversion followed by envelope detector Phase-shift discriminator Zero-crossing detection (tutorials) PLL-detector (tutorials) Effect of additive interference on FM and PM analytical expressions and phasor diagrams implications for demodulator design FM preemphases and deemphases filters

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**Linear and exponential modulation**

In linear CW (carrier wave) modulation: transmitted spectra resembles modulating spectra spectral width does not exceed twice the modulating spectral width destination SNR can not be better than the baseband transmission SNR (lecture: Noise in CW systems) In exponential CW modulation: baseband and transmitted spectra does not carry a simple relationship usually transmission BW>>baseband BW bandwidth-power trade-off (channel adaptation-suppression of in-band noise)

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Assignment

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**Phase modulation (PM) Carrier Wave (CW) signal:**

In exponential modulation the modulation is “in the exponent” or “in the angle” Note that in exponential modulation superposition does not apply: In phase modulation (PM) carrier phase is linearly proportional to the modulation x(t): Angular phasor has the instantaneous frequency (phasor rate)

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**Instantaneous frequency**

Constant frequency carrier: Angle modulated carrier Angular frequency w (rate) is the derivative of the phase (the same way as the velocity v(t) is the derivative of distance s(t)) For continuously changing frequency instantaneous frequency is defined by differential changes: Compare to linear motion:

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Assignment

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**Frequency modulation (FM)**

In frequency modulation carrier’s instantaneous frequency is linearly proportional to modulation frequency: Hence the FM waveform can be written as Therefore for FM and for PM integrate

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**AM, FM and PM waveforms Instantaneous frequency directly**

proportional to modulation waveform constant frequency follows derivative of the modulation waveform

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Assignment Briefly summarize what is the main difference between FM and PM ? How would you generate FM by using a PM modulator?

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**Narrowband FM and PM (small modulation index, arbitrary modulation waveform)**

The CW presentation: The quadrature CW presentation: Narrow band (small angle) condition: Hence the Fourier transform of XC(t) is

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**Narrow band FM and PM spectra**

Remember the instantaneous phase in CW presentation: The small angle assumption produces compact spectral presentation both for FM and AM: What does it mean to set this component to zero?

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Example Assume:

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Assignment

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Solution

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**Tone modulation with PM and FM: modulation index b**

Remember the FM and PM waveforms: Assume tone modulation Then

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**Tone modulation in frequency domain: Phasors and spectra for narrowband case**

Remember the quadrature presentation: For narrowband assume

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**Narrow band tone modulation: spectra and phasors**

Phasors and spectra resemble AM:

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Assignment AM NB-FM (i) Discuss the phasor diagrams and explain phasor positions based on analytical expressions (ii) Comment amplitude and phase modulation in both cases

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**FM and PM with tone modulation and arbitrary modulation index**

Time domain expression for FM and PM: Remember: Therefore: Jn is the first kind, order n Bessel function

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**Wideband FM and PM spectra**

After simplifications we can write:

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**Assignment Assuming other parameters fixed, what happens to spectral**

width and line spacing when Frequency deviation increases in FM Modulation frequency increases in FM Phase deviation decreases in PM

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**Determination of transmission bandwidth**

The goal is to determine the number of significant sidebands Thus consider again how Bessel functions behave as the function of b, e.g. we consider Significant sidebands: Minimum bandwidth includes 2 sidebands (why?): Generally:

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Assignment

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**Transmission bandwidth and deviation D**

Tone modulation is extrapolated into arbitrary modulating signal by defining deviation by Therefore transmission BW is also a function of deviation For very large D and small D with that can be combined into

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**Example: Commercial FM bandwidth**

Following commercial FM specifications High-quality FM radios RF bandwidth is about Note that under estimates the bandwidth slightly

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**A practical FM modulator circuit**

A tuned circuit oscillator biased varactor diode capacitance directly proportional to x(t) other parts: input transformer RF-choke DC-block

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**Generating FM/PM De-tuned tank circuit**

Capacitance of a resonant circuit can be made to be a function of modulation voltage. That can be simplified by the series expansion Resonance frequency De-tuned resonance frequency Capacitance diode De-tuned tank circuit Note that this applies for a relatively small modulation index Remember that the instantaneous frequency is the derivative of the phase

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Assignment

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**Phase modulators: narrow band mixer modulator**

Integrating the input signal to a phase modulator produces frequency modulation! Thus PM modulator applicable for FM Also, PM can be produced by an FM modulator by differentiating its input Narrow band mixer modulator: Narrow band FM/PM spectra

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**Frequency detection Methods of frequency detection**

FM-AM conversion followed by envelope detector Phase-shift discriminator Zero-crossing detection (tutorials) PLL-detector (tutorials) FM-AM conversion is produced by a transfer function having magnitude distortion, as the time derivative (other possibilities?): As for example

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Assignment

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**Indirect FM transmitter**

FM/PM modulator with high linearity and modulation index difficult to realize One can first generate a small modulation index signal that is then applied into a nonlinear circuit Therefore applying FM/PM wave into non-linearity increases modulation index Mathematica®-expressions

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**Indirect FM transmitter:circuit realization**

The frequency multiplier produces n-fold multiplication of instantaneous frequency Frequency multiplication of tone modulation increases modulation index but the line spacing remains the same

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