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"Digital-Communications Analog-Communication Noise Complex-Baseband Discrete-Time Error-Analysis"

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# Analog Communication

Module by: Phil Schniter. E-mail the author

Summary: This module describes basic analog modulation techniques, including amplitude modulation (AM) with suppressed carrier, AM with a pilot tone or carrier tone, quadrature AM (QAM), vestigial sideband modulation (VSB), and frequency modulation (FM). Various demodulation techniques are also discussed, including envelope detection and the discriminator. Application examples include NTSC television and FM radio (both mono and stereo).

1. Amplitude modulation (AM)
3. Vestigial sideband modulation (VSB)
4. Frequency modulation (FM)

## AM with “suppressed carrier”

AM of real-valued message m(t)m(t) (e.g., music) is

Euler's cos(2πfct)=12ej2πfct+e-j2πfctcos(2πfct)=12ej2πfct+e-j2πfct then implies

S ( f ) = - m ( t ) cos ( 2 π f c t ) e - j 2 π f t d t = 1 2 - m ( t ) e - j 2 π ( f - f c ) t d t + 1 2 - m ( t ) e - j 2 π ( f + f c ) t d t = 1 2 M ( f - f c ) + 1 2 M ( f + f c ) . S ( f ) = - m ( t ) cos ( 2 π f c t ) e - j 2 π f t d t = 1 2 - m ( t ) e - j 2 π ( f - f c ) t d t + 1 2 - m ( t ) e - j 2 π ( f + f c ) t d t = 1 2 M ( f - f c ) + 1 2 M ( f + f c ) .
(1)

Because m(t)Rm(t)R, know |M(f)||M(f)| symmetric around f=0f=0, implying the AM transmitted spectrum below fc is redundant! This motivates the QAM and VSB modulation schemes...

With fc known, AM demodulation can be accomplished by:

For a trivial noiseless channel, we have r(t)=s(t)r(t)=s(t), so that

v ( t ) = LPF { s ( t ) · 2 cos ( 2 π f c t ) } = LPF { m ( t ) · 2 cos 2 ( 2 π f c t ) 1 + cos ( 2 π · 2 f c t ) } = LPF { m ( t ) + m ( t ) cos ( 2 π · 2 f c t ) } = m ( t ) , v ( t ) = LPF { s ( t ) · 2 cos ( 2 π f c t ) } = LPF { m ( t ) · 2 cos 2 ( 2 π f c t ) 1 + cos ( 2 π · 2 f c t ) } = LPF { m ( t ) + m ( t ) cos ( 2 π · 2 f c t ) } = m ( t ) ,
(2)

assuming a LPF with passband cutoff BpWBpW Hz and stopband cutoff Bs2fc-WBs2fc-W Hz:

Note that we've assumed perfectly synchronized oscillators!

When the receiver oscillator has {freq,phase} offset {γ,φ}{γ,φ}:

v ( t ) = LPF m ( t ) cos ( 2 π f c t ) · 2 cos ( 2 π ( f c + γ ) t + φ ) cos ( 2 π γ t + φ ) + cos ( 2 π ( 2 f c + γ ) t + φ ) = m ( t ) cos ( 2 π γ t + φ ) time-varying attenuation! . v ( t ) = LPF m ( t ) cos ( 2 π f c t ) · 2 cos ( 2 π ( f c + γ ) t + φ ) cos ( 2 π γ t + φ ) + cos ( 2 π ( 2 f c + γ ) t + φ ) = m ( t ) cos ( 2 π γ t + φ ) time-varying attenuation! .
(3)

### Note:

a freq offset of λ=νfccλ=νfcc Hz can occur when there is relative velocity of ν m/s between transmitter and receiver.

## AM with “pilot tone” or “carrier tone”

It's common to include a pilot/carrier tone with frequency| fc:

s ( t ) = m ( t ) cos ( 2 π f c t ) + A cos ( 2 π f c t ) pilot/carrier tone = [ m ( t ) + A ] cos ( 2 π f c t ) S ( f ) = 1 2 M ( f - f c ) + M ( f + f c ) + A δ ( f - f c ) + A δ ( f + f c ) s ( t ) = m ( t ) cos ( 2 π f c t ) + A cos ( 2 π f c t ) pilot/carrier tone = [ m ( t ) + A ] cos ( 2 π f c t ) S ( f ) = 1 2 M ( f - f c ) + M ( f + f c ) + A δ ( f - f c ) + A δ ( f + f c )
(4)

While modern systems choose Amax|m(t)|Amax|m(t)|, many older systems use A>max|m(t)|A>max|m(t)|, known as “large carrier AM,” allowing reception based on envelope detection:

v ( t ) = π 2 LPF { | r ( t ) | } - A m ( t ) (with a trivial channel) v ( t ) = π 2 LPF { | r ( t ) | } - A m ( t ) (with a trivial channel)
(5)

where |·||·| can be easily implemented using a diode.

The gain π2π2 above makes up for the loss incurred when LPFing the rectified signal:

QAM is motivated by unwanted redundancy in the AM spectrum, which was symmetric around fc.

QAM sends two real-valued signals {mI(t),mQ(t)}{mI(t),mQ(t)} simultaneously, resulting in a non-symmetric spectrum.

QAM demodulation is accomplished by:

where the LPF specs are the same as in AM, i.e., passband edge BpWBpW Hz and stopband edge Bs2fc-WBs2fc-W Hz.

For a trivial channel, we have r(t)=s(t)r(t)=s(t), so that

v I ( t ) = LPF { r ( t ) · 2 cos ( 2 π f c t ) } = LPF { m I ( t ) 2 cos 2 ( 2 π f c t ) 1 + cos ( 4 π f c t ) - m Q ( t ) 2 sin ( 2 π f c t ) cos ( 2 π f c t ) sin ( 4 π f c t ) } = m I ( t ) v Q ( t ) = LPF { - r ( t ) · 2 sin ( 2 π f c t ) } = LPF { - m I ( t ) 2 cos ( 2 π f c t ) sin ( 2 π f c t ) sin ( 4 π f c t ) + m Q ( t ) 2 sin 2 ( 2 π f c t ) 1 - cos ( 4 π f c t ) } = m Q ( t ) , v I ( t ) = LPF { r ( t ) · 2 cos ( 2 π f c t ) } = LPF { m I ( t ) 2 cos 2 ( 2 π f c t ) 1 + cos ( 4 π f c t ) - m Q ( t ) 2 sin ( 2 π f c t ) cos ( 2 π f c t ) sin ( 4 π f c t ) } = m I ( t ) v Q ( t ) = LPF { - r ( t ) · 2 sin ( 2 π f c t ) } = LPF { - m I ( t ) 2 cos ( 2 π f c t ) sin ( 2 π f c t ) sin ( 4 π f c t ) + m Q ( t ) 2 sin 2 ( 2 π f c t ) 1 - cos ( 4 π f c t ) } = m Q ( t ) ,
(6)

assuming synchronized oscillators.

When the oscillators are not synchronized, one gets coupling between the I&Q components as well as attenuation of each. Writing the I&Q signals in the “complex-baseband” form

m ˜ ( t ) = m I ( t ) + j m Q ( t ) v ˜ ( t ) = v I ( t ) + j v Q ( t ) m ˜ ( t ) = m I ( t ) + j m Q ( t ) v ˜ ( t ) = v I ( t ) + j v Q ( t )
(7)

yields a much simpler description of QAM:

### Note:

Re{u(t)}=12u(t)+u*(t)F12U(f)+U*(-f)Re{u(t)}=12u(t)+u*(t)F12U(f)+U*(-f).

We now verify the complex-baseband model for modulation:

Re { m ˜ ( t ) e j 2 π f c t } = Re m I ( t ) + j m Q ( t ) cos ( 2 π f c t ) + j sin ( 2 π f c t ) = m I ( t ) cos ( 2 π f c t ) - m Q ( t ) sin ( 2 π f c t ) = s ( t ) , Re { m ˜ ( t ) e j 2 π f c t } = Re m I ( t ) + j m Q ( t ) cos ( 2 π f c t ) + j sin ( 2 π f c t ) = m I ( t ) cos ( 2 π f c t ) - m Q ( t ) sin ( 2 π f c t ) = s ( t ) ,
(8)

as well as for demodulation (assuming r(t)=s(t)r(t)=s(t)):

v ˜ ( t ) = LPF { s ( t ) · 2 e - j 2 π f c t } = LPF { m I ( t ) cos ( 2 π f c t ) - m Q ( t ) sin ( 2 π f c t ) · 2 e - j 2 π f c t } = LPF { m I ( t ) e j 2 π f c t + e - j 2 π f c t e - j 2 π f c t - m Q ( t ) j e - j 2 π f c t - j e j 2 π f c t e - j 2 π f c t } = LPF m I ( t ) 1 + e - j 4 π f c t - m Q ( t ) j e - j 4 π f c t - j = m I ( t ) + j m Q ( t ) . v ˜ ( t ) = LPF { s ( t ) · 2 e - j 2 π f c t } = LPF { m I ( t ) cos ( 2 π f c t ) - m Q ( t ) sin ( 2 π f c t ) · 2 e - j 2 π f c t } = LPF { m I ( t ) e j 2 π f c t + e - j 2 π f c t e - j 2 π f c t - m Q ( t ) j e - j 2 π f c t - j e j 2 π f c t e - j 2 π f c t } = LPF m I ( t ) 1 + e - j 4 π f c t - m Q ( t ) j e - j 4 π f c t - j = m I ( t ) + j m Q ( t ) .
(9)

The convenience of complex-baseband results in widespread use of complex-valued signals for comm systems!

### Note:

To get the complex baseband formulation for AM, we simply set mQ(t)=0mQ(t)=0 and mI(t)=m(t)mI(t)=m(t).

## Vestigial Sideband Modulation (VSB)

VSB is another way to restore regain the spectral efficiency lost in AM. It's used to transmit North American terrestrial TV, both analog (NTSC) and digital (ATSC) formats.

Like AM, it can operate with or without a carrier tone.

Basically, VSB suppresses most of the redundant AM spectrum by filtering it:

The passband VSB filter is a BPF C(f)C(f) where

C ( f - f c ) + C ( f + f c ) = 2 for | f | W , C ( f - f c ) + C ( f + f c ) = 2 for | f | W ,
(10)

which implies its inside rolloff is symmetric around f=fcf=fc:

For VSB modulation, we have

s ( t ) = m ( t ) cos ( 2 π f c t ) * c ( t ) S ( f ) = 1 2 M ( f + f c ) + M ( f - f c ) C ( f ) . s ( t ) = m ( t ) cos ( 2 π f c t ) * c ( t ) S ( f ) = 1 2 M ( f + f c ) + M ( f - f c ) C ( f ) .
(11)

It turns out that VSB demod is identical to AM demod:

v ( t ) = LPF r ( t ) · 2 cos ( 2 π f c t ) = LPF s ( t ) · 2 cos ( 2 π f c t ) (trivial channel) V ( f ) = LPF S ( f - f c ) + S ( f + f c ) = 1 2 LPF { M ( f ) + M ( f - 2 f c ) C ( f - f c ) + M ( f + 2 f c ) + M ( f ) C ( f + f c ) } = M ( f ) 1 2 C ( f - f c ) + C ( f + f c ) = 1 for f [ - W , W ] = M ( f ) . v ( t ) = LPF r ( t ) · 2 cos ( 2 π f c t ) = LPF s ( t ) · 2 cos ( 2 π f c t ) (trivial channel) V ( f ) = LPF S ( f - f c ) + S ( f + f c ) = 1 2 LPF { M ( f ) + M ( f - 2 f c ) C ( f - f c ) + M ( f + 2 f c ) + M ( f ) C ( f + f c ) } = M ( f ) 1 2 C ( f - f c ) + C ( f + f c ) = 1 for f [ - W , W ] = M ( f ) .
(12)

We note that the property

F cos ( 2 π f c t ) c ( t ) = 1 2 C ( f - f c ) + C ( f + f c ) F cos ( 2 π f c t ) c ( t ) = 1 2 C ( f - f c ) + C ( f + f c )
(13)

may be convenient, e.g., for testing whether a given filter c(t)c(t) satisfies the passband VSB criterion. VSB ﬁltering can also be implemented at baseband using a complex-valued ﬁlter response c~(t)c~(t) which satisﬁes

C ˜ ( f ) + C ˜ * ( - f ) = 2 for | f | W , C ˜ ( f ) + C ˜ * ( - f ) = 2 for | f | W ,
(14)

generating the complex-baseband message signal m˜(t)m˜(t). The message can be recovered by simplying ignoring the imaginary part of the complex-baseband output v˜(t)v˜(t).

Motivation: filtering at baseband is usually much cheaper than filtering at passband.

## Frequency Modulation (FM)

While AM modulated the carrier amplitude, FM modulates the carrier frequency.

In particular, FM modulates the real-valued message m(t)m(t) via

s ( t ) = cos 2 π f c t + 2 π k f 0 t m ( τ ) d τ ϕ ( t ) "instantaneous modulation phase" . s ( t ) = cos 2 π f c t + 2 π k f 0 t m ( τ ) d τ ϕ ( t ) "instantaneous modulation phase" .
(15)

where kfkf is called the “frequency-sensitivity factor.” Since the instantaneous modulation frequency

d ϕ ( t ) d t = 2 π k f m ( t ) d ϕ ( t ) d t = 2 π k f m ( t )
(16)

is a scaled version of the message m(t)m(t), it is fitting to call this scheme “frequency modulation.”

Using the peak frequency deviation Δf=kfmax|m(t)|Δf=kfmax|m(t)|, the “modulation index” D is defined as

D = Δ f W denominator is the single-sided BW of m ( t ) . D = Δ f W denominator is the single-sided BW of m ( t ) .
(17)

Increasing D decreases spectral efficiency but increases robustness to noise/interference.

D 1 : narrowband FM'' , D 1 : wideband FM'' . D 1 : narrowband FM'' , D 1 : wideband FM'' .
(18)

Carson's Rule approximates the FM passband signal-BW as

BW 99 2 ( Δ f + W ) = 2 ( D + 1 ) W . BW 99 2 ( Δ f + W ) = 2 ( D + 1 ) W .
(19)

• Message signal filtered to freq interval [30,15k] Hz.
• FCC limits Δf75Δf75 kHz (channels 200 kHz apart). D=7515=5D=7515=5

FM stereo uses smaller D due to message spectrum:

There are various FM demodulators, but the “discriminator” is one of the best known. Recalling that

d d t cos ϕ ( t ) = - d ϕ ( t ) d t sin ϕ ( t ) , d d t cos ϕ ( t ) = - d ϕ ( t ) d t sin ϕ ( t ) ,
(20)

we see that

d d t s ( t ) = d d t cos 2 π f c t + 2 π k f 0 t m ( τ ) d τ = - 2 π f c + 2 π k f m ( t ) sin 2 π f c t + 2 π 0 t m ( τ ) d τ d d t s ( t ) = d d t cos 2 π f c t + 2 π k f 0 t m ( τ ) d τ = - 2 π f c + 2 π k f m ( t ) sin 2 π f c t + 2 π 0 t m ( τ ) d τ
(21)

is a form of large-carrier AM (assuming fc>kfm(t)fc>kfm(t)), which can be demodulated using an envelope detector as follows:

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