Signals and Linear System (PowerPoint) by ert554898

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									Signals and Linear System

 Fourier Transforms
 Sampling Theorem
Fourier Transform
   Is the extension of Fourier series
      to non-periodic signal
   Definition of Fourier transform
       Fourier transform
          
             X ( f )  F[ x(t )]   x(t )e j 2 ft dt
                                     


       Inverse Fourier transform     
            x(t )  F 1[ X ( f )]   X ( f )e j 2 ft df
                                     


       From Fourier series (T0  ) 
               1  T
           xn       x(t )e j 2 nf t dt x(t )   xn e j 2 nf t
                            0
                                         0                       0

               T0                                 n 
Properties of FT
   For a real signal x(t)
       X(f) is Hermitian Symmetry
            X ( f )  X * ( f )
                  Magnitude spectrum is even about the origin (f=0)
                  Phase spectrum is odd about the origin


       f, called frequency (units of Hz), is just a parameter of FT
        that specifies what frequency we are interested in
        looking for in the x(t)
       The FT looks for frequency f in the x(t) over - < t < 
       F(f) can be complex even though x(t) is real
       If x(t) is real, then Hermitian symmetry
Properties of FT
   Linearity
       F[ x1 (t )   x2 (t )]   F[ x1 (t )]   F[ x2 (t )]

   Duality
       If   X ( f )  F[ x(t )]        , Then         F[ X (t )]  x( f )

   Time Shift
       A shift in the time domain results in a phase shift
        in the frequency domain
        F ( x(t  t ))  e  X ( f ) ( e  X ( f )  X ( f ) )
                    0
                             j 2 ft0                j 2 ft0
Properties of FT
   Scaling
       An expansion in the time domain results in a
        contraction in the frequency domain, and vice
        versa
                        1   f
       F [ x(at )]      X( ) , a  0
                        a   a
Properties of FT
   Modulation
          Multiplication by an exponential in the time
           domain corresponds to a frequency shift in the
           frequency domain
          F[e j 2 f0t x(t )]  X ( f  f 0 )
                                    1
           F[ x(t ) cos(2 f 0t )]  [ X ( f  f 0 )  X ( f  f 0 )]
                                    2

               x(t)

                   cos
    X(f)                                                  -f0           f0
               -f0           f0
Properties of FT
   Differentiation
       Differentiation in the time domain corresponds to
        multiplication by j2f in the frequency domain
           d
       F[    x(t )]  j 2 f  X ( f )
           dt
           dn
        F [ n x(t )]  ( j 2 f ) n  X ( f )
           dt
Properties of FT
   Convolution
       Convolution in the time domain is equivalent to
        multiplication in the frequency domain, and vice
        versa
       F[ x(t )  y(t )]  X ( f )Y ( f )
        F[ x(t ) y(t )]  X ( f )  Y ( f )
Properties of FT
   Parseval’s relation
                                    
       
                x(t ) y* (t )dt   X ( f )Y * ( f )df
                                     

       Energy can be evaluated in the frequency
        domain instead of the time domain
   Rayleigh’s relation
                                                       
               x(t ) x (t )dt          x(t ) dt  
                      *                       2                    2
                                                            X ( f ) df
                                                 
More on FT pairs
   See Table 1.1 at page 20
       Delta function  Flat
       Time / Frequency shift
       Sin / cos input
            Periodic signal  impulses in the frequency domain
       sgn / unit step input
       Rectangular  sinc
       Lambda  sinc2
       Differentiation
       Pulse train with period T0
            Periodic signal  impulses in the frequency domain
FT of periodic signals
   For a periodic signal with period T0
   x(t) can be expressed with FS coefficient
                     
          x(t )    xe
                    n 
                            n
                                j 2 nt / T0




   Take FT                                                     
                       X ( f )  F [ x(t )]  F[  xn e j 2 nt / T0 ]
                                                             n 
                                                                             
                                                                                                 n
                                      
                                      n 
                                               xn F [e   j 2 nt / T0
                                                                        ]   
                                                                             n 
                                                                                     xn ( f 
                                                                                                 T0
                                                                                                    )


       FT of periodic signal consists of impulses at
        harmonics of the original signal
FS with Truncated signal
   FS coefficient can be expressed using FT
       Define truncated signal
                           T0      T0
                       ,  t 
                         x(t )
             xT0 (t )     2       2
                        0
                        
                        , otherwise
       FT of  truncated signal: X T0 ( f )  F[ xT0 (t )]
       Expression of FS coefficient
                         1        n
                  xn       X T0 ( )
                         T0       T0
Spectrum of the signal
   Fourier transform of the signal is called the
    Spectrum of the signal
       Generally complex
            Magnitude spectrum
            Phase spectrum
   Illustrative problem 1.5
       Time shifted signal
            Same magnitude, Different phase
       Try it by yourself with Matlab !
   Sampling Theorem
      Basis for the relation between continuous-
       time signal and discrete-time signals
      A bandlimited signal can be completely
       described in terms of its sample values taken
       at intervals Ts as long as Ts  1/(2W)
x(t)                                   X(f)
                                       1


                              -W              W   f
                 Ts
   Impulse sampling
       Sampled waveform
                                                
            x (t )  x(t )   (t  nTs )     x(nT ) (t  nT )
                                                             s            s
                           n                 n 

       Take FT                                                                 
        X  ( f )  F [ x (t )]  F [ x(t )   (t  nTs )]  X ( f )  F [   (t  nTs )]
                                        n                                    n 
                                                        
                           1                 n    1                      n
                 X ( f )
                           Ts
                                  
                                 n 
                                        (f  ) 
                                             Ts   Ts
                                                        
                                                        n 
                                                                 X(f 
                                                                         Ts
                                                                            )

x(t)                                                                     X(f)
                                                                         1/Ts

                                                                                               f
                                                                 -W           W       1/Ts
                           Ts                                                 1/(2Ts)
Reconstruction of signal
   Low pass filter
       With Bandwidth 1/(2Ts) and Gain of Ts

                        X(f)         Low pass filter



                                        f
                  -W     W       1/Ts
                         1/(2Ts)
Reconstruction from sampled signal

   If we have sampled values
       {… x(-2Ts), x(-Ts), 0, x(Ts), x(2Ts) …}
       With Nyquist interval (or Nyquist rate)
            Ts = 1/(2W)
   Then we can reconstruct x(t) using
                    
        x(t )     x(nT )sinc(2W (t  nT ))
                   n 
                           s              s




   Example: Figure 1.17 at page 24
Aliasing or Spectral folding
    If sampling rate is Ts > 1/(2W)
        Spectrum is overlapped
        We can not reconstruct original signal with
         under-sampled values
        Anti-aliasing methods are needed

           X(f)                           X(f)
                                         1/Ts

                                                       f
-W                W     f           -W      W
                                             1/Ts
Discrete Fourier Transform
   DFT of discrete time sequence x[n]
                     

      X d ( f )   x[n]e
                            j 2 fnT
                                    s


                   n 

   Relation between FT and DFT
       X ( f )  Ts X d ( f ),   for f  W


   FFT(Fast Fourier Transform)
       Efficient numerical method to compute DFT
       See fft.m and fftseq.m, for more information
   Example: Try Illustrative problem 1.6 by yourself
DFT in Matlab
   fft.m and ifft.m with finite samples
                                            N
       Definition of DFT: X (n)   x[k ]e j 2 nk / N
                                           k 1

                                         1 N
       Definition of IDFT:       x[k ]   X (n)e j 2 nk / N
                                         N n 1

       Time and frequency is not appeared explicitly
            Just definition implemented on a computer to compute N values for
             the DFT and IDFT
       N is chosen to be N=2m
            Zero padding is used if N is not power of 2
FFT in matlab
   A sequence of length N=2m of samples of x(t)
    taken at Ts
       Ts satisfies Nyquist condition
       Ts is called time resolution
   FFT gives a sequence of length N of sampled
    Xd(f) in the frequency interval [0, fs=1/Ts]
       The samples are apart f  fs / N
       f is called frequency resolution
            Frequency resolution is improved by increasing N
Some remarks on short signal
   FT works on signals of infinite duration
   But, we only measure the signal for a short time




   FFT works as if the data is periodic all the time
Some remarks on short signal
   Sometimes this is correct
Some remarks on short signal
   Sometimes wrong
Frequency leakage
   If the period exactly fits the measurement time, the
    frequency spectrum is correct
   If not, frequency spectrum is incorrect




       It is broadened
Frequency domain analysis of LTI system

   The output of LTI system
       y(t )  x(t )  h(t )

   Take FT (using convolution theorem)
       Y ( f )  X ( f )H ( f )
            Where the Transfer Function of the system
                                           
                  H ( f )  F[h(t )]   h(t )e j 2 ft dt
                                          

   The relation between input-output spectra
        Y( f )  X ( f ) H( f )
         Y ( f )  X ( f )  H ( f )
Homeworks
   Illustrative problem 1.7
   Problems
       1.10, 1.12, 1.14, 1.15
More on Sampling
   Most real signals are analog
       The analog signal has to be converted to digital




   Information lost during this procedure (Quantization
    Error)
       Inaccuracies in measurement
       Uncertainty in timing
       Limits on the duration of the measurement
More on Sampling
   Continuous analog signal has to be held
    before it can be sampled
More on Sampling
   The sampling take place at equal interval of
    time after the hold
       Need fast ADC
       Need fast hold circuit
            Signal is not changing during the time the circuit is
             acquiring the signal value
            Unless, ADC has all the time that the signal is held to
             make its conversion


   We don’t know what we don’t measure
More on Sampling
   In the process of measuring signal, some
    information is lost
Aliasing
   We only sample the signal at intervals
       We don’t know what happens between the
        samples
Aliasing
   We must sample fast enough to see the most
    rapid changes in the signal
       This is Sampling theorem


   If we do not sample fast enough
       Some higher frequencies can be incorrectly
        interpreted as lower ones
Aliasing
   Called “aliasing” because one frequency
    looks like another
Aliasing
   Nyquist frequency
       We must sample faster than twice the frequency
        of the highest frequency component
Antialiasing
   We simply filter out all the high frequency
    components before sampling
       Antialias filters must be analog
            It is too lte once you have done the sampling
More on sampling Theorem
   The sampling theorem does not say the
    samples will look like the signal
More on Sampling Theorem
   Sampling theorem says there is enough
    information to reconstruct the signal
       Correct reconstruction is not just draw straight
        lines between samples
More on Sampling Theorem
   The impulse response of the reconstruction
    filter has sinc (sinx/x) shape
       The input to the filter is the series of discrete
        impulses which are samples
       Every time an impulse hits the filter, we get
        ‘ringing’
       Superposition of all these rings reconstruct the
        proper signal
Frequency resolution
   We only sample the signal for a certain time
       We must sample for at least one complete cycle
        of the lowest frequency we want to resolve
Quantization
   When the signal is converted to digital form
       Precision is limited by the number of bits
        available
Uncertainty in the clock
   Uncertainty in the clock timing leads to
    errors in the sampled signal
Digitization errors
   The errors introduced by digitization are both
    nonlinear and signal dependent
       Nonlinear
            We can not calculate their effect using normal maths.
       Signal dependent
            The errors are coherent and so cannot be reduced by
             simple means
Digitization errors
   The effect of quantiztion error is often similar
    to an injection of random noise

								
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