LCS_7_notes - 1 LINEAR CIRCUITS AND SIGNALS Modulation...

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Unformatted text preview: 1 LINEAR CIRCUITS AND SIGNALS Modulation Department of Electrical and Computer Engineering University of Miami I. PRELIMINARIES In amplitude modulated (AM) radio, an audio signal f ( t ) is used to modulate the amplitude of a carrier signal cos[ ω c t ] . This generates the AM signal f ( t ) cos[ ω c t ] . The signal that is being modulated (i.e., cos[ ω c t ] ) is the carrier signal. Typically, the bandwidth of the modulating signal (i.e., f ( t ) ) is much smaller than the carrier frequency ω c . In this chapter, we will use FT to study AM radio. II. RELEVANT FT PROPERTIES A. Time Shift Claim 1: f ( t ) ↔ F ( ω ) = ⇒ f ( t- t o ) ↔ F ( ω ) e- jωt o . Proof: Note that FT [ f ( t- t o )] = Z t f ( t- t o ) e- jωt dt = Z τ f ( τ ) e- jω ( t o + τ ) dτ = e- jωt o Z τ f ( τ ) e- jωτ dτ = F ( ω ) e- jωt o . B. Frequency Shift Claim 2: f ( t ) ↔ F ( ω ) = ⇒ f ( t ) e jω o t ↔ F ( ω- ω o ) . 2 Proof: Note that FT [ f ( t ) e jω o t ] = Z t f ( t ) e jω o t e- jωt dt = Z t f ( t ) e- j ( ω- ω o ) t dt = F ( ω- ω o ) . C. Modulation Claim 3: f ( t ) ↔ F ( ω ) = ⇒ f ( t ) cos[ ω c t ] ↔ 1 2 [ F ( ω- ω c ) + F ( ω + ω c )] Proof: This follows by noting that cos[ ω c t ] = 1 2 e jω c t + e- jω c t , and then applying the frequency shift property. The implications of this modulation property appears in Figure 1. There are mainly two reasons for choosing to transmit AM signals instead of the audio signal directly: • For satisfactory performance, antennas must transmit signals at a high frequency; at lower frequencies, they have negligible amplitude response. • By choosing different carrier frequencies, signals can be transmitted and received with little or no interference. Figure 2 shows a basic AM transmitter. The multiplier unit is called the mixer; the mixing operation is also referred to as heterodyning. Example Consider the mixer output m ( t ) = f ( t ) " 1 + ∞ X n =1 1 n cos[ nω o t ] # ↔ M ( ω ) = F ( ω ) + ∞ X n =1 1 2 n [ F ( ω- nω o ) + F ( ω + nω o )] . 3 F( ω ) 1 0.5 ω ω- ω c + ω c (a) FT of f ( t ) cos( ω c t ) where f ( t ) is low-pass. F( ω ) 1 0.5 ω ω + ω o- ω c- ω o + ω o + ω o + ω c- ω o + ω c- ω o- ω c (b) FT of f ( t ) cos( ω c t ) where f ( t ) is band-pass. Fig. 1. FT of AM signals. III. COHERENT DEMODULATION OF AM SIGNALS Consider Figure 3 which represents a typical AM transmitter, a propagation channel, and a coherent demodulator as the AM receiver. Note that the propagation channel has been modeled as ideal in the sense that it only delays (by t o s ) and scales (by k units) the signal: H c ( ω ) = k e- jωt o . (1) 4 cos[ ω c t] x f(t) f(t).cos[ ω c t] Fig. 2. A basic AM transmitter....
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This note was uploaded on 10/28/2010 for the course EEN 307 taught by Professor Kamalpremeratne during the Spring '10 term at University of Miami.

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LCS_7_notes - 1 LINEAR CIRCUITS AND SIGNALS Modulation...

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