Signal Transmission and Modulation

Prof. Dr.-Ing. I. Willms Transmission and Modulation S. 1

FachgebietNachrichtentechnische Systeme

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Signal Transmission and Modulation

Prof. Dr.-Ing. Ingolf Willms

Partly based on the script of Prof. Thomas Kaiser



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Chapter 4

Digital TM

4.1 Basics of detection theory



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4.1 Basics of Detection TheoryDigital TM:

- required is to transmit data streams with lowest bit error rate (BER)

- thus optimal distinguishment of binäry data on the base of a sequence of predefined functions are needed

- combining M binary data leads to distinguishing 2M symbols zu unterscheiden



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TM method applied now: Signal x(t) is transmitted in case (H1) of „logical 1“, no signal in case (H0) of „logical 0“.

Signal x(t) has a duration of T and an energy of Ex.

Additive Gausssian noise n(t) is assumed Für die beiden Fälle H0 und H1 gilt damit:

4.1 Basics of Detection Theory

2

0

( )T

xE x t dt

0 1: ( ) ( ) : ( ) ( ) ( )H y t n t H y t x t n t



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The receiver has to decide on the base of y(t) for the case of H0 or H1 .

Instead of y(t) now coefficients of a set of orthonormal functions fn(t)describing it are evaluated for which hold:


0 0

( ) ( ) mit ( ) ( ) und 1n

T

n n n n fn t

y t Y f t Y y t f t dt E n

Due to linear relations of coefficients and the functions hold:

n n nY X N 0 0

0 0

( ) ( ) mit ( ) ( )

( ) ( ) mit ( ) ( )

T

n n n nn t

T

n n n nn t

x t X f t X x t f t dt

n t N f t N n t f t dt



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The best detection situation can be obtained by choosing a proper set of functions fn(t) with:


0 ( ) ( ) / xf t x t E

This gives:

0 0 00

0 00 0

0 0

( ) ( ) ( ) ( )

( )due to X ( ) ( ) ( )

which leads to . Important is: 0

n n xn

T Tx

t t x x

x

x t X f t X f t E f t

Ex tx t f t dt x t dtE E

X E X



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For making the decision now the follwing data have to becompared:


0 0 0 1 1 2 2

1 0 0 0 1 1 1 1 2 2

: ...: ...

H Y N Y N Y NH Y X N Y X N N Y N

0 00 0

1 0 0

Thus only on the basis of

1( ) ( ) ( ) ( )

the decision can be made.In case of , is always larger than in case of !

T T

t tx

Y y t f t dt y t x t dtE

H Y H



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• Example with separation into sine functions and matching x(t)• The Xn here represent the Fourier coefficients bn

0

0 0

2 00

0

0 0 00

0 00

( ) ( / - 0.5)sin( )

sin ( )2

2( ) ( ) / ( / - 0.5)sin( )

2( ) ( / - 0.5)sin( )

T

x

x

n

x t rect t T t

TE t dt

f t x t E rect t T tT

f t rect t T n tT

0

0 0

( ) ( ) mit ( ) ( )T

n n n nn t

x t X f t X x t f t dt



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For M orthonormal functions xm(t) and for possibly different signal energiesthe following block diagram for AWGN channels results.


0

. . .T

d t

0

. . .T

d t

0

. . .T

d t

Choose the Largest

1( )x t

2 ( )x t

( )Mx t

1

12 xE

2

12 xE

12 MxE

( )y t ( )truemx t



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Alternative method for realising the multiplication&integration:The operations are realised by a „matched“ filter.


00

1 ( ) ( ) .T

x

Y y t x t dtE

00

00

1The operation replaces ( ) ( )

1 by ( ) ( ) with ( ) ( ) or h( ) ( )

T

x

T

x

Y y x dE

Y y h T d x h T x TE

00 0

This is achieved by a filter and sampling the output signal at :

1 1( ) ( ) ( ) ( ) ( )t T

t T t Tx x

t T

Y g t y h t d y h T dE E



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1 ( )x T t

Choose the Largest

1

12 xE

2

12 xE

12 MxE

( )y t( )y t

( )truemx t

2 ( )x T t

( )Mx T t

sample T = t

+

+

+




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Chapter 4

Digital TM

4.2 Pulse Amplitude Modulation (PAM)

Mit freundlicher Unterstützung von Pearson-Studium –Die Abbildungen sind z.T. entnommen aus dem Buch

„Grundlagen der Kommunikationstechnik“von J.G. Proakis, M. Salehi



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Criteria for good transmission• Robustness concerning noise• Low bandwith usage• Low Tx power

4.2 PAM



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Transmission of data stream by amplitude modulated signals PAM with 0 and A as signal amplitude

• Bipolar (binary antipodal) with amplitudes –A and +A • Non binary PAM methods use M amplitude values

4.2 PAM

Binary and quaternary signals



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Binary Modulation• Each binary value corresponds to a signal with duration TD

• Thus no additional coding required

4.2 PAM

Non binary Modulation• L bits are combined to one symbol• Thus 2L symbols result • Transmission time for one symbol is TS = L * TD

• L less changes of signal levels• For equal maximum level L = 2L smaller amplitude values

need to be detected• Compromise of S/N ratio and bandwith usage is required



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4.2 PAM1 1 0 1 0 0 0 1 0 1 1 0

2 ASK BPSKBinary Modulation x t x t

DT

4 ASKQuaternary Modulation x t

S DT 2T

8 ASKOctonary Modulation x t

S DT 3T

d 2

d 2

d 23d 2

d 23d 25d 27 d 2

0



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Minimisation of bandwith usage • Avoiding of rectangular signal form due to si-form of spektrum• Alternatives of signal forms in time-domain: si-Function, Gauss function

4.2 PAM



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Bandpass signals• Modulation is typically required (matching channel properties)• Sequence of Gauss funktions form the signal in base band • Modulation of this signal by AM, PM or FM

4.2 PAM

Gauss signal and its Spectrum of baseband (a) andAM-Modulator AM signal (b)



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Chapter 4

Digital Transmission

4.3 Transmission usingorthogonal signals



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4.3 Transmission using orthogonal signals

Ein

• Extended example of S.8 with separation into sine and cosine functions and corresponding xm(t) functions

• Xim correspond to Fourier coefficients an and bn

0

1 0 0

2 01 0

0

01 1 1 0 00

1 0 00

( ) ( / - 0.5)cos( )

cos ( )2

2( ) ( ) / ( / - 0.5)cos( )

2( ) ( / - 0.5)cos( )

T

x

x

n

x t rect t T t

TE t dt

f t x t E rect t T tT

f t rect t T n tT

0 0( ) ( ) mit m 1, 2m m mx t X f t

0

2 0 0

2 02 0

0

02 2 2

2 0 00

( ) ( / - 0.5)sin( )

sin ( )2

( ) ( ) /

2( ) ( / - 0.5)sin( )

T

x

x

n

x t rect t T t

TE t dt

f t x t E

f t rect t T n tT



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M orthogonal functions xm(t) are used in detection

0

. . .T

d t

0

. . .T

d t

0

. . .T

d t

Choose the Largest

1( )x t

2 ( )x t

( )Mx t

1

12 xE

2

12 xE

12 MxE

( )y t ( )truemx t




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Example for coding in the transmitter• For binary value “0” x1(t) is transmitted• For binary value “1” x2(t) is transmitted

2

0 0

1 1( ) ( ) = ( ) mit 1, 2m

m m

T T

K m m xt tx x

Y y t x t dt x t dt E mE E

For noise free transmission it results: • For binary value “0” : Y01 = YK > 0, Y02 = 0

• For binary value “1” : Y01 = 0, Y02 = YK > 0 with


0 00 0

The value of

1( ) ( ) ( ) ( ) mit 1, 2

gives the criteria for choosing one hypothesism

T T

m m mt tx

Y y t f t dt y t x t dt mE



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Constellation diagram• The values Werte für Y01 und Y02 can be represented in a graphic for

good • Thus the degree of robustness can be figured out

Distance between marked points indicates degree of robustness against noise

02Y

01Y

K(0,Y )K(Y ,0)




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Other orthogonal base band signals• Sine and cosine functions are

orthogonal• This also holds for sine/cosine

signals of n times the fundamental frequency

• Walsh functions are orthogonal• Other functions are known and can

be specified

2 examples of 4 orthogonal functions




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Chapter 4

Digital TM

4.4 Quadrature AM



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4.4 QAMPrinciple of QAM• QM uses different amplitude and phase levels of the carrier• The number of M symbols depends on possibilities of combing

amplitude and phase levels

2 2

( ) cos( ) sin( )

cos( arctan( / ) mit 1, 2, ...

QAM m c m c

m m c m m

x t a t b t

a b t b a m M



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Example with 2 amplitude and 2 phase values• 4 different base band signals result• This QAM version can also be understood as phase shift keying

(4-PSK) verstanden werden• The constellation diagram describes here real and imaginary part

of the equivalent LP signal of the carrier

4.4 QAM

a) Constellationsdiagramm for M = 4

02Y

01Y

K(0, Y )

K( Y ,0)

K(0, Y )

K(Y ,0)

b) The same degree of robustness results for a rotated diagram, see next slide



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4.4 QAM4 Signals of the constellation diagram a) und b):

1

2

3

4

Zu a):( ) cos( ) ( / 0.5)( ) sin( ) ( / 0.5)( ) cos( ) ( / 0.5)( ) sin( ) ( / 0.5)

c

c

c

c

x t t rect t Tx t t rect t Tx t t rect t Tx t t rect t T

1

2

3

Zu b):

( ) 1 / 2(cos( ) sin( )) ( / 0.5) cos( / 4) ( / 0.5)

( ) 1 / 2( cos( ) sin( )) ( / 0.5) cos( 3 / 4) ( / 0.5)

( ) 1 / 2( cos( ) sin( )) ( / 0.5)

c c c

c c c

c c

x t t t rect t T t rect t T


x t t t rect t T

4

cos( 3 / 4) ( / 0.5)

( ) 1 / 2(cos( ) sin( )) ( / 0.5) cos( / 4) ( / 0.5)

c

c c c

t rect t T




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0 1 1 0 0 1

k

4 PSKx t

i,4 PSKx t

q,4 PSKx t

/ 4

1

1

1

1

1

1

3 45 47 4

t

t

t

t

k



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4.4 QAMAdditional extensions for M > 4• Additional amplitude and phase levels give more symbols

Constellation diagram for M = 16

0

0

Example for M = 16 ( ) ( 5 / 3 2 / 3) ( / - 0.5) cos( ) 1 4( ) ( 5 / 3 2 / 3) ( / - 0.5) sin( ) 5 8

i c

i c

x t i rect t T t ix t i rect t T t i



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• The graphics show the constellation diagrams for comparison of 4-ASK and 8-ASK

4.4 QAMM=4

00 10 11 011x 2x 3x 4x

33 1 1M=8

000 101001 100 110 111 011 0101x 2x 3x 4x 5x 6x 7x 8x

5 711357 3



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Further variants• In principle there is no limit for the

number of symbols • Different constellation diagrams can

be selected• Large M values lead to high

sensitivity to noise• Reason: Low distances between

points in constellation diagram

4.4 QAM



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4.4 QAM

Modulator for QAM The 2 filters specify the behaviour of impulses in time domain



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Receiver for QAM signals

4.4 QAM



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Chapter 4

Digital TM

4.5 PSK and FSK



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Principles for PSK/FSK• Constant amplitude for all symbols bei suitable modulation

of I/Q carrier • Constellation diagram thus always includes a circle• Again distance of points corresponds to robustness

4.5 PSK and FSK

Constellation diagrams for different number of symbols for PSK



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Properties• Similar to PM/FM of analog signals (e,g. conc. power efficiency)• Problem: Jumps due to chnges in the phase enlarge the bandwidth

4.5 PSK and FSK

Modulated carrier signal for M = 4 (4-PSK or QPSK)



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4.5 PSK and FSKGray-Coding• In principle freedom is given conceerning relation of

coed and symbols• Gray coding reduces also here bit errors

Example for a relation of bit combinations and phase values of the modulated carrier

for D n 0 and D n 1 043 for D n 1and D n 1 04k5 for D n 1and D n 1 14

7 for D n 0 and D n 1 14



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PSK-Demodulation• For that I- and Q-signal have to be determined• A PLL takes care of carrier phase estimation

4.5 PSK and FSK

Demodulator for PSK signals



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Offset-PSK• Phase changes lead to short-time drop of carrier amplitude• An “offset” (delay ) between I- and Q-Signal prevents this

4.5 PSK and FSK

Possible signal changes of 4-PSK Signal changes of offset-PSK

12

3 4

d2

d2

d2

d2

1f

0f

d2

d2

d2

d2



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0 1 1 0 0 1

4

d2

d2

d 2

d 2

d 2

d 2

3 45 47 4

t

t

t

t

k

k

4 OPSKx t

i,4 OPSKx t

q,4 OPSKx t

DT



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4.5 PSK and FSKDifferential PSK (DPSK)• In DPSK only phase chnages are evaluated • In DQPSK theer are only phase jumps of ±45° and ±135°• Phase jumps of 180° are avoided• This is achieved by alternating angle values (different for even and odd

clocks)

For phase differences hold:

k k 1 k

For DQPSK holds:32k 0, , ,

2 2

3 5 72k 1 , , , .4 4 4 4



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Advantages of DPSK• Almost constant amplitude• DPSK needs no knowledge of exact carrier phase!

4.5 PSK and FSK

The constellation diagram with possible phase values of DQPSK

qx

ix

d2

d2d

2

d2

Even K

Odd K



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4.5 PSK and FSKFSK signals• Modulation comparable to FM• In FSK also phase jumps can happen due to symbol changes• Only discrete frequencies are transmitted• Signals are almost orthogonal • Für Tx signal (with non-symmetrical frequency bands with

regard to carrier frequency holds:

By changing of frequencies by (M-1)ωd /2 a symmetrical spektrum can berealised.

Sm C d m

S

d

t T 2x t cos m t rectT

with as minimum difference frequency and m = 0, 1, ... M-1



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4.5 PSK and FSK

The first expression drops close to zero for large argument, the second expression is zero for:

Orthogonality property of FSK signals• Here 2 signals of different symbols are tested for

othogonalityt• This test leads to:

m kx t x t dt

ST

C d C d0

cos m t cos k t dt

C d SS

C d S

sin 2 m k TT2 2 m k T

S ST T

C d d0 0

1 1cos 2 m k t dt cos 2 m k t dt2 2

d SS

d S

sin 2 m k TT .2 2 m k T

dS

.2T



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Demodulation of M-FSK signals

4.5 PSK and FSK



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4.5 PSK and FSKContinuos PSK and FSK• An FSK modulator can be realised by switching M different

oszillators• Avoiding phase jumps reduces bandwidth• The same holds for jumps of instantanous frequency• Realisation by memories and by• Continuos phase jumps



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4.5 PSK and FSK

Carrier phase for binary CPFSK

• Points in constellation diagram moves with time!

• FSK gives movement at constant radius

• In CPFSK phase does not jump

• Shown points represent state at end of a clock



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4.5 PSK and FSK

Carrier phase for quaternary CPFSK



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4.5 PSK and FSKMSK signals (Special case of CPFSK)• Orthogonality for smallest frequency change is looked for• 2 equivalent LP carrier signal s are consideerd• Phase values at end of the clock are mI times of π

I I

S S

m mj t j tT T

T1 T2x t e x t e

S S

I

T T !2 j m*T1 T2 T1 S I

0 0

Due to complex valued LP signals holds:

x t x t dt x t dt T e si 2 m 0 I

1m is minimum value for orthogonality2

S

dS

Phasen rising about /2 in period T

gives instantaneous frequency of .2T



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4.5 PSK and FSK

a) I-Signal (real part of equiv. LP signal of MSK-modulated carrier)b) Q-Signal (imaginary part of it)c) MSK-Signal



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Power spectrum desnity of MSK compared to offset QPSK [Gronemeyer und McBride, 1976]

4.5 PSK and FSK



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4.5 PSK and FSKGMSK signals• Also MSK still produces frequency jumps• Further reduction of bandwidth results by LP filterung• A Gaussian filter is typically used

For its impulse response holds:

2

g

ln 22

GAUSS g

Transfer function of this filter:

H e with as cut-off frequency

2gt

g 2ln 2GAUSSh t e

2 ln 2



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4.5 PSK and FSK

Impulse response of Gaussian filters for different cut-off frequencies with ω3dB = ωg



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4.5 PSK and FSK

Comparison of phase of binary MSK and GMSK

0 1 1 0 0 1

MSK GMSK

t

t

GMSK t

DT

2

2



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4.5 PSK and FSK

Application of GMSK in mobile phones



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Chapter 4

Digital Transmission

4.6 Aspects of Symbol Interference



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4.6 SymbolinterferenzRelation bandwidth to data rate• For the AWGN situation an optimal matched filter reception is

assumed (see chap. 4.1)• ASK is the chosen transmission method

00

Without noise with ( ) ( ) for each symbol the value

1 ( ) ( ) = with ( ) ( ) is determined which corresponds to filtering

(correlation) of the signal ( ) with ( ) ( - ) .

For a sym

T

x

y t x t

Y y h T d x t h T tE

y t y t x T t

0 0

bol sequence ( ) at the matched filter output the signal results:

( ( ) ( - )) * ( - ) ( ) ( - - ) with the ACF ( ) ( ) ( )xx xxk k

S k

S k x t kT x T t S k t T kT t x x t d



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4.6 Symbolinterferenz

0 0

xx xx xxk 0

xx xx xx xx

S(k) (k) (k 1) For that the following output signals result:

S(k) (t - T - kT) S(0) (t - T) S(1) (t - 2T)

For the sample values therefore hold:At t T: S(k) (T - T) (T 2T) (0) (-T)

At

xx xx xx xx t 2T: S(k) (2T - T) (2T 2T) (T) (0)

Due to sampling the following sequence of values results:

xxk 0

xx xx

S(k) ( kT)

For preventing an overlapping of neighboured symbol values (i.e. no symbol interference) and because of the even ACF it must hold:

( kT) (kT) 0 k 0 (1st Nyquist criteria)

Example: Binary sequence with only 2 non-zero values



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Possible ACF’s:• Signals limited to symbol duration such as the rect-funktion• Rectangular signals give a ACF’s in triangular form• Disadvantage: For a perfect transmission in principle an infinitely

large bandwidth is needed• The ideal frequency limited signal is the si function • Disadvantage of the si function: Symbol interference in case of

unprecise sampling points

xx S

g

t Tsi( t / T )

This signal demands a bandwidth of (50% of the clock rate)T



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4.6 Symbolinterferenz Problems in the application of the si-function• High demands on exact sampling points• The problem is eased by a modification of the si-function • Disadvantge of this method: Not optimised use of the bandwidth

Original and modified si-function and its spectra



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4.6 Symbolinterferenz Other effects• In practice non-ideal channels are often given with strong changes in the

frequency response – even in the case of transmission at a small bandwidth

• This has similar effects as a non-si-form of the ACF• Counter measure: Application of a bandwith efficient method called

“Orthogonal Frequency Division Multiplex” (OFDM)• Basic idea: Separation of the whole channel into many sub-channels with

a constant frequency response and equal bandwidth



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4.6 Symbolinterferenz FSK signals in the subchannels• All K subchannels will be used simultaneously• Each subchannel has its own carrier• The symbolrate and the corresponding period T = KTS of each subchannel

will be adjusted to the K-times lower bandwidth (in comparison to the transmission without OFDM)

• Also the different S/N ratios can be taken into account

For K > 25 in the Tx and Rx devices FFT algorithms are applied instead of a bank of band-pass filters..

k k m

T

k k i i0

Carriers in the k's subchannel:x t cos t with k 0,1, ... K -1All carriers are orthogonal to each other due to:

cos t cos t dt 0 i k



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Block diagram of an OFDM transmission system


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