11/16/2010 The Inner Product.doc 1/9. The Inner Product. So we now know that a continuous, analog signal v t can be expressed as:

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1 11/16/2010 The Ier Product.doc 1/9 The Ier Product So we ow kow tht cotiuous, log sigl v t c be epressed s: v t t So tht cotiuous, log sigl c be (lmost) completel specified b discrete set of umbers:,,,,,, Q: But do t these umbers likewise deped o the bsis fuctios t?? How is this esier or simpler th just specifig v t. A: Remember, the sigl v t is rbitrr, but the bsis fuctios t re tpicll well-kow d frequetl used. We c thik of the bsis fuctios s t stdrd set of prts, from which we c costruct rbitrr fuctio v t! A: Not t ll! Q: We thik ou ve goe off the deep ed. Prts used for costructig fuctios? Is t this discussio imprcticl, ephemerl, esoteric d didctic? Jim Stiles The Uiv. of Kss Dept. of EECS

2 11/16/2010 The Ier Product.doc 2/9 The cocept of costructig mssive, comple thigs out of smll d simple elemets is pervsive ot ol i egieerig, but i other scieces d hum ctivit s well!! Some emples: 1. The geetic code i hum DNA is mde up of ver log sequece of just four purie compouds: deie (A), thmie (T), guie (G), d ctosie (C). 2. Almost ll hum kowledge d emotio c be epressed i Eglish usig the smbols: ABCDEFGHIJKLMNOPQRSTU VWXYZ !,.:; All mtter i the uiverse is costructed with reltive smll umber of elemets. H 2 O C 6 H 12 O 6 CO 2 Jim Stiles The Uiv. of Kss Dept. of EECS

3 11/16/2010 The Ier Product.doc 3/9 Thus, the set of bsis fuctios t re the prts we use to costruct our sigl v t. The vlues,,, tell us the specific blueprit (or recipe, code, prgrph choose our log) for puttig these prts together to crete the uique fuctio v t. Similr to other spects of life, we must choose which set of bsis fuctios re most useful to us. However, isted of choosig betwee lego blocks d tiker tos, or Eglish, Chiese d Spish, we must choose betwee (for emple) siusoids, sic fuctio, d wvelets! Q: But fter we choose bsis t, how do we determie the vlues? How do we fid the recipe for costructig fuctio v t?? A: First, we must uderstd wht the vlues tell us bout the sigl v t. Essetill, the vlues tell us how much of ech bsis fuctio t eists withi v t. For emple, if the vlue 1 is smll, the vlue 2 is moderte, d 3 is big, or recipe (metphoricll spekig) might be: To crete v t, dd pich of bsis fuctio 1t, cup of bsis fuctio 2t, d bout gllo of bsis fuctio 3t. Plce i hot ove for bout 45 miutes*. * This lst setece is ot prt of the log. Jim Stiles The Uiv. of Kss Dept. of EECS

4 11/16/2010 The Ier Product.doc 4/9 Q: I thought I would ever s this, but c ou be more mthemticll specific? A: The vlues respect to bsis re the compoets of sigl t. v t, with Q: I EECS 220 we ejoed lerig bout the compoets of vector ( A,A,A), z with respect to some set of bsis vectors ( ˆ, ˆ, ˆ z ). E.G.,: A ˆ A ˆ A ˆ A z z Is this the sme thig? A: Precisel the sme thig! The sclr compoets ( A,A,A ) of vector tells us how much of ech bse vector (ˆ, ˆ, ˆ ) eists withi vector A The sclr compoets ( A,A,A ) thus provide the recipe for costructig vector A from our fudmetl igrediets (ˆ, ˆ, ˆ ) Q: But wit! I remember tht we determied these compoets ( A,A,A ) b usig the dot product, e.g.,: A Aˆ A Aˆ A A ˆ z z Jim Stiles The Uiv. of Kss Dept. of EECS

5 11/16/2010 The Ier Product.doc 5/9 A: Tht s correct! For emple, cosider vector A, d bsis vectors ˆ d ˆ. A ˆ B usig the dot product, we fid the compoet of vector A i the directio of bsis vector ˆ. ˆ ˆ ˆ A A A ˆ A Likewise for the compoet of vector A i the directio of bsis vector ˆ. Α cos Α ˆ cos Α ˆ A A A A Α cos A ˆ cos A ˆ A ˆ A A Α ˆ ˆ Thus vector A c be epressed s the sum of two vector compoets: A ˆ A A ˆ ˆ A ˆ ˆ A ˆ A ˆ ˆ ˆ A ˆ Jim Stiles The Uiv. of Kss Dept. of EECS

6 11/16/2010 The Ier Product.doc 6/9 Q: But we re ot tlkig bout vectors, we re tlkig bout cotiuous sigls like v t. Surel there s o w to use the dot product to fid its compoets? A: Essetill there is! The mthemticl cousi of the dot product is mthemticl opertio kow s the ier product.* The ier product of two sigls t d bt is defied s: * t,b t t b t dt where * idictes comple cojugte. If t,bt 0, we s tht the two sigls t d bt re orthogol (just like if A B 0!). If this is the cse, the two sigls t d bt re cosidered to be completel dissimilr the hve o commo compoet. The eerg of some sigl is determied b tkig the ier 2 product of tht sigl with itself (just like if AA A!): E * t, t t t dt t 2 dt * Ad stop cllig me Shirle. Jim Stiles The Uiv. of Kss Dept. of EECS

7 11/16/2010 The Ier Product.doc 7/9 A sigl whose eerg is E 1 sid to hve uit eerg (just like uit vectors where ˆ ˆ 1!). Fill, cosider set of sigls ( ech hve uit eerg. t, t 1 t for ll, s). These sigls Likewise, these sigls re ll orthogol to ech other: t, t 0 for ll m m These sigls re kow s mutull orthogol. A set of mutull orthogol sigls with uit eerg is kow s orthoorml set of bsis fuctios (just like ˆ, ˆ, ˆ ). z Orthoorml bsis fuctios hve m eceptioll ttrctive mthemticl properties. As result, we fid tht we frequetl use them i sigl epsios of the form: v t t Q: So does tht me tht the Fourier d sic bsis fuctios re orthoorml? A: Absolutel! As re m (but ot ll) wvelet bsis fuctios. Jim Stiles The Uiv. of Kss Dept. of EECS

8 11/16/2010 The Ier Product.doc 8/9 The ke propert of orthoorml bsis fuctios is tht it llows us to determie the sigl compoets b use of the ier product: vt, t Of course, this is perfectl logous to our vector compoet lsis: A A ˆ So ow we c determie the vlues for our most populr bsis fuctios! 1. Fourier 2 j t T v t t e for 0 t T Therefore: 2 T 2 j t j t T T v t, t v t,e v t e dt V 0 2. Sic Fuctio t v t t sic Jim Stiles The Uiv. of Kss Dept. of EECS

9 11/16/2010 The Ier Product.doc 9/9 Therefore: t t v t, t v t, sic v t sic dt Q: Yikes! This itegrl looks s ugl d uplest! A: It does look tht w but it s ot! ICBST (It C Be Show Tht) the solutio to this itegrl is simple d stright forwrd: t v t sic dt v t The compoet vlue is simpl the vlue of fuctio v t t the specific time t. Thus: t t v t sic v t sic This is the reso wh sic bsis fuctios re so populr it is etremel es to determie ll the sigl compoets! All we eed is device tht smples the sigl v t t specific times t d we hve such device! Jim Stiles The Uiv. of Kss Dept. of EECS