EE4512 Analog and Digital Communications Chapter 6. Chapter 6 Analog Modulation and Demodulation

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1 Chapter 6 Analog Modulation and Demodulation

2 Chapter 6 Analog Modulation and Demodulation Amplitude Modulation Pages

3 The analytical signal for double sideband, large carrier amplitude modulation (DSB-LC AM) is: s DSB-LC AM AM(t) ) = A C (c + s(t)) cos (2π f C t) where c is the DC bias or offset and A C is the carrier amplitude. The continuous analog signal s(t) is a baseband signal with the information content (voice or music) to be transmitted.

4 The baseband power spectral density (PSD) spectrum of the information signal s(t) ) or S(f) ) for voice has significant components below 500 Hz and a bandwidth of < 8 khz: S(f) ) = F(s(t)) The single-sided sided spectrum of the modulated signal is: F(A C (c + s(t)) cos (2π f C t)) = S(f f C ) db Power Spectral Density of s(t) 500 Hz 8 khz

5 The single-sided sided (positive frequency axis) spectrum of the modulated signal replicates the baseband spectrum as a double-sided spectrum about the carrier frequency. Double-sided spectrum Carrier 25 khz Baseband spectrum

6 The double-sided modulated spectrum about the carrier frequency has an lower (LSB)) and upper (USB)) sideband.

7 The modulated DSB-LC AM signal shows an outer envelope that follows the polar baseband signal s(t).

8 The analytical signal for double sideband, suppressed carrier amplitude modulation (DSB-SC SC AM) is: s DSB-SC SC AM AM(t) ) = A C s(t) cos (2π f C t) where A C is the carrier amplitude. The single-sided sided spectrum of the modulated signal replicates the baseband spectrum as a double-sided spectrum about the carrier frequency but without a carrier component.

9 The analytical signal for double sideband, suppressed carrier amplitude modulation (DSB-SC SC AM) is: s DSB-SC SC AM AM(t) ) = A C s(t) cos (2π f C t) where A C is the carrier amplitude. The modulated signal s DSB-SC SC AM(t (t)) looks similar to s(t) but has a temporal but not spectral carrier component.

10 The DSB-LC AM and the DSB-SC SC AM modulated signals have the same sidebands. DSB-LC AM Carrier 25 khz DSB-SC AM No carrier

11 The modulated DSB-LC AM and the DSC-SC AM signals are different.

12 The modulated DSB-SC SC AM signal has an envelope that follows the polar baseband signal s(t) but not an outer envelope.

13 Chapter 6 Analog Modulation and Demodulation Coherent Demodulation of AM Signals Pages

14 The DSB-SC SC AM signal can be simulated in SystemVue. SVU Figure 1.64 modified

15 The DSB-SC SC AM coherent receiver has a bandpass filter centered at f C and with a bandwidth of twice the bandwidth of s(t) because of the SVU Figure 1.64 modified LSB and USB.. The output of the multiplier is lowpass filtered with a bandwidth equal to z(t) the bandwidth of s(t). r(t) = γ s DSB DSB-SC SC(t) ) + n(t) Bandpass filter Lowpass filter S&M Figure 6-46

16 EE4512 Analog and Digital Communications Chapter 5 The DSB-SC SC AM received signal is r(t) = γ s DSB-SC SC(t) ) + n(t). The bandpass filter passes the modulated signal but filters the noise: z(t) ) = γ s DSB DSB-SC SC(t) ) + n o (t) S&M Eq. 6.3 n o (t) has a Gaussian distribution. The bandpass filter has a center frequency of f C = 25 khz and a -33 db bandwidth of 8 khz (25 ± 4 khz). Bandpass filter n o (t) Gaussian noise

17 EE4512 Analog and Digital Communications Chapter 5 The filter noise n o (t) has a flat power spectral density within the bandwidth of the bandpass filter: n o (t) PSD 21 khz 29 khz f C = 25 khz

18 The filter noise n o (t) can be described as a quadrature representation: n o (t) = W(t) cos (2π f C t) ) + Z(t) sin (2π( f C t) ) S&M Eq. 5.62R In the coherent receiver the noise is processed: n o (t) cos (2π f C t) = W(t) cos 2 (2π f C t) ) + S&M Eq. 6.5 Z(t) cos (2π f C t) sin (2π f C t) PSD 21 khz 29 khz f C = 25 khz

19 Applying the trignometric identity the filter noise n o (t) is: n o (t) cos (2π f C t) = ½ W(t) + ½ W(t) cos (4π f C t) ) + ½ Z(t) sin (4π f C t) ) S&M Eq. 6.5 After the lowpass filter in the receiver the demodulated signal is: s demod (t) = ½ γ A C s(t) + ½ W(t) S&M Eq. 6.7 PSD 21 khz 29 khz f C = 25 khz

20 The transmitted DSB-SC SC AM signal is: s DSB-SC SC AM AM(t) ) = A C s(t) cos (2π f C t) The average normalized bi-sided power of s DSB-SC SC(t) ) is found in the spectral domain with S(f) ) = F (s(t)): 2 1 P trans =A [ S(f f C) + S(f+f C) ] df 2 2 S&M Eq. 6.8

21 The dual-sided spectral do not overlap (at zero frequency) and the cross terms are zero so that: P trans =A [ S DSB-SC(f f C) + S DSB-SC(f+f C )] df 2 2 A P trans = P 2 s where P s is the average normalized power of s(t). S&M Eq. 6.9

22 The average normalized power of s(t) is found in the spectral domain: 2 2 P s = S(f) df = S(f + f C) df S&M Eq In a noiseless channel the power in the demodulated DSB-SC SC AM signal is: γ P = γ A P= P 4 2 demod, noiseless s trans S&M Eq. 6.11

23 The average normalized power of the processed noise is: 1 P processed noise = N o(2 B) 4 The signal-to to-noise power ratio then is: 2 γ P trans 2 γ Ptrans SNRcoherent DSB-SC = 2 = 1 N(2 B) N o B 4 o 2 B S&M Eq. 6.12

24 The DSB-SC SC AM coherent receiver requires a phase and frequency synchronous reference signal. If the reference signal has a SVU Figure 1.64 modified phase error φ then: SNR coherent DSB-SC phase error 2 2 γ cos ϕ P N B o trans = S&M Eq cos (2π f C t + φ) S&M Figure 6-76

25 The DSB-SC SC AM coherent receiver requires a phase and frequency synchronous reference signal. If the reference signal has a SVU Figure 1.64 modified frequency error f then: S demod frequency error (t (t)) = ½ γ A C s(t) cos (2π f f t) + ½ X(t) cos (2π f f t) + ½ Y(t) ) sin (2π f f t) S&M Eq cos (2π f C t + φ) S&M Figure 6-76

26 Although the noise component remains the same, the amplitude of the demodulated signal varies with f: S demod frequency error (t (t)) = ½ γ A C s(t) cos (2π f f t) + ½ X(t) cos (2π f f t) + ½ Y(t) ) sin (2π f f t) S&M Eq cos (2π f C t + φ) SVU Figure 1.64 modified S&M Figure 6-76

27 The frequency error DSB-SC SC AM signal can be simulated in SystemVue. SVU Figure 1.64 modified Frequency sweep

28 Chapter 6 Analog Modulation and Demodulation Non-coherent Demodulation of AM Signals Pages

29 The non-coherent AM (DSB-LC) receiver uses an envelope detector implemented as a semiconductor diode and a low- pass filter: The DSB-LC AM analytical signal is: s DSB-LC AM AM(t) ) = A C (c + s(t)) cos (2π f C t) where c is the DC bias (offset).

30 EE4512 Analog and Digital Communications Chapter 5 The envelope detector is a half-wave rectifier and provides a DC bias (c)) to the processed DSB-LC AM signal : c = DC bias

low-pass filtered to remove the carrier frequency

31 EE4512 Analog and Digital Communications Chapter 5 The output of the half-wave diode rectifier is low-pass filtered to remove the carrier frequency and outputs the envelope which is the information:

32 The DSB-LC AM signal can be decomposed as: s DSB-LC AM AM(t) ) = s(t) cos (2π f C t) + A C c cos (2π f C t) S&M Eq. 6.20R The average normalized power of the information term: 2 AC P info term = P 2 S S&M Eq. 6.23

33 The average normalized transmitted power is: P P T 1 = carrier term T 0 C C 2 2 AC c carrier term = 2 [ ] A c cos(2πf t) dt Since s(t) ) + c must be >= 0 to avoid distortion in the DSB-LC AM signal: c min [s(t[ s(t)] or c 2 s 2 (t) for all t. 2 S&M Eq. 6.24

34 Therefore c 2 P s and for DSB-LC AM: P P S&M Eq carrier term info term The power efficiency η of a DSB-LC AM signal is: Pinfo term Pinfo term η = = 0.5 P + P P carrier term info term trans DSB-LC AM term S&M Eq. 6.29

35 The DSB-LC AM signal wastes at least half the transmitted power because the power in the carrier term has no information: P P η 0.5 carrier term info term The modulation index m is defined as: [ ] min[ s(t) + c] [ ] [ ] max s(t) + c m = S&M Eq max s(t) + c + min s(t) + c

36 The modulation index m defines the power efficiency but m must be less than 1. If m > 1 then min [s(t[ s(t) ) + c] < 0 and distortion occurs. [ ] min[ s(t) + c] [ ] [ ] max s(t) + c m = S&M Eq max s(t) + c + min s(t) + c

37 The average normalized power of the demodulation noiseless DSB-LC AM signal is: P demod, noiseless = 2 2 γ Pinfo term Then the signal-to to-noise power ratio for the DSB-LC AM signal is: S&M Eq γ Pinfo term γ Ptrans DSB-LC SNRnoncoherent DSB-LC = = N (2 B) N B o 2 B o S&M Eq. 6.39

38 Chapter 6 Analog Modulation and Demodulation Coherent and Non-Coherent AM Demodulation Pages

39 The coherent AM (DSB-LC) analog communication system can be simulated in SystemVue. SVU Figure 1.64

40 The non-coherent AM (DSB-LC) analog communication system can also be simulated in SystemVue. SVU Figure 1.67

41 The non-coherent AM (DSB-LC) receiver is the crystal radio which needs no batteries! Power for the high- impedance ceramic earphone is obtained directly from the transmitted signal.. For simplicity, the RF BPF is omitted and the audio frequency filter is a simple RC network. SVU Figure 1.67

42 Chapter 6 Analog Modulation and Demodulation Frequency Modulation and Phase Modulation Pages

43 The analytical signal for an analog phase modulated (PM) signal is: (t)) = A C cos [2π f C t + α s(t)] S&M Eq s PM (t where α is the phase modulation constant rad/v and A C is the carrier amplitude. The continuous analog signal s(t) is a baseband signal with the information content (voice or music) to be transmitted.

The analytical signal for an analog frequency modulated (FM) signal is: s FM (t (t)) = A C cos{ { 2π 2 [f C + k s(t)] t + φ] ] S&M Eq. 6.

44 The analytical signal for an analog frequency modulated (FM) signal is: s FM (t (t)) = A C cos{ { 2π 2 [f C + k s(t)] t + φ] ] S&M Eq where k is the frequency modulation constant Hz / V, A C is the carrier amplitude and φ is the initial phase angle at t = 0. The continuous analog signal s(t) is a baseband signal with the information content.

45 The instantaneous phase of the PM signal is: Ψ PM (t (t)) = 2π2 f C t + α s(t) ) S&M Eq The instantaneous phase of the FM signal is: Ψ FM (t (t)) = 2π [f C + k s(t)] t + φ] ] S&M Eq The instantaneous phase is also call the angle of the signal. The instantaneous frequency is the time rate of change of the angle: f(t) ) = (1/2π) dψ(t) / dt S&M Eq. 6.58

46 The instantaneous frequency of the unmodulated carrier signal is: f carrier (t (t)) = dψd carrier (t) ) / dt = d/dt {2π f C t + φ} } S&M Eq The instantaneous phase is also: t t Ψ(t) = f (λ)( ) dλ d = f (λ)( ) dλ d + φ S&M Eq There are practical limits on instantaneous frequency and instantaneous phase. To avoid ambiguity and distortion in FM signals due to phase wrapping: k s(t) f C for all t S&M Eq. 6.61

47 To avoid ambiguity and distortion in PM signals due to phase wrapping: -π < α s(t) π radians for all t S&M Eq Since FM and PM are both change the angle of the carrier signal as a function of the analog information signal s(t), FM and PM are called angle modulation. For example, is this signal FM, PM or neither: t x(t) ) = A C cos { 2π2 f C t + k s(λ) ) dλ d + φ} S&M Eq

48 The instantaneous phase of the signal is: t Ψ x (t) ) = 2π2 f C t + k s(λ) ) dλ d + φ S&M Eq. p which is not a linear function of s(t) ) so the signal is not PM. The instantaneous frequency of the signal is: f x (t) ) = (1/2π) dψ x (t) ) / dt = f C + k s(t) ) / 2π2 and the frequency difference f x f C is a linear function of s(t) so the signal is FM. The maximum phase deviation of a PM signal is max αs(t) ). The maximum frequency deviation of a FM signal is f f = max k s(t) ).

49 The spectrum of a PM or FM signal can be developed as follows: S&M Eqs through 6.71 v(t) = A C sin(2π ft C + β sin 2π ft) m v(t) = Re { exp(j 2π ft C +j β sin 2π ft) m } now exp(j 2π ft C +j β sin 2π ft) m = cos (2π ft C + β sin 2π ft) m + j sin (2π ft C + β sin 2π ft) m v(t) = Im { A exp(2π ft + jβ sin 2π ft) } now C C exp(j β sin 2π f t) = c exp(j 2π n f t) after further development m n m n = - exp(j β sin 2π ft) = J( β) exp(j 2π n f t) m n m n = - m Bessel function of the first kind

50 Bessel functions of the first kind J n (β) are tabulated for FM with single tone f m angle modulation (S&M Table 6.1): β n

51 For single tone f m angle modulation the spectrum is periodic and infinite in extent: C n m C n = - v(t) = A J ( β) sin[2π (n f + f ) t] S&M Eq n β

52 The complexity of the Bessel function solution for the spectrum of a single tone angle modulation can be simplified by the Carson s s Rule approximation for the bandwidth B.. Since β = f f / f m : B = 2 (β( + 1) f m = 2 ( f( f + f m ) Hz S&M Eq n β

Prior to this in 1915 he presaged the concept of bandwidth efficiency in AM by

53 Carson s s Rule for the approximate bandwidth of an angle modulated signal was developed by John R. Carson in 1922 while he worked at AT&T. Prior to this in 1915 he presaged the concept of bandwidth efficiency in AM by proposing the suppression of a sideband (see S&M p ) 333): B = 2 (β( + 1) f m = 2 ( f( f + f m ) Hz

54 The normalized power within the Carson s s Rule bandwidth for a single tone angle modulated signals is: P in-band, sinusoid 2 β+1 AC = 2 n = -(β+1) 2 J ( β ) Note that J -n (β)) = ± J n (β)) so that J -n2 (β)) = J n2 (β)) and for the normalized power calculation the sign of J(β) is not used. n S&M Eq Spectrum of single tone FM modulation

55 Chapter 6 Analog Modulation and Demodulation Frequency Modulation Pages

56 The analog FM transmitter and receiver can be simulated in SystemVue.. A bandpass audio filter removes the low frequency components in the voice signal for clarity. SVU Figure 1.68

57 A phase-locked loop (PLL) token has a frequency output which tracks the frequency deviation f f which is proportional to the voice signal. SVU Figure 1.68

58 The PLL token is somewhat complex. SVU Figure 1.68

59 The analog FM power spectral density PSD of the voice signal has a bandwidth predicted only by Carson s s Rule since it is not a single tone. Voice PSD

60 Here f max = 4 khz, k = 25 Hz/V and f max = 40(25) = 1 khz. The Carson s s Rule approximate maximum bandwidth B = 2 ( f( f + f m ) = 10 khz or ± 5 khz (but seems wrong!) Voice 40 f C PSD Bandwidth

61 A 200 Hz single tone FM signal has a PSD with periodic terms at f C ± n f m = 25 ± 0.2 n khz. PSD f C 200 Hz

62 Here f m = 200 Hz, k = 25 Hz/V and f max = 40(25) = 1 khz. The Carson s s Rule approximate maximum bandwidth B = 2 ( f( f + f m ) = 2.4 khz or ± 1.2 khz: PSD f C 200 Hz Bandwidth

63 Since β = f f / f m = 1 khz / 0.2 khz = 5 and the Bessel function predicts a bandwidth of 2 n f m = 2(12)(200) = 4.8 khz (since n = 12 for β = 5 from Table 6.1): PSD f C 200 Hz Bandwidth

64 Chapter 6 Analog Modulation and Demodulation Noise in FM and PM Systems Pages

65 A general angle modulated transmitted signal, where Ψ(t) is the instantaneous phase, is: s angle-modulated modulated(t The received signals is: r angle-modulated modulated(t (t)) = A C cos [Ψ(t)] S&M Eq (t)) = γ A C cos [Ψ(t)] + n(t) ) S&M Eq. 6.87

66 The analytical signal for PM is: s PM (t) ) = A C cos [Ψ(t)] = A C cos [2π f C t + α s(t)] S&M Eq After development the SNR for demodulated PM is: SNR PM = (αγ( A C ) 2 P S / (2 N o f max ) S&M Eq where π < α s(t) π for all t.

The analytical signal for FM is: s FM (t) ) = A C cos [Ψ(t)] = A C cos [2π f C t + k s(λ) ) dλ] d S&M Eq. 6.

67 The analytical signal for FM is: s FM (t) ) = A C cos [Ψ(t)] = A C cos [2π f C t + k s(λ) ) dλ] d S&M Eq After development the SNR for demodulated FM is: SNR FM = 1.5 (k γ A C /(2π) ) 2 P S / (N o f max3 ) S&M Eq where k s(t) f C for all t.

68 End of Chapter 6 Analog Modulation and Demodulation

Outline. Communications Engineering 1

Outline. Communications Engineering 1 Outline Introduction Signal, random variable, random process and spectra Analog modulation Analog to digital conversion Digital transmission through baseband channels Signal space representation Optimal