Signals and Linear System by HwEsQZ


									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

       Inverse Fourier transform

       From Fourier series (T0  )
Properties of FT
   For a real signal x(t)
       X(f) is Hermitian Symmetry

                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

   Duality
       If               , Then
   Time Shift
       A shift in the time domain results in a phase shift
        in the frequency domain
                             (                   )
Properties of FT
   Scaling
       An expansion in the time domain results in a
        contraction in the frequency domain, and vice
Properties of FT
   Modulation
          Multiplication by an exponential in the time
           domain corresponds to a frequency shift in the
           frequency domain


    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
Properties of FT
   Convolution
       Convolution in the time domain is equivalent to
        multiplication in the frequency domain, and vice
Properties of FT
   Parseval‟s relation

       Energy can be evaluated in the frequency
        domain instead of the time domain
   Rayleigh‟s relation
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

   Take FT

       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

       FT of truncated signal:
       Expression of FS coefficient
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)

                              -W              W   f
   Impulse sampling
       Sampled waveform

       Take FT

x(t)                            X(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

                  -W     W       1/Ts
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

   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)

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

   Relation between FT and DFT

   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
       Definition of DFT:

       Definition of IDFT:

       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
          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

   Take FT (using convolution theorem)

           Where the Transfer Function of the system

   The relation between input-output spectra
   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
       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
   We only sample the signal at intervals
       We don‟t know what happens between the
   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
   Called “aliasing” because one frequency
    looks like another
   Nyquist frequency
       We must sample faster than twice the frequency
        of the highest frequency component
   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
       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
   When the signal is converted to digital form
       Precision is limited by the number of bits
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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