Signal Detectability and Pulse Compression by U6SJEYO6

VIEWS: 26 PAGES: 68

									Signal Detection and Processing Techniques for
             Atmospheric Radars




                Dr. V.K. Anandan
           National MST Radar Facility
              Department of Space
                   IITM-WPW-VKA            1
                 Gadanki - India
• The essence of frequency analysis is the representation of a signal
  as superposition of sinusoidal components.
• In practical applications, where only a finite length of data is
  available, we cannot obtain a complete description of the adopted
  signal model.
• Therefore, an approximation (estimate) of the spectrum of the
  adopted signal model is computed.
   The quality of the estimate depends on
   How well the assumed signal model represents the data.
   What values we assign to the unavailable signal samples.
   Which spectrum estimation method we use.
• Clearly, a meaningful application of spectrum estimation to
  practical problems requires sufficient apriori information,
  understanding of the signal generation process, knowledge of
  theoretical concepts, and experience.

                             IITM-WPW-VKA                               2
Signal Detectability and Pulse Compression

•   The efficiency of the radar system depends on how best it can identify the echoes in
    the presence of noise and unwanted clutter.

•   The important parameters from the system point of view influence the radar
    returns are the average power of transmission and the antenna aperture size.

•   Signal detectability is a measure of the radar performance in terms of transmission
    parameters.




                                    IITM-WPW-VKA                                       3
•   Received signal power Psig to the uncertainty Pn in the estimate of the noise power
    after averaging

         Psig        Psig
                             N c N1 / 2
         Pn        Ts Brec        inc

                     A e Pt 
                          h 2 (PRF   )( t )1 / 2
                   Ts Brec
                                       c      c

                 A e h 2Ts1Pave (h)( t      )1 / 2
                                            Bsig
    Ae       effective antenna area              Pt       peak power transmitted
            pulse length,                      PRF       pulse repetition frequency
     Pave = PtPRF       average Tx power        Brec     receiver band-width
    Ts       effective system noise temp.       Nc        No. of samples coherently added
    Ninc     number of resulting sums which are incoherently averaged
     c  1/Bsig correlation time of the scattering medium for the wavelength used
     t      total integration time              h        range or height
    h       height resolution.



                                                     IITM-WPW-VKA                           4
•   Average power is the important parameter for the strong returns and this is function of
    pulse length.
•   Short pulses are required for good range resolution, and the shorter length of Inter pulse
    period (IPP) generates the problem of range ambiguity.
    (Therefore maximum limit on the PRF is restricted due to the above problems)

•   Pulse compression and frequency stepping are techniques which allow more of the
    transmitter average power capacity to be used without sacrificing range resolution.
•   A pulse of power P and duration  is in a certain sense converted into one of power nP
    and duration /n.
•   In the frequency domain compression involves manipulating the phases of the different
    frequency components of the pulse.

•   In the time domain a pulse can be compressed via phase coding, especially binary phase
    coding, a technique which is particularly amenable to digital processing techniques.
•   Since frequency is just the time derivative of phase, either can be manipulated to
    produce compression.
•   Phase coding has been used extensively in atmospheric radars and in commercial &
    military applications.


                                       IITM-WPW-VKA                                              5
                 Phase coded waveform                   Pulse compression
                 .
+1                                                      Barker codes
      +      +      +                    +              •   These were first discussed by
                             -       -       -              Barker (1953) and have been
 -1
                                                            used in Ionospheric
                                                            incoherent scatter
                                                            measurements.

                                                            The distinguishing feature of
                                                            these codes is that, the range
                                                            side-lobes have a uniform
      Binary phase coded signal                             amplitude of unity.

                                                        •   The compression process
                                                            only works, if the correlation
                                                            time of the scattering
ACF
                                                            medium is substantially
                                                            longer than the full-
                                                            uncompressed length of the
                                                            transmitted pulse
                              14 
                                         IITM-WPW-VKA                                    6
                                                    16            +++- ++-+ +++- --+-
                                                                  EDE2
Complementary code pairs
•  Barker codes have range side lobes                   8         ACF of pulse 1
   which are small, but which may still
   cause problems in MST applications.
•  Ideally a codes which supports high
   compression ratios (long codes) to get the
   possible altitude resolution.                    -8
                                                                  +++- ++-+ ---+ +++-+
                                                    16
•  Complementary phase codes are binary                           ED1D
   in their simplest form and they usually
   come in pairs.                                                 ACF of pulse 1
                                                    8
•  They are coded exactly as Barker codes,
   by a matched filter whose impulse
   response is the time reverse of the pulse.
•  The range side lobes of the resulting                    32
   ACF output for each pulse will generally         -8
   be larger for a barker code of
   comparable length.                               32
•  when the two pulses are
                                                                     Sum of ACFs
   complementary pair have the property
   that their side lobes are equal in
   magnitude but opposite in sign, so that          16
   when outputs are added the side lobes
   exactly cancel, leaving only the central
   peak.
                                                0
                                                            32
                                          IITM-WPW-VKA                             7
Decoding
  Input



     C(31)           C(30)              C(1)        C(0)




                                                           Output
                   Z-1                              Z-1


IMS A100 Modified transversal filter architecture




                                  IITM-WPW-VKA                      8
Signal and Data Processing


                      Signal Processor                    On-line/Off-line Processing
     I-Channel
                     Decoder             Coherent
                      (I&Q)              Integrator     Normalization    Windowing

     Q-Channel

                                    Time Series

                                                                    Fourier Analysis
                                                      Incoherent           &
                  Noise level     Spectrum            Averaging     Power Spectrum
                  Estimation      Cleaning




                                                      Power Spectrum

                          Moments               UVW           Zonal, Meridional, Vertical
                                                                     wind velocity

                                   Total Power, Mean Doppler, Doppler Width
         Off-line Processing


     Processing steps for extraction of parameters



                                         IITM-WPW-VKA                                       9
Coherent Integration
•   The detected quadrature signals are coherently integrated for many pulse returns
    which lead to an appreciable reduction in the volume of the data to be processed
    and an improvement in the SNR.

•   The coherent integration is made possible because of the over sampling of the
    Doppler signal resulting from the high PRF relative to the Doppler frequency.

•   In other words, the coherence time of the scattering process c is much greater than
    the sampling interval given by the inter pulse period tp.

•   The operation of coherent integration amounts to applying a low pass filter, whose
    time-domain representation is a rectangular window of Ti duration.

•   The signal spectrum is weighted by that of the integration filter sin2x/x2, where x =
    fTi and f is the Doppler shift in Hz.

•   The sampling operation at the integration time interval of Ti leads to frequency
    aliasing with signal power at frequencies f  (m/Ti), where m is any integer, added
    to that at f.


                                     IITM-WPW-VKA                                       10
•   In the case of a flat spectrum, the filtering and aliasing balance each other and
    white noise still looks white, with no tapering at window edges.

•   On the other hand, a signal peak with Doppler shift of 0.44/Ti Hz, near the edge of
    the aliasing window, will be attenuated by 3 dB by the filter function, whereas a
    peak near the center of the spectrum will be almost unaffected.

•   One should, therefore, be conservative in choosing Ni for coherent integration so as
    to ensure that all signals of interest are in the central portion of the post-
    integration spectrum.

•   The objective is to measure a signal x(n) of duration of N samples, n = 0,1, . . . . N-1.
    The measurement can be performed repeatedly. A total of M such measurements
    are performed and the results are averaged by the signal averaging. Let the results
    of the mth measurements, for m = 1,2, . . . M, are the samples.

    y m (n)  x(n)  (n) for   n  0,1,..., N  1


    where x(n) and (n) corresponds to signal and noise respectively.



                                       IITM-WPW-VKA                                        11
•   Integrates (average) the results of the M measurements

                  1 M
       ˆ( n ) 
       x             y m (n) for
                  M m 1
                                        n  0,1,.......N  1


                                                                            1
    ym(t)                                ˆ m (n)
                                         x                       ˆ( n ) 
                                                                 x            ˆ m (n )
                                                                              x
                            ym(n)                                           M
            A/D Converter                           Memory
                                                                  After measurements

                                                     ˆ m 1(n)
                                                     x

•   The result of the averaging operation may be expressed as

               1 M                                                                         1 M
      x (n ) 
      ˆ             y m (n ) 
                                1
                                     x(n)  (n)  x(n)   (n)              where (n)     m(n)
                                                                                             M m 1
               M m 1           M m 1

•   Assuming m (n) to be mutually uncorrelated; that is,                        E[ m(n) l(n) ] = s2 ml,
•   The variance of the averaged noise

                                       M                                   M
                                1                                  1
      2  E[
     s     m (n) ] 
                  2
                                    2    E[vm(n)v1(n)]             2       s2 ml
                                                                               v
                              M m,l 0                           M m,l o

                                                   IITM-WPW-VKA                                                12
                  1                               1             1 2
                      (s 2  s 2   s 2) 
                          v     v          v           M s2 
                                                          v       sv
                  M2                              M2            M

•   therefore signal to noise ratio (SNR) is improved by a factor of M.

•   To increase the SNR, the number of coherent integration should be
    selected as large as possible within which the received signals are phase
    coherent with each other.

•   There are two cases that make the integration time finite

•   1. movement of scatterers relative to each other within the radar sampling
    volume.
•   2. the mean motion of scatterers relative to the radar due to background wind
    fields.

•   The relative motion of scatterers is estimated by a correlation time, which is
    defined as a half-power width of an auto correlation function of the received signal.



                                       IITM-WPW-VKA                                    13
•   It depends on the radar wavelength, antenna beamwidth and altitude.

•   Inverse of the coherent integration time corresponds to half of the
    maximum frequency range of the Doppler spectra.

•   Therefore integration time should be so selected as to unambiguously
    determine the maximum radial wind velocity.

•   This limits the length of coherent integration.

Normalization
•   The input data is to be normalized by applying a scaling factor corresponding to
    the operation done on it.


•   This will reduce the chance of data overflowing due to any other succeeding
    operation.




                                    IITM-WPW-VKA                                       14
Normalization
•   The Normalization has following components.

•          a. sampling resolution of ADC
•          b. scaling due to pulse compression in decoder
•          c. scaling due to coherent integration
•          d. scaling due to number of FFT points.

•   if     v - ADC bit resolution ( 10/16384),
           w - Pulse width in microsecond,
           M -Number of IPP integrated = Integrated time /inter pulse period,
           N - Number of FFT points,
                                                v
    then the Normalization factor           s
                                               w MN

•   The complex time series { Ii , Qi where i = 0, . . ,N-1} at the output of the signal
    processor is scaled as
                                        ~
                                        Ii  s  Ii
                                        ~
                                        Qi  s  Qi

                                      IITM-WPW-VKA                                         15
  Windowing
  •   It is well known that the application of FFT to a finite length data gives rise to
      leakage and picket fence effects.

                Amp                   F()
Leakage                                                            Picket fence effect

                                                            Amp
                               f1
                                                             1
                      Cosine wave impulse
                                                            0.6

                                                                        0       1   2       3   4      5   6    7
                                                                                                                    freq
                                                  W()
                                                                                Independent Filters


                                                                                    Power response
           -f                                        +f    Power
                  FT of Rectangular Window
                                                             1
                                                             0.4

                                            F() * W()             0       1       2   3       4      5    6   7
                                                                                                    Frequency


           -f                    f1                   +f
                                                   IITM-WPW-VKA                                                       16
•   Weighting the data with suitable windows can reduce these effects.
•   However the use of the data windows other than the rectangular window affects the bias,
    variance and frequency resolution of the spectral estimates.
•   In general variance of the estimate increases with the uses of a window. An estimate is
    said to be consistent if the bias and the variance both tend to zero as the number of
    observations is increased.
•   Thus, the problem associated with the spectral estimation of a finite length data by the
    FFT techniques is the problem of establishing efficient data windows or data smoothing
    schemes.
•   Characteristics of a window
•   It is desired that a window, f(t), has the following properties.
•   1. f(t) is real and non-negative.
•   2. f(t) is an even function, i.e., f(t) = f(-t).
•   3. f(t) should attain its maximum at t=0, i.e., | f(t) | < f(0) for all t.
•   4. Main lobe width should be as small as possible.
•   5. The Maximum sidelobe level should be as small as possible relative to the main lobe
    peak.
•   6. The mainlobe should contain a large part of the total energy.
•   7. If the mth derivative of f(t) is impulsive, then the peak of the side-lobes of | F() |
    decays asymptotically as 6m dB/octave.



                                      IITM-WPW-VKA                                         17
•   Rectangular ( Box Car ) window
    Window is defined by
     f (t)  1           t    
              0         Elsewhere                              and its Fourier transform
                                                                                2 sin( )
                                                                  F()      
•   Bartlet Window                                                                  
    Window is defined by
                         t                                     and its Fourier transform
      f (t)        1                  t 
                         
                   0                   Elsewhere                                4 sin 2 ()
                                                                  F()      
                                                                                     2

•   Hanning Window
    Window is defined by
                                                               and its Fourier transform
                                   t
     f (t)        0.5  0.5 cos               t                          sin( )         sin(    ) sin(    ) 
                                                                  F()                0.5               
                  0                        Elsewhere                                                            




                                                           IITM-WPW-VKA                                                         18
•   Hamming Window
    Window is defined by
    f (t)           0.54  0.46 cos(t ),             t         
                    0                                 Elsewhere

    and its Fourier transform

                      sin( )        sin(    ) sin(    ) 
     F()  1.08                0.46               
                                                          

•   COS3 Window
    window is defined by

     f (t)  0.75 cos ( t 2)  0.25 cos ( 3t 2),          t 
             0                                        Elsewhere

    and its Fourier transform
                            sin (   2)  sin (   2) 
      F()           0.75                                           
                               (   2)      (   2)  
                               sin (  3 )  sin (  3 )  
                         0.25                  
                               (  3 )        (  3 )   

                                                    IITM-WPW-VKA           19
•   Blackman Window
    Window is defined by
     f (t)  0.42  0.5 cos (t )  0.25 cos (2t )                  t             
            0                                                Elsewhere

    and its Fourier transform is
                 0.84 sin( )             sin (   ) sin (   )  
     F()                           0.5                
                                          (   )        (   )   
                                             sin (  2  ) sin (  2  )  
                                      0.08                  
                                             (  2  )        (  2  )   

•   Computation of window parameters
    The important window parameters which are useful in selecting an appropriate window
    for a particular application, which are
                                                                   
                                                              1
                                                                    f (t) dt
                                                                            2
1. Variance compensation factor                        Q
                                                              2
                                                                   

                                                                
                                                                        2
                                                                 f ( t) dt
2. Dispersion factor                                         
                                                                                  2
                                                           1                
                                                                 f ( t )2dt 
                                                            2  
                                                                             
                                                                              

                                                IITM-WPW-VKA                              20
                                      
 3. Total energy                  E   f ( t)2 dt
                                     

 4. Major lobe energy contents (MLE).
 5. Half power bandwidth (HPBW)
 6. W = major lobewidth/ major lobe width of rectangular window.
 7. BW = Half power bandwidth/ Half power bandwidth of rectangular window.
 8. coherent gain           1  2
                        G     f ( t ) dt
                                    2  

 9. Rate of fall of sidelobe levels (RFSL).
 10. Peak sidelobe level (PSLL).
 11. Degradation loss L, is the reciprocal of the dispersion in dB.
Data window           Q                     E            MLE        SLL      W      BW     G       L      RFSL
              (dB)     (dB/Oct)                           (%)


Rectangular   1.0      1.0                   2.0          90.282     -13.26   1.0    1.0    1.0     0.0     -6

Hanning       1.50     0.375                 0.75         99.9485    -31.48   2.0    1.63   0.5    -1.76    -6

Hamming       1.363    0.397                 0.795        99.9632    -42.62   2.0    1.48   0.54   -1.34    -6

cosine3       1.735    0.313                 0.625        99.9925    -39.30   2.50   1.81   0.42 -2.39      -24

Bartlet       1.333    0.333                 0.6667       99.7057    -26.53   2.0    1.40   0.5    -1.25    -12


Blackman      1.727    0.305                 0.609        99.9989    -58.12   3.0    1.86   0.42   -2.37    -18




                                                      IITM-WPW-VKA                                                21
•   A window that yields

•   small values of variance compensation factor, dispersion factor, total
    energy, peak sidelobe level, W and BW, and a large value of major lobe
    energy content is desirable in spectral estimation via FFT.

•   However a decrease in peak sidelobe level is associated with an increase in
    major lobewidth and, hence, a corresponding increase in the loss of
    frequency resolution.

•   The energy of window is an important parameter since the variance of the
    smoothed spectral estimate is proportional to E.

•   But the variance of the estimate is a measure of its reliability; smaller the
    value, higher is the reliability of the estimate.

•   Thus, the selection of data window for spectral estimation is a judicious
    compromise among the various parameters described above.

                                     IITM-WPW-VKA                               22
• Weighting the data with suitable windows can reduce leakage.

• Tapering is another name for the data windowing operation in the time
  domain.

• Single taper smoothed spectrum estimates are plagued by a trade-off
  between the variance of the estimate and the bias caused by spectral
  leakage.
• Applying a taper to reduce bias discards data, increasing the variance
  of the estimate.

• Using a taper also unevenly samples the record. Single taper
  estimators, which are less affected by leakage, not only have increased
  variance but also can misrepresent the spectra of non-stationary data.

• So as long as only a single data taper is used, there will be a trade-off
  between the resistance to spectral leakage and the variance of a
  spectral estimate.

                               IITM-WPW-VKA                               23
• Single taper spectral estimates have relatively large variance
  (increasing as a large fraction of data is discarded and the bias of the
  estimate is reduced) and are inconsistent estimates (i.e., the variance of
  the estimate does not drop as one increases the number of data).

• To counteract this, it is conventional to smooth the single taper spectral
  estimate by applying a moving average to the estimate.

• This reduces the variance of the estimate but results in a short - range
  loss of frequency resolution and therefore an increase in the bias of the
  estimate.

• An estimate is to be consistent if the bias and the variance both tend to
  zero as the number of observations is increased.

• Thus, the problem associated with the spectral estimation of a finite
  length data by the FFT techniques is the problem of establishing
  efficient data windows or data smoothing schemes.

                               IITM-WPW-VKA                               24
• Multitaper spectral analysis technique find wider applications in the
  signal analysis

• First, the data are multiplied by not one, but several leakage - resistant
  tapers. This yields several tapered time series from one record.

• Taking the DFTs of each of these time series, several “eigen spectra”
  are produced which are averaged to form a single spectral estimate.

• There are a number of Multitapers that have been proposed.

• Some of them are Slepian tapers, Discrete Prolate Spheroidal
  sequences, Sinusoidal Tapers, etc,

• The central premise of this multitaper approach is that if the data tapers
  are properly designed orthogonal functions, then, under mild
  conditions, the spectral estimates would be independent of each other
  at every frequency.


                               IITM-WPW-VKA                                25
• Averaging would reduce the variance while proper design of full -
  length windows would reduce bias and loss of resolution.

• The pictorial description of the multitaper approach to power
  spectrum estimation is as shown in Figure.

                               1                 N
                                   Data Record



             1             N                         1                 N
                 Taper 1                                 Periodogram         A
                                                                             V
                                                                             E
                                                                             R
             1             N                         1                 N
                                                                             A
                 Taper 2                                 Periodogram
                                                                             G
                                                                             E
                                                                             R




             1             N                         1                 N     Final
                 Taper M                                 Periodogram       Estimate




                                      IITM-WPW-VKA                                    26
• The multiple tapers are constructed so that each taper samples the time
  series in a different manner while optimizing resistance to spectral
  leakage.

• The statistical information discarded by the first taper is partially
  recovered by the second taper, the information discarded by the first
  two tapers is partially retrieved by the third taper, and so on.

• Only a few lower-order tapers are employed, as the higher - order
  tapers allow an unacceptable level of spectral leakage.

• One can use these tapers to produce an estimate that is not hampered
  by the trade - off between leakage and variance that plagues single-
  taper estimates.




                              IITM-WPW-VKA                                27
Sinusoidal Multitaper

• The continuous time minimum bias tapers are given as
        vk (t )  2 sin(kt  ) (k  1,2,...)


   and its Fourier Transform as
                                k          k  
                       sin      sin      
        Vk ( )  2 j                         
                             k            k 
                                         
                      
                                                  
                                                     




• The discrete analogs of the continuous time minimum bias tapers are
  called sinusoidal tapers. The nth sinusoidal taper is given by
                  2      kn 
      vk (n)        sin            ; n = 1,2,…,N ; k=1,2,…,K
                 N 1  N 1




                                                IITM-WPW-VKA            28
Where the amplitude term on the right is normalization factor that ensures
  orthonormality of the tapers.

These sine tapers have much narrower main-lobe and much higher side-
   lobes.

Thus they achieve a smaller bias due to smoothing by the main lobe, but
  at the expense of side-lobe suppression.

Clearly this performance is acceptable if the spectrum is varying slowly.

The kth sinusoidal taper has its spectral energy concentrated in the
   frequency bands.


         .(k  1)           (k  1)
                                     k=1,2,..., K
          N 1               N 1




                                           IITM-WPW-VKA                 29
• Sinusoidal Multitaper




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Fourier analysis
•   Spectral analysis is connected with characterizing the frequency content of a signal.
•   A large number of spectral analysis techniques are available in the literature. This
    can be broadly classified in to non-parametric or Fourier analysis based method
    and parametric or model based methods.
•   Fourier proposed that any finite duration signal, even a signal with discontinuities,
    can be expressed as an infinite summation of harmonically related sinusoidal
    component
                  
      x( t )     (A k cos (k0t)  Bk sin (k0 t))
                 k 0

•   FFT applied to complex time series {(Ii, Qi), i = 0,1, . . . ,N-1} to obtain complex
    frequency domain spectrum { (Xi, Yi), i = 0, . . . . , N-1}
                         N 1
                   1
      Xi  Yi 
                   N
                          (Ik  jQ k)      exp (2ik / N)   i  0,  N  1
                         k 0

•   Power Spectrum
    Power spectrum is calculated from the complex spectrum as
      Pi  Xi2  Yi2 ,    i  0,1,......., N  1



                                                    IITM-WPW-VKA                      36
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Incoherent Integration ( Spectral averaging)

•   Incoherent integration is the averaging the power spectrum number of times.
             1 m
      Pi       P       i  0,  N  1
             m k  1ik
    where m is the number of spectra integrated.
•   The advantage of incoherent integration is that it improves the detectability of Doppler
    spectrum. The detectability is defined as
               PS
      D
             sS  N

    where PS is peak spectral density of the signal spectrum, and sS+N is the standard
    deviation of spectral densities.

•   When fluctuations of the signal spectral density is much smaller than that for noise, then
         P
      D s
             sN


•   The noise spectral density has a c2 distribution, because the noise spectral density
    is a summation of square of real and imaginary components of amplitude spectrum
    which are assumed to have Gaussian distribution.

                                            IITM-WPW-VKA                                    39
The mean value m and standard deviation s of the c2 distribution becomes

      m      m        and      s       m

•   For a single spectrum sN is equal to PN.

•   When Doppler spectra are averaged m times, the mean values of spectral densities of
    both the signal and noise are not changed.

•   But sN / PN becomes 1/(m).

•    as m times integration of the noise produces a c2 distribution and as a result D is
    increased by (m).




                                        IITM-WPW-VKA                                 40
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Power spectrum cleaning

•   Due to various reasons the radar echoes may get corrupted by ground clutter,
    system bias, interference, image formation etc..

•   The data is to be cleaned from these problems before going for analysis.




    -fmax                                   0                     fmax
                        p   N 1  p N
                            2          1
            p   N
                                    2

                2
                                2

 N/2 corresponds to zero frequency.
• Spikes (glitches) in the time series will generate a constant amplitude band all over
    the frequency bandwidth.
• Constant frequency bands will form in the power spectrum by the interference
    generated in the system or due to extraneous signal.

                                                IITM-WPW-VKA                          42
Noise level estimation
•   There are many methods adapted to find out the noise level estimation.
•   Basically all methods are statistical approximation to the near values.
•   The method implemented here is based on the variance decided by a threshold criterion,
    Hildebrand and Sekhon (1974).
•   This method makes use of the observed Doppler spectrum and of the physical
    properties of white noise; it does not involve knowledge of the noise level of the
    radar instrument system and is now widely used in atmospheric radar noise
    threshold estimation and removal.
    Variance(S)
                     1   over number of spectra averaaged
     mean(S)2




                                            IITM-WPW-VKA                                43
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•   The noise level threshold shall be estimated to the maximum level L, such that the set of
    Spectral points below the level S, nearly satisfies the criterion,
Step 1:
    Reorder the spectrum { Pi, i = 0, . . . N-1} in ascending order to form. Let this sequence
    be written as{ Ai, i = 0, . . . N-1} and Ai < Aj for i < j
Step 2: compute
          n A                                    n      2
                  i                                  Ai                 2
    Pn                                   Qn                   Pn
        i  0 (n  i)                          i  0 n 1


                                        P2
     and     if   Q n  0,    Rn 
                                         n    ,     for     n  1,  , N
                                     (Qn
                                          M)



Where M is the number of spectra that were averaged for obtaining the data.

Step 3:
                                                                  n   such that R  1
          Noise   level (L)          Pk    where         k  min 1. if no n meetsnthe above criterion




                                                          IITM-WPW-VKA                                   46
Moments Estimation

•   The extraction of zeroth, first and second moments is the key reason for on doing all the
    signal processing and there by finding out the various atmospheric and turbulence
    parameters in the region of radar sounding. The basic steps involved in the estimation of
    moments, Woodman (1985) are given below.
•   Step 1.
•   Reorder the spectrum to its correct index of frequency (ie. -fmaximum to +fmaximum) in the
    following manner.
•   Spectral index      0          1                      N/2                    N-1
           ambiguous freq.       -fmaximum             Zero freq.             +fmaximum
•   Step 2:
•   Subtract noise level L from spectrum
•   Step 3:
•   i) Find the index l of the peak value in the spectrum,
                  ~ ~
       ie         P1  Pi      for all i  0, N  1
•   ii) Find m, the lower Doppler point of index from the peak point.
                  ~
            ie    pi  0    for all m  i  l
•   iii) Find n the upper Doppler point of index from the peak point
                   ~
             ie    pi  0    for all l  i  n

                                                 IITM-WPW-VKA                              47
•   Step 4:
    The moments are computed as
                        n ~
       i)         M0     
                          Pi
                       im

•   represents zeroth moment or Total Power in the Doppler spectrum.

                       1 n ~                             (i  N 2)
      ii )       M1       Pifi
                      M0 i  m
                                      where     fi 
                                                       (IPP  n  N)

•   represents the first moment or mean Doppler in Hz
                          1 n ~             2
         iii )      M2        Pi (f  M1)
                         M 0 im i
•   represents the second moment or variance, a measure of dispersion from central
    frequency.                                                       P
                                                         M0 
             v)     Signal to Noise Ratio (SNR)  10 log                dB        P
                                                         ( N  L) 
                                                                               s
             iv )    Doppler width (full )  2 M 2                   Hz

                                                                               fd
                                                 IITM-WPW-VKA                           48
Sample Doppler spectra for a few range gates showing 5 candidate-peaks per range gate that form
                    the basis for the adaptive technique of signal detection.

                                       IITM-WPW-VKA                                    49
 (a) Height profiles of Doppler power spectra observed on 10 July 2002 using the 10 o east radar
beam when the SNR is low. (b) Mean Doppler Velocity-Height profile extracted from the spectra
    shown in (a) using the conventional peak detection method (dotted line) and the adaptive
                            moments extraction technique (solid line).
                                        IITM-WPW-VKA                                     50
  Eight-profile average mean Doppler velocity-Height profiles and corresponding
standard deviations observed on July 10, 2002 when the SNR is low for (a) 10 deg
  east radar beam using adaptive moments estimation technique and (b) 10 deg
           east radar beam using conventional peak detection method.
                                 IITM-WPW-VKA                             51
(a) Height profiles of Doppler power spectra observed 10 May 2002 using the 10o
east radar beam when the SNR is high. (b) Mean Doppler-Height profile extracted
  from the spectra shown in (a) using the conventional peak detection method
    (dotted line) and the adaptive moments extraction technique (solid line).
                                IITM-WPW-VKA                             52
 Eight-profile average mean Doppler velocity-Height profiles and corresponding
 standard deviations observed on 10 May 2002 for the (a) 10o east radar beam
using adaptive moments extraction technique and (c) 10o east radar beam using
                      conventional peak detection method.
                                IITM-WPW-VKA                            53
                               Press)
Anandan et al., JAOT, 2004 (In IITM-WPW-VKA   54
UVW Computation
• The prime objective of atmospheric radar is to obtain the vector wind
   velocity.
• Velocity measured by a radar with the Doppler technique is a line of sight
   velocity, which is the projection of velocity vector in the radial direction.
• There are two different techniques of determining the three components
   of the velocity vector: the Doppler Beam Swinging (DBS) method and
   Spaced Antenna (SA) method.
• The DBS method uses a minimum of three radar beam orientations
   (Vertical, East-West, and North-South) to derive the three components of
   the wind vector (Vertical, Zonal and Meriodional)




                                 IITM-WPW-VKA                                 55
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•   Calculation of radial velocity and height:
•   For representing the observation results in physical parameters, the Doppler
    frequency and range bin have to be expressed in terms of corresponding radial
    velocity and vertical height.

                    c  t R  cos                                  c  fD         fD  
     Height, H                      meters         Velocity, V             or              m / sec
                         sin                                       2  fc           2




                                                                                  w
                      v
            v hor    rad
                     sin                                                                    u

            vrad  v hor sin  T  w cos T
                                                                    T

                    v  wcos T      
            vhor   rad
                                     
                                      
                        sin  T      
                                                                                         E




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Change in phase of the wave as a measure of velocity
                                         c                    r
                t  2ft  2              t  2
                                                             
                                                     ( 2r )
              considering two way  2
                                                      
                                             r
                        (t )  2(2            )
                                             
                       d       2 dr
                           2 ( )
                       dt         dt
                              4
                       d       v
                                   
                                   4
                       2f d           v
                                     
                                 2v                        fd
                        fd             or v 
                                                             2


                           IITM-WPW-VKA                           58
•   Computation of absolute Wind velocity vectors (UVW):

•   After computing the radial velocity for different beam positions, the absolute
    velocity (UVW) can be calculated.

•   To compute the UVW, at least three non-coplanar beam radial velocity data is
    required. If higher number of different beam data are available, then the
    computation will give an optimum result in the least square method.

•   Line of sight component of the wind vector V (Vx, VY, Vz) is

    VD = V . i = Vx cosx + Vy cosy + Vz cosz

    where X, Y, and Z directions are aligned to East-West, North-South and Zenith
    respectively.




                                     IITM-WPW-VKA                               59
•   Applying least square method, residual

           2 = (Vx cosx + Vy cosy + Vz cosz - VD i)2

                            where VD i = fD i * /2 and i represents the beam number

•   To satisfy the minimum residual

    2 /Vk = 0

    k corresponding to X,Y, and Z leads to


                                                                           1
                                                                      
                cos2 Xi           cos Xi cos Yi  cos Xi cos Zi
     Vx         i                i                 i                          VDi cos Xi 
      Vy    cos Xi cos Yi                                                  VDi cos Yi 
                                   cos  2 Yi
                                                       cos Yi cos Zi
                                                                                              
      Vz   i
      
                                       i              i                           VDi cos Zi 
                                                                                               
                cos Xi cos zi    cos Yi cos Zi    cos  2 Zi
                                                                       
               i
                                   i                    i             
                                                                       


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             20




             15
Height(km)




             10




              5



                  -30 -20 -10   0   10 20   -15 -10 -5   0   5 10 15 -0.4 -0.2   0.0   0.2   0.4
                      Zonal (m/s)             Meridional (m/s)           Vertical (m/s)


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                20
                                                       GPS                                        GPS
                18                                     MST                                        MST


                16


                14
Height ( km )




                12


                10


                 8


                 6


                 4


                 2


                 0
                     0   8      16      24        32         40   0   90        180         270         360
                             Wind Speed ( m/s )                            Wind Direction




                                                  IITM-WPW-VKA                                                62
Cases of special interest




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Cases of special interest




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Cases of special interest




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Cases of special interest




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Cases of special interest




                            IITM-WPW-VKA   67
THANK YOU



  IITM-WPW-VKA   68

								
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