Electrical Communications Systems ECE.09.331 Spring 2007 by bvm20830

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									                                        S. Mandayam/ ECOMMS/ECE Dept./Rowan University




     Electrical
Communications Systems
                  ECE.09.331
                 Spring 2007

                Lecture 3a
             January 30, 2007
        Shreekanth Mandayam
                  ECE Department
                  Rowan University


 http://engineering.rowan.edu/~shreek/spring07/ecomms/
                                             S. Mandayam/ ECOMMS/ECE Dept./Rowan University




                             Plan
• Recall: CFT‟s (spectra) of common waveforms
       • Impulse
       • Sinusoid
       • Rectangular Pulse
• CFT‟s for periodic waveforms
• Sampling
   •   Time-limited and Band-limited waveforms
   •   Nyquist Sampling
   •   Impulse Sampling
   •   Dimensionality Theorem
• Discrete Fourier Transform (DFT)
   • Fast Fourier Transform (FFT)
                                                                                                              S. Mandayam/ ECOMMS/ECE Dept./Rowan University




                                   ECOMMS: Topics
                                                              Electrical Communication Systems



                    Signals                                                                        Systems


Discrete                          Continuous                             Analog                                                     Digital
                                                                                                                          Digital Comm Transceiver


    Probability                  Power & Energy Signals                         AM                             Baseband                              Bandpass
                                                                        Switching Modulator                     CODEC                                 MODEM
                                                                         Envelop Detector


    Information                Continuous Fourier Transform                   DSB-SC
                                                                         Product Modulator                   Source Encoding                            ASK
                                                                         Coherent Detector                    Huffman codes                             PSK
                                                                            Costas Loop                                                                 FSK

     Entropy                    Discrete Fourier Transform                     SSB                     Error-control Encoding                          BPSK
                                                                          Weaver's Method                 Hamming Codes
                                                                          Phasing Method
                                                                         Frequency Method
                                                                                                                Sampling                               QPSK
 Channel Capacity             Baseband and Bandpass Signals         Frequency & Phase Modulation                  PAM
                                                                        Narrowband/Wideband
                                                                        VCO & Slope Detector
                                                                                PLL                           Quantization                           M-ary PSK
                                                                                                                 PCM
                                    Complex Envelope

                                                                                                              Line Encoding                            QAM


                                  Gaussian Noise & SNR
                                                                                                         Time Division Mux
                                                                                                         T1 (DS1) Standards


                                    Random Variables                                                      Packet Switching
                                    Noise Calculations                                                        Ethernet
                                                                                                        ISO 7-Layer Protocol
                                                      S. Mandayam/ ECOMMS/ECE Dept./Rowan University




                       Definitions
Continuous Fourier Transform (CFT)
                           
 W (f )  Fw ( t )   w ( t ) e  j2ft dt
                          
 W (f )  X (f )  j Y (f )
                                                Frequency, [Hz]
 W ( f )  W ( f ) e j ( f )
                                Phase
                                Spectrum
          Amplitude
          Spectrum


Inverse Fourier Transform (IFT)
                                
 w (t)  F    W(f )   W(f ) e  j2ft df
             -1
                                                          See p. 45
                                                            Dirichlet Conditions
                                     S. Mandayam/ ECOMMS/ECE Dept./Rowan University




 CFT’s of Common Waveforms
• Impulse (Dirac Delta)

• Sinusoid

• Rectangular Pulse
                      Matlab Demo:
                      recpulse.m
                                                   S. Mandayam/ ECOMMS/ECE Dept./Rowan University




      CFT for Periodic Signals
 Recall:                           FS: Periodic Signals
CFT: Aperiodic Signals                        
         
                   j2ft           w ( t )   Wn e j2nf0 t
W (f )   w ( t ) e        dt              n  
                                  where
                                          1  T0 / 2      j2 nf0 t
                                    Wn         w (t) e             dt
                                         T0 T / 2
                                                   0
  • We want to get the CFT for a periodic signal

  • What is F   e j 2nf t  ?
                         0
                                         S. Mandayam/ ECOMMS/ECE Dept./Rowan University




    CFT for Periodic Signals
• Sine Wave                • Square Wave
    w(t) = A sin (2f0t)
                             A

                             -A
                                  T0/2   T0




                                  Instrument Demo
                                        S. Mandayam/ ECOMMS/ECE Dept./Rowan University




                   Sampling
 • Time-limited waveform     • Band-limited waveform
      w(t) = 0; |t| > T        W(f)=F{(w(t)}=0; |f| > B

           w(t)                          W(f)


      -T      T      t             -B          B               f




• Can a waveform be both time-limited and band-limited?
                                        S. Mandayam/ ECOMMS/ECE Dept./Rowan University




Nyquist Sampling Theorem
• Any physical waveform can be represented by
                                        n 
                         sin f s  t  
                                   
                                        f s 
                                              
           w(t )   an 
                  n                 n
                            f s  t  
                                 
                                       fs 
• where
                            
                   an  f s  w(t )
                           
• If w(t) is band-limited to B Hz and f s  2 B
                             
                      an  w fn
                                s
                                             S. Mandayam/ ECOMMS/ECE Dept./Rowan University




      What does this mean?
• If f s  2 B then we can reconstruct w(t) without error
  by summing weighted, delayed sinc pulses
   • weight = w(n/fs)
   • delay = n/fs                                           3 
                                                      sin  fs  t   

• We need to store only                                     fs  
  “samples” of w(t),                                           3
                                 a3 =                     fs  t  
  i.e., w(n/fs)                  w(3/fs)                       fs 
                                                                                 w(t)
• The sinc pulses
  can be generated
  as needed (How?)
                                 1/fs 2/fs   3/fs 4/fs        5/fs                 t

                                                           Matlab Demo:
                                                           sampling.m
                           S. Mandayam/ ECOMMS/ECE Dept./Rowan University




       Impulse Sampling
• How do we mathematically represent a
  sampled waveform in the
    • Time Domain?
    • Frequency Domain?
                                                S. Mandayam/ ECOMMS/ECE Dept./Rowan University




             Sampling: Spectral Effect
Original
                                                            |W(f)|
           w(t)
                              F



                      t                         -B 0 B                                f
Sampled
                          F                                 |Ws(f)
      ws(t                                                  |
      )




                              -2fs        -fs           0             fs            2 fs
                      t                                                                    f
                                  (-fs-B) -(fs +B) -B        B (fs -B)      (fs +B)
                    S. Mandayam/ ECOMMS/ECE Dept./Rowan University




Spectral Effect of Sampling

  Spectrum        Spectrum
     of a           of the
  “sampled”   =   “original”
  waveform        waveform
                  replicated
                  every fs Hz
                                           S. Mandayam/ ECOMMS/ECE Dept./Rowan University




                     Aliasing
• If fs < 2B, the waveform is “undersampled”
        • “aliasing” or “spectral folding”

• How can we avoid aliasing?
     • Increase fs
     • “Pre-filter” the signal so that it is bandlimited to
       2B < fs
                                          S. Mandayam/ ECOMMS/ECE Dept./Rowan University




   Dimensionality Theorem
• A real waveform can be completely specified by
                 N = 2BT0
  independent pieces of information over a
  time interval T0
         • N: Dimension of the waveform
         • B: Bandwidth

• BT0: Time-Bandwidth Product

• Memory calculation for storing the waveform
         • fs >= 2B
         • At least N numbers must be stored over the
           time interval T0 = n/fs
                                                S. Mandayam/ ECOMMS/ECE Dept./Rowan University




Discrete Fourier Transform (DFT)
                                         Equal time intervals
• Discrete Domains
   • Discrete Time:             k = 0, 1, 2, 3, …………, N-1
   • Discrete Frequency:        n = 0, 1, 2, 3, …………, N-1

                                       Equal frequency intervals

• Discrete Fourier Transform

                                 2  nk
                              j 
                   N 1
            X[n ]   x[k ] e  N  ;         n = 0, 1, 2,….., N-1
                     k 0
• Inverse DFT
                                   2  nk
                                 j 
                     1 N 1
           x[k ]        X[n ] e  N  ; k = 0, 1, 2,….., N-1
                     N n 0
                                              S. Mandayam/ ECOMMS/ECE Dept./Rowan University




     Importance of the DFT
• Allows time domain / spectral domain
  transformations using discrete arithmetic operations

• Computational Complexity
   • Raw DFT: N2 complex operations (= 2N2 real operations)
   • Fast Fourier Transform (FFT): N log2 N real operations

• Fast Fourier Transform (FFT)
   • Cooley and Tukey (1965), „Butterfly Algorithm”, exploits the
     periodicity and symmetry of e-j2kn/N
   • VLSI implementations: FFT chips
   • Modern DSP
                                                 S. Mandayam/ ECOMMS/ECE Dept./Rowan University




  How to get the frequency axis in the DFT
• The DFT operation just converts one set of number,
  x[k] into another set of numbers X[n] - there is no
  explicit definition of time or frequency
               x0                          X0 
      x[k ]   .                  X[n ]   . 
                                                 
               x N 1 
                                          X N 1 
                                                             (N-point FFT)


• How can we relate the DFT to the CFT and obtain
  spectral amplitudes for discrete frequencies?
         n=0    1    2     3    4        n=N
                                                                  Need to
         f=0                            f = fs                    know fs
                               fs
                               N
          S. Mandayam/ ECOMMS/ECE Dept./Rowan University




Summary

								
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