mywbut.com Chapter 5 Amplitude Modulation Amplitude modulation was historically the first modulation developed and conceptually the simplest to understand. Consequently this modulation is developed first. 5.1 Linear Modulation The simplest analog modulation is to make Γm (m(t)) = Acm(t), i.e., a linear function of the message signal. The complex envelope and the spectrum of this modulated signal are given as 2 xz(t)=Acm(t) Gxz (f)=Ac Gm(f). This modulation has xI (t)=Acm(t) and xQ(t) = 0, so the imaginary portion of the complex envelope is not used in a linear analog modulation. The resulting bandpass signal and spectrum are given as √ √ xc(t)= 2xz(t) exp[j2πfct] = Acm(t) 2 cos(2πfct) (5.1) A2 A2 A2 A2 G (f)= c G (f − f )+ c G (−f − f )= c G (f − f )+ c G (f + f ) (5.2) xc 2 m c 2 m c 2 m c 2 m c where the fact that m(t) was real was used to simplify (5.2). Fig. 5.1 shows the√ complex envelope and an example bandpass signal for the message signal shown in Fig. 4.1 with Ac =1/ 2. It is quite obvious from Fig. 5.1 that the amplitude of the carrier signal is modulated directly proportional to the absolute value of the message signal. Fig. 5.2 shows the resulting energy spectrum of the linearly modulated signal for the message signal shown in Fig. 4.2. A very important characteristic of this modulation is that if the message signal has a bandwidth of W Hz then the bandpass signal will have a transmission bandwidth of BT =2W . This implies EB=50%. Because of this characteristic the modulation is often known as double sideband-amplitude modulation (DSB-AM). An efficiency of 50% is wasteful of the precious spectral resources but obviously the simplicity of the modulator is a positive attribute. The output power of a DSB-AM modulator is often of interest (e.g., to specify the characteristics of amplifiers or to calculate the received signal-to-noise ratio). To this end the power is given as T T 1 2 1 2 2 2 2 Pxc = lim xc (t)dt = lim Ac m (t)2 cos (2πfct)dt = PmAc (5.3) T →∞ 2T −T T →∞ 2T −T For a DSB-AM modulated waveform the output power is usually given as the product of the power 2 associated with the carrier amplitude (Ac ) and the power in the message signal (Pm). 1 mywbut.com 1.5 x (t) 1 z x (t) c 0.5 0 -0.5 -1 -1.5 0 5 10 15 20 Time Figure 5.1: Example waveforms for linear analog modulation for the message signal in Fig. 4.1. Gf() xc 2W − f f c f c Figure 5.2: The energy spectrum of the linearly modulated signal for the message signal spectrum in Fig. 4.2. 2 mywbut.com Example 5.1: Linear modulation with β2 β2 m(t)=β sin (2πf t) G (f)= δ(f − f )+ δ(f + f ) m m 4 m 4 m produces √ xc(t)=Acβ sin (2πfmt) 2 cos(2πfct) and A2β2 G (f)= c (δ(f − f − f )+δ(f + f − f )+δ(f − f + f )+δ(f + f + f )) . xc 8 m c m c m c m c The output power is A2β2 P = c xc 2 5.1.1 Modulator and Demodulator The modulator for a DSB-AM signal is simply the structure in Fig. 3.4 except with DSB-AM there is no imaginary part to the complex envelope. Fig. 5.3 shows the simplicity of this modulator. m(t) A c ()π 2 cos 2 f ct Figure 5.3: A DSB-AM modulator. Example 5.2: The computer generated voice signal given in Example 1.15(W =2.5KHz) is used to DSB- AM modulate a 7KHz carrier. A short time record of the message signal and the resulting modulated output signal is shown in Fig. 5.5-a). The energy spectrum of the signal is shown in Fig. 5.5-b). Note the bandwidth of the carrier modulated signal is 5KHz. Demodulation can be accomplished in a very simple configuration for DSB-AM. Given the channel model in Fig. 4.6 a straightforward demodulator is seen in Fig. 5.4. This demodulator simply derotates the received complex envelope by the phase induced by the propagation delay and uses the real part of this derotated signal as the estimate of the message signal. A lowpass filter is added to give noise immunity and the effects of this filter will be discussed later. Note output of the demodulator is given as mˆ (t)=Acm(t)+NI (t) so that ET = 100%. An advantage of DSB-AM is that all transmitted power is directly utilized at the output of the demod- ulator. 3 mywbut.com () () ytz Re[] • LPF mtˆ − φ exp[]j p Figure 5.4: The block diagram of an DSB-AM demodulator. Message signal and the modulated signal Energy spectrum of the DSB-AM signal 0.3 20 0.2 10 0 0.1 -10 0 -20 -0.1 -30 -0.2 -40 -0.3 -50 -0.4 -60 0.07 0.075 0.08 0.085 0.09 0.095 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 time, seconds Frequency, Hertz x 10 a) A short time record of the message signal, b) An energy spectrum of the DSB-AM signal. m(t), and the corresponding modulated signal. Figure 5.5: Example of DSB-AM with the message signal given in Example 1.15. fc=7KHz. 5.1.2 Coherent Demodulation An important function of a DSB-AM demodulator is producing the appropriate value of φp for good message reconstruction. Demodulators that require an accurate phase reference like DSB-AM requires are often call phase coherent demodulators. Often in practice this phase reference is obtained manually with a tunable phase shifter. This is unsatisfactory if one or both ends of the link are moving (hence a drifting phase) or if automatic operation is desired. Automatic phase tracking can be accomplished in a variety of ways. The techniques available for automatic phase tracking are easily divided into two sets of techniques: a phase reference derived from a transmitted reference and a phase reference derived from the received modulated signal. Note a transmitted reference technique will reduce ET since the transmitted power used in the reference signal is not available at the output of the demodulator. Though a transmitted reference signal is wasteful of transmitted power it is often necessary for more complex modulation schemes (e.g., see Section 5.3.3). For each of these above mentioned techniques two methodologies are typically followed in deriving a coherent phase reference; open loop estimation and closed loop or phase-locked estimation. Consequently four possible architectures are available for coherent demodulation in analog communications. An additional advantage of DSB-AM is that the coherent reference can easily be derived from the received modulated signal. Consequently in the remainder of this section the focus of the discussion will be on architectures that enable automatic phase tracking from the received modulated signal for DSB-AM. The block diagram of a typical open loop phase estimator for DSB-AM is shown in Fig. 5.6. The essential idea in open loop phase estimation for DSB-AM is that any channel induced phase rotation 4 mywbut.com ()= 22() φ + () VtzcAmt[] j2exp pV N t ()2 () {} φˆ Hfz arg • ÷2 p ()= () φ + () φˆ Ytzzxtexp[] jpz Nt 2 p Figure 5.6: An open loop phase estimator for DSB-AM. can easily be detected since DSB-AM only uses the real part of the complex envelope. Note that the received DSB-AM signal has the form yz(t)=xz(t) exp [jφp]=Acm(t) exp [jφp] . (5.4) The phase of yz(t) (in the absence of noise) will either take value of φp (when m(t) > 0) or φp + π (when m(t) < 0). Squaring the signal gets rid of this bi-modal phase characteristic as can be seen by examining the signal 2 2 2 vz(t)=(xz(t) exp [jφp]) = Ac m (t) exp [j2φp] (5.5) 2 2 because Ac m (t) > 0 so the arg (vz(t))=2φp. In Fig. 5.6 the filtering, Hz(f), is used to smooth the phase estimate in the presence of noise. Example 5.3: Consider the DSB-AM with the computer generated voice signal given in Example 5.2 with a carrier frequency of 7KHz and a propagation delay in the channel of 45.6µs. This results in a ◦ φp = −114 (see Example 5.0.) The vector diagram which is a plot of xI (t) versus xQ(t) is a useful tool for understanding the operation of the carrier phase recovery system. The vector diagram was first introduced in Problem 3.6. The vector diagram of the transmitted signal will be entirely on the x-axis since xQ(t)=0. The plot of the vector diagram for the channel output (in the absence of noise) is shown in Fig. 5.7-a). The −114◦ phase shift is evident from this vector diagram. The output vector diagram from the squaring device is shown in Fig. 5.7-b). This signal now has only one phase angle (2 ×−114◦) and the coherent phase reference for demodulation can now be easily obtained. A phase-locked type of phase tracking can be implemented in a DSB-AM demodulator with a Costas loop which has a block diagram given in Fig.