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									                        Communications Part II

4. Optimal Receiver for AGN                       3. Error Probability at Viterbi
       1. Optimal AWGN Receiver                      Detection
       2. Generalization for coloured             4. Influence of Channel Impulse
          noise                                      Response
       3. Symbol / Bit Error Probability          5. Channel Estimation (Least
                                                     Squares)
5. Equalization
                                             7. Mobile Radio Channels
       1. Linear Equalization
                                                  1. Multipath Propagation
       2. Decision Feedback
                                                  2. Doppler Spreading
       3. Adaptive Equalization
                                                  3. Multiple Access
6. Optimal Receiver for ISI
   Conditions                                8. Mobile Radio Transmission
       1. Forney-Receiver (MLSE)                Concepts
       2. Viterbi-Algorithm                       1. The OFDM System
                                                  2. Principles of Code Multiplex
                                                     (CDMA)

 Universität
Bremen                                 Contents                                     8-1
              Introduction: The Multi Carrier Philosophy
                           Tsc
|C(f)|              f

                                                    single carrier (SC)
                B                                    equalization problems


                                                t
                            |ca(t)|

                                           t
                    f

                                                    multi carrier (MC)
                B                                    non-selective
                                                      subchannels

                                  N .Tsc        t
Universität
Bremen                           Introduction                            8-2
                  History of Multi Carrier Technique
1957     Doelz: Early multi carrier system ”KINEPLEX“
1967     Saltzberg: Multi carrier system with ”soft impulse shaping“
1971     Weinstein, Ebert: DFT based multi carrier concept (a ”OFDM“)
1981     Kolb (Ph.D.Thesis, Erlangen): Re-invention of OFDM
1983     Schüßler (FTZ-Prof.-Conf.): Presentation of OFDM (without
         consequences)
1990     Fliege, Kammeyer: Re-invention of orthogonal Nyquist impulse
         shaping
1992     Bingham: “Multi carrier modulation, an idea whose time has come”
1993     Kammeyer, Tuisel, Schulze: Multi carrier concept for DAB
1998     Standards: OFDM proposals for DAB, DVB, WLAN, DRM
today annual Workshops (e.g. Rohling, TU-Hamburg-Harburg)
    Universität
    Bremen                        Introduction                          8-3
              Single Carrier Transmission System (SC)

      Transmitter                                                    Channel
                                                          j2 f0 t
                                                      e
                  b(ib)              d(i )                            sa (t )
        Source              Mod.             g(t )


                                                                      channel
                                                                        ca (t)



      Receiver                                                       (t)
                   ^                  ^               e -j2f0 t
                   b(i b)             d(i)
        Dest.               Demod.           h(t)
                                                                      ^ ( t)
                                                                      sa

Universität
Bremen                                 Introduction                              8-4
     Influence of Multipath Propagation on SC Transmission
     |ca(t)|                                      f
                                                               Tc




                                           B


                                                  1 2 3 4 5 6 7 8 9 10 11 12 13 14




                                               }
                                   t              TSC                                t
            Tc
     a) Channel impulse response                      b) ISI at SC transmission
Limits for the raise of transmission rate:
Increasing the bandwidth leads to a reduced symbol duration. In case of multipath
channels the influence of Inter Symbol Interference (ISI) is amplified.
 The equalization effort increases dramatically!

   Universität
   Bremen                          Introduction                                          8-5
               Multi Carrier Transmission (MC)
              Transmitter                                                    j2 f 0 t              Channel
                                      ld(M)                              e
                                                       d0 (i)
                                              Mod.                g(t)
                                                                             j2 f 1 t
                                                                         e
                       b(ib )                          d1 (i)
                                              Mod.                g(t)                               sa (t)
              Source            S/P                                                        
                                                                             j2 f N-1 t
                                                                         e
                                                       dN-1(i)
                                              Mod.                g(t)
                                                                                                     channel
                                                                                                       ca(t)
              Receiver                                                       -j2f0 t
                                      ld(M)             ^                e
                                                        d0 (i)
                                              Demod.              h(t)                              (t)
                                                                             -j2f1 t
                                                        ^                e
                       ^                                d1 (i)
                       b(i b)                 Demod.              h(t)                     ^a (t)
                                                                                           s
               Dest.            P/S

                                                                             -j2fN-1 t
                                                        ^                e
                                                        dN-1(i)
                                              Demod.              h(t)


Universität
Bremen                                           Introduction                                                  8-6
                Channel influence on MC transmission
                f
                                           Tc




       B


                        1                   2                      3
                                                                                  t
                    TMC = N TSC

Advantages of MC over SC:
Spreading of data over multiple subcarriers reduces the data rate on each sub channel.
This leads to an increased symbol duration which reduces the influence of ISI.
 The necessary equalization effort can be reduced dramatically!


  Universität
  Bremen                             Introduction                                        8-7
              Influence of the Channel in Frequency Domain
                 f                                                   f




( f )|          1 2 3 4 5 6 7 8 9 10 11 12 13 14          |H( f )|               1                       2
                }




                TSC                                  t                          TMC                            t

                  a) Single carrier transmission                         b) Multi carrier transmission

         Advantage of multi carrier over single carrier transmission:
         Increasing the number of subcarriers by reducing the frequency spacing leads to a lower
         bandwidth of the corresponding subchannels. With a sufficient number of sub-carriers each
         subchannel can be considered as non frequency selective. In this case the equalization
         only consists of a multiplicative correction on each subcarrier.
          The equalization effort can be reduced dramatically!

           Universität
           Bremen                                  Introduction                                          8-8
                     Inter Carrier Interference (ICI)
                                                    SMC ( f )




                            f-2       f-1      f0           f1    f2              f

                                              ICI

Problem of MC:
If the frequency bands of different subcarriers overlap, Inter Carrier Interference (ICI)
appears.
Solution:
A special design of transmit and receive filter leads to orthogonality of the subcarriers.

   Universität
   Bremen                               Introduction                                         8-9
     OFDM transmission system (time continuous)
              Transmitter                                                                            Channel
                                       ld(M)             d0 (i)
                                               Mod.                               g (t)
                                                                                   0

                                                         d1 (i)
                       b(i b )                 Mod.                               g (t)               sa (t)
                                                                                            
                                                                                   1
              Source             S/P


                                                         dN-1(i)
                                               Mod.                              gN-1(t)
                                                                                                      channel
                                                                                                       ca(t)
              Receiver                                             T
                                       ld(M)            ^
                                                        d0 (i)
                                               Demod.                             h0 (t )            (t)
                                                                       ^0(t )
                                                                       s
                       ^                                ^
                       b(i b )                          d1 (i)
                                               Demod.                             h1 (t )   ^a (t)
                                                                                            s
                                                                       ^
                                                                       s1 (t )
               Dest.             P/S

                                                        ^
                                                        dN-1 (i)
                                               Demod.                            hN-1 (t)
                                                                   ^N-1(t)
                                                                   s

Universität
Bremen                                                OFDM Basics                                               8-10
                          Orthogonal Subcarriers

    |g (t)|                                                                   |G ( f )|




0                    Ts            t                         -1/Ts        0      1/Ts         f



                                          |G ( f )|   |G    ( f )|
                           |G   ( f )|                               |G       ( f )|




                 0                       fn-1   fn    fn+1 fn+2                           f

       Universität
       Bremen                          OFDM Basics                                            8-11
  Mathematical Description of an OFDM System 1/2
                                N 1
 time continuous s(t )   dn (i) g  t  iT  e j 2 f nt
                                       
                                       
                                                  
                                              S 
  representation of an    n0
  OFDM transmitter:     g (t )  rect  t /TS  , f n  nf  n /TS
                                       
                                       
                                               
                                               
                                                    

                                N 1         j 2 nt /TS
                               dn (i) e                  , iTS  t   i 1TS
                                                                       
                                                                            
                                                                             
                                                                           
                                n0


                                                    N 1           j 2 nkTA /TS
                       sk (i)  s(t) t iT
                                  kT A
                                                   dn(i) e                       ,   k [0,1,2,..., N 1]
 time discrete                 S       n0
  representation of
  an OFDM transmitter: N  TS /TA
                               N 1
                              dn (i) e j 2 nk / N = IDFT d0(i), d1(i),..., d N 1(i)
                                                            
                                                            
                                                                                        
                                                                                        
                                                                   
                                                                                                 
                                                                                                  
                               n0

    Universität
    Bremen                             OFDM Basics                                                    8-12
 Mathematical Description of an OFDM System 2/2

                              N 1
 time discrete       dn (i)   rk (i) e j 2 kn/ N
                      ˆ
  representation of           k 0
  an OFDM receiver:         = DFT r0(i), r1(i),..., rN 1(i)
                                  
                                  
                                                             
                                                             
                                      
                                                            
                                                             




                   ˆ
 Complete System: d  DFTN IDFTN (r)c  
                            
                                  
                                            
                                                        
                                  
                                                       
                                                        




   Universität
   Bremen                      OFDM Basics                       8-13
              Symbol Rate Model of an OFDM System
                 Transmitter                                                                      Discrete
                                  ld(M)            d0 (i)               s0 (i)                    Channel
                                          Mod.                0     0


                                                   d1 (i)               s1 (i)
                                          Mod.                1     1
                                                                                         s(i,k)
                 Source     S/P                               IDFT                 P/S

                                                                                                        ga (t)
                                                   dN-1 (i)             sN-1(i)
                                          Mod.                N-1 N-1                                       sa (t)

                                                                                                    channel
                                                                                                      ca (t)
                 Receiver
                                  ld(M)             d0 (i)               r0 (i)
                                          Demod.              0     0                              (t)
                                                                                                            ra (t)
                                                    d1 (i)               r1 (i)                         ha (t)
                                          Demod.              1     1
                                                                                         r(i,k)
                  Dest.     P/S                                   DFT              S/P

                                                                                                   TA
                                                    dN-1(i)              rN-1(i)
                                          Demod.              N-1 N-1




Universität
Bremen                                      OFDM Basics                                                              8-14
   Inter-Symbol- (ISI) and Inter-Carrier-Interference (ICI)

    (i-2)                  (i-1)                 (i+0)           (i+1)            (i+2)
OFDM symbol           OFDM symbol       OFDM symbol          OFDM symbol       OFDM symbol
                                                                                                 t
                                       |ca(t)|
           magnitude of channel
           impulse response:
                                                         t

                                                              fade out (ISI)
              ...       symbol(i -1)

                    fade in (ICI)
                                             symbol i


     Universität
     Bremen                              OFDM Basics                                      8-15
                 The OFDM Cyclic Prefix / Guard Interval
                                         cyclic prefix


             G         (i-1)        G         (i+0)      G            (i+1)             G
                                                                                                 t
                                  |ca (t)|
           magnitude of channel                                        Why cyclic?
           impulse response:
                                                                      y(k )  x(k )    * c(k )
                                                                                      circ

             ...    symbol(i-1)
                                                                  Y (n)  DFT{x(k )} DFT{c(k )}
                      fade in
                                              symbol i             X (n) =Y (n) /C (n)

                                                           fade out
                                    Tg            Ts
 The OFDM cyclic prefix serves for the suppression of ISI and ICI !
   Universität
   Bremen                                    OFDM Basics                                             8-16
                          OFDM Transmitter
                     (HIPERLAN/2 / IEEE802.11a)
                                         d0(i)
                                  Map.
                                  Mod.

                                         d1(i)
                                  Map.
                                                          N
  source        CC      S/P              d2(i)     IDFT       PS/GI   DAC
                                  Map.
                              .
                              .
                              .
                                         dN-1(i)
                                  Map.


 channel coding (convolutional codes with Viterbi decoding)
 IDFT: discrete realized filter bank (very efficient FFT)
 cyclic prefix / guard interval (GI) prevents intersymbol interference (ISI)

  Universität
  Bremen                            OFDM Basics                             8-17
        OFDM Receiver (HIPERLAN/2 / IEEE802.11a)
                            e                           0
                                             d0(i)
                                                               Mod.
                                                              Demap.
                                                       e1
                                             d1(i)
                                                              Demap.
                         SYNC     N                                                  -1
ADC                 PE       -1
                                      DFT              e2                  P/S    CC       dest.
                          GI                 d2(i)
                                                              Demap.
                                                                       .
                                                       eN-1            .         Viterbi
                                                                       .
                                             dN-1(i)                             decoder
 Synchronization
        FFT window position (time domain)
        sample and modulation frequency correction

 Pre equalizer (PE) for impulse compression
 OFDM: Orthogonal Frequency Division Multiplexing
           separate multiplicative channel correction on each subcarrier
           equalizer coefficient design: en = 1 / Cn      circular convolution

      Universität
      Bremen                                OFDM Basics                                     8-18
               Channel Estimation (CE): Training Symbols
 burst structure of HIPERLAN/2 and IEEE802.11a
                  short symbols for AGC and raw synchronization
                  training sequence (TS): 2 identical symbols per subcarrier (52)
                  data OFDM symbols with 48 user data and 4 pilot symbols each
                  pilot symbols for fine synchronization (insufficient for channel
                   estimation)
           f
16.5 MHz




             AGC                                             …
           0 SYNC            TS



               0         8        16    24                                     t in s
           Universität
           Bremen                       Channel Estimation                            8-19
    Nonblind (reference-based) Channel Estimation

                            ~
                 y0 (i)     d0,ref (i)                               N               d0(i)
                            ~              ~            ~
          y1 (i) disc.      d1,ref (i)     dn,ref (0) + dn,ref (1)                   d1(i)
y(k)             Prefix           C(n) =                                   1
     S/P                                            2dn,ref
                   & ~                                                    C(n)
         yP-1(i) FFT dN-1,ref (i)      dn,ref                                    dN-1(i)
                                                  N
                                     Channel estimator                   Equalizer

 Averaging over only two identical training symbols
   • 2 dB loss in SNR compared to „estimator“ with ideal channel knowledge
        • Perform additional noise reduction (NR) to increase estimation quality



   Universität
   Bremen                          Channel Estimation                                   8-20
                      Noise Reduction Algorithm (NR)
 Background
    a-priori knowledge: limited channel impulse response in time domain
     a channel impulse response fits into guard interval
    lowpass filtering in frequency domain
 NR algorithm (required operations)
    transform the estimated channel transfer function into time domain (IDFT)
    truncate the estimated impulse response (rectangular window)
    re-transform into frequency domain (DFT)



                                   noise reduction (NR)
                                                       ~            ~
                        C          c                   c            C
                 CE         IDFT                              DFT
                                           NC
                                   windowing in time domain


   Universität
   Bremen                          Channel Estimation                        8-21
              Noise Reduction Algorithm – Example
 • estimated and real channel transfer functions (frequency domain)


                   0 dB

                  -10 dB
                                          | Cn |
                                          | Cn |
                  -20 dB
                           0 4 8    16              32      64   n
 • ... in time domain


                                   | c(k) |
                                   | c(k) |



      0       4     8        16                    32                64   k
Universität
Bremen                                 Channel Estimation                     8-22
              Noise Reduction Algorithm – Example
     • smoothed and real transfer functions (in frequency domain)


                   0 dB

                  -10 dB                  ~
                                        | Cn |
                                        | Cn |
                  -20 dB
                           0 4 8   16            32        64   n

     • time limited (windowed) impulse response

                                     ~
                                   | c(k) |




       0      4     8        16                  32                 64   k
Universität
Bremen                                Channel Estimation                     8-23
    Noise Reduction Algorithm – Simulation Results
• simulation of a HIPERLAN/2 system (27 Mbit/s)

• time invariant                  10
                                       0

                                                                     only CE
  Rayleigh distributed                                               CE+NR
                                                                     ideal
  multipath channel
                                       -1
                                  10
• (only CE)                 PER

 Eb/N0 loss: about 1.8 dB

                                       -2
                                  10
• (CE+NR)
 Eb/N0 loss: about 0.5 dB

                                       -3
• (ideal)                         10
                                       10    12          14     16             18
 perfectly known channel                                Eb/N0

   Universität
   Bremen                          Channel Estimation                          8-24
              Channel Tracking – Motivation
   PHY burst length < 2 ms (IEEE802.11a: < 5 ms)
   Jakes distributed Doppler shift
   object speed: 3 m/s
    (figure: 10 m/s)
                                             ms
   time variant channel             th: 1.6
                              t leng
    transfer function     burs
                                                    channel transfer functio
                                                              |Cf (t)|




                                                                         t

a channel tracking is required 20 MHz


                                            f

Universität
Bremen                   Channel Tracking                        8-25
        Block Diagram: “Turbo Channel Estimation“
                                                                            vector of length 52

s                                  received OFDM symbols
                      d
        FFT                 …                                                …
                                                              4 s
                                                                                                       t
                                                      i0 +1                      demodulation
                                                 -1                                                        b
                                             z
                                                                            dem.              dec.
                            1-a0                                      ~                                        decided data
                        ~                                             Ctr
         CE             C
        NRA                                                                                 optional
                                                              ~
                            a0         initial CE             Cinit
                      re
                    d
                                             z -(1+i0)                      mod.             cod.

                     1st order loop filter
                  (time domain smoothing)                                          remodulation



    Universität
    Bremen                                    Channel Tracking                                                   8-26
               Example without Channel Tracking
 time variant channel (100 ns, 30 m/s)
        HIPERLAN/2 (12 Mbit/s)
        burst length: 180 OFDM symbols (720 s)
        CE+NR (no tracking)




 Universität
 Bremen                     Channel Tracking       8-27
               Example with Channel Tracking
 time variant channel (100 ns, 30 m/s)
        HIPERLAN/2 (12 Mbit/s)
        burst length: 180 OFDM symbols (720 s)
        CE+NR+Tracking




 Universität
 Bremen                     Channel Tracking       8-28
                            Simulation Results
 time variant multipath channel (Jakes distributed Doppler shift)
                                      0
                                 10
                                                     no tracking
       channel noise                                uncod.
                                                     cod.
        Eb / N 0  15 dB         10
                                      -1




       HIPERLAN/2
                           BER

                                      -2
        (27 Mbit/s)              10

       burst length
                                      -3
        320 s                   10

       CE+NR
                                      -4
                                 10              0                      1             2
                                            10                       10          10
                                                                   vmax in m/s
    Universität
    Bremen                                 Channel Tracking                               8-29
              Parameters of an OFDM System

Meaning                 Time continuous    Time discrete

Sampling frequency              -            fA 1  N
                                                TA TS
Core symbol duration           TS                N T A
Guard time                     TG                N G T A

Total symbol duration      T  TS  TG     T  ( N G  N ) T A

                            f  1            f      1
Subcarrier spacing              TS                   N TA
Bandwidth                  B  N  f          B  N  f

                         Rb  N ld( M )   Rb  N ld( M )
Data rate                     TS  TG          T A  ( N  NG )


Universität
Bremen                  OFDM Summary                              8-30
                        Communications Part II

4. Optimal Receiver for AGN                       3. Error Probability at Viterbi
       1. Optimal AWGN Receiver                      Detection
       2. Generalization for coloured             4. Influence of Channel Impulse
          noise                                      Response
       3. Symbol / Bit Error Probability          5. Channel Estimation (Least
                                                     Squares)
5. Equalization
                                             7. Mobile Radio Channels
       1. Linear Equalization
                                                  1. Multipath Propagation
       2. Decision Feedback
                                                  2. Doppler Spreading
       3. Adaptive Equalization
                                                  3. Multiple Access
6. Optimal Receiver for ISI
   Conditions                                8. Mobile Radio Transmission
       1. Forney-Receiver (MLSE)                Concepts
       2. Viterbi-Algorithm                       1. The OFDM System
                                                  2. Principles of Code Multiplex
                                                     (CDMA)

 Universität
Bremen                                 Contents                                 8-31
                      Multiple Access Schemes
                           Code Division Multiple Access
                           (CDMA):       Code




                                                     Time



                                     Frequency
Frequency Division Multiple Access               Time Division Multiple Access
(FDMA):        Code                              (TDMA):      Code




                         Time                                           Time



          Frequency                                   Frequency

  Universität
  Bremen                                                                         8-32
              Principles of Code Division Multiple Access:
                             Block Diagram

                                                                b(t )



                                             +



              p1 (t )            p 2 (t )        pU (t )



                  d1 (t )        d2 (t )           dU ( t )
                        user 1      user 2                 user U


Universität
Bremen                                                                  8-33
              Principles of Code Division Multiple Access:
                               Spreading


                                                TS
Data Signal:                     du (t )
                                           1

  *                                        -1

                                                TC
                                           1
Chip sequence:                   pu (t )
                                           -1
  =

                                           1
Spreaded data signal:            bu (t )
                                           -1


Universität
Bremen                                                       8-34
                   Principles of Code Division Multiple Access:
                                   Despreading
                          TS                            TS

              1                                    1
 bu (t )                                bn (t )
            -1                                     -1

  *                                          *
              1                                    1
pu (t )                                  pu (t )
            -1                                     -1
  =                                          =
             1                                     1
du (t )                                 d n (t )
            -1                                     -1

 T          TS                           T         TS
 
t TS
                                         
                                        t TS
            TS                                     TS

     Universität
     Bremen                                                       8-35
                        Spread Spectrum

                                          d (t )   D( j )
                                          b(t )    B( j )

                             1/ TC
              D( j )
                        2

                                                   B( j)
                                                             2




                                     1                      
                                     TS
Universität
Bremen                                                   8-36
                 How to choose spreading codes?

Walsh Hadamard codes:

                    H k 1 H k 1 
              Hk  
                    H k 1 H k 1 
                                    

Orthogonality:                                                                        t

                                T
           H k 1     H k 1   H k 1    H k 1   2HT 1H k 1        0       
  Hk Hk  
   T
                                                    
                                                         k
                                                                                     2 I 2k
                                                                                        k

           H k 1     H k 1   H k 1
                                          H k 1      0          2HT 1H k 1 
                                                                        k



                                                               1 1 1 1 
Example:                                                       1 1 1 1
                                          1 1           H2             
                      H0  1         H1                     1 1 1 1
                                          1 1                          
                                                               1  1 1 1 

Universität
Bremen                                                                                          8-37
                           Orthogonal Codes?

  Problem: Synchronisation (Uplink)
                      At the transmitter        Receiver of user 1


                 1                         1
chip sequence
of user 1
                 -1                        -1


                 1                         1
chip sequence
of user 2
                 -1                        -1
                                                worst case




   Universität
   Bremen                                                            8-38
                              Orthogonal Codes?

Problem: Multipath channel (Up- and Downlink)


              chip sequence                1
              of user 1           Path 1
                                           -1
       1

      -1                                   1
                                  Path 2
                                           -1




Universität
Bremen                                            8-39
                      Pseudo Random Codes
                                                         T
                                                       1
Cross correlation function:                  run ( )   pu (t ) pn (t   )
                                                       T0
                                         (t ) : i  j
Ideal Case:           run ( )  
                                        0        : otherwise

Random code (e.g. m-sequence, Gold-code)

 mother
 code 1
              1   2    3   4   5     6                          Gold code

              1   2    3   4   5     6          z-n
  mother
  code 2


Universität
Bremen                                                                          8-40
               CDMA System and mobile radio channel
                 1   narrow band signal                           1                   wideband signal
                 T                                               TC
                                        p 0 (t )                  TC

   d0 (t )
                       T                p1 (t )

   d1 (t )                              pU (t )




                                                                                                   mobile radio channel 2

                                                                                                                            mobile radio channel 1
    dU ( t )
                                                   time variant mobile radio channel




                                                                                C ( j , t )
                                                                                               2
                                    t

                                                   
                                                                       y (t )
Universität
Bremen                                                                                                                                8-41
                                   Rake Receiver


                                                                                    
                y (t )



              pu (t   L 1 )
               
                                        pu (t   L 1 )
                                         
                                                                         pu (t   L 1 )
                                                                          




                                   
                                   TS
                                                           
                                                           TS
                                                                                            
                                                                                            TS



                                                                                     
                          c L 1              cL  2                                  c0




                                                                +


                                                                ˆ
                                                                du (i)

Universität
Bremen                                                                                           8-42
                                                        Rake Receiver
                                                                                L1

th branch of RAKE: yu (t    )   c                                                             du (t      ) pu (t      )  n(t    )
                                                                               0 channel
                                                                                                             data               code
                                                                                        coefficient

Correlation with code:
  iTS  L1                                           L 1         iTS  L 1

                   yu (t    ) pu (t   L 1 )dt   c
                                    
                                                                                      d u (t      ) pu (t      ) pu (t   L 1 )dt
                                                                                                                                

( i 1)TS    L 1
                                                        0       ( i 1)TS     L 1


                                                                                for   L 1    du (i);                  else 0
                                                                    iTS  L1

                                                                                       n(t    ) pu (t   L 1 )dt
                                                                                                     
                                                              
                                                                  ( i 1)TS  L1




                                                      du (i )cL 1  n(i )

Multiplication with conjugate channel coefficient:
                       iTS  L1

                           
                                                                                                                     2
 c  
    L 1                               yu (t    ) p (t   L 1 )dt  du (i ) cL 1  n(i )cL 1
                                                       
                                                       u
                                                                                                   

                     ( i 1)TS    L1


 Universität
 Bremen                                                                                                                                              8-43

								
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