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An angle modulated signal: Different Example
- 1. ' & $ % Angle Modulation In this type of modulation, the frequency or phase of carrier is varied in proportion to the amplitude of the modulating signal. t c(t) Ac Figure 1: An angle modulated signal If s(t) = Ac cos(θi(t)) is an angle modulated signal, then 1. Phase modulation:
- 2. ' & $ % θi(t) = ωct + kpm(t) where ωc = 2πfc. 2. Frequency Modulation: ωi(t) = ωc + kf m(t) θi(t) = Z t 0 ωi(t) dt = 2π Z t 0 fi(t) dt + Z t 0 kf m(t) dt • Phase Modulation If m(t) = Am cos(2πfmt) is the message signal, then the phase modulated signal is given by
- 3. ' & $ % s(t) = Ac cos(ωct + kpm(t)) Here, kp is phase sensitivity or phase modulation index. • Frequency Modulation If m(t) = Am cos(2πfmt) is the message signal, then the Frequency modulated signal is given by 2πfi(t) = ωc + kf Am cos(2πfmt) θi(t) = ωct + kf Am 2πfm sin(2πfmt) here, kf Am 2π is called frequency deviation (∆f) and ∆f fm is called modulation index (β). The Frequency modulated signal is given by
- 4. ' & $ % s(t) = Ac cos(2πfct + β sin(2πfmt)) Depending on how small β is FM is either Narrowband FM(β << 1) or Wideband FM(β ≈ 1). – Narrow-Band FM (NBFM) In NBFM β << 1, therefor s(t) reduces as follows: s(t) = Ac cos(2πfct + β sin(2πfmt)) = Ac cos(2πfct) cos(β sin(2πfmt)) − Ac sin(2πfct) sin(β sin(2πfmt)) Since, β is very small, the above equation reduces to s(t) = Ac cos(2πfct) − Acβ sin(2πfmt) sin(2πfct)
- 5. ' & $ % The above equation is similar to AM. Hence, for NBFM the bandwidth is same as that of AM i.e., 2 × message bandwidth(2 × B). A NBFM signal is generated as shown in Figure ??. DSB−SC oscillator + m(t) A cos( t) ω ω −Asin( t) NBFM signal /2 π Phase shifter c c Figure 2: Generation of NBFM signal
- 6. ' & $ % – Wide-Band FM (WBFM) A WBFM signal has theoritically infinite bandwidth. Spectrum calculation of WBFM signal is a tedious process. For, practical applications however the Bandwidth of a WBFM signal is calculated as follows: Let m(t) be bandlimited to BHz and sampled adequately at 2BHz. If time period T = 1/2B is too small, the signal can be approximated by sequence of pulses as shown in Figure ??
- 7. ' & $ % t m p T Figure 3: Approximation of message signal If tone modulation is considered, and the peak amplitude of the sinusoid is mp, the minimum and maximum frequency deviations will be ωc − kf mp and ωc + kf mp respectively. The spread of pulses in frequency domain will be 2π T = 4πB
- 8. ' & $ % as shown in Figure ?? ω − k m k m ω + 4 B π c p f c f p Figure 4: Bandwidth calculation of WBFM signal Therefore, total BW is 2kf mp + 8πB and if frequency deviation is considered BWfm = 1 2π (2kf mp + 8πB) BWfm = 2(∆f + 2B)
- 9. ' & $ % ∗ The bandwidth obtained is higher than the actual value. This is due to the staircase approximation of m(t). ∗ The bandwidth needs to be readjusted. For NBFM, kf is very small an d hence ∆f is very small compared to B. This implies Bfm ≈ 4B But the bandwidth for NBFM is the same as that of AM which is 2B ∗ A better bandwidth estimate is therefore: BWfm = 2(∆f + B) BWfm = 2( kf mp 2π + B) This is also called Carson’s Rule
- 10. ' & $ % – Demodulation of FM signals Let Φfm(t) be an FM signal. Φfm(t) = A cos(ωct + kf Z t 0 m(α) dα) This signal is passed through a differentiator to get Φ̇fm(t) = A (ωc + kf m(t)) sin ωct + kf Z t 0 m(α) dα If we observe the above equation carefully, it is both amplitude and frequency modulated. Hence, to recover the original signal back an envelope detector can be used. The envelope takes the form (see Figure ??):
- 11. ' $ % Envelope = A (ωc + kf m(t)) FM signal Envelope of FM signal Figure 5: FM signal - both Amplitude and Frequency Modulation The block diagram of the demodulator is shown in Figure ??
- 12. ' $ % Φ t ( ) Φ (t) fm fm . d/dt Detector Envelope A( + k m(t)) ωc f Figure 6: Demodulation of an FM signal • The analysis for Phase Modulation is identical. – Analysis of bandwidth in PM
- 13. ' $ % ωi = ωc + kpm0 (t) m0 p = [m0 (t)]max ∆ω = kpmp BWpm = 2(∆f + B) BWpm = 2( kpm0 p 2π + B) – The difference between FM and PM is that the bandwidth is independent of signal bandwidth in FM while it is strongly dependent on signal bandwidth in PM. a aowing to the bandwidth being dependent on the peak of the derivative of m(t) rather than m(t) itself
- 14. ' $ % Angle Modulation: An Example • An angle-modulated signal with carrier frequency ωc = 2π × 106 is described by the equation: φEM (t) = 12 cos(ωct + 5 sin 1500t + 10 sin 2000πt) 1. Determine the power of the modulating signal. 2. What is ∆f? 3. What is β? 4. Determine ∆φ, the phase deviation. 5. Estimate the bandwidth of φEM(t)? 1. P = 122 /2 = 72 units 2. Frequency deviation ∆f, we need to estimate the instantaneous frequency:
- 15. ' $ % ωi = d dt θ(t) = ωc + 7, 500 cos 1500t + 20, 000πt The deviation of the carrier is ∆ω = 7, 500 cos 1500t + 20, 000πt. When the two sinusoids add in phase, the maximum value will be 7, 500 + 20, 000π Hence ∆f = ∆ω 2π = 11, 193.66Hz 3. β = ∆f B = 11,193.66 1000 = 11.193 4. The angle θ(t) = ωct + 5 sin 1500t + 10 sin 2000πt). The maximum angle deviation is 15, which is the phase deviation. 5. BEM = 2(∆f + B) = 24, 387.32 Hz