The Communications Channel (Ch.11):

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1 ECE-5 Phil Schniter February 5, 8 The Communications Channel (Ch.): The eects o signal propagation are usually modeled as: ECE-5 Phil Schniter February 5, 8 Filtering due to Multipath Propagation: The signal may propagate along paths with dierent lengths: transmitted signal S() s(t) the communications channel w(t) x(t) H() + r(t) received signal R() s(t) x(t) where H() : linear iltering due to multipath propagation w(t) : additive noise/intererence. Noise/intererence sources include: electronic circuitry ( thermal/shot noise or quantization noise ); usually broadband, other comm systems ( multi-access intererence or co-channel intererence ); broadband or narrowband. SNR can be improved with appropriate iltering at receiver: signal-to-noise ratio (SNR) = signal energy noise energy = E s E n. roughly speaking, average SNR can be improved by iltering out the requencies dominated by noise: beore iltering: ater iltering: Since dierent path lengths imply dierent path delays: x(t) = c s(t τ ) + c s(t τ ) + + c N s(t τ N ), which can be written as x(t) = s(t) h(t) or h(τ) = c δ(τ τ ) + c δ(τ τ ) + + c N δ(τ τ N ). The result is an H() that varies with, implying requency-dependent signal attenuation: t_max = ; Ts = /; c = ; tau =.; c = -.5; tau =.8; c3 =.; tau3 =.; h = zeros(,t_max/ts); h(tau/ts) = c/ts; h(tau/ts) = c/ts; h(tau3/ts) = c3/ts; plott(h,ts); amplitude magnitude 5! time.5.5!5!4!3!! requency 3 4

2 ECE-5 Phil Schniter February 5, 8 Analog Communication (Ch.3-4):. Amplitude modulation (AM). Quadrature amplitude modulation (QAM) 3. Vestigial sideband modulation (VSB) 4. Frequency modulation (FM) AM with suppressed carrier : AM o real-valued message m(t) (e.g., music) is m(t) s(t) s(t) = m(t) cos(π c t), cos(π c t) c = carrier req. Euler s cos(π c t) = [ e jπ c t + e jπ ct ] then implies S() = m(t) cos(π c t)e jπt dt = m(t)e jπ( c)t dt + = M( c) + M( + c). m(t)e jπ(+ c)t dt ECE-5 Phil Schniter February 5, 8 Because m(t) R, know M() symmetric around =, implying the AM transmitted spectrum below c is redundant! This motivates the QAM and VSB modulation schemes... With c known, AM demodulation can be accomplished by:. r(t) LPF v(t). v(t) = LPF{r(t) cos(π c t)}. cos(π c t) For a trivial noiseless channel, we have r(t) = s(t), so that v(t) = LPF{s(t) cos(π c t)} = LPF{m(t) cos (π c t) } +cos(π c t) = LPF{m(t) + m(t) cos(π c t)} = m(t), assuming a LPF with passband cuto B p W Hz and stopband cuto B s c W Hz: M() S() W W c c B c c+w c B s W W B pb s c c W c Note that we ve assumed perectly synchronized oscillators! 5 6

3 ECE-5 Phil Schniter February 5, 8 When the receiver oscillator has {req,phase} oset {γ, φ}: v(t) = LPF { } m(t) cos(π c t) cos(π( c + γ)t + φ) cos(πγt+φ) + cos(π( c +γ)t+φ) = m(t) cos(πγt + φ). time-varying attenuation! Note: a req oset o λ = ν c Hz can occur when there is c relative velocity o ν m/s between transmitter and receiver. AM with pilot tone or carrier tone : It s common to include a pilot/carrier tone with requency c : s(t) = m(t) cos(π c t) + A cos(π c t) pilot/carrier tone = [m(t) + A] cos(π c t) [ ] S() = M( c )+M( + c )+Aδ( c )+Aδ( + c ) ECE-5 Phil Schniter February 5, 8 While modern systems choose A max m(t), many older systems use A > max m(t), known as large carrier AM, allowing reception based on envelope detection: v(t) = π LPF{ r(t) } A m(t) (with a trivial channel) where can be easily implemented using a diode. The gain π above makes up or the loss incurred when LPFing the rectiied signal: desired output level LPF output level 4c amplitude 3! t 4c cos(π c t)dt 4 c = π. large!carrier AM! M() S() W MATLAB code here! time [sec] envelope!detected signal (and original message).5 W W c c amplitude!.5! Advantage: aids receiver with carrier synchronization. Disadvantage: consumes transmission power.! time [sec] 7 8

4 ECE-5 Phil Schniter February 5, 8 Quadrature Amplitude Modulation (QAM): QAM is motivated by unwanted redundancy in the AM spectrum, which was symmetric around c. QAM sends two real-valued signals {m I (t), m Q (t)} simultaneously, resulting in a non-symmetric spectrum. m I (t) m Q (t) cos(π c t) sin(π c t) + in phase s(t) = m I (t) cos(π c t) m Q (t) sin(π c t) quadrature QAM demodulation is accomplished by: r(t) LPF cos(π c t) sin(π c t) LPF v I (t) v Q (t) where the LPF specs are the same as in AM, i.e., passband edge B p W Hz and stopband edge B s c W Hz. ECE-5 Phil Schniter February 5, 8 For a trivial channel, we have r(t) = s(t), so that v I (t) = LPF{r(t) cos(π c t)} = LPF { m I (t) cos (π c t) +cos(4π c t) m Q (t) sin(π c t) cos(π c t) sin(4π c t) = m I (t) v Q (t) = LPF{ r(t) sin(π c t)} = LPF { m I (t) cos(π c t) sin(π c t) } + m Q (t) sin (π c t) cos(4π c t) = m Q (t), assuming synchronized oscillators. sin(4π c t) When the oscillators are not synchronized, one gets coupling between the I&Q components as well as attenuation o each. } 9

5 ECE-5 Phil Schniter February 5, 8 Writing the I&Q signals in the complex-baseband orm m(t) = m I (t) + jm Q (t) ṽ(t) = v I (t) + jv Q (t) yields a much simpler description o QAM: QAM modulation QAM demodulation m(t) Re s(t) r(t) LPF ṽ(t) e jπct M() M( c) e jπct R() R( + c) ECE-5 Phil Schniter February 5, 8 as well as or demodulation (assuming r(t) = s(t)): ṽ(t) = LPF{s(t) e jπ ct } = LPF{ ( m I (t) cos(π c t) m Q (t) sin(π c t) ) e jπ ct } = LPF { m I (t) ( e jπ ct + e jπ ct ) e jπ ct m Q (t) ( je jπ ct je jπ ct ) e jπ ct } = LPF { m I (t) ( + e j4π ct ) m Q (t) ( je j4π ct j )} = m I (t) + jm Q (t). The convenience o complex-baseband results in widespread use o complex-valued signals or comm systems! Note: To get the complex baseband ormulation or AM, we simply set m Q (t) = and m I (t) = m(t). S() Ṽ () Vestigial Sideband Modulation (VSB): Note: Re{u(t)} = [ u(t) + u (t) ] F [ U() + U ( ) ]. We now veriy the complex-baseband model or modulation: Re{ m(t)e jπ ct } = Re {( m I (t) + jm Q (t) )( cos(π c t) + j sin(π c t) )} = m I (t) cos(π c t) m Q (t) sin(π c t) = s(t), VSB is another way to restore regain the spectral eiciency lost in AM. It s used to transmit North American terrestrial TV, both analog (NTSC) and digital (ATSC) ormats. Like AM, it can operate with or without a carrier tone. Basically, VSB suppresses most o the redundant AM spectrum by iltering it:

6 ECE-5 Phil Schniter February 5, 8 ECE-5 Phil Schniter February 5, 8 AM: S() W VSB: S() For VSB modulation, we have s(t) = m(t) cos(π c t) c(t) c c c c S() = [ M( + c ) + M( c ) ] C(). m(t) s(t) m(t) cos(π c t) cos(π c t) The passband VSB ilter is a BPF C() where passband VSB ilter C( c ) + C( + c ) = or W, s(t) It turns out that VSB demod is identical to AM demod: v(t) = LPF { r(t) cos(π c t) } = LPF { s(t) cos(π c t) } (trivial channel) V () = LPF { S( c ) + S( + c ) } which implies its inside rollo is symmetric around = c : c c c c c c c W C() W c C( + c) c C( c) c c c c C( + c) + C( c) c = { [M() LPF + M( c ) ] C( c ) + [ M( + c ) + M() ] } C( + c ) = M() [ C( c ) + C( + c ) ] } {{} = or [ W,W ] = M(). We note that the property { } F cos(π c t)c(t) = [ ] C( c ) + C( + c ) may be convenient, e.g., or testing whether a given ilter c(t) satisies the passband VSB criterion. 3 4

7 ECE-5 Phil Schniter February 5, 8 VSB iltering can also be implemented at baseband using a complex-valued ilter response c(t) which satisies C() + C ( ) = or W, generating the complex-baseband message signal m(t). The message can be recovered by simplying ignoring the imaginary part o the complex-baseband output ṽ(t). m(t) VSB modulation baseband m(t) ṽ(t) VSB ilter Re s(t) r(t) LPF Re v(t) M() M() e jπct M( c) S() VSB demodulation e jπct R() R( + c) Ṽ () V () Motivation: iltering at baseband is usually much cheaper than iltering at passband. ECE-5 Phil Schniter February 5, 8 Frequency Modulation (FM): While AM modulated the carrier amplitude, FM modulates the carrier requency. t_max =.; W = ; % message params Ts = /; t = :Ts:t_max; m = sin(*pi*w*t); % message signal c = ; % carrier req D = 5; % FM mod index k = D*W/max(abs(m)); % req sensitivity s_am = m.*cos(*pi*c*t); s_m = cos(*pi*c*t+*pi*k*cumsum(m)*ts); subplot(3,,) plot(t,m); grid on; title( message ); subplot(3,,) plot(t,s_am); grid on; title( AM ); subplot(3,,3) plot(t,s_m); grid on; title( FM );.5!.5 message! !.5 AM! !.5 FM! In particular, FM modulates the real-valued message m(t) via s(t) = cos ( t π c t + πk m(τ)dτ ). ϕ(t) instantaneous modulation phase where k is called the requency-sensitivity actor. Since the instantaneous modulation requency dϕ(t) dt = πk m(t) is a scaled version o the message m(t), it is itting to call this scheme requency modulation. 5 6

8 ECE-5 Phil Schniter February 5, 8 Using the peak requency deviation = k max m(t), the modulation index D is deined as D = W single-sided BW o m(t). Increasing D decreases spectral eiciency but increases robustness to noise/intererence. D : narrowband FM, D : wideband FM. Carson s Rule approximates the FM passband signal-bw as BW 99 Example: Mono FM radio: ( + W ) = (D + )W. Message signal iltered to req interval [3,5k] Hz. FCC limits 75 khz (channels khz apart). D = 75 5 = 5 FM stereo uses smaller D due to message spectrum: ECE-5 Phil Schniter February 5, 8 There are various FM demodulators, but the discriminator is one o the best known. Recalling that we see that d dt cos( ϕ(t) ) = dϕ(t) sin ( ϕ(t) ), dt d dt s(t) = d dt cos( π c t + πk t m(τ)dτ) = [ π c + πk m(t) ] sin ( π c t + π t m(τ)dτ) is a orm o large-carrier AM (assuming c > k m(t)), which can be demodulated using an envelope detector as ollows: r(t) d dt MATLAB code here envelope detector DC block v(t)! !.5 message! FM modulated discriminator demodulated sum L+R AM-modulated di L-R other optional services (e.g., text, muzak)!! khz 7 8

Outline. Communications Engineering 1

Outline. Communications Engineering 1 Outline Introduction Signal, random variable, random process and spectra Analog modulation Analog to digital conversion Digital transmission through baseband channels Signal space representation Optimal