{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

LCS_7_notes

# LCS_7_notes - 1 LINEAR CIRCUITS AND SIGNALS Modulation...

This preview shows pages 1–5. Sign up to view the full content.

1 LINEAR CIRCUITS AND SIGNALS Modulation Department of Electrical and Computer Engineering University of Miami I. P RELIMINARIES 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. R ELEVANT FT P ROPERTIES 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 - ( t o + τ ) = e - jωt o Z τ f ( τ ) e - jωτ = F ( ω ) e - jωt o . B. Frequency Shift Claim 2: f ( t ) F ( ω ) = f ( t ) e o t F ( ω - ω o ) .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
2 Proof: Note that FT [ f ( t ) e o t ] = Z t f ( t ) e 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 c t + e - 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[ o t ] # M ( ω ) = F ( ω ) + X n =1 1 2 n [ F ( ω - o ) + F ( ω + o )] .
3 F( ω ) 1 0.5 ω ω - ω c + ω c 0 0 (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 0 0 - 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. C OHERENT D EMODULATION OF AM S IGNALS 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)

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
4 cos[ ω c t] x f(t) f(t).cos[ ω c t] Fig. 2. A basic AM transmitter. cos[ ω c t] x f(t) cos[ ω c (t-t o )] H c ( ω ) x H LPF ( ω ) m T (t) m R (t) r(t) y(t) Tx Rx Fig. 3. A typical AM transmitter-receiver communication system. We then have the following: m T ( t ) = f ( t ) cos[ ω c t ]; r ( t ) = k m T ( t - t o ); = k f ( t - t o ) cos[ ω c ( t - t o )]; m R ( t ) = r ( t ) cos[ ω c ( t - t o )] = k f ( t - t o ) cos 2 [ ω c ( t - t o )] = k 2 f ( t - t o ) [1 + cos[2 ω c ( t - t o )] . (2) Therefore M R ( ω ) = k 2 e - jωt o F ( ω ) + 1 2 [ F ( ω - 2 ω c ) + F ( ω + 2 ω c )] . (3) The recovery of the desired signal f ( t )
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 12

LCS_7_notes - 1 LINEAR CIRCUITS AND SIGNALS Modulation...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online