PHASE AND FREQUENCY MODULATION

Phase modulation

Phase modulation (PM) is a modulation pattern that encodes information as variations in the instantaneous phase of a carrier wave.

The phase of a carrier signal is modulated to follow the changing voltage level (amplitude) of modulation signal. The peak amplitude and frequency of the carrier signal remain constant, but as the amplitude of the information signal changes, the phase of the carrier changes correspondingly. The analysis and the final result (modulated signal) are similar to those of frequency modulation.

Phase modulation is widely used for transmitting radio waves and is an integral part of many digital transmission coding schemes that underlie a wide range of technologies like Wi-Fi, GSM and satellite television.

Phase modulation is closely related to frequency modulation (FM); it is often used as an intermediate step to achieve FM. Mathematically both phase and frequency modulation can be considered a special case of quadrature amplitude modulation (QAM).

Theory 

The modulating wave (blue) is modulating the carrier wave (red), resulting the PM signal (green).
                         g(x) = π/2 * sin(2*2πt+ π/2*sin(3*2πt))

PM changes the phase angle of the complex envelope in direct proportion to the message signal.

Suppose that the signal to be sent (called the modulating or message signal) is  m(t) and the carrier onto which the signal is to be modulated is

                        c(t)=A_{c}\sin \left(\omega _{\mathrm {c} }t+\phi _{\mathrm {c} }\right).

Annotated:

carrier(time) = (carrier amplitude)*sin(carrier frequency*time + phase shift)

This makes the modulated signal

                    y(t)=A_{c}\sin \left(\omega _{\mathrm {c} }t+m(t)+\phi _{\mathrm {c} }\right).

This shows how  m(t) modulates the phase – the greater m(t) is at a point in time, the greater the phase shift of the modulated signal at that point. It can also be viewed as a change of the frequency of the carrier signal, and phase modulation can thus be considered a special case of FM in which the carrier frequency modulation is given by the time derivative of the phase modulation.

The modulation signal could here be

                         {\displaystyle m(t)=\cos \left(\omega _{\mathrm {c} }t+h\omega _{\mathrm {m} }(t)\right)\ }

The mathematics of the spectral behavior reveals that there are two regions of particular interest:

  • For small amplitude signals, PM is similar to amplitude modulation (AM) and exhibits its unfortunate doubling of basebandbandwidthand poor efficiency.
  • For a single large sinusoidal signal, PM is similar to FM, and its bandwidth is approximately
                                 2\left(h+1\right)f_{\mathrm {M} },
where  f_{\mathrm {M} }=\omega _{\mathrm {m} }/2\pi  and  h is the modulation index defined below. This is also known as Carson’s Rule for PM.

 

Frequency modulation

Theory 

If the information to be transmitted (i.e., the baseband signal) is  x_{m}(t) and the sinusoidal carrier is  x_{c}(t)=A_{c}\cos(2\pi f_{c}t)\,, where fc is the carrier’s base frequency, and Ac is the carrier’s amplitude, the modulator combines the carrier with the baseband data signal to get the transmitted signal: 

             {\displaystyle {\begin{aligned}y(t)&=A_{c}\cos \left(2\pi \int _{0}^{t}f(\tau )d\tau \right)\\&=A_{c}\cos \left(2\pi \int _{0}^{t}\left[f_{c}+f_{\Delta }x_{m}(\tau )\right]d\tau \right)\\&=A_{c}\cos \left(2\pi f_{c}t+2\pi f_{\Delta }\int _{0}^{t}x_{m}(\tau )d\tau \right)\\\end{aligned}}}

where  f_{\Delta }\, =  K_{f} A_{m} , K_{f} being the sensitivity of the frequency modulator and  A_{m} being the amplitude of the modulating signal or base band signal.

In this equation,  f(\tau )\, is the instantaneous frequency of the oscillator and  f_{\Delta }\, is the frequency deviation, which represents the maximum shift away from fc in one direction, assuming xm(t) is limited to the range ±1.

While most of the energy of the signal is contained within fc ± fΔ, it can be shown by Fourier analysis that a wider range of frequencies is required to precisely represent an FM signal. The frequency spectrum of an actual FM signal has components extending infinitely, although their amplitude decreases and higher-order components are often neglected in practical design problems. 

Sinusoidal baseband signal 

Mathematically, a baseband modulated signal may be approximated by a sinusoidalcontinuous wave signal with a frequency fm . This method is also named as Single-tone Modulation.The integral of such a signal is:

            {\displaystyle \int _{0}^{t}x_{m}(\tau )d\tau ={\frac {A_{m}\sin(2\pi f_{m}t)}{2\pi f_{m}}}\,}

In this case, the expression for y(t) above simplifies to:

             {\displaystyle y(t)=A_{c}\cos \left(2\pi f_{c}t+{\frac {A_{m}f_{\Delta }}{f_{m}}}\sin \left(2\pi f_{m}t\right)\right)\,}

where the amplitude  A_{m}\, of the modulating sinusoid is represented by the peak deviation  f_{\Delta }\, (see frequency deviation).

Modulation index 

As in other modulation systems, the modulation index indicates by how much the modulated variable varies around its unmodulated level. It relates to variations in the carrier frequency:

 h={\frac {\Delta {}f}{f_{m}}}={\frac {f_{\Delta }|x_{m}(t)|}{f_{m}}}\

where  f_{m}\, is the highest frequency component present in the modulating signal xm(t), and  \Delta {}f\, is the peak frequency-deviation—i.e. the maximum deviation of the instantaneous frequency from the carrier frequency. For a sine wave modulation, the modulation index is seen to be the ratio of the peak frequency deviation of the carrier wave to the frequency of the modulating sine wave.

If  h\ll 1, the modulation is called narrow band FM, and its bandwidth is approximately  2f_{m}\,. Sometimes modulation index h<0.3 rad is considered as Narrow band FM otherwise Wide band FM.

For digital modulation systems, for example Binary Frequency Shift Keying (BFSK), where a binary signal modulates the carrier, the modulation index is given by:

                h={\frac {\Delta {}f}{f_{m}}}={\frac {\Delta {}f}{\frac {1}{2T_{s}}}}=2\Delta {}fT_{s}\

where  T_{s}\, is the symbol period, and  f_{m}={\frac {1}{2T_{s}}}\, is used as the highest frequency of the modulating binary waveform by convention, even though it would be more accurate to say it is the highest fundamental of the modulating binary waveform. In the case of digital modulation, the carrier  f_{c}\, is never transmitted. Rather, one of two frequencies is transmitted, either  f_{c}+\Delta {}f or   f_{c}-\Delta {}f, depending on the binary state 0 or 1 of the modulation signal.

If  h\gg 1, the modulation is called wide band FM and its bandwidth is approximately  2f_{\Delta }\,. While wide band FM uses more bandwidth, it can improve the signal-to-noise ratio significantly; for example, doubling the value of  \Delta {}f\,, while keeping  f_{m} constant, results in an eight-fold improvement in the signal-to-noise ratio. (Compare this with Chirp spread spectrum , which uses extremely wide frequency deviations to achieve processing gains comparable to traditional, better-known spread-spectrum modes).

 

ADVANTAGES OF FM OVER AM

FM is more clear in transmission than AM.Its wave length is short whereas the frequency is high and vise verse for AM. Here the frequency is modulated and in AM amplitude is modulated. Natural or human activity like traffic etc doesn’t effect the FM transmission whereas AM transmission gets effected.
1. Lesser distortion. Frequency modulated wave is less susceptible to intereferences from      buildings, traffic etc which provides improved signal to noise ratio (about 25dB) w.r.t.        To man made interference.
2. Waves at higher frequencies can carry more data than the waves at low frequency.
3. Smaller geographical interference between neighboring stations.
4. Less radiated power.
5. Well defined service areas for given transmitter power.