B1. Fourier Analysis of Discrete Time Signals


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1 B. Fourier Aalysis of Discrete Time Sigals Objectives Itroduce discrete time periodic sigals Defie the Discrete Fourier Series (DFS) expasio of periodic sigals Defie the Discrete Fourier Trasform (DFT) of sigals with fiite legth Determie the Discrete Fourier Trasform of a complex expoetial. Itroductio I the previous chapter we defied the cocept of a sigal both i cotiuous time (aalog) ad discrete time (digital). Although the time domai is the most atural, sice everythig (icludig our ow lives) evolves i time, it is ot the oly possible represetatio. I this chapter we itroduce the cocept of expadig a geeric sigal i terms of elemetary sigals, such as complex expoetials ad siusoids. This leads to the frequecy domai represetatio of a sigal i terms of its Fourier Trasform ad the cocept of frequecy spectrum so that we characterize a sigal i terms of its frequecy compoets. First we begi with the itroductio of periodic sigals, which eep repeatig i time. For these sigals it is fairly easy to determie a expasio i terms of siusoids ad complex expoetials, sice these are just particular cases of periodic sigals. This is exteded to sigals of a fiite duratio which becomes the Discrete Fourier Trasform (DFT), oe of the most widely used algorithms i Sigal Processig. The cocepts itroduced i this chapter are at the basis of spectral estimatio of sigals addressed i the ext two chapters.. Periodic Sigals VIDEO: Periodic Sigals (9:45) I this sectio we defie a class of discrete time sigals called Periodic Sigals. There are two reasos why these sigals are importat: A umber of sigals i ature exhibit periodic repetitio. Thi of vibratios, ocea waves, seismic waves or electromagetic waves;
2 It is pretty easy to believe that periodic sigals are made of (periodic) siusoids of differet frequecies, which is the goal of the Fourier aalysis of the rest of the chapter. Defiitio: a discrete time sigal x [ ] is periodic if ad oly if there exists a positive iteger such that x [ ] = x [ + ] for all () I other words, a periodic sigal eeps repeatig itself for all values of the idex from to +. The smallest positive iteger satisfyig () is called the period of the sigal. Example: cosider the sigal Figure : periodic sigal ( π π) x [ ] = cos. +.9 It is periodic with period = sice x [ + ] = cos. π ( + ) +.9π for all. ( ) ( π π π) = cos = x [ ] I geeral a siusoidal sigal of the form x [ ] = Acos + α with, itegers is periodic with period, sice x[ + ] = Acos π ( + ) + α = Acos + α + π = x [ ] for all. () (3) Example: cosider the siusoid This ca be writte as ( π π) x [ ] = 5cos.3. 3 x [ ] = 5cos.π
3 Comparig this with () we ca see that the sigal is periodic with period =. I fact we ca easily write x [ + ] = 5cos.3 π ( + ).π ( ) ( π π π) = 5cos = x [ ] for all. I a very similar way a complex expoetial of the form is periodic with period sice x[ + ] = Ae j x[ ] = Ae (4) j ( + ) j j Ae e x = = Example: cosider the complex expoetial. [ ] ( ) j π x = + je This ca be writte as j x [ ] = ( + je ) ad it is periodic with period =. [ ] 3. Expasio of Periodic Sigals: the Discrete Fourier Series (DFS). VIDEO: Referece Frames (:5) ow the problem we wat to address is to determie a referece frame for discrete time periodic sigals. I other words the issue is to determie a set of periodic sigals which serve as a basis of ay other periodic sigal. The cocept of a referece frame is very importat i a umber of fields, icludig geometry, avigatio ad sigal processig. It exteds simple, ituitive geometric ideas to more complex abstract problems. Goig bac to geometry, recall that, give a vector x we ca represet it i terms of a referece frame defied by: A origi ; Referece vectors e, e,... Figure below shows a example of a vector i the plae.
4 Figure : a vector ad a referece frame i the plae Give this defiitio we ca see that a vector (say) i the plae is always represeted by the compoets alog the directios of the basis vectors. I this case, as show i figure 3, the vector x is represeted as show i figure 3 as x = ae + ae where a is the projectio of the vector x alog e a is the projectio of the vector x alog e Figure 3: vector represetatio i terms of its projectios o the referece frame For example you wat to locate yourself with respect to a referece poit ad you say that you are 3m East ad m orth of your referece. I this case the referece vectors e, e are i the East ad orth directios respectively, of uit legth ( meter) ad the compoets a, a of your positio are 3m ad m respectively. This is show i figure 4. Figure 4: example of a geographical positio as a vector i the Earth referece frame. For sigals we use a very similar cocept. We wat to express ay arbitrary sigal [ ] x as the sum of referece sigals havig appropriate physical properties, sigificat for our problems. I our case ad i a lot of applicatios, the referece sigals are siusoids or complex expoetials, as show i figure 5.
5 Figure 5: a sigal as a sum of referece sigals Give ay periodic sigal with period a very attractive set of referece cadidates is the set of complex expoetials with the same period. These are defied as j e [ ] = e, =,..., (5) It turs out that we ca show the followig Fact: ay discrete time periodic sigal with period ca be writte as for some costats a, a,... a. x [ ] = ae [ ] Example: tae the periodic sigal show i figure 6 below. It is easy to see that it is periodic with period =, sice it eeps repeatig the value,,,, periodically. = Figure 6: periodic sigal with period = Sice the period is = the referece expoetials e [ ] i (5) are give by j e [ ] = e = = j e[ ] = e = ( ) We ca easily verify (try to believe!) that the give periodic sigal i figure 6 ca be writte as x [ ] =.5 e[ ] +.5 e[ ] =.5.5 ( ) I fact this expressio yields x [ ] =.5.5 = for eve ad x [ ] = = for odd.
6 Example: cosider aother example of a periodic sigal with period = show i figure 7. Figure 7: periodic sigal with period = Agai we use the same referece sigals e[ ], =, defied as j e [ ] = e = = j e[ ] = e = ( ) It ca be easily verified that this sigal ca be writte as x [ ] =.5 e[ ] +.8 e[ ] = ( ) 4. Orthogoality of the Referece Sigals VIDEO: Orthogoal Referece Sigals (:5) From the simple geometric examples i the previous sectio (figures,3,4) we see that the vectors i the referece frames are orthogoal to each other. It turs out that this maes it easier i determiig the expasio of the vectors i terms of referece frames. A very similar approach exteds to referece sigals. I particular we ca verify that the j complex expoetials e [ ] = e are all orthogoal to each other i the sese that * if m e[ e ] m[ ] = = if m= (6) This is easy to show just by applyig the defiitio of complex expoetials as follows: m m j π * j π e[ e ] m[ ] = e = e = = = ow we ca apply the geometric sum, defied as (7)
7 j m e π α if α α = α (8) = if α = with α =. The applyig (7) to (8) we obtai the orthogoality coditio i (6). The fact that the complex expoetials e[ ] are orthogoal as i (6) maes the computatio of the expasio coefficiets a a very simple matter. I fact, let x [ ] be a periodic sigal with period ad let s write it agai as x [ ] = ae [ ] The, the coefficiets a, a,..., a of the expasio ca be easily computed from the orthogoality of the complex expoetials as * * * x[ ] e[ ] = amem[ ] e[ ] = am e[ ] em[ ] = a = = m= m= = x [ ] = VIDEO: Discrete Fourier Series (6:8) The rightmost expressio comes from the orthogoality property, for which the summatio is ozero oly whe = m. It is customary to defie X[ ] = a for =,..., ad call it the Discrete Fourier Series (DFS) of the periodic sequece x [ ]. This ca be computed as = j X[ ] = DFS{ x [ ]} = xe [ ], =,..., (9) Coversely, the origial the sigal x [ ] is called the Iverse Discrete Fourier Series (IDFS) ad it ca be writte as j x [ ] = IDFS{ X [ ]} = Xe [ ] () = Example Revisited: cosider the example we have see above (figure 6), with period =. The the Discrete Fourier Series (DFS) of the give sigal is computed as = j X[ ] = xe [ ] = x[] + x[]( ) = + ( ), =, Therefore this yields X [] = 3 ad X [] =. The give sigal ca the be expressed as j x [ ] = IDFS{ X [ ]} = Xe [ ] =.5.5 ( ) = same as i the example above.
8 Example: Cosider the periodic sigal x [ ] show i figure 8 below. Figure 8: a periodic sigal with period =. The we compute its Discrete Fourier Series (DFS) as j e j j if,,...,9 X[ ] = = π xe [ ] = e = j = = e 5 if = This sequece ca be plotted i terms of magitude ad phase as Figure 9: DFS of the sigal i figure 8 Example: Let the sigal x [ ] be defied as x [ ] = cos(.5 π ) It is periodic ad let s see what the period is. Just compute a few values to obtai x[] =, x[] =, x[] =, x[3] =,... This shows that the period is =, which yields its DFS as = j X[ ] = xe [ ] = x[] + x[] ( ), =, Substitutig for x[], x [] we obtai X[] = X[] =.
9 5. Sigals of a Fiite Legth ad the Discrete Fourier Trasform (DFT) VIDEO: the Discrete Fourier Trasform (:) I all data processig experimets we aalyze data of a fiite legth. A typical situatio is show i figure below, where a segmet of data of legth T MAX (i secods, for example) is digitized at a rate F S samples/secod, thus yieldig a vector of = TMAX FS samples. For example we have TMAX = 3msec (ie.3sec ) of data, sampled at FS = Hz, ie, samples/sec. The we obtai a umerical file with legth =.3, = 6 samples. Figure : a sampled sigal of fiite legth As a cosequece we obtai a data vector x [ ], =,... with data poits. I may applicatios we wat the frequecy cotet of this data which allows us to extract relevat iformatio. Thi about a music sigal: the frequecies are related to the otes i the music score. I geeral ay sort of audio sigal, be it speech or from a hydrophoe, has a sigature i frequecy which tells a lot about the ature of the sigal itself. We ca use the frequecy sigature i speech recogitio, for example. I other words it is very importat to determie the spectrum of a sigal umerically directly from the samples. I order to do this, we ca mae use of what we have developed for periodic sigals. I particular we ca loo at a sigal x [ ], =,... of legth samples as oe period of a periodic sigal with period. I this way we ca use exactly the same formulas of the Discrete Fourier Series (DFS) ad defie the Discrete Fourier Trasform (DFT) i exactly the same way: { } = j X[ ] = DFT x [ ] = xe [ ], =,..., j x [ ] = IDFT{ X [ ]} = Xe [ ], =,..., =
10 These expressios are the same as the DFS with the oly differece that the DFS is for periodic sigals, thus x [ ] is periodic ad defied for all samples, while the DFT is for fiite legth sigals, thus x [ ] is defied oly for =,...,. The meaig of this expasio is that ay fiite sequece ca be expaded i terms of complex expoetials with frequecies π /. Example. Cosider the sigal x [ ], =,..,9 of legth = show i figure. Figure : example of a sigal of legth Its DFT ca be computed aalytically usig the geometric series as j e j j if,,...,9 X[ ] = = π xe [ ] = e = j = = e 5 if = This ca be plotted i terms of magitude ad phase as show i figure below. Figure : DFT of the sigal i figure. 6. DFT of Complex Expoetials VIDEO: DFT of Complex Expoetials (:35) Sice we use the DFT to detect spectral compoets, it is particularly useful to see the DFT of a complex expoetial sigal of fiite legth defied as
11 x Ae jω [ ] =, =,... where ω represets the digital frequecy i radias. Applyig the defiitio of its DFT we obtai the followig expressio { } X[ ] = DFT x [ ] = xe [ ] = j j j j ω ω = Ae e = Ae, =,..., = = otice that this expressio has a particular structure ad it ca be writte i the followig way: X [ ] A W π = ω where the fuctio W ( ω ) depeds oly o the data legth ad it is defied as W jω e ( ω) = e = e = jω jω Agai the variable ω idicates digital frequecy ad it is i radias. As we will show below, the same expressio ca be writte i a differet way as ω si j ( )/ W ( ω) e ω = ω si The plot of its magitude for = is show i figure 3. Figure 3: magitude of W ( ω ) otice that the largest values are cocetrated aroud ω =. Also we ca distiguish the mai lobe withi the iterval < ω < ad all the other sidelobes. Most of the eergy is o the mai lobe. Example: cosider the sequece x e j.3π [ ] =, =,...,3
12 I this caseω =.3 π, = 3. The its DFT becomes 3 ( ω π) X [ ] = W.3, =,...,3 π ω= 3 I order to see how this loos lie, let s plot first W ( ω π) 3.3 as i figure 4 The, to see its DFT, we tae samples as This is show i figure 5. Figure 4: plot of W ( ω π) 3.3 { } ( ω π) X [ ] = DFT x [ ] = W.3, =,...,3 π ω= 3 Figure 5: DFT of the example. otice that the maximum correspods to the idex = 5 which yields the digital frequecy ω = 5 π / 3 =.3.3π 7. The Fast Fourier Trasform i Matlab Oe of the most importat reaso (if ot the most importat) reasos why the DFT has become a very popular algorithm is the extremely high efficiecy of its implemetatio. I particular the Fast Fourier Trasform (FFT) ca compute the DFT of a large data set i a very short time. Just to give a idea, if we have data poits, ad is a power of, we ca compute its DFT i a amout of time proportioal to log, usig the FFT, versus by brute force
13 computatio. For example let (say) = = 4. The the time tae by the FFT is 3 proportioal to 4log 4, while the brute force approach would tae a amout of 6 time proportioal to 4. As a cosequece, all computer applicatios, icludig Matlab, use the FFT rather the the DFT. The results of both operatios are idetical, but the FFT computes it i a fractio of the time tae by the brute force DFT. VIDEO: FFT i Matlab (:43) seg4.wmv Agai we wat to determie the DFT of the sequece x e j.3π [ ] = with =,..., 3. The: % Data Legth: =3; %. geerate data =:; x=exp(j.3*pi*); %. tae the FFT X=fft(x); % 3. plot it (choose oe) stem(abs(x)) % if you wat to see the idividual values plot(abs(x)) % to see a cotiuous plot Example: let us tae the same sequece, but with more data poits. Say, let =56. The the same Matlab program yields the graph show i the figure below. Figure 6. DFT of the same sequece, with =56 samples.
14 The pea ca be verified to be at = 38 so that the estimated frequecy is as expected. ω = 38 =.969π. 3π 56 I the followig example, we loo at the opposite problem. Give a sigal, we wat to determie the frequecy cotet looig at its DFT. Each pea yields a estimate of the domiat frequecies. Cosider a sigal x [ ], =,..., 55 of legth = 56. Its DFT X [] has its magitude as show i the figure below. So the problem is what ca we say about this sigal. Sice it has two distict ad well defied peas, at = 7 ad = 8 the sigal has two frequecies at ω = 7 =. 9π 56 ω = 8 =. 638π 56 Figure 7. Give DFT for the Example. 8. DFT of Siusoidal Sigals. VIDEO: DFT of a Siusoid (8:54) Recall that a siusoid is the sum of two complex expoetials, as
15 A A x[ ] = Acos jα jω jα jω ( ω + α ) = e e + e e I the spectral domai it is represeted as two frequecies as show i the figure below. Figure 8. Spectral Represetatio of a Siusoid ow we wat to see what is its DFT whe we tae a fiite data legth x [ ], =,...,. Defie agai X [ ] = DFT{ x[ ] } ad sice it is the sum of two complex expoetials, its DFT is the sum of the respective DFT s, as A jα jω A jα jω X[ ] = DFT e e + DFT e e The rightmost term, with egative frequecy ω requires some cosideratio sice the frequecies /, =,..., computed by the DFT caot be egative. We ca easily see jω j( ω ) that e = e, with ω so that the above DFT becomes A jα jω A jα j X[ ] = DFT e e + DFT e e The two frequecies ω ad ω are show i the figure below. ( ω ) Figure 9. Frequecy domai represetatio of a siusoidal sigal, usig positive frequecies oly.
16 As a cosequece of this, the DFT of a siusoidal sigal at frequecy < ω < π is goig to show two peas: oe at ω ad oe at ω. Both compoets have the same magitude A / ad phase ± α of opposite sig. Example. Suppose we have a sigal x ] = cos(. 3π + α ) [ with =,..., 3. I terms of complex expoetials, it has two frequecies, at ω =. 3π ad ω =. 3π. The egative frequecy is equivalet to ω = π.3π =. 7π so that its DFT becomes X [ ] = W3 ( ω.3π ) + W3 ( ω.7π ) ω= / 3 The magitude of ( ω.3π ) + W ( ω. 7π ) W is show i the figure below. 3 3 W. Figure. Magitude Plot of ( ω.3π ) + W ( ω. 7π ) 3 3 otice the two peas, as expected, ad the maximum value / = 6 at the peas. The DFT ow is computed as samples of the values at ω = / 3 with =,..., 3. This yields the 3 values show below. DFT of the Siusoid i the example. We ca easily see the two peas at = 5 ad = 3 5 = 7. The correspodig frequecies are ω = 5 / 3. 3π ad ω = 7 / 3. 7π as expected. By the defiitio of the DFT ad the example we see a importat property called Symmetry.
17 VIDEO: Symmetry of the DFT (9:) Symmetry of the DFT. Give a real sigal x [ ], =,...,, its DFT X [ ] = DFT x[ ], =,..., is such that { } Proof. From the defiitio of the DFT replacig the idex with X[ ] = X [ ] = X * [ ] we obtai = x[ ] e j ( ) = = x[ ] e j Sice x [] is real by assumptio, the rightmost expressio is the complex cojugate of X [] which shows the property. The cosequece of this property is that, whe the sigal is real, the whole iformatio of the frequecy spectrum is i the first half as X [ ], =,...,( / ), while the iformatio i the secod half of the DFT is redudat. Example. Referrig to the previous example, sice the sigal is real the whole iformatio of its frequecy spectrum is show i the plot for =,..., 5 as show below. Figure. DFT of a Siusoid: first half of the plot. 9. FFT of a Siusoid i Matlab This sectio (o video oly) shows a implemetatio of the FFT of a siusoid i Matlab. VIDEO: FFT of a Siusoid i Matlab (3:)
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