Eeng Chapter 5 AM, FM, and Digital Modulated Systems  Phase Modulation (PM)  Frequency Modulation (FM)  Generation of PM and FM  Spectrum of PM and FM  Carson’s Rule  Narrowband FM Huseyin Bilgekul Eeng360 Communication Systems I Department of Electrical and Electronic Engineering Eastern Mediterranean University

Eeng AM and FM Modulation (a) Carrier wave. (b) Sinusoidal modulating signal. (c) Amplitude-modulated signal. (d) Frequency modulated signal.

Eeng Angle Modulation  We have seen that an AM signal can be represented as  Now we will see that information can also be carried in the angle of the signal as Note that in this type of modulation the amplitude of signal carries information. Here the amplitude A c remains constant and the angle is modulated. This Modulation Technique is called the Angle Modulation Angle modulation: Vary either the Phase or the Frequency of the carrier signal  Phase Modulation and Frequency Modulation are special cases of Angle Modulation

Eeng Angle Modulation Representation of PM and FM signals: The Complex Envelope for an Angle Modulation is given by Is a constant Real envelope, θ(t) - linear function of the modulating signal m(t) The Angle-modulated Signal in time domain is given by g(t) - Nonlinear function of the modulation. Special Case 1: For PM the phase is directly proportional to the modulating signal. i.e.; Where D p is the Phase sensitivity of the phase modulator, having units of radians/volt. Special Case 2: For FM, the phase is proportional to the integral of m(t) so that where the frequency deviation constant D f has units of radians/volt-sec.

Eeng Angle Modulation Resulting PM wave:  Phase Modulation occurs when the instantaneous phase varied in proportion to that of the message signal. Dp is the phase sensitivity of the modulator  Frequency Modulation occurs when the instantaneous frequency is varied linearly with the message signal. Resulting FM wave: D f is the frequency deviation constant  Instantaneous Frequency (f i ) of a signal is defined by

Eeng Phase and Frequency Modulations Phase Modulation Frequency Modulation Comparing above two equations, we see that if we have a PM signal modulated by m p (t), there is also FM on the signal, corresponding to a different modulation wave shape that is given by: Similarly if we have a FM signal modulated by m f (t), the corresponding phase modulation on this signal is: Where f and p denote frequency and phase respectively.

Eeng Integrator Phase Modulator (Carrier Frequency fc) Differentiator Frequency Modulator (Carrier Frequency fc) Generation of FM from PM and vice versa FM Signal PM signal Generation of FM using a Phase Modulator: Generation of PM using a Frequency Modulator:

Eeng FM with sinusoidal modulating signal  The Instantaneous Frequency of the FM signal is given by:  The Peak Frequency Deviation is given by:  The Frequency Deviation from the carrier frequency:  The Peak-to-peak Deviation is given by ∆F is related to the peak modulating voltage by Where If a bandpass signal is represented by:

Eeng FM with sinusoidal modulating signal But,  VpVp BW   Average Power does not change with modulation

Eeng Angle Modulation Advantages:  Constant amplitude means Efficient Non-linear Power Amplifiers can be used.  Superior signal-to-noise ratio can be achieved (compared to AM) if bandwidth is sufficiently high. Disadvantages:  Usually require more bandwidth than AM  More complicated hardware

Eeng Modulation Index  The Peak Phase Deviation is given by: ∆θ is related to the peak modulating voltage by: Where  The Phase Modulation Index is given by: Where ∆θ is the peak phase deviation  The Frequency Modulation Index is given by: ∆F Peak Frequency Deviation B Bandwidth of the modulating signal

Eeng Spectra of Angle modulated signals  Spectra for AM, DSB-SC, and SSB can be obtained with simple formulas relating S(f) to M(f).  But for angle modulation signaling, because g(t) is a nonlinear function of m(t). Thus, a general formula relating G(f) to M(f) cannot be obtained.  To evaluate the spectrum for angle-modulated signal, G(f) must be evaluated on a case-by-case basis for particular modulating waveshape of interest. Where Spectrum of Angle modulated signal

Eeng Spectrum of PM or FM Signal with Sinusoidal Modulating Signal  Assume that the modulation on the PM signal is Then Where is the phase Modulation Index. Same θ(t) could also be obtained if FM were used where The Complex Envelope is: and The peak frequency deviation would be which is periodic with period

Eeng Using discrete Fourier series that is valid over all time, g(t) can be written as Where Which reduces to J n (β) – Bessel function of the first kind of the nth order Taking the fourier transform of the complex envelope g(t), we get Is a special property of Bessel Functions Spectrum of PM or FM Signal with Sinusoidal Modulating Signal or

Eeng Bessel Functions of the First Kind J 0 (β)=0 at β=2.4, 5.52 & so on

Eeng Bessel Functions of the First Kind

Eeng  The FM modulated signal in time domain  From this equation it can be seen that the frequency spectrum of an FM waveform with a sinusoidal modulating signal is a discrete frequency spectrum made up of components spaced at frequencies of  c ± n  m.  By analogy with AM modulation, these frequency components are called sidebands.  We can see that the expression for s(t) is an infinite series. Therefore the frequency spectrum of an FM signal has an infinite number of sidebands.  The amplitudes of the carrier and sidebands of an FM signal are given by the corresponding Bessel functions, which are themselves functions of the modulation index Observations: Frequency spectrum of FM

Eeng Spectra of an FM Signal with Sinusoidal Modulation BTBT f 1.0  The following spectra show the effect of modulation index, , on the bandwidth of an FM signal, and the relative amplitudes of the carrier and sidebands

Eeng BTBT J 0 (1.0) J 1 (1.0) J 2 (1.0) f 1.0 Spectra of an FM Signal with Sinusoidal Modulation

Eeng BTBT f 1.0 Spectra of an FM Signal with Sinusoidal Modulation

Eeng  Although the sidebands of an FM signal extend to infinity, it has been found experimentally that signal distortion is negligible for a bandlimited FM signal if 98% of the signal power is transmitted.  Based on the Bessel Functions, 98% of the power will be transmitted when the number of sidebands transmitted is 1+  on each side. Carson’s rule (1+  f m

Eeng Carson’s rule  Therefore the Bandwidth required is given by β – phase modulation index/ frequency modulation index B – bandwidth of the modulating signal For sinusoidal modulation  Carson’s rule : Bandwidth of an FM signal is given by Note: When β =0 i.e. baseband signals

Eeng Narrowband Angle Modulation  Narrowband Angle Modulation is a special case of angle modulation where θ(t) is restricted to a small value.  The complex envelope can be approximated by a Taylor's series in which only first two terms are used. becomes  The Narrowband Angle Modulated Signal is  The Spectrum of Narrowband Angle Modulated Signal is where PM FM

Eeng Indirect method of generating WBFM Balanced Modulator

Eeng Wideband Frequency modulation

Eeng FM Stero System

Eeng FM Stero System