7. Discrete Fourier Transform (DFT)
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1 7 Discrete ourier Trasor (DT) 7 Deiitio ad soe properties Discrete ourier series ivolves to seueces o ubers aely the aliased coeiciets ĉ ad the saples (T) It relates the aliased coeiciets to the saples ad its iverse expresses the aliased coeiciets i ters o the saples o e deie a ore eeral cocept called the discrete ourier trasor (DT) Deiitio ive a seuece { o ubers real or coplex The discrete ourier trasor o { ubers deied by the euatio is a seuece { o (7) here π e
2 65 I order to express the ubers i ters o e use the iversio orula (7) To prove euatio (7) e use (7) chai the idex ro to or ay ixed beloi to the set { e obtai ( ) ( ) (7) o e reer to atheatical aalysis Recall that the su o a eoetric series coposed o ters is ( ) i i a a a a a a (74) I the applyi (74) e obtai ( ) ( ) This euatio holds or or relatio (74) ives ( ) Usi the above results e have
3 66 Euatios (7) ad (7) sho that there is a oe to oe correspodece betee the seueces { ad { These to seueces or a discrete ourier trasor pair ote The seuece { ubers does ot have to be periodic ad its eleets ay be coplex Let us cosider a seuece { cosisti o real ubers Let us assue that is a ixed iteer beloi to set { The the euatio ˆ ( ˆ ) ˆ ˆ ( ) ˆ holds here the asteris deotes the coplex couate Thus or a real seuece { the eleets o the seuece { satisy the relatio ˆ { ˆ ˆ ˆ (75) ˆ We sho that the seuece { ive by (7) is periodic ith period ( ) ( ) Siilarly orula (7) deies a periodic seuece ith period ( ) I the seuece { is periodic ith period the also the seuece { or ay ixed is periodic ith period ( )
4 67 As a result the liits o suatio i (7) ca be chaed as lo as the total iterval is (76) Siilarly sice ad or ay ixed are periodic seueces ith period the liits o suatios i (7) ca be chaed as lo as the total iterval is { (77) Exaple 7 ive a seuece { or or Deterie the DT o this seuece We apply euatio (7) ith 4 ad e e 4 π π ( ) Setti e obtai: e e 6
5 68 The seuece { is sho i i7 hereas the aitude ad phase spectra o { are sho i is7 ad 7 4 i 7 Seuece { or Exaple i 7 Maitude spectru o the sial sho i i 7 ( ) i 7 Phase spectru o the sial sho i i 7
6 69 7 Properties o the DT applied to periodic seueces We ill study soe properties o the DT applied to periodic seueces ith the period Liearity { { be the DT o to periodic seueces { Let ad the sae period The the DT o the cobied seuece { h { α β here α ad β are arbitrary costats is ive by Proo or ay ixed the euatio holds Shiti H β { H { α β ( α β ) α β α { { ad ith I is the DT o a periodic seuece ith period the the DT o the shited seuece { here is a positive or eative iteer is ive by the seuece Proo We apply (7) or a arbitrary ixed { { { H (78)
7 7 H ν ν ν ( ) here ν Sice { is a periodic seuece ith period e obtai ν ν ν ν ad H Exaple 7 ive to periodic seueces: { { { { deterie the DT o these seueces We apply (7) ith 4 ad π e 4 Setti e obtai: Siilarly e have: { Seuece { ca be obtaied ro the periodic seuece uits to the riht or by shiti to
8 7 Hece the DT o { ca be oud usi the shiti property (78) Setti e obtai: 6 Periodic (circular) covolutio Let { ad { be to periodic seueces ith the sae period The periodic covolutio o { ad is deied as { h (79) Observe that { h is a periodic seuece ith period Exaple 7 Cosider the periodic seueces { ad { seueces have the sae period 4 sho i is74 ad 75 Both i 74 Periodic seuece { i 75 Periodic seuece { cosidered i Exaple 7 cosidered i Exaple 7
9 7 To deterie the periodic covolutio h e peror our steps: oldi traslati ultiplyi ad addi These steps or are illustrated i is p i 76 Periodic seuece { i 77 oldi the seuece { (-) p i 78 Traslati the seuece { i 79 Multiplyi sials Usi the sial o i79 e obtai h ad 9 Repeati this procedure or e id: h 9 h 7 h The periodic covolutio { h is illustrated i i 7
10 7 h i 7 Periodic covolutio o the sials cosidered i Exaple 7 Periodic covolutio theore Let { ad { be the DTs o the periodic seueces { ad { h is the DT o the periodic covolutio { Proo The H (7) H h p p ( p ) here p Hece e have
11 74 Exaple 74 p H p p Let us copute DT o the seueces { { ad { Exaple 7 ote that 4 ad e π Hece e rite () ( ) h deied i Setti e obtai 6 Thus DT o { is { { 6 o e deterie DT o the seuece { ( ) Hece e id Thus DT o { Siilarly e obtai 6 is { { 6
12 75 Thus DT o { h is H ( ) H 6 H 4 H H 4 { { H ad H holds 7 Copariso o the DS ith the DT The pair o DS deied i Sectio 6 is: ( T ) ĉ (7) ĉ ( T ) (7) here π e Let us cosider the seuece { here ( ) ad deterie the DT o this seuece ( ) T (7) Usi (7) ad (7) e obtai: or ĉ (74)
13 76 ĉ (75) Thus e coclude that the aliasi coeiciets o the DS reerri to the seuece { ( T ) ca be deteried o the basis o the DT o the seuece T usi euatio (75) { ( )
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