Complex envelope analysis, Lecture notes of Communication

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Bandpass Signalling

Definitions

Complex Envelope Representation

Representation of Modulated Signals

Spectrum of Bandpass Signals

Power of Bandpass Signals

Examples

 Energy spectrum of a bandpass signal is concentrated around the carrier frequency fc.

 (^) A time portion of a bandpass signal. Notice the carrier and the baseband envelope.

Bandpass Signals

Bandpass Signal Spectrum

Time Waveform of Bandpass Signal

Complex Envelope Representation

 The waveformsg(t) ,x(t),R(t), and are all baseband waveforms. Additionally all of them exceptg(t) are real andg(t) is the Complex Envelope.

g(t) is the Complex Envelope ofv(t)

x(t) is said to be the In-phase modulation associated withv(t)

y(t) is said to be the Quadrature modulation associated withv(t)

R(t) is said to be the Amplitude modulation (AM) onv(t)

(t) is said to be the Phase modulation (PM) onv(t)

In communications, frequencies in the baseband signalg(t) are said to be heterodyned up tofc  THEOREM: Any physical bandpass waveformv(t) can be represented as below

wherefc is the CARRIER frequency and c 2  fc

( )  

j g t^ j^ t g t x t j y t g t e R t e

^ 

R e cos

= cos si n

j (^) ct c

c c

v t g t e R t t t

x t t y t t



  ^  

Generalized transmitter using the AM–PM

generation technique.

v(t)– bandpass waveform with non-zero spectrum concentrated nearf=f c =>c n – non-zero for ‘n’ in the range

The physical waveform is real, and using , Thus we have:

Complex Envelope Representation

PROOF: Any physical waveform may be represented by the Complex Fourier Series

c n - negligible magnitudes for n in the vicinity of 0 and, in particular, cIntroducing an arbitrary parameter 0 =0 fc , we get

=>g(t)– has a spectrum concentrated nearf=0 (i.e.,g(t) - baseband waveform)

R e    12    12  *

 THEOREM: Any physical bandpass waveformv(t) can be represented by

wherefc is the CARRIER frequency and c 2  fc

v (^)  t R e g (^)  t (^) ej^ ^ ct

 Converting from one form to the other form

 Equivalent representations of the Bandpass

signals:

Complex Envelope Representation

v  t   x  t  cos  ct  y  t si n ct I n p h ase an d Q u ad ratu re (I Q ) form

g  t   x  t   j y  t   g ( )t e j^ g^ (^ t)^  R ( )t e j^ (^ t) C om p l ex E n v el op e of v t( )

Inphase and Quadrature (IQ) Components.

  (^)   

  (^)   

R e ( ) cos ( )

I m ( ) si n ( )

x t g t R t t

y t g t R t t





 

 

Envelope and Phase Components

 

2 2

1

( ) ( ) ( )

( ) ( ) ( ) tan ( ) ( )

R t g t x t y t

y t t g t x t

 

  

  

  R e^   ^ c    cos^   E n v el op e an d P h ase form

j t v t g t e R t ct t

Representation of Modulated Signals

The complex envelope g(t) is a function of the modulating signal m(t) and is

given by: g(t)=g[m(t)] where g[• ] performs a mapping operation on m(t).

The g[m] functions that are easy to implement and that will give desirable

spectral properties for different modulations are given by the TABLE 4.

At receiver the inverse function m[g] will be implemented to recover the

message.

Mapping should suppress as much noise as possible during the recovery.

 (^) Modulation is the process of encoding the source informationm(t) into a bandpass signal s(t). Modulated signal is just a special application of the

bandpass representation. The modulated signal is given by:

  R e^  ( )^ c ^2

j t s t g t e c fc

 ^   

Bandpass Signal Conversion

g( t )

s( t )

1 0 1 0 1 2

Ac 2

0

 Ac 2

X n

Unipolar X Line Coder

cos(ct)

Xn g(t) A c

s( t )

 On off Keying (Amplitude Modulation) of a unipolar line

coded signal for bandpass conversion.

Mapping Functions for Various Modulations

Eeng 360 14

Envelope and Phase for Various Modulations

PSD of Bandpass Signals

,  

 (^) PSD is obtained by first evaluating the autocorrelation forv(t):

Using the identity where (^) and

• Linear operators

but

AC reduces to PSD =>

We get

o r

f (^) c frequencie sin g(t)

Evaluation of Power

g (^)  t

Theorem: Total average normalized power of a bandpass waveformv(t) is

Proof:

But

So,

or

But is always real So,

        2 1 2 0 v v v 2 P v t P f d f R g t





Example : Amplitude-Modulated Signal

Spectrum of AM signal.

Example : Amplitude-Modulated Signal

Total average power: