Exponential Carrier Wave Modulation S Transmission Methods in Telecommunication Systems (5 cr)

2 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen Exponential modulation: Frequency (FM) and phase (PM) modulation n FM and PM waveforms n Instantaneous frequency and phase n Spectral properties –narrow band arbitrary modulating waveform tone modulation - phasor diagram –wideband tone modulation –Transmission BW n Generating FM: de-tuned tank circuit n Generating PM: narrow band mixer modulator n Generating FM/PM: indirect modulators (more linear operation - What linearity means here?)

3 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen Contents (cont.) n Detecting FM/PM –FM-AM conversion followed by envelope detector –Phase-shift discriminator –Zero-crossing detection (tutorials) –PLL-detector (tutorials) n Effect of additive interference on FM and PM –analytical expressions and phasor diagrams –implications for demodulator design n FM preemphases and deemphases filters

4 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen Linear and exponential modulation n 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) n In exponential CW modulation: –usually transmission BW>>baseband BW –bandwidth-power trade-off (channel adaptation) –baseband and transmitted spectra does not carry a simple relationship

5 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen Phase modulation (PM) n Carrier Wave (CW) signal: n In exponential modulation the modulation is “in the exponent” or “in the angle” n Note that in exponential modulation superposition does not apply: n In phase modulation (PM) carrier phase is linearly proportional to the modulation amplitude: n Angular phasor has the instantaneous frequency (phasor rate)

6 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen Instantaneous frequency Angular frequency  (rate) is the derivative of the phase (the same way as the velocity v(t) is the derivative of distance s(t)) n For continuously changing frequency instantaneous frequency is defined by differential changes: Angle modulated carrier Constant frequency carrier: Compare to linear motion:

7 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen Frequency modulation (FM) n In frequency modulation carrier instantaneous frequency is linearly proportional to modulation frequency: n Hence the FM waveform can be written as n Note that for FM and for PM integrate

8 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen AM, FM and PM waveforms Constant frequency: follows the derivative of the modulation waveform

9 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen Narrowband FM and PM (small modulation index, arbitrary modulation waveform) n The CW presentation: n The quadrature CW presentation: n The narrow band condition: n Hence the Fourier transform of X C (t) is in this case

10 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen Narrow band FM and PM spectra n Remember the instantaneous phase in CW presentation: n The small angle assumption produces compact spectral presentation for both FM and AM: What does it mean to set this component to zero?

11 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen n Assume: Example

12 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen Tone modulation with PM and FM: modulation index  n Remember the FM and PM waveforms: n Assume tone modulation n Then

13 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen Tone modulation in frequency domain: Phasors and spectra for narrowband case n Remember the quadrature presentation: n For narrowband assume

14 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen Narrow band tone modulation: spectra and phasors n Phasors and spectra resemble AM:

15 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen FM and PM with tone modulation and arbitrary modulation index n Time domain expression for FM and PM: n Remember: n Therefore: J n is the first kind, order n Bessel function

16 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen Wideband FM and PM spectra n After simplifications we can write:

17 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen Bessel Functions

18 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen Resulting Spectrum … 1 –Bessel functions determine the amplitude of the carrier and its sidebands. Carrier (J 0 ) falls from 1 to zero as the modulation index increases and the sidebands rise from zero following J 1, J 2, J 3 etc. –We commonly ignore sidebands of amplitude less than 1% of the carrier ( beyond the maximum amplitude sideband for any m p )

19 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen Resulting Spectrum … 2

20 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen Bandwidth Requirements for FM At low modulation index bandwidth is twice modulating frequency, at high modulation index bandwidth is slightly more than twice frequency deviation.

21 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen Determination of transmission bandwidth n The goal is to determine the number of significant sidebands Thus consider again how Bessel functions behave as the function of , e.g. we consider n Significant sidebands: n Minimum bandwidth includes 2 sidebands (why?): n Generally:

22 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen Transmission bandwidth and deviation D n Tone modulation is extrapolated into arbitrary modulating signal by defining deviation by n Therefore transmission BW is also a function of deviation n For very large D and small D with n that can be combined into

23 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen Example: Commercial FM bandwidth n Following commercial FM specifications n High-quality FM radios RF bandwidth is about n Note that under estimates the bandwidth slightly

24 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen A practical FM modulator circuit realization n A tuned circuit oscillator –biased varactor diode capacitance directly proportional to x(t) –other parts: input transformer RF-choke DC-block

25 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen Generating FM/PM n Capacitance of a resonant circuit can be made to be a function of modulation voltage. n That can be simplified by the series expansion Remember that the instantaneous frequency is the derivative of the phase Note that this applies for a relatively small modulation index De-tuned tank circuit Capacitance diode De-tuned resonance frequency Resonance frequency

26 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen n Integrating the input signal to a phase modulator produces frequency modulation! Thus PM modulator applicable for FM n Also, PM can be produced by an FM modulator by differentiating its input n Narrow band mixer modulator: Phase modulators: narrow band mixer modulator Narrow band FM/PM spectra

27 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen Indirect FM transmitter n FM/PM modulator with high linearity and modulation index difficult to realize n One can first generate a small modulation index signal that is then applied into a nonlinear circuit n Therefore applying FM/PM wave into non-linearity increases modulation index Mathematica ® -expressions

28 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen Indirect FM transmitter:circuit realization n The frequency multiplier produces n-fold multiplication of instantaneous frequency n Frequency multiplication of tone modulation increases modulation index but the line spacing remains the same

29 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen Frequency detection n Methods of frequency detection –FM-AM conversion followed by envelope detector –Phase-shift discriminator –Zero-crossing detection (tutorials) –PLL-detector (tutorials) n FM-AM conversion is produced by a transfer function having magnitude distortion, as the time derivative (other possibilities?): n As for example