Introductory Notions

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Luke Brooks
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1 Introdutory Notions - he blok diagram of a transmission link, whih onveys information by means of eletromagneti signals, is depited in the figure below. Message Signal aqusition blok Information ransmitter signal ransmitted Modulated signal ransmission hannel Reeived modulated signal Reeiver Reeived information signal Message rendering unit Reeived message Distortions & interferenes Blok diagram of a transmission link - the original message (sound, image,..) is aquired and translated into eletromagneti signals by the message aquisition blok. his signal represents the signal that has to be onveyed at destination. - the information (modulating) signal (usually a time-variable voltage) may be expressed as: g t g g f t () where the real onstant from the average value), and M g M denotes the amplitude (def. amplitude the maximum deviation (of the voltage) f t is a real funtion whih has the following properties: f t ; () f t dt (3) Note Relation () and (3) shows that f t is zero-mean value and unitary amplitude. - he energy of the informational signal is defined as: s M (4) E g t dt g f t dt Note: the definition of the energy (4) is atually the energy dissipated on a unitary load E Z - he power of the informational signal: P lim s g t dt gm f t dt gm f t lim (5) Ps Note: (5) defines the power dissipated on a unitary load P Z - the spetrum of the informational signal is desribed by the omplex funtion G whih is obtained by applying the Fourier transform on equation (): G Y gt; (6) - If the modulating signal is analog (e.g. voal, audio, video signals) its spetrum has usually a finite bandwidth, e.g.: ; m; M G (7) ; m; M Def. the frequeny band (FB) of a signal is the frequeny range whih ontains the spetral omponents with a non-zero (or non-negligible) modulus. m M FB ; ; fm fm (8) def. frequeny bandwidth (BW) is the width of the frequeny band Es

2 BW fm f m (9) - the digital information signal have their bandwidths of infinite lengths BF ;, but usually their energy is onentrated in finite-width band (see leture 4 on Baseband transmissions). ransmitter using the informational signal, it generates the modulated signal, whih is adapted to the harateristis of the employed hannel. he modulated signal should be generated suh that the reeiver would be able to extrat the information (modulating) signal out of it, while having a transmission system as effiient as possible. Communiation hannel (e.g. radio hannel, twisted wires (UP able), telephony system (PSN), ) might be regarded as a iruit whih distorts the transmitted signal and adds to it a random signal alled noise. s t maybe expressed as: - the signal obtained at the hannel output (i.e. the reeived modulated signal) where () s t s t h d n t s t h t n t () r t t s t denotes the transmitted modulated signal, t ht is the hannel s impulse response, r nt is the noise signal, while is the onvolutional produt. - the used ommuniations hannel also imposes some additional onstraints to the transmitted hannel, suh as the average and maximum power, bandwidth, spetral distribution, et. Reeiver is intended to extrat the informational signal out of the reeived modulated signal, whih is affeted by the interferenes and distortions inserted by the hannel. he message rendering blok (e.g. speaker, display,...) is the devie that transforms the eletromagneti informational signal reeived into the reeived message. he Modulation ehniques ourse presents some of the basi tehniques and methods to generate the transmitted signal, in terms of various informational signals, and the basi tehniques of extrating the informational signals out of the reeived signals. Def. Modulation the modifiation, in aordane to a ertain rule, of some magnitudes that are harateristi to the arrier signal, in order to failitate the transmission of the information-arrying signals. - he arrier signal is usually a osine signal, defined by three harateristi magnitudes: amplitude, frequeny and phase. Depending on the parameter that is modified within the modulation proess, the modulation tehniques (modulations) an be divided into three basi types: Amplitude Modulations the information is transmitted by varying the amplitude of the arrier signal Frequeny Modulation the information is transmitted by varying the momentary frequeny of the arrier signal Phase Modulation the information is transmitted by varying the momentary phase of the arrier signal Amplitude+Phase Modulation - the information is transmitted by jointly varying the momentary phase and amplitude of the arrier signal - Aording to the type of proessing required to generate the modulated signal, the modulations an be divided in two ategories: Linear Modulations the modulated signal an be generated by using linear proesses (addition, multipliation) in this ategory we find the amplitude modulation. Nonlinear (or Exponential) Modulations in this ase the modulated signal annot be obtained by linear proesses. In this ategory we may inlude the phase and frequeny modulations, as well as all the ombined modulations (e.g. amplitude+phase). - Aording to the nature of the modulating (informational) signal the modulations an be lassified as: Analog Modulations the informational signal is ontinuous in amplitude and time, whose voltage (or urrent) level may take infinity of values. Digital Modulations the informational signal is digital (e.g. a stream of bits), its voltage level taking disreet values out of a finite set. he informational signals are also disreet in time, the level values of the signals being onstant during time intervals equaling integer multiples of an elementary period.

Linear Modulations (LM) - the useful information is ontained in the amplitude of the modulated arrier signal s A(t) osf t A(t) os t () LM - the modulating signal g(t) may take several forms,

3 Linear Modulations (LM) - the useful information is ontained in the amplitude of the modulated arrier signal s A(t) osf t A(t) os t () LM - the modulating signal g(t) may take several forms, depending on the type of LM modulation: g(t) g gmf(t); f(t) [, ]; g gm DSBC (AM); a. g(t) gm f(t); f(t) [, ]; DSBSC, b. "speial" forms for SSB, VSB; to be disussed later;. - the arrier signal is: () s(t) Vos t; (3) he Amplitude modulation (AM) - Double Sideband with Carrier DSB-C - the expression of the modulated signal: g() t V ost sml () t (4) V whih an be partiularized for DSB-C (AM) by using ().a: V g gm s AM ( m f (t)) os t; m mod ulation index ; (5) Vref g - for V = V ref the average power of the AM signal is: ~ ~ g gm f (t) gm f (t) g P ; (6) - the power of the informational part of the signal is smaller than the power of teh arrier signal; ref AM modulating signal entered on g C AM modulated signal - the DSB-C (AM) signal is the only LM signal for whih the envelope of the modulated signal is diretly proportional to the modulating signal, allowing for a very simple demodulation. Spetral Composition of the AM Signal - taking relation (7) into aount, relation (5) an be rewritten as (8), if we assume V ref = : ix ix e e osx (7) i ft i ft e e sam Vg m f t os ftvg m f t V i ft V i ft g gm f t e g gm f t e (8) - the spetrum of the modulated signal is obtained by applying the Fourier transform to relation (5) or (8): S Y s t (9) AM AM 3

- he spetrum of the AM modulated signal an be omputed by the following steps: assuming that the spetrum of the modulating signal is expressed by funtion G Y gt; and that the modulating signal has a

4 - he spetrum of the AM modulated signal an be omputed by the following steps: assuming that the spetrum of the modulating signal is expressed by funtion G Y gt; and that the modulating signal has a limited frequeny bandwidth, i.e.: ; m m; mm G () ; m m; mm using the property of the Fourier transform expressed by (), one ould ompute the spetrum of the DSB- C (AM) signal f x, h x integrable funtions F Y f x; () H Y h x ; ix if h x e f x then H F for real knowing that f - then equation (9) beomes: Vg V S GG AM () where x denotes the Dira funtion: ; x x ; x (3) - note: sine there are one spetral omponent on the arrier frequeny f and two sidebands the modulation is named double sideband with arrier DSB-C (BLD-P) see the figure below Modulating Signal G(f) Vg Modulated Signal Vg MA Spetrum a. to relation () -f mm=-w f mm =W V G V G f -f -W -f -f +W f -W f f +W f - rewriting (5) for V = V ref and for g(t) = g M osω m t we get: gm gm s AM g ( m os mt) os t gos t os( m )t os( m)t (4) - equation (4) indiates sidebands plaed symmetrially around the arrier - the frequeny band FB and the bandwidth BW of the AM signal are: FB = [f f mm, f + f mm ]; BW = f mm ; (5) Spetrum of the AM for f m = 4 Hz and f = Hz - the major disadvantage of AM is the signifiant amount of power ontained in the arrier signal; also note that BW AM =BW g(t) - the major advantage is the very simple demodulation to be disussed later 4

. omponent inserted - advantage: the modulated signal has smaller power; still the BW DSB-SC = BW g(t) DSB-SC modulated signal - the envelope of the DSB-SC signal no longer follows the modulating

5 Linear Modulation with double sideband and suppressed arrier DSB-SC - the expression of DSB-SC, is obtained from ().b and (3): V sbldps gm f(t) ost; for V Vref slm gm f(t) ost; V ~ M ref g f (t) P ; (6) LM modulating signal no d.. omponent inserted - advantage: the modulated signal has smaller power; still the BW DSB-SC = BW g(t) DSB-SC modulated signal - the envelope of the DSB-SC signal no longer follows the modulating signal it inserts a 8º unertainty - requires a more elaborated demodulation Spetral Composition of the DSB-SC Signal - the spetrum of the DSB-SC signal is obtained in a manner similar to () and is expressed by (7), see the figure below: V S GG BLDPS (7) Modulating Signal G(f) Modulated Signal DSB-SC Spetrum a. to (7) -fmm=-w fmm=w V G f -f-w -f -f+w f-w f f+w f 3 3 i m i i we get: i i AM i i i i i i i i i - rewriting (6) for V = V ref and for g(t) g sin(it) g sin( t) V G s os t g sin t g sin( )t g sin( )t (8) - (7) and (8) indiate sidebands plaed symmetrially around the arrier, but no omponent inserted intentionally on the arrier frequeny the modulation is named Double SideBand with Suppressed Carrier DSB-SC (BLD-PS) Spetrum of the DSB-SC for retangular signal with f m = 4Hz and f = 4 Hz 5

6 - the frequeny band FB and the bandwidth BW of the above signal are: FB = [f f mm, f + f mm ]; BW = f mm ; (9) - Advantage of DSB-SC no power inserted on the arrier frequeny f ; still the transmitted power is high - Disadvantages:. More elaborate demodulation;. Poor spetral effiieny, sine the DSB requires a BW= f m to transmit a modulating signal with a BW = f m. Quadrature Amplitude Modulation - QAM - the spetral effiieny of the DSB modulations is rather small, sine they require a bandwidth of BW to transmit a modulating signal with a BW bandwidth, where BW = [, f mm ]. - to use more effiiently the frequeny band, the QAM transmits two independent modulating signals in the same frequeny band, by modulating them on two orthogonal arrier signals. - the orthogonality of the two arrier signals allows for the separation of the two modulating signals at the reeiving end. - two real funtions are orthogonal if: if f x g x f xgxdx (3) t. if f x g x - it an be easily shown that the signals s I (t)=v os( t) and s Q (t)=v sin( t) observe the ondition (3) when integrated over a period = /f. - assuming that the signals g I (t) and g Q (t) are two real modulating signals with a limited bandwidth, the expression of the QAM signal modulated on the orthogonal arriers s I (t) and s Q (t) is : gi t si t gqt sqt smaq t (3) V V ref I ref Q - imposing the onditions (3) to simplify the expressions: Vref I Vref Q V (3) the expression (3) of the MAQ signal beomes: smaq t gi tostgq t sin t (33) - the blok diagram of the QAM modulator is: g I (t) LPF S I (t) g Q (t) LPF + os( t) sin( t ) - S Q (t) s MAQ (t) Blok diagram of the QAM modulator for two analogue modulating signals - the QAM modulation may be regarded as a sum of two DSB-SC signals within whih, if the two modulating signals g I (t) and g Q (t) have the same frequeny band, the modulated signals S I (t) and S Q (t) would oupy the same frequeny band (not the same as the modulating signals!). - assuming that the modulating signals g I (t) and g Q (t) have non-null spetral omponents in the frequeny band (, f mm ], the frequeny band FB and the bandwidth BW of the QAM signal would be: FBMAQ f fmm ; f fmm (34) BWMAQ fmm - though the BW QAM equals the BW DSB, within it one an transmit two modulating signals with BW = ω mm, and so the BW of the modulated signal an be onsidered equal to ω mm /modulating signal - the LP filters plaed at the inputs of the QAM modulator ensure that the frequeny spetra of the modulating signals are upper-limited to ω mm. Linear Modulation with Single Sideband SSB - the whole information of the DSB-SC modulated signal is ontained in one sideband - the DSB-SC uses redundantly the seond sideband and transmits a signifiant amount of additional power - in order to derease the frequeny band employed and the transmitted power, only one sideband is transmitted, see the figure below 6

Modulating Signal Modulated Signal G(f) SSB-inf Spetrum -f mm=-w f mm=w V G V G f -f -W -f -f +W f -W f f +W f - the SSB signal ould be obtained

the filtering method: the DSB-SC signal is BP filtered suppressing the undesired sideband - the FB and the BW of the SSB are: FB = [f - f mm, f ]

7 Modulating Signal Modulated Signal G(f) SSB-inf Spetrum -f mm=-w f mm=w V G V G f -f -W -f -f +W f -W f f +W f - the SSB signal ould be obtained by two methods: a. by filtering the DSB-SC signal; b. by phase-shifting the modulating signal a. the filtering method: the DSB-SC signal is BP filtered suppressing the undesired sideband - the FB and the BW of the SSB are: FB = [f - f mm, f ] the inferior-ssb; FB = [f, f + f mm ] the superior-ssb; BW = f mm ~ g M f () t P ; (35) 4 Filtering method for SSB-superior; the modulating signal has no d.. SSB sup the modulating signal has no low-frequeny omponents the modulating signal has signifiant low-frequeny omponents - the required filter should have a higher slope; more ompliated to implement and sometimes impossible b. SSB by phase-shifting the modulating signal - it employs the Hilbert transform of the modulating signal 7

8 - Hilbert transform transfer funtion: j H ( ) j j denotes a phase shift of /; H ( ) (36) - the SSB signal is expressed by: ^ s SSB(t) g(t)ost g(t)sint; for sup SB; for inf SB; g(t) ˆ H(g(t)) (37) f t t f f Generating the SSB-sup by using the Hilbert transform - for an aurate implementation the modulating signal should have no d.. omponent and small lowfrequeny omponents - the SSB signal (37) is similar to a QAM signal (33), but the modulating signals are not independent, but are versions of the same signal - the QAM approah is used only at the transmitter - the blok diagram of the SSB generation is similar to the one of the QAM modulator (see figure above) - SSB requires smaller BW and less power than both DSB-(S)C to transmit the same modulating signal Linear Modulation with Vestigial Sideband - VSB - it is employed for modulating signals that have low frequeny omponents, e.g. analog V signals - the modulated signals have a vestige of the undesired sideband (see the drawing on the blakboard) - the modulating signal is filtered with a low-pass harateristi K β and the modulated signal is: s VSB(t) g(t)ost g (t)sint; for sup SB; for inf SB; (38) - the FB and BW of the VSB are: FB = [f - f mm, f + β] the inferior-vsb; FB = [f - β, f + f mm ] the superior-vsb; BW = f mm + β (39) General expression of LM modulated signals: slm (t) g(t) ost gq (t)sin t; (4) Partiular ases: α = ; g q (t) = DSB: if g(t) = g + g M f(t) DSB-C; if g(t) = g M f(t) DSB-SC; (4) ; g q(t) h(t) g(t); h(t) F (H( )); SSB; - for SSB sup ; + for SSB inf ; (4) ; g q (t) K (t) g(t) - VSB (43) - the general expression employs the Quadrature Amplitude Modulation (QAM) - only at the transmitter end - the SSB and VSB an be expressed easier using this approah. LM modulation Methods a. modulators with analog multipliers 8

9 b. modulators with hoppers. modulators with non-linear iruits see referene Ed.Niolau d. modulators operating diretly on the tuned iruit see referene Ed. Niolau a. LM modulator with analog multiplier s x (t) - partiular ases will be disussed on the blakboard s e (t) - the analog multipliers are not available for very high F(ω) g(t) frequenies V osω t b. modulators with hoppers b.. Unbalaned Chopper - the modulating signal is hopped with the interruption funtion with the frequeny f : if V ost ; FB fi() t fi() t sint sin3 t... (44) s I if V ost ; 3 g(t) os(? t) R L C Unbalaned hopper DSB modulator - the multipliation between the f i (t) signal and the modulating signal g(t) is aomplished by means of a transistor (see the neighbouring figure): s i i i x(t) k g(t) k g(t) sin k g(t) t sin 3 t the spetrum of the modulated signal ontains spetral omponents in the baseband - by BP filtering only the desired bandwidth is seleted: (45) ω ω mm ω mm ω -ω mm ω ω +ω mm 3ω -ω mm 3ω 3ω +ω mm - requirements to make the filtering possible: ω ω mm > ω mm ; 3ω ω mm > ω + ω mm ω > ω mm (46) b.. Balaned Chopper - the balaned hopper performs the multipliation of the g(t) with the swithing funtion f s (t): if V ost ; 4 4 f s(t) f i(t) ; f s(t) f s(t) sin t sin 3t... (47) if V os t ; - the multipliation is implemented with two omplementary transistors and the output signal is expressed by; 4kig(t) 4kig(t) s x(t) sin t sin 3 t... 3 (48) g(t) os(? t) R FB L C Balaned hopper DSB modulator - it does not generate spetral omponents in the baseband and generates a double amplitude of the modulated signal - requirements to make the filtering possible: 3ω ω mm > ω + ω mm ω > ω mm (49) Effets of the imperfet implementation of the Hilbert transform upon the phase-shifting method for SSB - the SSB signal is: ^ sssb (t) g(t) ost g(t)sin t; for sup SB; for inf SB; (5) - onsidering g(t) = osω m t, if due imperfet implementation of the Hilbert transform the phase shift is expressed by (5) instead of π/: Φ(ω m ) = - π/ + φ (5) then the SSB-inf would be expressed by (5) as shown in Annex. 9

10 I(t) Q(t) os os[( m)t ] sin sin[( m)t ]; (5) - the SSB-inf is attenuated by os(φ/)/ and phase-shifted by φ/, whih distorts the signal - the SSB-sup (whih should be suppressed) has a level that depends proportionally on φ/ - Conlusion: φ should be as small as possible φ Considerations regarding the Peak to Average Power Ratio (PAPR) of the LM signals a. Comparison between the powers of AM (DSB-C), DSB-SC and SSB ~ ~ ~ ~ g g m f g m f gmf gmf P AM ; P DSBSC ; P SSB ; 4 (53) PAM P ; DSBSC ; P ~ DSBSC P SSB m f b. PAPR onsiderations - PAPR [db] = lg(p p /P av ) (54) - the Peak to Average Power Ratio is depending of the rest-fator of the modulating signal - for random signals SSB is to be employed sine it has the smallest PAPR value - for deterministi signals DSB-SC should be employed - the signal BW should also be onsidered - a high PAPR leads to spetral and signal distortions in HP-RF amplifiers this issue will be disussed later Annex Effets of the imperfet implementation of the Hilbert transform upon the phase-shifting method for SSB - not required for the exam - we prove expression (5) - the SSB signal is: ^ sssb (t) g(t) ost g(t)sin t; for sup SB; for inf SB; (55) - if due imperfet implementation of the Hilbert transform the phase shift is expressed by (5) instead of π/: - onsider g(t) = osω m t; on the in-phase branh we would have: I(t) os mt os t [os( t mt) os( t mt)] (56) 4 - on the quadrature branh the phase shift equals (5) instead of π/; - so the signal on this branh would be: Q(t) os( mt ) sint (57) os sin [os( t mt) os( t mt)] [sin( t mt) sin( t mt)] by adding the signals of the two branhes, (56) and (57), to get the SSB-inf, we obtain: os os sin sin I(t) Q(t) os( t mt) os( t mt)] sin( t mt) sin( t mt)] (58) (58) an be easily shown to be equal to (5) by some trigonometri manipulations.

6. Amplitude Modulation

6. Amplitude Modulation Modulation is a proess by whih some parameter of a arrier signal is varied in aordane with a message signal. The message signal is alled a modulating signal. Definitions A bandpass