Modulation Principles

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EGR 544 Communication Theory

Z. Aliyazicioglu

Electrical and Computer Engineering Department Cal Poly Pomona

5. Characterization of Communication Signals and Systems

Modulation Principles

• Almost all communication systems transmit digital data using s sinusoidal carrier waveform

• The transmitted channel has limited band-width and – is centered about the carrier for double side modulation – is next to carrier signal for single side.

Physical modulation system implementation – Process digital information at baseband – Pulse shaping and filtering of digital waveform – Mixing with carrier signal

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Cal Poly Pomona Electrical & Computer Engineering Dept. EGR 544-5 3

Modulation Signals representation

We can modify amplitude, phase , or frequency of baseband signal Amplitude Shift Keying (ASK) or On/Off Keying (OOK)

Frequency Shift Keying (FSK)

1 cos(2 ) 0 0 c A π f t ⇒ ⇒ 1 2 1 cos(2 ) 0 cos(2 ) A f t A f t π π ⇒ ⇒

Phase Shift Keying (PSK)

1 cos(2 ) 0 cos(2 ) cos(2 ) c c c A f t A f t A f t π π π π π ⇒ ⇒ + = − +

Representation of Band-Pass Signal

• The transmitted signal is usually a real valued band-pass signal and let’s call s(t)

• Mathematical model of a real-valued narrowband band-pass signal is

( ) 0 for c B c B and c B

S f ≠ f − f ≤ f ≤ f + f f f

S(f) is Fourier transform of s(t). u(f) is the unit step function in frequency domain f_c -f_c f S( f ) ½ S+(f)=u(f)S(f) ½ S_-(f)=u(-f)S(f)

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Representation of Band-Pass Signal

Goal is to develop a mathematical representation in time domain of S+(f ) and S-(f )

The time domain representation of S₊(f ) is s₊(t), which is called

pre-envelope of s(t)

[

]

[

]

[

]

2 2 1 1 ( ) ( ) 2 ( ) ( ) 2 ( ) ( ) ( ) ( ) ( ) ( ) ˆ ( ) ( ) j ft j ft s t S f e df u f S f e df F u f F S f j t s t t j s t s t t s t js t π π δ π π ∞ + _−∞ + ∞ −∞ − − = = = ∗   =_ + _∗   = + ∗ = +

∫

1 ˆ( ) ( ) 1 ( ) s t s t t s d t π τ _τ π τ ∞ −∞ = ∗ = −

∫

where

may be considered the output of the filter such as

• The frequency response of the filter is

ˆ( ) s t Hilbert Transform ˆ( ) s t ( ) s t ₁ t π 2 1 1 2 ( ) ( ) j ft j ft H f h t e dt e dt t π π π ∞ ₋ ∞ ₋ −∞ −∞ =

∫

=

∫

0 ( ) sgn( ) 0 0 0 j f H f j f f j f − >   = − =_ =  _<  ( ) 1 for 0 H f = f ≠ for 0 2 ( ) f f π π − >  Θ _{= } sgn( ) 1 _{sgn( )} j F f t F j f t π π   =       = −     Fourier Transform of Hilbert Transform

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Cal Poly Pomona Electrical & Computer Engineering Dept. EGR 544-5 7 j

The low-pass representation of S₊(f ) is in

time domain Hilbert Transform ˆ( ) s t ( ) s t ₁ t π X + s t+( ) f_c -f_c f S( f ) _S +(f)=2u(f)S(f) ( ) ( ) l c S f =S f₊ + f

[

]

2 _ˆ 2 ( ) ( ) j f tc ( ) ( ) j f tc l s t =s t e+ − π = s t + js t e− π ( ) ( ) ( ) l s t =x t + jy t In complex form ( ) ( )cos(2 ) ( )sin(2 ) ˆ( ) ( )sin(2 ) ( )cos(2 ) c c c c s t x t f t y t f t s t x t f t y t f t π π π π = − = + f

x(t)and y(t):quadrature components of s_l(t)

( ) ( )

l c

S f =S f+ + f

Representation of Band-Pass Signal

• Another representation of the signal

Or we can represent

a(t) is called envelope of s(t) and θ(t) is called the phase of s(t)

[

]

{

2

}

2 ( ) Re ( ) ( ) Re ( ) c c j f t j f t l s t x t jy t e s t e π π = +   = _ _ ( ) ( ) ( ) j t l s t =a t eθ _{a t}_{( )}₌ _{x t}2_{( )}₊_{y t}2_{( )} ( ) tan 1 ( ) ( ) y t t x t θ ₌ −

[

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2 2 ( ) ( ) Re ( ) Re ( ) ( )cos 2 ( ) c c j f t l j f t j t c s t s t e a t e e a t f t t π π θ π θ   = _ _   = _ _ = +

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• The energy in the signal s(t) is defined as

• Using representation of s(t) in cosine form

Then

a(t) is the envelope and varies slowly relative to cosine function

{

₂

}

2 2_{( )} _Re _{( )} j f tc l s t dt s t e π dt ε=

∫

_−∞∞ =

∫

_−∞∞ _ _

[

]

{

}

2 2_{( )} _{( )cos 2} _{( )} c s t dt a t f t t dt ε=

∫

_−∞∞ =

∫

_−∞∞ π +θ

[

]

{

}

2 2 1 1 ( ) ( )cos 4 2 ( ) 2 a t dt 2 a t f tc t dt ε ∞ ∞ π θ −∞ −∞ =

∫

+

∫

+ 2 2 1 1 ( ) ( ) 2 a t dt 2 s t dtl ε ∞ ∞ −∞ −∞ =

∫

=

∫

Representation of Linear Band-Pass signal

• A linear filter or system can be represented either h(t)or of H(f).

• Since h(t)is real

*

( ) ( )

H f =H −f

Then, we have

The Inverse transform of H(f) Let’s define Hl( f – fc) ( ) 0 ( ) 0 0 l c H f f H f f f >  − _{= } <  And H_l*_{( - f – f} c) * * 0 0 ( ) ( ) 0 l c f H f f H f f >  − − _{= } − <  * ( ) _l( _c) _l( _c) H f =H f − f +H − −f f 2 * 2 2 ( ) ( ) ( ) 2 Re ( ) c c c j f t j f t l l j f t l h t h t e h t e h t e π π π − = +   = _ _

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The output of the band-pass filter is r(t)also band-pass signal Let’s have

•s(t)narrowband band-pass signal and is the equivalent low-pass signal s_l(t).

•Band-pass filter (system) the impulse response h(t)and its equivalent low-pass impulse response h_l(t)

2 ( ) Re ( ) j f tc l r t = _r t e π _ h(t) r(t) s(t)

Response of Band-pass System to a Band-pass signal

r(t) can be given as ( ) ( ) ( ) r t =

∫

_−∞∞sτ h t−τ τd In frequency domain R f( )=S f H f( ) ( ) * * 1 ( ) ( ) ( ) ( ) ( ) 2 l c l c l c l c R f = _S f− f +S − −f f _{ } H f − f +H − −f f _ * * 1 ( ) ( ) ( ) ( ) ( ) 2 l c l c l c l c R f = _S f− f H f − f +S − −f f H − −f f _ * 1 ( ) ( ) ( ) 2 l c l c R f = _R f − f +R − −f f _ R fl( )=S f H fl( ) l( ) * ( ) ( ) 0 l c l c S f − f H − −f f = *₍ ₎ ₍ _{) 0} l c l c S − −f f H f − f =

For narrow band signal s(t) and narrow band system h(t)

( ) ( ) ( )

l l l

r t ∞s τ h t τ τd

−∞

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Bandpass Stationary Stochastic Process

Let’s have n(t) as narrowband band-pass process with spectral

density is much smaller than f_c, zero mean and power

spectral density Φ_nn(f).

And we can represent it as

[

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2 ( ) ( )cos 2 ( ) ( )cos(2 ) ( )sin(2 ) Re ( ) c c c c j f t n t a t f t t x t f t y t f t z t e π π θ π π = + = −   = _ _

a(t) is the envelope and θ(t) is the phase of the real valued

signal. x(t) and y(t) are the quadrature component of n(t). z(t) is the complex envelope of n(t)

n(t) is zero mean, therefore x(t) and y(t) will be zero mean , The autocorrelation and cross-correlation function satisfy

( ) ( ) ( ) ( ) xx yy xy yx φ τ φ τ φ τ φ τ = = −

Bandpass Stationary Stochastic Process

• The autocorrelation function φnn(τ) of n(t)

• Using trigonometric identities

[

]

{

[

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}

( ) [ ( ) ( )] ( )cos2 ( )sin 2 ( )cos2 ( ) ( )sin 2 ( ) nn c c c c E n t n t E x t f t y t f t x t f t y t f t φ τ τ π π τ π τ τ π τ = + = − + + − + + 1 1 ( ) [ ( ) ( )]cos2 [ ( ) ( )]cos2 (2 ) 2 2 1 1 [ ( ) ( )]sin 2 [ ( ) ( )]sin 2 (2 ) nn xx yy fc xx yy fc t f f t φ τ φ τ φ τ π τ φ τ φ τ π τ φ τ φ τ π τ φ τ φ τ π τ = + + − + − − − + +

( ) ( )cos2 cos2 ( ) ( )sin 2 sin 2 ( ) ( )]sin 2 cos2 ( ) cos2 sin 2 ( )

nn xx c c yy c c xy c c yx c c f t f t f t f t f t f t f t f t φ τ φ τ π π τ φ τ π π τ φ τ π π τ φ π π τ = + + + − + − +

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Bandpass Stationary Stochastic Process

• z(t) can be given complex-valued as z(t)=x(t)+jy(t)

• The autocorrelation function of z(t) is given as

* 1 ( ) [ ( ) ( )] 2 1_[ _{( )} _{( )} _{( )} _{( )]} 2 ( ) ( ) zz xx yy xy yx xx yy E z t z t j j φ τ τ φ τ φ τ φ τ φ τ φ τ φ τ = + = + − + = +

• The autocorrelation functionφnn(τ) of n(t) can be obtained

by φzz(τ) and the carrier frequency fc

2 ( ) Re ( ) j f tc nn zz e π φ τ = _φ τ _ ( ) ( )cos2 ( )sin 2 nn xx fc yx fc φ τ =φ τ π τ φ τ+ π τ

• Using zero mean properties

Bandpass Stationary Stochastic Process

• The Fourier transform of the autocorrelation functionφnn(τ) gives

The power spectral density Φnn(f)

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2 2 ( ) Re ( ) 1 ( ) ( ) 2 c c j f t j f nn zz zz c zz c f e e d f f f f π π τ φ τ τ ∞ ₋ −∞   Φ = _ _ = Φ − + Φ − −

∫

• Where Φ_zz(f) is the power density spectrum of equivalent

low-pass process z(t).

• Φzz(f) is real-valued function and the autocorrelation function of

z(t) satisfy that

*

( ) ( )

zz zz