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Note for Digital Communication Techniques - DCT By Kishan Borra

  • Digital Communication Techniques - DCT
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Rajalakshmi Engineering College, Thandalam Prepared by J.Saranya, Lecturer/ECE DIGITAL COMMUNICATION UNIT I III YEAR ECE A &B DIGITAL COMMUNICATION SYSTEM SAMPLING: Sampling Theorem for strictly band - limited signals 1.a signal which is limited to  W  f  W , can be completely n   described by  g ( ).  2W  n   2.The signal can be completely recovered from  g ( )  2W  Nyquist rate  2W Nyquist interval  1 2W When the signal is not band - limited (under sampling) aliasing occurs .To avoid aliasing, we may limit the signal bandwidth or have higher sampling rate. Let g (t ) denote the ideal sampled signal g ( t )    g (nT )  (t  nT ) n   s s where Ts : sampling period f s  1 Ts : sampling rate (3.1)

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From Table A6.3 we have  g( t )   (t  nTs )  n   1 G( f )  Ts    m     f G( f m   ( f  s m ) Ts  mf s ) g ( t )  f s   G( f m    mf s ) (3.2) or we may apply Fourier Transform on (3.1) to obtain G ( f )    g (nT ) exp(  j 2 nf T ) n   s or G ( f )  f sG ( f )  f s s   G( f m   m 0  mf s ) (3.3) (3.5) If G ( f )  0 for f  W and Ts  1 2W n j n f G ( f )   g ( ) exp(  ) 2W W n    (3.4) With 1.G ( f )  0 for f W 2. f s  2W we find from Equation (3.5) that 1 G ( f ) ,  W  f  W (3.6) 2W Substituting (3.4) into (3.6) we may rewrite G ( f ) as G( f )  1  n jnf

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n   To reconstruct g (t ) from  g ( )  , we may have 2 W    g (t )   G ( f ) exp( j 2ft )df  W  W 1 2W   n  g ( 2W ) exp(  n   j n f ) exp( j 2 f t )df W n   exp j 2  f ( t  ) df (3.8)    W 2 W  n    n sin( 2 Wt  n )   g( ) 2 W 2 Wt  n n    g( n 1 ) 2W 2W W  n ) sin c( 2Wt  n ) , -   t    2W n   (3.9) is an interpolation formula of g (t )  g( (3.9)

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Figure 3.3 (a) Spectrum of a signal. (b) Spectrum of an undersampled version of the signal exhibiting the aliasing phenomenon. Figure 3.4 (a) Anti-alias filtered spectrum of an information-bearing signal. (b) Spectrum of instantaneously sampled version of the signal, assuming the use of a sampling rate greater than the Nyquist rate. (c) Magnitude response of reconstruction filter. Pulse-Amplitude Modulation : Let s ( t ) denote the sequence of flat - top p ulses as s (t )    m( nT s n   ) h ( t  nTs ) (3.10) 0 t T  1, 1 h (t )   , t  0, t  T 2  0, otherwise  The instantaneously samp led version of m ( t ) is m ( t )    m( nT s n   m ( t )  h ( t )        (3.12) m ( )h ( t   )d    ) ( t  nTs ) (3.11)  m( nT ) ( s  nTs ) h ( t   ) d n     m( nT )  s n      (  nTs )h ( t   )d (3.13)

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