29
TOPIC 5 ZETA TRANSFORM PART 2

TOPIC&5&& ZETA&TRANSFORM · RECALL:%ZETATRANSFORM X (z)= X+1 n=1 x[n]zn AAACKHicbVDNSsNAGNz4W

  • Upload
    others

  • View
    0

  • Download
    0

Embed Size (px)

Citation preview

Page 1: TOPIC&5&& ZETA&TRANSFORM · RECALL:%ZETATRANSFORM X (z)= X+1 n=1 x[n]zn AAACKHicbVDNSsNAGNz4W

TOPIC  5    ZETA  TRANSFORM  

PART  2  

Page 2: TOPIC&5&& ZETA&TRANSFORM · RECALL:%ZETATRANSFORM X (z)= X+1 n=1 x[n]zn AAACKHicbVDNSsNAGNz4W

RECALL:  ZETA  TRANSFORM  

X(z) =+1X

n=�1x[n]z�n

<latexit sha1_base64="z3DmogIykn8ZGmKolxbDMDsopzo=">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</latexit>

z = rej⌦

<latexit sha1_base64="ZvlB0RzP1sGgequLEINahKXTQRw=">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</latexit>

We  will  use  the  polar  representa?on    for  the  variable  z.   FREQUENCY  

z 2 ROC

<latexit sha1_base64="2Dug1z6kONctejT3T1DswOnlQQY=">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</latexit>

Region  of  Convergence  

Page 3: TOPIC&5&& ZETA&TRANSFORM · RECALL:%ZETATRANSFORM X (z)= X+1 n=1 x[n]zn AAACKHicbVDNSsNAGNz4W

RECALL:  Example  1  

x[n] = a

nu[n]

<latexit sha1_base64="l9thd/Lf9TAODrOvE6+bnawO6LA=">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</latexit>

ROC  !!!!!  Region  of  convergence  

The  ROC  is  depicted  in  “grey”  

Page 4: TOPIC&5&& ZETA&TRANSFORM · RECALL:%ZETATRANSFORM X (z)= X+1 n=1 x[n]zn AAACKHicbVDNSsNAGNz4W

RECALL:  Example  2  

x[n] = �a

nu[�n� 1]

<latexit sha1_base64="M36EQ2y3Y9hFix3OV+X/nJumD40=">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</latexit>

X(z) =z

z � a

<latexit sha1_base64="gQc6vp6SX9lUNijJ5RR90LmPC84=">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</latexit>

The  ROC  is  depicted  in  “grey”  

|z| < a

<latexit sha1_base64="2+TUcOC0qiGhdaCrEZTSGWh8sVM=">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</latexit>

ROC  

|      |  

Page 5: TOPIC&5&& ZETA&TRANSFORM · RECALL:%ZETATRANSFORM X (z)= X+1 n=1 x[n]zn AAACKHicbVDNSsNAGNz4W

Example  3  

0 < b < 1

<latexit sha1_base64="2uSM9ngFrFQmSPEOJCXzleTzygU=">AAAB7HicbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0SJF0MYygvmA5Ah7m7lkyd7esbsnhCO/wcZCEVt/kJ3/xk1yhSY+GHi8N8PMvCARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d3Mbz+h0jyWj2aSoB/RoeQhZ9RYqenWgprXL1fcqjsHWSVeTiqQo9Evf/UGMUsjlIYJqnXXcxPjZ1QZzgROS71UY0LZmA6xa6mkEWo/mx87JWdWGZAwVrakIXP190RGI60nUWA7I2pGetmbif953dSEN37GZZIalGyxKEwFMTGZfU4GXCEzYmIJZYrbWwkbUUWZsfmUbAje8surpHVR9S6rVw+XlfptHkcRTuAUzsGDa6jDPTSgCQw4PMMrvDnSeXHenY9Fa8HJZ47hD5zPH6aLje8=</latexit>

b > 1

<latexit sha1_base64="uS6OHvAdAvWQZCsI8CkQVjeb/bY=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0ZMUvXisaD+gDWWz3bRLN5uwOxFK6E/w4kERr/4ib/4bt20O2vpg4PHeDDPzgkQKg6777RRWVtfWN4qbpa3tnd298v5B08SpZrzBYhnrdkANl0LxBgqUvJ1oTqNA8lYwup36rSeujYjVI44T7kd0oEQoGEUrPQTXXq9ccavuDGSZeDmpQI56r/zV7ccsjbhCJqkxHc9N0M+oRsEkn5S6qeEJZSM64B1LFY248bPZqRNyYpU+CWNtSyGZqb8nMhoZM44C2xlRHJpFbyr+53VSDK/8TKgkRa7YfFGYSoIxmf5N+kJzhnJsCWVa2FsJG1JNGdp0SjYEb/HlZdI8q3rn1Yv780rtJo+jCEdwDKfgwSXU4A7q0AAGA3iGV3hzpPPivDsf89aCk88cwh84nz+6kI1x</latexit>

Page 6: TOPIC&5&& ZETA&TRANSFORM · RECALL:%ZETATRANSFORM X (z)= X+1 n=1 x[n]zn AAACKHicbVDNSsNAGNz4W

Example  3  

FROM  Example  1  and  2  (replacing  “a”  with  “b”):  

ROC  

ROC  

Page 7: TOPIC&5&& ZETA&TRANSFORM · RECALL:%ZETATRANSFORM X (z)= X+1 n=1 x[n]zn AAACKHicbVDNSsNAGNz4W

Example  3:  ROC  

|z| < 1/b

<latexit sha1_base64="6pHd9QT/UuKqw/h+jk6nX/VO2mo=">AAAB7nicbVA9SwNBEJ2LXzF+RS1tFoNgFe8kooVFwEa7COYDkiPsbfaSJXu7x+6eEC/5ETYWitj6e+z8N26SKzTxwcDjvRlm5gUxZ9q47reTW1ldW9/Ibxa2tnd294r7Bw0tE0VonUguVSvAmnImaN0ww2krVhRHAafNYHgz9ZuPVGkmxYMZxdSPcF+wkBFsrNQcP42vvbOgWyy5ZXcGtEy8jJQgQ61b/Or0JEkiKgzhWOu258bGT7EyjHA6KXQSTWNMhrhP25YKHFHtp7NzJ+jEKj0USmVLGDRTf0+kONJ6FAW2M8JmoBe9qfif105MeOWnTMSJoYLMF4UJR0ai6e+oxxQlho8swUQxeysiA6wwMTahgg3BW3x5mTTOy16lfHFfKVXvsjjycATHcAoeXEIVbqEGdSAwhGd4hTcndl6cd+dj3ppzsplD+APn8wfM/Y8/</latexit>

|z| > b

<latexit sha1_base64="jFC6sQ6NXh5FQgE2CylI14T6Dq0=">AAAB7HicbVBNSwMxEJ2tX7V+VT16CRbBU9kVRU9S8KK3Cm5baJeSTbNtaDZZkqxQt/0NXjwo4tUf5M1/Y9ruQVsfDDzem2FmXphwpo3rfjuFldW19Y3iZmlre2d3r7x/0NAyVYT6RHKpWiHWlDNBfcMMp61EURyHnDbD4c3Ubz5SpZkUD2aU0CDGfcEiRrCxkj9+Gl+H3XLFrbozoGXi5aQCOerd8lenJ0kaU2EIx1q3PTcxQYaVYYTTSamTappgMsR92rZU4JjqIJsdO0EnVumhSCpbwqCZ+nsiw7HWozi0nTE2A73oTcX/vHZqoqsgYyJJDRVkvihKOTISTT9HPaYoMXxkCSaK2VsRGWCFibH5lGwI3uLLy6RxVvXOqxf355XaXR5HEY7gGE7Bg0uowS3UwQcCDJ7hFd4c4bw4787HvLXg5DOH8AfO5w/wsY7N</latexit>

ROC  of  this  example:    we  have  to  sa?sfy  SIMULTANEOUSLY    the  two  ROCs  below  (of  the  previous  examples)    

ROC-­‐1:   ROC-­‐2:  

The  only  way  to  obtain  that  is:   b < 1

<latexit sha1_base64="Ta7IV/c2snZ2+kxoHrJJeSl9scE=">AAAB6nicbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0cIiYKNdRPMByRH2NnvJkr29Y3dOCEd+go2FIrb+Ijv/jZvkCk18MPB4b4aZeUEihUHX/XYKK6tr6xvFzdLW9s7uXnn/oGniVDPeYLGMdTughkuheAMFSt5ONKdRIHkrGN1M/dYT10bE6hHHCfcjOlAiFIyilR6Ca69XrrhVdwayTLycVCBHvVf+6vZjlkZcIZPUmI7nJuhnVKNgkk9K3dTwhLIRHfCOpYpG3PjZ7NQJObFKn4SxtqWQzNTfExmNjBlHge2MKA7NojcV//M6KYZXfiZUkiJXbL4oTCXBmEz/Jn2hOUM5toQyLeythA2ppgxtOiUbgrf48jJpnlW98+rF/XmldpfHUYQjOIZT8OASanALdWgAgwE8wyu8OdJ5cd6dj3lrwclnDuEPnM8fuaGNdg==</latexit>

In  this  case,  the  ROC  is:  

b < |z| < 1/b

<latexit sha1_base64="7y0cW5tZKPL9nDkqYXtBu1Hnk2s=">AAAB8HicbVDLSgNBEOyNrxhfUY9eBoPgKe5KRA85BLzoLYJ5SLKE2clsMmRmdpmZFeImX+HFgyJe/Rxv/o2Tx0ETCxqKqm66u4KYM21c99vJrKyurW9kN3Nb2zu7e/n9g7qOEkVojUQ8Us0Aa8qZpDXDDKfNWFEsAk4bweB64jceqdIskvdmGFNf4J5kISPYWOkhKI+eRmXvLOjkC27RnQItE29OCjBHtZP/ancjkggqDeFY65bnxsZPsTKMcDrOtRNNY0wGuEdblkosqPbT6cFjdGKVLgojZUsaNFV/T6RYaD0Uge0U2PT1ojcR//NaiQmv/JTJODFUktmiMOHIRGjyPeoyRYnhQ0swUczeikgfK0yMzShnQ/AWX14m9fOiVype3JUKldt5HFk4gmM4BQ8uoQI3UIUaEBDwDK/w5ijnxXl3PmatGWc+cwh/4Hz+AAspj/E=</latexit>

Page 8: TOPIC&5&& ZETA&TRANSFORM · RECALL:%ZETATRANSFORM X (z)= X+1 n=1 x[n]zn AAACKHicbVDNSsNAGNz4W

Solu?on  of  Example  3  for  b<1  Solu?on  for  b<1:  

! The  FT  always  exists!  The  unit  circle  is  always  within  The  ROC  !!  

!   b  and  1/b  are  the  two  poles  !  !   we  have  a  “DONUT”  ROC  !  

…it  can  be  also  wri\en  with  a  unique  frac?on….  

Page 9: TOPIC&5&& ZETA&TRANSFORM · RECALL:%ZETATRANSFORM X (z)= X+1 n=1 x[n]zn AAACKHicbVDNSsNAGNz4W

Solu?on  of  Example  3  for  b>1  

Solu?on  for  b>1:  

! No  solu?on.  ! The  ROC  is  empty.  

Page 10: TOPIC&5&& ZETA&TRANSFORM · RECALL:%ZETATRANSFORM X (z)= X+1 n=1 x[n]zn AAACKHicbVDNSsNAGNz4W

Recalling  the  signal…  

0 < b < 1

<latexit sha1_base64="2uSM9ngFrFQmSPEOJCXzleTzygU=">AAAB7HicbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0SJF0MYygvmA5Ah7m7lkyd7esbsnhCO/wcZCEVt/kJ3/xk1yhSY+GHi8N8PMvCARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d3Mbz+h0jyWj2aSoB/RoeQhZ9RYqenWgprXL1fcqjsHWSVeTiqQo9Evf/UGMUsjlIYJqnXXcxPjZ1QZzgROS71UY0LZmA6xa6mkEWo/mx87JWdWGZAwVrakIXP190RGI60nUWA7I2pGetmbif953dSEN37GZZIalGyxKEwFMTGZfU4GXCEzYmIJZYrbWwkbUUWZsfmUbAje8surpHVR9S6rVw+XlfptHkcRTuAUzsGDa6jDPTSgCQw4PMMrvDnSeXHenY9Fa8HJZ47hD5zPH6aLje8=</latexit>

b > 1

<latexit sha1_base64="uS6OHvAdAvWQZCsI8CkQVjeb/bY=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0ZMUvXisaD+gDWWz3bRLN5uwOxFK6E/w4kERr/4ib/4bt20O2vpg4PHeDDPzgkQKg6777RRWVtfWN4qbpa3tnd298v5B08SpZrzBYhnrdkANl0LxBgqUvJ1oTqNA8lYwup36rSeujYjVI44T7kd0oEQoGEUrPQTXXq9ccavuDGSZeDmpQI56r/zV7ccsjbhCJqkxHc9N0M+oRsEkn5S6qeEJZSM64B1LFY248bPZqRNyYpU+CWNtSyGZqb8nMhoZM44C2xlRHJpFbyr+53VSDK/8TKgkRa7YfFGYSoIxmf5N+kJzhnJsCWVa2FsJG1JNGdp0SjYEb/HlZdI8q3rn1Yv780rtJo+jCEdwDKfgwSXU4A7q0AAGA3iGV3hzpPPivDsf89aCk88cwh84nz+6kI1x</latexit>

The  result  “makes  sense”:  for  b>1,  the  signal  is  unbounded  in  both  sides  (infinite  energy  in  both  size),  and  the  factor  “r”  in  ZT  cannot  help  simultaneously  both  sides  to  converge.    

Page 11: TOPIC&5&& ZETA&TRANSFORM · RECALL:%ZETATRANSFORM X (z)= X+1 n=1 x[n]zn AAACKHicbVDNSsNAGNz4W

RECALL:  about  ROCs    

The  ROCs  are  always  “circular  pieces/por?ons”  of  the  complex  plane,  possibly  

infinite  pieces.    

A  ROC  does  not  contain  poles  (never!)    

Page 12: TOPIC&5&& ZETA&TRANSFORM · RECALL:%ZETATRANSFORM X (z)= X+1 n=1 x[n]zn AAACKHicbVDNSsNAGNz4W

RECALL:  about  ROCs    

POLES  

Page 13: TOPIC&5&& ZETA&TRANSFORM · RECALL:%ZETATRANSFORM X (z)= X+1 n=1 x[n]zn AAACKHicbVDNSsNAGNz4W

About  ROCs    

The  ROCs  always  are  defined  considering  the  module  of  z,  i.e.,  |z|=r,  

which  is  the  variable  r  !!  

Page 14: TOPIC&5&& ZETA&TRANSFORM · RECALL:%ZETATRANSFORM X (z)= X+1 n=1 x[n]zn AAACKHicbVDNSsNAGNz4W

Example  4  

x[n] = �[n]

<latexit sha1_base64="ZqtscU7WD2YKDElnGMht2OQqmtY=">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</latexit>

X(z) =+1X

n=�1x[n]z�n

<latexit sha1_base64="dN9L+r6fqjhSFf/Wh4lou/n8EwM=">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</latexit>

X(z) =+1X

n=�1�[n]z�n

<latexit sha1_base64="r6aHyoRxOEPbjJ1x0seD0HqtU7g=">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</latexit>

X(z) = 1

<latexit sha1_base64="3fZvqyOOXQPdpTgOxbmrHFSGkuU=">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</latexit>

ROC:  all  the  complex  plane  !!!!  

Page 15: TOPIC&5&& ZETA&TRANSFORM · RECALL:%ZETATRANSFORM X (z)= X+1 n=1 x[n]zn AAACKHicbVDNSsNAGNz4W

Example  5  

x[n] = �[n� 1]

<latexit sha1_base64="O2ISfz16DylVwlWJ8bWg1UbXzIc=">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</latexit>

X(z) =+1X

n=�1x[n]z�n

<latexit sha1_base64="dN9L+r6fqjhSFf/Wh4lou/n8EwM=">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</latexit>

X(z) =+1X

n=�1�[n� 1]z�n

<latexit sha1_base64="NnTnFETEP50dMjNUK4DNHyTH9bs=">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</latexit>

X(z) = z�1

<latexit sha1_base64="JjisZ5xzzjxmlp5i+r4YEDhfXqk=">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</latexit>

ROC:  all  the  complex  plane  but  except  zero  !!  There  is  a  pole  at  0.  

Page 16: TOPIC&5&& ZETA&TRANSFORM · RECALL:%ZETATRANSFORM X (z)= X+1 n=1 x[n]zn AAACKHicbVDNSsNAGNz4W

Example  6  X(z) =

+1X

n=�1x[n]z�n

<latexit sha1_base64="dN9L+r6fqjhSFf/Wh4lou/n8EwM=">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</latexit>

X(z) = z

<latexit sha1_base64="u7CgSMPlLmV7HXWaMHkt/HQ6Z+g=">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</latexit>

x[n] = �[n + 1]

<latexit sha1_base64="qpFSMM4BLSlgfXx1c0CKnabsLs0=">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</latexit>

X(z) =+1X

n=�1�[n + 1]z�n

<latexit sha1_base64="dtJHfYca0N6crT/ymu4nUd8/R+k=">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</latexit>

ROC:  all  the  complex  plane  (but  except  infinity!!  There  is  a  pole  at  Infinity).  **  Generally,  also  “infinity”  is  included  in  this  analysis….      

Page 17: TOPIC&5&& ZETA&TRANSFORM · RECALL:%ZETATRANSFORM X (z)= X+1 n=1 x[n]zn AAACKHicbVDNSsNAGNz4W

Example  7  X(z) =

+1X

n=�1x[n]z�n

<latexit sha1_base64="dN9L+r6fqjhSFf/Wh4lou/n8EwM=">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</latexit>

X(z) =+1X

n=�1�[n� n0]z�n

<latexit sha1_base64="NxrRXf29lVyxZE1XnmWxhfxhE2M=">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</latexit>

X(z) = z�n0

<latexit sha1_base64="79+630Jku7P7cOZCFtev5B0AFHU=">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</latexit>

x[n] = �[n� n0]

<latexit sha1_base64="MzJHXtG3Rd4waBobcTRam6CepvQ=">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</latexit>

ROC:  all  the  complex  plane  but  except  zero  !!  There  is  a  (mul?ple)  pole  at  0  (with  mul?plicity  n0,  i.e.,  n0  coincident  poles  at  0).  

n0 > 0<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Page 18: TOPIC&5&& ZETA&TRANSFORM · RECALL:%ZETATRANSFORM X (z)= X+1 n=1 x[n]zn AAACKHicbVDNSsNAGNz4W

Example  8:  property  

Q(z) =1X

n=�1q[n]z�n

<latexit sha1_base64="dUCtzoyrfZ3XjoqObV18Sw+8a4U=">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</latexit>

q[n] = x[n� n0]

<latexit sha1_base64="vZP0RdwWMfUFN8m4CwLfvouGwrU=">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</latexit>

Q(z) =1X

n=�1x[n� n0]z�n

<latexit sha1_base64="A1a9cYzqbRtSFJgDfkVdLsvKBno=">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</latexit>

Change  of  variable:    k = n� n0

<latexit sha1_base64="9bU6gwgLosVxNnBO7Sx1zJlrOc8=">AAAE13icjVTPb9MwFM7WACP86uDIJaKK1GlqlaIiuFSaxGUHDhtat44mjVzXaa0kTrAdtDayOIAQV/41bvwR/A/YSSrapmVYivTyve97n/P8nHESYsZt+9fefk2/c/fewX3jwcNHj5/UD59esjilEPVhHMZ0MAYMhZigPsc8RIOEIhCNQ3Q1Dt6q/NUnRBmOyQWfJ8iNwJRgH0PAJeQd7v8OeqRFPNuwzpuLo57D0sjLSK/lYOLzuRhlZXAzzGnuYpS1iFhjB9vYs5zZDI6l6Og/+CTnB8pjyc6t5Os/VEGhkoqChHu2GL0zxx4253LD2C0tqcLfm9CjpvoQ6kpBBPgMgjD7IJwQ+dzJdpRwKJ7OuCN6uxSV4kvFll1Va1RcNmtWJbe6qKZgcf235SvlVI6KgcwZ1k1ubViloABPc5lPAcxUBZEpWJTIDitRbUXhI3ITeQwTFHIgI8MabJ+z441RUEN2C3VZdG0wB6ujY1gflftyeA2v3rDbdr7MatApg4ZWrjOv/tOZxDCNEOEwBIwNO3bC3QxQjmGIhOGkDCUABmCKhjIkIELMzfJ7KUxLIhPTj6l8CDdzdFWRgYixeTSWTHXGbDOnwG25Ycr9N26GSZJyRGBh5KehyWNTXXJzgimCPJzLAECK5V5NOAPy/Lj8FagmdDY/uRpcvmx3uu1X593GSbdsx4H2XHuhNbWO9lo70U61M62vwdpFbVH7UvuqX+uf9W/694K6v1dqnmlrS//xB364m2U=</latexit>

n = k + n0

<latexit sha1_base64="9GS8Ue0VEFX0tb6ZnSZA+QJFu3A=">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</latexit>

Page 19: TOPIC&5&& ZETA&TRANSFORM · RECALL:%ZETATRANSFORM X (z)= X+1 n=1 x[n]zn AAACKHicbVDNSsNAGNz4W

Example  8:  property  

Q(z) = z

�n0

1X

k=�1x[k]z�k

<latexit sha1_base64="oFUGnQaK3fjow5COcQg11XWcKqY=">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</latexit>

Q(z) =1X

k=�1x[n]z�k�n0

<latexit sha1_base64="Gg0rTBiDcW8BMKuRJhwUgQCs4mU=">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</latexit>

Q(z) = z�n0X(z)

<latexit sha1_base64="wDpzcxUmfDrBUt4PfMUnxZ2LUJo=">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</latexit>

X(z)

<latexit sha1_base64="WE0KOdJux1ASjJNi2npQzky2W3c=">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</latexit>

ROC:  note  that  we  adding  a  (mul?ple)  pole  at  0  (or  at  Infinity  of  n0<0)  to  the    possible  poles  of  X(z).  If  X(z)  has  a  (mul?ple)  zero  at  0  this  pole  is  removed.  

Page 20: TOPIC&5&& ZETA&TRANSFORM · RECALL:%ZETATRANSFORM X (z)= X+1 n=1 x[n]zn AAACKHicbVDNSsNAGNz4W

Example  8:  property  

Q(z) = z�n0X(z)

<latexit sha1_base64="wDpzcxUmfDrBUt4PfMUnxZ2LUJo=">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</latexit>

q[n] = x[n� n0]

<latexit sha1_base64="vZP0RdwWMfUFN8m4CwLfvouGwrU=">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</latexit>

Shil  in  ?me  !  mul?plica?on  by  a  power  of  z  in  the  transformed  domain    

x[n� i]

<latexit sha1_base64="WRn587pwNveH9KXOG5RpiwzzBww=">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</latexit>

z�iX(z)

<latexit sha1_base64="GgniimjM3HoEc589eTs249q+Ogk=">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</latexit>

Page 21: TOPIC&5&& ZETA&TRANSFORM · RECALL:%ZETATRANSFORM X (z)= X+1 n=1 x[n]zn AAACKHicbVDNSsNAGNz4W

Example  9  

x[n] = �3�[n] + 2�[n� 1] + �[n� 4]

<latexit sha1_base64="NSSOUn3mDomoWmq6yJ7JUvdC7uE=">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</latexit>

x[n]

<latexit sha1_base64="eErXTvxu/0viQxII5fBt2KuUMf4=">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</latexit>

n

<latexit sha1_base64="61YPRXYruGo8TsRueX/FYWWhfJA=">AAAFKHicjVRNb9MwGM6WACN8dXDkElFF6lR1SkYRXDJN4rIDhw3RrdCkkes6rZXECY6D1kX5OVz4K1wQAqFd+SXYSVq6pmVYSvTk9fO8r/36iUdxgBNmGFdb27Jy6/adnbvqvfsPHj5q7D4+S6KUQtSDURDR/ggkKMAE9RhmAerHFIFwFKDzkf9azJ9/QjTBEXnHZjFyQjAh2MMQMB5yd+VD3bdIh7iGqhPLbwtAVP1iQByr89weo4ABjtsHc9gxnc4Cdx1VP21d7ll2koZuRqyOjYnHZvkwqwBPJJI7l8OsQ/JrbH8de1owW8VC9v6DTwq+L2rM2UUp/vkPlV+qVhV9/qHq5btUY8vIh2+0kYu1Gd8Jdqq1UBF/q0GXamKHlPfBDgGbQhBkH3I7QB6zsw0pbIonU2bn1iZFLflcsWZV9RxFFX25zGrSuubGMqJDOH//9zCW0ok5WvXuotihqleCMnhcyDwKYCYy5JkI51VkQ6m83ouyTl6Zc2HN8sDWOLC9YhJhvxuoC2MvW7a/bBFV/yiqz22tuo2msW8UQ6sDswJNqRonbuO7PY5gGiLCYACSZGAaMXMyQBmGAcpVO01QDKAPJmjAIQEhSpys+NFzTeeRseZFlD+EaUV0WZGBMElm4YgzxRknq3MiuG5ukDLvlZNhEqcMEVgW8tJAY5Embg1tjCmCLJhxACDFfK0anAJ+fozfLaIJ5uqW6+DsYN/s7r847TaPDqt27EhPpWdSSzKll9KRdCydSD0Jyp/lr/IP+afyRfmm/FKuSur2VqV5Il0byu8/dIm2Fw==</latexit>

�3

<latexit sha1_base64="oxlBxP6dxrBrvkzTfeH6t0b5C+Y=">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</latexit>

2

<latexit sha1_base64="uT84atibnlKNsLpA8rdBJf1EqIA=">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</latexit>

1

<latexit sha1_base64="j4XVxpIoDXViIG0Jp0f50MxZs94=">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</latexit>

x[0] = �3, x[1] = 2, x[4] = 1,

otherwise 0

<latexit sha1_base64="vbWlAShiV2EJLQ/gXxNTMI/jx+k=">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</latexit>

1

<latexit sha1_base64="CrhDCJec2zgbrJR/MuuphEhWrs4=">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</latexit>

2

<latexit sha1_base64="fLHCSo2dlG8fPsCQmuEcGqu9Kk0=">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</latexit>

3

<latexit sha1_base64="HFKaV9T4UTy4YzYDg4bv3KHUQ1A=">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</latexit>

4

<latexit sha1_base64="Tjry01IT0zoaXyPjaYg3PRvMwpg=">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</latexit>

�1

<latexit sha1_base64="oyuJvaM2Q7SnegwfwcIkw5m/Hxw=">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</latexit>

�2

<latexit sha1_base64="mzze0hqriKjRWVpp7hbs56lHcKI=">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</latexit>

5

<latexit sha1_base64="IkJYmvyLKLU0uMkRZJLruzM8mtc=">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</latexit>

Easy  to  express  with  deltas    

Page 22: TOPIC&5&& ZETA&TRANSFORM · RECALL:%ZETATRANSFORM X (z)= X+1 n=1 x[n]zn AAACKHicbVDNSsNAGNz4W

Example  9  x[n] = �3�[n] + 2�[n� 1] + �[n� 4]

<latexit sha1_base64="NSSOUn3mDomoWmq6yJ7JUvdC7uE=">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</latexit>

X(z) = �3 + 2z�1 + z�4

<latexit sha1_base64="aev4nFI8qjsxxTv2OVYA/ZHoBLA=">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</latexit>

X(z) = z�n0

<latexit sha1_base64="79+630Jku7P7cOZCFtev5B0AFHU=">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</latexit>

x[n] = �[n� n0]

<latexit sha1_base64="MzJHXtG3Rd4waBobcTRam6CepvQ=">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</latexit>

Using  the  result  of  example  7:  

X(z) =�3z4 + 2z3 + 1

z4

<latexit sha1_base64="CE37IDXIf09PY4IabjKsyKVL/UY=">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</latexit>

ROC:    all  the  complex  plane  except  0!  We  have  a  (mul?ple)  pole  at  0.  At  infinity,  the  pole  at  0  help  us….  

Page 23: TOPIC&5&& ZETA&TRANSFORM · RECALL:%ZETATRANSFORM X (z)= X+1 n=1 x[n]zn AAACKHicbVDNSsNAGNz4W

ZT  of  finite  length  signals    

x[n] =L�1X

j=0

an�[n� j]

<latexit sha1_base64="QHnKK1AJ4Fq3wucZHVCgD69tsPc=">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</latexit>

For  instance:  

=x[0]zL�1 + x[1]zL�2 + x[2]zL�3 + ... + x[L� 1]

z

L�1

<latexit sha1_base64="cwLHHtk7tN3RkMm3ZNpTOL0cKT8=">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</latexit>

X(z) =1X

n=�1x[n]z�n =

L�1X

n=0

x[n]z�n

= x[0] + x[1]z�1 + x[2]z�2 + ... + x[L� 1]z�(L�1)

<latexit sha1_base64="XC6wQgIUW75iqTShS9bjmyJD8cM=">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</latexit>

ROC=  ?  !  

Page 24: TOPIC&5&& ZETA&TRANSFORM · RECALL:%ZETATRANSFORM X (z)= X+1 n=1 x[n]zn AAACKHicbVDNSsNAGNz4W

ROCs  of  ZT    of  finite  length  signals    

If  x[n]  has  finite  length,  the  ROC  is  all  the  complex  plane  except  POSSIBLY  the  

points  z=0  and/or  z=Infinity.  

The  ROC  is  all  the  complex  plane  just  for  x[n]=Delta[n].  

Page 25: TOPIC&5&& ZETA&TRANSFORM · RECALL:%ZETATRANSFORM X (z)= X+1 n=1 x[n]zn AAACKHicbVDNSsNAGNz4W

Zeta  Transform  for  LTI  systems  

The  Zeta  Transform  of  an  impulse  response  of  an  LTI  system  is  always  

ra?onal:  namely,  a  frac?on  of  polynomials  !  

(see  proof  in  the  next  slides)  

For  the  proof  we  need  to  use  the  property:    

x[n� i]

<latexit sha1_base64="WRn587pwNveH9KXOG5RpiwzzBww=">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</latexit>

z�iX(z)

<latexit sha1_base64="GgniimjM3HoEc589eTs249q+Ogk=">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</latexit>

Page 26: TOPIC&5&& ZETA&TRANSFORM · RECALL:%ZETATRANSFORM X (z)= X+1 n=1 x[n]zn AAACKHicbVDNSsNAGNz4W

LX

i=0

biy[n� i] =RX

r=0

crx[n� r]

<latexit sha1_base64="JiV48GpuAm9O2674TdbjJhMTaHs=">AAACGHicbZC7TsMwFIYdrqXcAowsFhUSCyVBRbBUqmBhYCiIXqQ0RI7rtFYdJ7IdRBX1MVh4FRYGEGLtxtvgtBmg5Zcs/frOOTo+vx8zKpVlfRsLi0vLK6uFteL6xubWtrmz25RRIjBp4IhFou0jSRjlpKGoYqQdC4JCn5GWP7jK6q1HIiSN+L0axsQNUY/TgGKkNPLMk45MQi+lVWv0cAN9j8Khw4+pW51ykfE7iD0BnzQXrmeWrLI1EZw3dm5KIFfdM8edboSTkHCFGZLSsa1YuSkSimJGRsVOIkmM8AD1iKMtRyGRbjo5bAQPNenCIBL6cQUn9PdEikIph6GvO0Ok+nK2lsH/ak6iggs3pTxOFOF4uihIGFQRzFKCXSoIVmyoDcKC6r9C3EcCYaWzLOoQ7NmT503ztGxXyme3lVLtMo+jAPbBATgCNjgHNXAN6qABMHgGr+AdfBgvxpvxaXxNWxeMfGYP/JEx/gEjJ581</latexit>

LINEAR  DIFFERENCE  EQUATION  WITH  COSTANT  COEFFICIENTS  (AND  INITIAL  CONDITIONS  =  0)  

The  output  of  a  LTI  system  can  be  expressed  with  the  convolu?on  sum  and:  

Zeta  Transform  of  the  impulse  response  of  a  LTI  system  

Page 27: TOPIC&5&& ZETA&TRANSFORM · RECALL:%ZETATRANSFORM X (z)= X+1 n=1 x[n]zn AAACKHicbVDNSsNAGNz4W

Z(

LX

i=0

biy[n� i]

)= Z

(RX

r=0

crx[n� r]

)

<latexit sha1_base64="9JIXHu1D8BUCQAJWCwJlL2r2rxM=">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</latexit>

LX

i=0

biZ {y[n� i]} =RX

r=0

crZ {x[n� r]}

<latexit sha1_base64="/RLSHsjA2sSXXVykx+yZFAfqOBo=">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</latexit>

LX

i=0

biz�iY (z) =

RX

r=0

crz�rX(z)

<latexit sha1_base64="Hju46oFl8nLqDDBW03OvIc/57zc=">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</latexit>

Zeta  Transform  of  the  impulse  response  of  a  LTI  system  

Page 28: TOPIC&5&& ZETA&TRANSFORM · RECALL:%ZETATRANSFORM X (z)= X+1 n=1 x[n]zn AAACKHicbVDNSsNAGNz4W

Zeta  Transform  of  the  impulse  response  of  a  LTI  system  LX

i=0

biz�iY (z) =

RX

r=0

crz�rX(z)

<latexit sha1_base64="Hju46oFl8nLqDDBW03OvIc/57zc=">AAADHXicdVLPT9swGHXCxljZWIHjLtaqSuxAlWxM41IJwWWHHRhaodCEyHGd1sJxIvsLokT5R3bZv7LLDpumHXZB+2/mtJ0ENXySpaf3/N73+UecC67B8/467tKjx8tPVp42Vp89X3vRXN841lmhKOvRTGSqHxPNBJesBxwE6+eKkTQW7CS+OKj1k0umNM/kZ5jkLEzJSPKEUwKGitadt+1AF2lU8q5XnX/EccTxZCC3edid8armjzCNFL4yvAob7SAlMKZElGdVIFgCQflARKD4aAxB1X3IYYX/dzTsqewMq8tipm1Z6GI1uT4vt3l1unX92k6rNVX1jRY1W17Hmxa2gT8HLTSvw6j5JxhmtEiZBCqI1gPfyyEsiQJOBTNzFJrlhF6QERsYKEnKdFhOX7fCbcMMcZIpsyTgKXvbUZJU60kam531efWiVpP3aYMCkt2w5DIvgEk6a5QUAkOG66+Ch1wxCmJiAKGKm1kxHRNFKJgP1TCX4C8e2QbHbzr+Tufdp53W3v78OlbQS/QKbSEfvUd76AM6RD1EnS/ON+eH89P96n53f7m/Z1tdZ+7ZRHfKvfkHMHf/sA==</latexit>

The  Zeta  Transform  of  an  impulse  response  of  an  LTI  system  is  always  ra?onal:  namely,  a  

frac?on  of  polynomials  !  

It  can  be  expressed  as  a  frac?on  of  polynomials  !!!  

H(z) =Y (z)X(z)

=PR

r=0 crz�r

PLi=0 biz�i

<latexit sha1_base64="uRVixUnmzkeFyjODFPHZHODWxZU=">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</latexit>

Page 29: TOPIC&5&& ZETA&TRANSFORM · RECALL:%ZETATRANSFORM X (z)= X+1 n=1 x[n]zn AAACKHicbVDNSsNAGNz4W

In  LTI  systems:  ROCs  and  poles  

When  X(z)  is  ra?onal  (a  frac?on  of  polynomials,  i.e.,  for  impulse  

response  of  LTI  systems)  the  poles  “define/determine”  the  ROC.