Modulation Principles

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1 EGR 544 Communiation Theory 5. Charaterization of Communiation Signas and Systems Z. Aiyaziiogu Eetria and Computer Engineering Department Ca Poy Pomona Moduation Prinipes Amost a ommuniation systems transmit digita data using s sinusoida arrier waveform The transmitted hanne has imited band-width and is entered about the arrier for doube side moduation is next to arrier signa for singe side. Physia moduation system impementation Proess digita information at baseband Puse shaping and fitering of digita waveform Mixing with arrier signa Fitering RF signa and ampifying to transmission Ca Poy Pomona Eetria & Computer Engineering Dept. EGR 544-5

waveform The transmitted hanne has imited band-width and is entered about the arrier for doube side moduation is next to arrier signa for singe side.

2 Moduation Signas representation We an modify ampitude, phase, or frequeny of baseband signa Ampitude Shift Keying (ASK) or On/Off Keying (OOK) Aos( π ft) 0 0 Frequeny Shift Keying (FSK) Phase Shift Keying (PSK) Aos( π ft) 0 Aos( π ft) Aos( π f t) 0 Aos( π f t+ π) = Aos( π f t+ π) Ca Poy Pomona Eetria & Computer Engineering Dept. EGR Representation of Band-Pass Signa The transmitted signa is usuay a rea vaued band-pass signa and et s a s(t) Mathematia mode of a rea-vaued narrowband band-pass signa is S( f) 0 for f f f f + f and f f B B B ½ S - (f)=u(-f)s(f) S( f ) ½ S + (f)=u(f)s(f) f -f f S(f) is Fourier transform of s(t). u(f) is the unit step funtion in frequeny domain Ca Poy Pomona Eetria & Computer Engineering Dept. EGR

EGR 544-5 3 Representation of Band-Pass Signa The transmitted signa is usuay a rea vaued band-pass signa and et s a s(t) Mathematia mode of a rea-vaued narrowband band-pass signa is S( f)

3 Representation of Band-Pass Signa Goa is to deveop a mathematia representation in time domain of S + (f ) and S - (f ) The time domain representation of S + (f ) is s + (t), whih is aed pre-enveope of s(t) ft s+ () t = S+ ( f) e df = [ ( ) ( )] ft u f S f e df [ ( )] [ ( )] F u f F S f = j = δ () t + s() t πt j = st () + st () πt = st () + jst ˆ() where st ˆ( ) = st () πt s( τ ) = dτ π t τ Ca Poy Pomona Eetria & Computer Engineering Dept. EGR Representation of Band-Pass Signa sˆ( t) may be onsidered the output of the fiter suh as s() t πt Hibert Transform sˆ( t) The frequeny response of the fiter is π t j f > 0 H( f) = jsgn( f) = 0 f = 0 j f < 0 ft ft H ( f) = h( t) e dt = e dt H( f) = for f 0 π for f > 0 Θ ( f ) = π for f 0 < Fourier Transform of Hibert Transform j F = sgn( f) πt F = jsgn( f) πt Ca Poy Pomona Eetria & Computer Engineering Dept. EGR

Poy Pomona Eetria & Computer Engineering Dept.

4 Representation of Band-Pass Signa s() t sˆ( t) s X + + () t πt Hibert Transform j S( f ) S + (f)=u(f)s(f) f -f f S ( f) = S ( f + f ) f The ow-pass representation of S + (f ) is S( f) = S+ ( f + f) in time domain s () () [ () ˆ t = s+ t e = s t + js() t ] e In ompex form s() t = x()os( t π ) y()sin( t π ) s () t = x() t + jy() t sˆ( t) = x()sin( t π ) + y()os( t π ) x(t) and y(t):quadrature omponents of s (t) Ca Poy Pomona Eetria & Computer Engineering Dept. EGR Representation of Band-Pass Signa Another representation of the signa Or we an represent {[ ] } st () = Re xt () + jyt () e = Re s ( t) e s () t () j t () = a t e θ st () = Re s () te at () = x() t + y() t jθ () t = Re ate ( ) e = at ()os ft+ () t [ π θ ] yt () θ () t = tan x() t a(t) is aed enveope of s(t) and θ(t) is aed the phase of s(t) Ca Poy Pomona Eetria & Computer Engineering Dept. EGR

omponents of s (t) Ca Poy Pomona Eetria & Computer Engineering Dept.

5 Representation of Band-Pass Signa The energy in the signa s(t) is defined as f t { } ε = s () t dt = Re s () t e dt Using representation of s(t) in osine form Then { [ ]} ε = s () t dt = a ()os t π f t+ θ () t dt ε = π θ + + { [ ]} a () t dt a ()os4 t f t () t dt a(t) is the enveope and varies sowy reative to osine funtion ε = a () t dt s () t dt = Ca Poy Pomona Eetria & Computer Engineering Dept. EGR Representation of Linear Band-Pass signa A inear fiter or system an be represented either h(t) or of H(f). Sine h(t) is rea H( f) = H ( f) Let s define H ( f f ) H( f) f > 0 H( f f) = 0 f < 0 Then, we have And H ( - f f ) H ( f f) = H f H f f H f f ( ) = ( ) + ( ) 0 f > 0 H f f ( ) < 0 The Inverse transform of H(f) π ht () = h() te + h () te = Re h ( t) e j f t h (t), inverse transform of H (f), is impuse response of ow-pass system and is omex Ca Poy Pomona Eetria & Computer Engineering Dept. EGR

EGR 544-5 9 Representation of Linear Band-Pass signa A inear fiter or system an be represented either h(t) or of H(f).

6 Response of Band-pass System to a Band-pass signa s(t) h(t) r(t) Let s have s(t) narrowband band-pass signa and is the equivaent ow-pass signa s (t). Band-pass fiter (system) the impuse response h(t) and its equivaent ow-pass impuse response h (t) The output of the band-pass fiter is r(t) aso band-pass signa rt = j () Re r () te π Ca Poy Pomona Eetria & Computer Engineering Dept. EGR Response of Band-pass System to a Band-pass signa r(t) an be given as rt () = s( τ ) ht ( dτ In frequeny domain R( f) = S( f) H( f) R( f) = S( f f) + S ( f f) H( f f) + H( f f) For narrow band signa s(t) and narrow band system h(t) S f f H f f ( ) ( ) = 0 R f S f f H f f S f f H f f ( ) = ( ) ( ) + ( ) ( ) R f R f f R f f ( ) = ( ) + ( ) S ( f f ) H ( f f ) = 0 R( f) = S ( f) H ( f) Ca Poy Pomona Eetria & Computer Engineering Dept. EGR r() t = s ( τ ) h( t dτ

7 Bandpass Stationary Stohasti Proess Let s have n(t) as narrowband band-pass proess with spetra density is muh smaer than f, zero mean and power spetra density Φ nn (f). And we an represent it as nt () = at ()os [ π ft + θ() t] = x()os( t π ) y()sin( t π ) j = Re zte ( ) π a(t) is the enveope and θ(t) is the phase of the rea vaued signa. x(t) and y(t) are the quadrature omponent of n(t). z(t) is the ompex enveope of n(t) n(t) is zero mean, therefore x(t) and y(t) wi be zero mean, The autoorreation and ross-orreation funtion satisfy φxx( τ ) = φyy( τ ) φ ( τ ) = φ ( τ ) xy yx Ca Poy Pomona Eetria & Computer Engineering Dept. EGR Bandpass Stationary Stohasti Proess The autoorreation funtion φ nn ( of n(t) {[ ] [ x( t+ os π f ( t+ y( t+ sin π f ( t+ ] } φ ( = Entnt [ ( ) ( + ] = E xt ( )os π ft yt ( )sinπ ft nn φ ( τ ) = φ ( osπ f tos π f ( t+ + φ ( sinπ f tsin π f ( t+ nn xx yy φ ( ]sin π f tos π f ( t+ φ osπ f tsin π f ( t+ xy yx Using trigonometri identities φnn( τ ) = [ φxx( τ ) + φyy( ]os π fτ + [ φxx( τ ) φyy( ]os π f( t+ [ φyx( τ ) φxy( ]sin π fτ [ φ yx( τ ) + φxy( ]sin π f( t+ Ca Poy Pomona Eetria & Computer Engineering Dept. EGR

x(t) and y(t) are the quadrature omponent of n(t).

8 Bandpass Stationary Stohasti Proess Using zero mean properties φ ( τ ) = φ ( os π fτ + φ ( sinπ fτ nn xx yx z(t) an be given ompex-vaued as z(t)=x(t)+jy(t) The autoorreation funtion of z(t) is given as φzz( = Ez [ ( tzt ) ( + ] = [ φxx ( τ ) + φyy ( τ ) jφ xy ( τ ) + jφ yx ( τ )] = φ ( + φ ( xx The autoorreation funtion φ nn ( of n(t) an be obtained by φ zz ( and the arrier frequeny f yy φ τ φ τ j nn( ) = Re zz( ) e π Ca Poy Pomona Eetria & Computer Engineering Dept. EGR Bandpass Stationary Stohasti Proess The Fourier transform of the autoorreation funtion φ nn ( gives The power spetra density Φ nn (f) j π f t j π f τ f zz e e d Φ ( ) = Re φ ( τ nn = Φ +Φ [ ( f f ) ( f f )] zz zz Where Φ zz (f) is the power density spetrum of equivaent owpass proess z(t). Φ zz (f) is rea-vaued funtion and the autoorreation funtion of z(t) satisfy that φ τ φ τ zz( ) = zz( ) Ca Poy Pomona Eetria & Computer Engineering Dept. EGR

j nn( ) = Re zz( ) e π Ca Poy Pomona Eetria & Computer Engineering Dept.

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