<< Chapter < Page Chapter >> Page >

{} {} If the signal is transmitted at a carrier frequency f c = 600 Mhz size 12{f rSub { size 8{c} } ="600" ital "Mhz"} {} , then the λ 3 × 10 8 m/s 6 × 10 8 Hz 0 . 5 m size 12{λ approx { {3 times "10" rSup { size 8{8} } " m/s"} over {6 times "10" rSup { size 8{8} } ` ital "Hz"} } approx 0 "." 5`m} {} so that an antenna whose length is λ / 10 5 size 12{λ/"10" approx 5} {} cm which is much more manageable for the pigeon.

5/ Narrow-band signals

The modulated transduced pigeon sound has a spectrum that is centered about the carrier frequency f c size 12{f rSub { size 8{c} } } {} and has a bandwidth of 2f m size 12{2f rSub { size 8{m} } } {} where f m size 12{f rSub { size 8{m} } } {} is the maximum frequency of the pigeon sound.

For the pigeon sound we have f m size 12{f rSub { size 8{m} } } {} = 3 kHz and f c size 12{f rSub { size 8{c} } } {} = 600 MHz. Thus, the bandwidth is only 10 5 size 12{"10" rSup { size 8{ - 5} } } {} of the carrier frequency — an example of a narrowband signal.

An arbitrary narrowband signal can be expressed as

x ( t ) = x c ( t ) cos ( 2πf c t ) + x s ( t ) sin ( 2πf c t ) size 12{x \( t \) =x rSub { size 8{c} } \( t \) "cos" \( 2πf rSub { size 8{c} } t \) +x rSub { size 8{s} } \( t \) "sin" \( 2πf rSub { size 8{c} } t \) } {}

where x c ( t ) size 12{x rSub { size 8{c} } \( t \) } {} and x s ( t ) size 12{x rSub { size 8{s} } \( t \) } {} are lowpass time functions. We can expand x(t) as follows

x ( t ) = 1 2 ( x c ( t ) + 1 j x s ( t ) ) e j2πf c t + 1 2 ( x c ( t ) 1 j x s ( t ) ) e j2πf c t , x ( t ) = R { ( x c ( t ) + 1 j x s ( t ) ) e j2πf c t } , x ( t ) = a ( t ) cos ( 2πf c t + ϕ ( t ) ) , alignl { stack { size 12{x \( t \) = { {1} over {2} } \( x rSub { size 8{c} } \( t \) + { {1} over {j} } x rSub { size 8{s} } \( t \) \) e rSup { size 8{j2πf rSub { size 6{c} } t} } + { {1} over {2} } \( x rSub {c} size 12{ \( t \) - { {1} over {j} } x rSub {s} } size 12{ \( t \) \) e rSup { - j2πf rSub { size 6{c} } t} } size 12{,}} {} #size 12{x \( t \) =R lbrace \( x rSub { size 8{c} } \( t \) + { {1} over {j} } x rSub { size 8{s} } \( t \) \) e rSup { size 8{j2πf rSub { size 6{c} } t} } rbrace ,} {} # size 12{x \( t \) =a \( t \) "cos" \( 2πf rSub { size 8{c} } t+ϕ \( t \) \) ,} {}} } {}

Where

a ( t ) = x c 2 ( t ) + x s 2 ( t ) and ϕ ( t ) = tan 1 x s ( t ) x c ( t ) . size 12{a \( t \) = sqrt {x rSub { size 8{c} } rSup { size 8{2} } \( t \) +x rSub { size 8{s} } rSup { size 8{2} } \( t \) } ~ matrix { {} # {}} ital "and"~ matrix { {} # {}} ϕ \( t \) = - "tan" rSup { size 8{ - 1} } { {x rSub { size 8{s} } \( t \) } over {x rSub { size 8{c} } \( t \) } } "." } {}

An arbitrary narrowband signal can be written as

x ( t ) = a ( t ) Cos ( 2πf c t + ϕ ( t ) ) size 12{x \( t \) =a \( t \) ` ital "Cos"` \( 2πf rSub { size 8{c} } t+ϕ \( t \) \) } {}

Thus, a general narrowband signal contains both amplitude and phase/frequency modulation. In amplitude modulation (AM) ϕ ( t ) size 12{ϕ \( t \) } {} is constant; in phase/frequency modulation (PM or FM) a(t) is constant.

II. AMPLITUDE MODULATION

1/ AM, suppressed carrier

Perhaps the simplest amplitude modulation scheme is the suppressed carrier scheme in which

x ( t ) = x m ( t ) × cos ( 2πf c t ) size 12{x \( t \) =x rSub { size 8{m} } \( t \) times "cos" \( 2πf rSub { size 8{c} } t \) } {}

Therefore, the Fourier transform is

x ( f ) = x m ( f ) F { cos ( 2πf c t ) } , x ( f ) = x m ( f ) 1 2 ( δ ( f f c ) + δ ( f + f c ) ) , x ( f ) = 1 2 ( x m ( f f c ) + x m ( f + f c ) ) . alignl { stack { size 12{x \( f \) =x rSub { size 8{m} } \( f \) *F lbrace "cos" \( 2πf rSub { size 8{c} } t \) rbrace ,} {} #x \( f \) =x rSub { size 8{m} } \( f \) * { {1} over {2} } \( δ \( f - f rSub { size 8{c} } \) +δ \( f+f rSub { size 8{c} } \) \) , {} # x \( f \) = { {1} over {2} } \( x rSub { size 8{m} } \( f - f rSub { size 8{c} } \) +x rSub { size 8{m} } \( f+f rSub { size 8{c} } \) \) "." {}} } {}

The Fourier transform of the modulated signal x(t) can be obtained graphically.

The Fourier transform X m ( f ) size 12{X rSub { size 8{m} } \( f \) } {} is repeated at ± f c size 12{ +- f rSub { size 8{c} } } {} .

2/ Demodulation (detection) of AM, suppressed carrier

The original signal x m ( t ) size 12{x rSub { size 8{m} } \( t \) } {} can be recovered by modulating the modulated signal and passing the result through a lowpass filter, a process called demodulation or detection.

3/ AM, suppressed carrier radio

A radio communication system that consists of a transmitter and receiver and which uses suppressed carrier AM is shown below.

Therefore,

x ( t ) = x m ( t ) cos ( 2πf c t ) , y ( t ) = x ( t ) cos ( 2πf c t ) , and y ( t ) = x m ( t ) cos 2 ( 2πf c t ) = x m ( t ) 2 ( 1 + cos ( 4πf c t ) ) . alignl { stack { size 12{x \( t \) =x rSub { size 8{m} } \( t \) "cos" \( 2πf rSub { size 8{c} } t \) ~,~y \( t \) =x \( t \) "cos" \( 2πf rSub { size 8{c} } t \) ,~ ital "and"} {} #y \( t \) =x rSub { size 8{m} } \( t \) "cos" rSup { size 8{2} } \( 2πf rSub { size 8{c} } t \) = { {x rSub { size 8{m} } \( t \) } over {2} } \( 1+"cos" \( 4πf rSub { size 8{c} } t \) \) "." {} } } {}

Hence,

Y ( f ) = 1 2 X m ( f ) + 1 4 X m ( f 2f c ) + 1 4 X m ( f + 2f c ) . size 12{Y \( f \) = { {1} over {2} } X rSub { size 8{m} } \( f \) + { {1} over {4} } X rSub { size 8{m} } \( f - 2f rSub { size 8{c} } \) + { {1} over {4} } X rSub { size 8{m} } \( f+2f rSub { size 8{c} } \) "." } {}

The spectrum of y(t) involves the spectrum of cos 2 ( 2πf c t ) size 12{"cos" rSup { size 8{2} } \( 2πf rSub { size 8{c} } t \) } {} which can be found by the trigonometric identity or as shown below.

y ( t ) = x m ( t ) cos 2 ( 2πf c t ) = x m ( t ) × cos ( 2πf c t ) × cos ( 2πf c t ) size 12{y \( t \) =x rSub { size 8{m} } \( t \) "cos" rSup { size 8{2} } \( 2πf rSub { size 8{c} } t \) =x rSub { size 8{m} } \( t \) times "cos" \( 2πf rSub { size 8{c} } t \) times "cos" \( 2πf rSub { size 8{c} } t \) } {}

can be written as

Y ( f ) = X m ( f ) F { cos ( 2πf c t ) } F { cos ( 2πf c t ) } size 12{Y \( f \) =X rSub { size 8{m} } \( f \) *F lbrace "cos" \( 2πf rSub { size 8{c} } t \) rbrace *F lbrace "cos" \( 2πf rSub { size 8{c} } t \) rbrace } {}

The Fourier transform of F { cos ( 2πf c t ) } F { cos ( 2πf c t ) } size 12{F lbrace "cos" \( 2πf rSub { size 8{c} } t \) rbrace *F lbrace "cos" \( 2πf rSub { size 8{c} } t \) rbrace } {} is shown below.

Thus, using Fourier transform properties it is easy to derive trigonometric identities.

Two-minute miniquiz problem

Problem 23-1 — AM, suppressed carrier radio

A slight alternative to the AM suppressed carrier radio system is shown below.

Using an appropriate low pass filter, H(f), to detect X m ( f ) size 12{X rSub { size 8{m} } \( f \) } {} , determine the spectrum Y m ( f ) size 12{Y rSub { size 8{m} } \( f \) } {} .

Solution

We need to convolve the spectrum of the modulated function X(f) with the Fourier transform of Sin ( 2πf c t ) size 12{ ital "Sin" \( 2πf rSub { size 8{c} } t \) } {} .

Thus, the output is zero.

The following reviews the results for suppressed carrier radio.

Suppressed carrier AM requires that the transmitter and receiver be perfectly synchronized. A small difference in frequency of transmitter and receiver oscillators results in a drift in the phase difference between the oscillators which causes variations in the amplitude of the detected signal, called signal strength fading.

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Signals and systems. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10803/1.1
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Signals and systems' conversation and receive update notifications?

Ask