Baseband equivalents of bandpass signals - TKK ... · Baseband equivalents of bandpass signals ......

© NOKIA FILENAME.PPT / 13.12.1997 / NN page: 1

Baseband equivalents of bandpassBaseband equivalents of bandpasssignalssignals

Jukka Henriksson


ContentsContents

• Background, motivation and goals

• Complex baseband representation of signals

• Complex representation of noise

• Channel models

• Examples of modulated signals

• Signal-to-noise considerationsIf time allows


Background, motivation and goalsBackground, motivation and goals

• Simulation requires digital representation of signals

• sampling in time domain

• efficient representation needed

• minimum amount of samples

• noise (and interference) should be correctly included

The way: complex baseband representationThe way: complex baseband representation


Background, motivation and goalsBackground, motivation and goals((cont’dcont’d))

• Direct application of sampling theorem

• ⇒ required sampling frequency fs > 2 ⋅ fm (??)

fm

f

ncs1.drw

0fc


Background, motivation and goalsBackground, motivation and goals((cont’dcont’d ))

• For complex bb-representation only fs> 2.fb is needed

fbf

ncs1.drw

0


Useful complex relationshipsUseful complex relationships

• Define complex number:

• Then

(1)

• and

(2)

{ } { } ( ) ( )

{ } { }( )*2121

*22

*1121

ReRe2

12

1

2

1ReRe

zzzz

zzzzzz

+=

+⋅+=⋅

{ } { } ( ) ( )

{ } { }( ) { } { }( )2*121

*2121

*22

*1121

ImIm2

1ImIm

2

1

2

1

2

1ImRe

zzzzzzzz

zzj

zzzz

+=−=

−⋅+=⋅

jyxz +=


Useful complex relationships (Useful complex relationships (cont’dcont’d))

• and

(3)

• and also

(4)

(5)

{ } { } ( ) ( )

{ } { }( )*2121

*22

*1121

ReRe2

1

2

1

2

1ImIm

zzzz

zzj

zzj

zz

+−=

−⋅−=⋅

{ } { } { } { } { }2121212121 ImImReReRe zzzzyyxxzz −=−=

{ } { } { } { } { }2121212121 ReImImReIm zzzzxyyxzz +=+=


Complex representation of bandpassComplex representation of bandpasssignalsignal

Assume bandpass signal

(6)

this is equivalent to

(7)

where the complex envelopecomplex envelope is defined

(8)

v t A t t tc( ) ( )cos( ( ))= +ω φ

{ } { }[ ]

v t A t e z t e

z t e z t e

j t j t j t

j t j t

c c

c c

( ) Re ( ) Re ~( )

~( ) ~( )

( )

*

= =

= +

+

−

ω φ ω

ω ω12

~( ) ( ) ( )z t A t e j t= φ


Alternative real representationAlternative real representation

v t x t t y t tc c( ) ( )cos( ) ( )sin( )= −ω ω

~( ) ( ) ( )z t x t jy t= +

BP-signal of (7) may be written also

(9)

where complex envelope has been given using

inphase (I) and quadrature (Q)

components

(10)


Spectral considerationsSpectral considerations

• If v(t) is bandpass => A(t) and φ(t) are slowly varying lowpass type

• If ωc is the center frequency of the bandpass signal and ωc> ωb =>complex envelope is unique

fc

f

ncs2.drw

0fc + f

b-f

c

V(f)

1/2


Spectral considerations (Spectral considerations (cont’dcont’d))

Fourier transform of (7) gives

(11)

so the bandpass spectrum can be given using only

information of

[ ]V f Z f f Z f fc c( )~

( )~

( )*= − + − −12

~( )Z f

!��


fb

f0

1Z(jω)∼

Spectrum of the complex low-passSpectrum of the complex low-passsignalsignal

Notice the spectrum height!


Recovery of quadrature component xRecovery of quadrature component x

(12)

Fourier transforming we get

(13)

The transform components are bandlimited Õ

lowpassfiltering the result we get 1/2 X(f) or

(14)

{ } { } { }[ ]

v t e z t z t e

x t z t e z t e

j t j t

j t j t

c c

c c

( )Re Re ~( ) Re ~( )

( ) ~( ) ~( )

−

∗ −

= +

= + +

ω ω

ω ω

12

12

2

12

14

2 2

[ ]12

14 2 2X f Z f f Z f fc c( )

~( )

~( )+ − + − − ∗

{ }[ ]x t v t e j t

LP

c( ) ( )Re= −2 ω

��


Recovery of quadrature component yRecovery of quadrature component y

(15)

Fourier transforming we get

(16)

and hence

(17)

{ } { } { }[ ]

v t e z t z t e

y t z t e z t e

j t j t

jj t j t

c c

c c

( )Im Im ~( ) Im ~( )

( ) ~( ) ~( )

−

∗ −

= −

= − −

ω ω

ω ω

12

12

2

12

14

2 2

[ ]12

14 2 2Y f Z f f Z f fj c c( )

~( )

~( )+ − − − − ∗

{ }[ ]y t v t e j t

LP

c( ) ( )Im= −2 ω


Spectra of low-pass componentsSpectra of low-pass components

Starting from (14) Fourier-transforming, using (11) we get

(18)

and similarly

(19)

The following figure demonstrates above.

(Note: x(t) and y(t) real => X(f) and Y(f)conjugate symmetric i.e. X(-f)=X(f)* )

(Note: x(t) and y(t) real => X(f) and Y(f)conjugate symmetric i.e. X(-f)=X(f)* )

[ ]X f Z f Z f( )~

( )~

( )= + − ∗12

[ ]Y f Z f Z fj( )~

( )~

( )= − − ∗12


Derivation of X(f) from BP spectrum V(f)Derivation of X(f) from BP spectrum V(f)

f0 fc-f

c

V(f)

1/2

ncs3.drwf0

1

X(f)

f0

1

∼Z(-f)* Z(f)

∼

⇒


Linear filtering of BP signalsLinear filtering of BP signals

Assume BP-filter impulse response

(20)

Let input be v(t) − then the output w(t) is

(21)

{ }tj cethth ω)(~

2Re)( =

{ }{ }

{ }

w t h t v d

h t e z e d

h t e z e d

h t z d e

j t j

j t j

j t

c c

c c

c

( ) ( ) ( )

Re~

( ) ~( )

Re~

( ) ~( )

Re~

( )~( )

( )

( )

= −

= − +

−

= −

∫∫

∫∫

−

− ∗ −

τ τ τ

τ τ τ

τ τ τ

τ τ τ

ω τ ω τ

ω τ ω τ

ω

12

12

2

2


((cont’dcont’d))

and hence the complex envelope of the output w(t) is

(22)

[However, in simulations this is usually not critical;

it is only constant scaling for signals and noise]

~( )~

( )~( )w t h t z d= −∫ τ τ τ

Note the use of number 2 in defining the complex envelope of filter impulse response

Note the use of number 2 in defining the complex envelope of filter impulse response


Bandpass random signals and noiseBandpass random signals and noise

Assume wide sense stationary (WSS) BP random signal

(23)

or for simplicity

(24)

Autocorrelation function

(25)

Due to WSS => second term must be ZERO

{ }n t n t e n t t n t tj tc c s c

c( ) Re ~( ) ( )cos( ) ( )sin( )= = −ω ω ω

n t x t t y t tc c( ) ( )cos( ) ( )sin( )= −ω ω

{ } { }R n t n t

n t n t e n t n t e

n

j j j tc c c

( ) ( ) ( )

Re ~( ) ~( ) Re ~( )~( )

τ τ

τ τω τ ω τ ω

= +

= + + +∗ +12

12

2


Bandpass random signals and noise (2)Bandpass random signals and noise (2)

Define

(26)

Then

(27)

WSS condition gives

(28)

which gives general properties for the inphase and quadrature correlations

R n t n t

x t x t y t y t j x t y t y t x t

R R j R R

n

x y xy yx

~( ) ~( ) ~( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )

( ) ( ) [ ( ) ( )]

τ ττ τ τ τ

τ τ τ τ

= += + + + + + − += + + −

∗

{ }R R en nj c( ) Re ( )~τ τ ω τ= 1

2

~( )~( ) ( ) ( ) [ ( ) ( )]n t n t R R j R Rx y xy yx+ = − + + =τ τ τ τ τ 0


Bandpass random signals and noise (3)Bandpass random signals and noise (3)

(29)

and hence

(30)

Note that the autocorrelation of the complex envelope may be complex (but thespectrum is real!)

R R

R R

x y

xy yx

( ) ( )

( ) ( )

τ ττ τ

=

= −

)(2)(2)(~ τττ xyxn jRRR +=

S f S f f S f fn n c n c( ) [ ( ) ( )]~ ~= − + − −14

Fourier transforming (27) we get

(31)

As before we find low-pass spectrum(U(f) is the "unilateral step function")


Bandpass random signals and noise (4)Bandpass random signals and noise (4) (32)S f S f f U f fn n c c~( ) ( ) ( )= + +4

f0 fc

-fc

S ( f)

No/2

n

ncs4.drw

f0

S ( f) 2Non∼


Spectra of quadrature componentsSpectra of quadrature components x(t) and y(t) x(t) and y(t)

From (30) we get

(33)

and F-transform gives

(34)

[see following picture]

R R R

R R R

x n n

xy j n n

( ) [ ( ) ( ) ]

( ) [ ( ) ( ) ]

~ ~

~ ~

τ τ τ

τ τ τ

= +

= −

∗

∗

14

14

)]()([)(

)]()([)(

~~41

~~41

fSfSfS

fSfSfS

nnjxy

nnx

−−=−+=


Spectra and cross-spectra of low-passSpectra and cross-spectra of low-passcomponents x and ycomponents x and y

f0

S (f) 2Nox

3No/4

f

S (f) 2Non∼

-S (-f)n∼

ncs5.drw

f

jS (f)

2No

xy

No/4

f0

S (f)

2Non∼S (-f)

n∼

No


Conclusions about low-pass spectraConclusions about low-pass spectra

Note that if BP-spectrum is symmetric (ref fc) then

(35)S f

R

S f S f S f f U f

R R

xy

xy

x n n c

x n

( )

( )

( ) ( ) ( ) ( )

( ) ( )

~

~

≡

≡

= = +

=

0

0

212

12

τ

τ τ

So• low-pass components are uncorrelated at any t• low-pass spectrum is twice the (two-sided) BP spectrum• Hey, keep eye on those damned "engineer’s 2s"


Symmetric spectraSymmetric spectra

f0 fc

-fc

S (f)No/2n

f0

S (f) 2Non∼

ncs6.drw

f0

S (f)

Nox


Power considerationsPower considerations

Power of the BP-process

(36)

From (27) we have

(37)

From (29), (30) and (37) we conclude (real variables!)

(38)

P Rn n= =( )0 2σ

R Rn n~ ( ) ( )0 2 0 2 2= = σ

R

R R P P

xy

x n x y

( )

( ) ( )~

0 0

0 012

2

=

= = = =σ


RemarksRemarks

• Power of each component is the same as total BP power (general result)

• In general case inphase and quadrature components are uncorrelated at theSAME time instant (but not necessarily if t ≠ 0)

• For Gaussian random process samples taken at the SAME instant are thenindependent

• Only for symmetric BP Gaussian process are also x(t) and y(t) independentGaussian processes (at any time)

So, if you are dealing with unsymmetric cases you might consider including correlation into your model

So, if you are dealing with unsymmetric cases you might consider including correlation into your model


Complex white Gaussian processComplex white Gaussian process

Assume that

• BP spectrum is constant No/2 (two-sided), Gaussian

• ideal rectangular filtering, bandwidth W

• system filtering much narrower than W

then the complex noise envelope is characterized (may be approximated as)

(39)S f N f W

R Nn o

n o

~

~

( ) , /

( ) ( )

= ≤

=

2 2

2τ δ τ


Complex white Gaussian process (2)Complex white Gaussian process (2)

and the joint density function for x(t) and y(t) is

(40)

where σ2 is the filtered noise power (e.g. NoW). This can also be written in theform:

(41)

EXTENSION RESULT:

General N-dimensional complex Gaussian

p x yx y

xy ( , ) exp= − +

12

2 2

22

2πσ σ

−=2

2

21

~2

~exp)~( 2 σπσ

nnpn


Assume zero mean and stationarity

(42)

where bold x means (column) vector, T stands for transpose and Λ forcovariance matrix.

N-dimensional complex GaussianN-dimensional complex Gaussiandensitydensity

E

E

E

T

T

[~]

[~~ ]

[~~ ]

~

x

xx

xxx

=

≡

=

∗

0

2

0

Λ


N-dimensional complex Gaussian densityN-dimensional complex Gaussian density(2)(2)

Probability density function for these joint complex Gaussian variables is

(43)

where

(44)

{ }pN

T~

~~(~)

( ) detexp ~ ~

xx

xx x x= − ∗ −1

212

1

π ΛΛ

[ ]~ ~ , ~ ,...~ ~

~ ~ ...~ ~

:

:

~

x

x

TN N

N

x x x x

x x

x x

=

=

−

∗

∗

1 2 1

12 1

2 1 2

12 2 1 2

2

2

Λ

σσ

σ


Channel modelingChannel modeling

We use complex low-pass representation

• if the signal BP bandwidth is W only samples at rate W are needed

• system filtering may be represented using complex samples 1/W apart

• also channel may be represented with complex samples1/W apart

In practice somewhat denser sampling is preferable

Note also that time varying channel expands bandwidth (Doppler spread) anddenser tapping is needed


Sampled channelSampled channel

• Generally coefficients hi are complex numbers (possibly time varying)

• Often channel is assumed "freezed" i.e. fixed for a number of transmittedsymbols

• Often coefficients complex Gaussian

+

1/W 1/W 1/W 1/W

....h0

~h1

~h2

~

ncs7.drw


ExamplesExamples

Fixed radio link channel

• normally only two taps needed

• freezing can be assumed

• main component may have nonzero mean i.e.

(45)

Note: magnitude of h1 follows Rayleigh law in fading if it is zero mean complexGaussian

~ ~

~

~ ~

h h h e

h

h h e

m mj

j

o0 0 0

1

1 1

0

1

= =

=

=

ϕ

ϕ


Mobile radio channelMobile radio channel

• normally a few taps needed (e.g. 5 or so)

• channel varies relatively fast compared to symbol lengths => Doppler spreads

• time variations are important => time varying coefficients often used in models

• in many cases all taps may be zero mean complex processes

• but also it may be possible that one or more taps have nonzero mean(specular reflections)


Modulated signals with complexModulated signals with complexenvelopesenvelopes

Many digital modulations can be represented in a general form with complexenvelope

(46)

where u(t) and w(t) are the basic pulse forms.

Example:

QPSK

an gets values +1+j and bn is zero

u(t) is, e.g., half Nyqvist pulse

∑ ∑ −+−=n n

nn nTtwbjnTtuatz )()()(~)(~ tz


Signal-to-noise ratioSignal-to-noise ratio

Let us consider QPSK signal with u(t) rectangular pulse [0,T]

(47)

Assume ideal T-integrator in the receiver

The output of the integrator at t=(k+1)T (compl.env)

(48)

{ }r t v t n t a Eu t nT e n t enj t

n

j tc c( ) ( ) ( ) Re ( ) Re ~( )= + = −

+∑ ω ω

~( ) ( ) ~[( ) ]

~

g kTT

a u T dT

n k T d

a E n

kET

T T

k

= − + + −

= +

∫ ∫1 1

10 0

1

τ τ τ τ


Signal-to-noise ratio (2)Signal-to-noise ratio (2)

Signal power is now

(49)

and noise power

(50)

Note that variance of each noise component (I and Q) is now No.

a E Ek

22=

~ ~[( ) ]~[( ) ]n n k T n k T d d

N d N

T

T o

T

o

1

2 11 2 1 2

11

0

1 1

2 2

= + − + −

= =

∫∫

∫

∗τ τ τ τ

τ


Probability of errorProbability of error

Probality of symbol error (appr) is now

(51)

where Eb is the bit energy and E is the real energy of one pulse.

P QE

NQ

E

Neo

b

o

= =( ) ( )2

I

Q

E√

ncs8.drw


FinallyFinally

• complex envelopes reduce simulation complexity

• complex envelopes make conceptually simple analysis of I-Q signals

• some care is needed when calculating signal-to-noise ratios (and may besignal-to-interference also) ["engineer’s 2’s"]

• WARNING: if system contains nonlinearities, one should watch what signalrepresentations are valid (bandwidth expansion!)

• also fast time varying channels may require special treatment


ReferencesReferences

1. Benedetto, Biglieri, Castellani: Digital Transmission Theory.Prentice Hall International Inc. New Jersey 1987, 639 p.

2. vanTrees H.,L.: Detection, Estimation and Modulation Theory,Part III. John Wiley & Sons, Inc., New York 1971, 626p.

Baseband equivalents of bandpass signals - TKK ... · Baseband equivalents of bandpass signals ......

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