¦pªG§Ú̱N¬Y¤@Ãþ¯S©wªº¿é¤J¨ç¼Æ°e¶i¬YÓ¨t²Î«á¡A¦pªG©Ò±o¨ìªº¿é¥X¨ç¼Æ¤´µM¨ã¦³¿é¤J¨ç¼Æªº®æ¦¡¡A«h¦¹¨ç¼ÆºÙ¬°¦¹¨t²Îªº¡u©T¦³¨ç¼Æ¡v¡]Eigen Function¡^¡C¥»¸`±N»¡©ú¡A«ü¼Æ¨ç¼Æ¡]§t©¶ªi¨ç¼Æ¡^´N¬O LTI ¨t²Îªº©T¦³¨ç¼Æ¡C¦pªG§Ú̪º¿é¤J¨ç¼Æ¬O¤@Ó«ü¼Æ¨ç¼Æ¡G
x[n]=epn ¨º»ò®Ú¾Ú¤§«e±À¾Éªº¤½¦¡¡A¥i±o¨ì¹ïÀ³ªº¿é¥X¬O¡G´«¥y¸Ü»¡¡A¿é¥X¨ç¼Æ y[n] µ¥©óì¿é¤J¨ç¼Æ¼¤W¤@Ó±`¼Æ H(ep)¡]= Ske-pk h[k]¡A¬O¤@Ó¤£ÀH®É¶¡¦ÓÅܤƪº¨ç¼Æ¡^¡A¦]¦¹§ÚÌ»¡¡u«ü¼Æ¨ç¼Æ¬O LTI ¨t²Îªº©T¦³¨ç¼Æ¡v¡C
y[n] = x[n]*h[n] = Skx[n-k] h[k] = Skep(n-k) h[k] = Skepn-pk h[k] = epnSke-pk h[k] = epnH(ep) ±q³o¸Ì±À¼s¡A¥ô¦ó¿é¤J¨ç¼Æ¡A¥un¯à°÷ªí¥Ü¦¨«ü¼Æ¨ç¼Æªº½u©Ê²Õ¦X¡A´N¥i¥H¨Ï¥ÎÅ|¥[ì²z¡A¥ýºâ¥X³o¨Ç«ü¼Æ¨ç¼Æªº¹ïÀ³¿é¥X¡A¦A¶i¦æ½u©Ê²Õ¦X¡A´N¥i¥H±o¨ìì¿é¤J¨ç¼Æªº¹ïÀ³¿é¥X¡A¥i¥Î¼Æ¾Ç¦¡ªí¥Ü¦p¤U¡G
x[n] = aepn + beqn ¡÷ y[n] = aH(ep) + bH(eq) ¹ï©ó©¶ªi¨ç¼Æ¡A§ÚÌ¥i¥H¨Ï¥Î Euler Identity ¨Ó¥N´«¦¨«ü¼Æ¨ç¼Æ¡G
ejq = cos(q) + j sin(q) ´«¥y¸Ü»¡¡Gcos(q) = (ejq + e-jq)/2 = Re{ejq} ¦]¦¹¡A·í¿é¤J¬O
sin(q) = (ejq - e-jq)/(2j) = Im{ejq}x[n] = cos(wn) = Re{ejwn} ¹ïÀ³ªº¿é¥X«h¬O´«¥y¸Ü»¡¡A¿é¥X¨ç¼ÆÁÙ¬O cos¡A¥u¤£¹L¬Û¦ì°¾²¾¤F q = ¡çH(ejw)¡A¾_´T¤]¼¤W¤F |H(ejw)|¡C´XÓ¬ÛÃö¦Wµü¡A»¡©ú¦p¤U¡C
y[n] = Re{ejwnH(ejw)} = Re{ejwn |H(ejw)| ejq}, q = ¡çH(ejw) = Re{|H(ejw)| ej(wn+q)} = |H(ejw)| cos(wn + q) ¦P²z¡A·í¿é¤J¬O
- ¦pªG§Ṳ́£¬Ý¬Û¦ì¡A¨º»ò¦¹¿é¥Xªº¼¼Æ |H(ejw)| ¥Nªí¦¹¨t²Î¹ï¤£¦P¨¤ÀW²v w ªº°T¸¹ªºÂX¤j©ÎÀ£ÁY¯S©Ê¡A¦]¦¹§Ú̳q±`ºÙ |H(ejw)| ¬°¡uÅT«×ÀW²vÅTÀ³¡v¡]Magnitude Frequency Response¡^¡C
- ¡çH(ejw) «hºÙ¬°¡u¬Û¦ìÀW²vÅTÀ³¡v¡]Phase Frequency Response¡^¡C¡]¤@¯ë¦Ó¨¥¡A¸û¤Ö³Q¥Î¨ì¡A¦]¬°¤H¦Õ¹ï©ó¬Û¦ìªº·Pı¡A¨Ã¤£±Ó¾U¡C¡^
- H(ejw) ²ÎºÙ¬O¦¹¨t²Îªº¡uÀW²vÅTÀ³¡v¡]Frequency Response¡^¡C
x[n] = sin(wn) = Im{ejwn} «h¿é¥X¬O¦]¦¹¡A§ÚÌ¥i¥H±o¨ì¤@Óµ²½×¡G¹ï©ó¤@Ó LTI ¨t²Î¦Ó¨¥¡A·í¿é¤J¨ç¼Æ¬O¤@Ó³æ¤@©¶ªi¨ç¼Æ®É¡A¿é¥X¤]¤@©w¬O¤@ÓÀW²v¬Û¦Pªº©¶ªi¨ç¼Æ¡A¥u¦³¾_´T©M¬Û¦ì·|³Q¨t²Î©Ò§ïÅÜ¡A¦Ó§ïÅܪº¤è¦¡§¹¥þ¨Ì·Ó¨t²ÎªºÀW²vÅTÀ³ H(ejw) ¦Ó©w¡C
y[n] = Im{ejwnH(ejw)} = Im{ejwn |H(ejw)| ejq}, q = ¡çH(ejw) = Im{|H(ejw)| ej(wn+q)} = |H(ejw)| sin(wn + q) ¤Wzªº¨âºØ©¶ªi¿é¤J¡A§ÚÌ¥i¥H§ï¥Î¤@Óµê¼Æªº«ü¼Æ¨ç¼Æ¨Ó·§¬A¡G
x[n] = ejwn ¨º»ò¹ïÀ³ªº¿é¥X¬Oy[n] = ejwnH(ejw) ¨ä¤¤ H(ejw) ºÙ¬° h[n] ªº¡uÂ÷´²®É¶¡³Å¥ß¸Âà´«¡v¡]Discrete-time Fourier Transform¡^¡A¥i¥Hªí¥Ü¦p¤U¡GH(ejw) = Skh[k]e-jwk ¨Æ¹ê¤W¡A¤Wzªº¿é¤J©M¿é¥X¦b²{¹ê¥@¬É¨Ã¤£¦s¦b¡A¦]¬°¤@Ó¨t²Îªº¿é¤J©M¿é¥X¡A¤£¥i¯à¬O½Æ¼Æ¡C¦ý¬O³o¨Ç¼Æ¾Ç¹Bºâ¦b²z½×¤W¬O¦¨¥ßªº¡A¦]¦¹§ÚÌ¥i¥H¨Ï¥Î³o¨Ç¦¡¤l¨ÓÅý§Ú̪º±À¾É§ó²¼ä¤@¨Ç¡C
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