Draft. Amplitude Modulation and Demodulation

Transcription

1 Chapter 8 Amplitude Modulation and Demodulation This chapter is dedicated to Amplitude Modulation (AM) and related aspects. It is the easiest form of modulation to grasp. 8. Amplitude Modulation AM results from the product of two signals: a carrier signal at frequency f Hz and a modulating signal at frequency g Hz, with f much greater than g. The following equation models AM s(t) = [ + sin(πgt)] sin(πft) Volts. (8.) The term sin(πgt) denotes the modulating signal. The term sin(πf t) is an abstraction of the carrier signal. The unit term represents a DC component added to the modulating signal to prevent loss of information. An example is shown in Figure 8.. The top subplot shows a carrier at frequency f = Hz and a modulating signal at frequency g = Hz. The bottom subplot is the corresponding AM signal. The bottom subplot also shows the envelope of the AM signal. The envelope is obtained by joining together with a line either the peaks of the positive half cycles or the troughs of the negative half cycles. The frequency of the signal within the envelope corresponds to the frequency of the carrier. The frequency of the envelope corresponds to the frequency of the modulating signal. The plots have been generated by the following Octave code: % Create an array of time instants, in steps of. s. time=[:999]*.; 3 % Create a carrier signal at frequency f 4 f=; % Hz 5 carrier=sin(*pi*f*time); 5

2 5 Michel Barbeau Software Defined Radio Carrier and Modulating Signal ( ) AM Signal and Envelope (:) Time (s) Figure 8.: Amplitude modulation. 6 % Create a modulating signal at frequency g 7 g=; % Hz 8 modulation=+sin(*pi*g*time); 9 % Modulation depth m=; % % modulation % Multiply the modulating signal by the carrier signal=m*modulation.*carrier; 3 % Plot the carrier 4 subplot(,,); 5 plot(time,carrier); 6 hold on; 7 % Plot the modulating signal 8 plot(time,modulation,'.r'); 9 % Annotations title('carrier and Modulating Signal ( )'); ylabel(''); grid on; 3 % Plot the AM signal 4 subplot(,,); 5 plot(time,signal); 6 hold on; 6CHAPTER 8. AMPLITUDE MODULATION AND DEMODULATION

3 Software Defined Radio 7 % Plot the envelope 8 plot(time,modulation,':b'); 9 % Annotationse 3 title('am Signal and Envelope (:)'); 3 xlabel('time (s)'); 3 ylabel(''); 33 grid on; 5 Michel Barbeau The carrier signal at frequency f is generated at line 5. The modulating signal at frequency g is produced at line 8. On line, the factor m is the modulation depth. It represents a percentage of modulation. % modulation is done in this example. The degree of variation of the envelope (minimum voltage versus maximum voltage) is determined by the modulation depth. Line implements the Equation 8. of AM. 8. AM Using the I and Q Signals I, Q( ) and Modulating Signal (.) I and Q ( ) AM Signals and Envelope (:) Time (s) Figure 8.: AM using the I and Q signals. Figure 8. shows an example where an in-phase carrier and its quadrature carrier are modulated in amplitude. The carrier frequency is Hz and the modulating signal frequency is Hz. The plots have been generated by the following Octave script: CHAPTER 8. AMPLITUDE MODULATION AND DEMODULATION7

4 5 Michel Barbeau Software Defined Radio % Create an array of time instants, in steps of. s. time=[:999]*.; 3 % Create a carrier signal at frequency f 4 f=; % Hz 5 % Create an in phase carrier 6 Icarrier=cos(*pi*f*time); 7 % Create a quadrature carrier 8 Qcarrier = sin(*pi*f*time); 9 % Create a modulating signal at frequency g g=; % Hz modulation=+sin(*pi*g*time); % Modulation depth 3 m=; % % modulation 4 % Multiply the modulation by the carrier 5 Isignal=m*modulation.*Icarrier; 6 Qsignal=m*modulation.*Qcarrier; 7 % Plot the carriers and modulating signal 8 subplot(,,); 9 plot(time,icarrier); hold on; plot(time,qcarrier,' k'); hold on; 3 plot(time,modulation,'.r'); 4 % Annotations 5 title('i, Q( ) and Modulating Signal (.)'); 6 ylabel(''); 7 grid on; 8 % Plot the AM signals 9 subplot(,,); 3 plot(time,isignal); 3 hold on; 3 plot(time,qsignal,' k'); 33 hold on; 34 % Plot the envelope 35 plot(time,modulation,':b'); 36 % Annotations 37 title('i and Q ( ) AM Signals and Envelope (:)'); 38 xlabel('time (s)'); 39 ylabel(''); 4 grid on; The in-phase and quadrature, both at frequency f, are constructed on lines 6 and 8. The modulating signal, of frequency g, is created on line. It is used to modulate the amplitude of both the in-phase and quadrature on lines 5 and 6, in accordance to Equation 8.. Note that both the in-phase AM signal and quadrature AM signal share the same envelope. In other words, they carry exactly the same information. On the air, it is only necessary to send one of them. But inside a receiver, it is convenient to have both of them because demodulation is a matter of simple calculation. 8CHAPTER 8. AMPLITUDE MODULATION AND DEMODULATION

5 Software Defined Radio 8.3 AM Demodulation 5 Michel Barbeau Given an in-phase AM signal I and quadrature Q, the modulating signal is recovered by applying Pythagora s theorem on the I and Q signals. The equation for the demodulation of an amplitude-modulated signal is I + Q. (8.) Using the variables Isignal and Qsignal, the following Octave script imple Amplitude Demodulation Time (s) ments amplitude demodulation: % Demodulation Figure 8.3: AM demodulation. demodulation = sqrt((isignal).ˆ + (Qsignal).ˆ); 3 % Plot the signal 4 plot(time,demodulation); 5 % Annotations 6 title('amplitude Demodulation'); 7 xlabel('time (s)'); 8 ylabel(''); Demodulation is implemented using the variables Isignal and Qsignal of the previous Octave script. Equation 8. of AM demodulation is implemented on line. CHAPTER 8. AMPLITUDE MODULATION AND DEMODULATION9

6 5 Michel Barbeau Software Defined Radio 8.4 M-ary Quadrature Amplitude Modulation Q Figure 8.4: 6-QAM constellation. M-ary Quadrature Amplitude Modulation (M-QAM) is used for data communications. An in-phase carrier and its quadrature are modulated in amplitude to encode bits. There are M different symbols. For each signal, i.e., the in-phase and quadrature, there are M amplitudes. An example, i.e., 6-QAM, is pictured in Figure 8.4. M is 6 and M is four. Each circle represents a symbol. The horizontal axis corresponds to the amplitude of the in-phase carrier. The vertical axis is the amplitude of the quadrature carrier. In this examples log M bits can be represented per symbol, i.e., four bits. For example, when the symbol is sent, the amplitude of both the in-phase I and quadrature Q is one. On the receiver side, when the amplitudes of the in-phase and quadrate signals both fall within the range.5 to.5 Volts the corresponding decoded symbol is. To model the signal sent over the air, we just have to leverage Equation 3.. Let f be the frequency of the carrier, in Hertz. For a given symbol, let I and CHAPTER 8. AMPLITUDE MODULATION AND DEMODULATION I

7 Software Defined Radio 5 Michel Barbeau Q be the corresponding required amplitudes of the in-phase and quadrature signals, in Volts. For that symbol, the value of the modulated signal at time t is s(t) = I cos(πft) + Q sin(πft) Volts. (8.3) Demodulation of a QAM signal implies separating the in-phase and quadrature from the modulated signal. Let us see how the in-phase is recovered. Let r(t) be a received QAM signal at frequency f Hertz. The received signal r(t) is multiplied by a cosine signal at the same frequency r(t) cos(πft). Assuming that r(t) is equal to s(t), this product can be expanded as which is equal to [I cos(πft) + Q sin(πft)] cos(πft). [I cos(πft) cos(πft)] + [Q sin(πft) cos(πft)]. Using the two trigonometric equivalences cos ω = (+cos ω)/ and sin ω cos ω = (sin ω)/, it can be rewritten as which is equal to I/ [ + cos(4πft)] + Q/ sin(4πft). I/ + I/ cos(4πft) + Q/ sin(4πft). A low pass filter is used to eliminate the second and fourth terms at frequency f Hertz. The first term is multiplied by two to yield the value I. Exercise 8. Starting from a received QAM signal r(t), develop the equation that selects the quadrature signal. CHAPTER 8. AMPLITUDE MODULATION AND DEMODULATION