6.003: Signal Processing

Size: px

Start display at page:

Download "6.003: Signal Processing"

Buck Skinner
5 years ago
Views:

1 6.003: Sigal Processig Discrete-Time Fourier Series orthogoality of harmoically related DT siusoids DT Fourier series relatios differeces betwee CT ad DT Fourier series properties of DT Fourier series September 20, 2018

2 Last Time: Cotiuous-Time Fourier Series Express a periodic sigal as a sum of harmoically related siusoids ( x(t) = ck cos kω o t + d k si kω o t ) k=0 where ω o represets the fudametal frequecy. Basis fuctios: cos 0t t 2π ωo si 0t t 2π ωo cos ωot t 2π ωo si ωot t 2π ωo cos 2ωot. t 2π ωo si 2ωot. t 2π ωo

3 Cotiuous-Time Fourier Series Separatig harmoic compoets relies o two key observatios. 1. Multiplyig two harmoics produces a ew harmoic with the same fudametal frequecy: e jkωot e jlωot = e j(k+l)ωot. 2. The itegral of a harmoic over ay time iterval with legth equal to the period T is zero uless the harmoic is at DC: t0 +T { T if k = 0 e jkωot dt e jkωot dt = t 0 T 0 if k 0 = T δ[k]. Fourier compoets are orthogoal.

4 Cotiuous-Time Fourier Series Assume that x(t) is periodic i T ad is composed of a weighted sum of harmoics of ω o = 2π/T. x(t) = x(t + T ) = a k e jωokt k= The sift out oe compoet: x(t)e jωolt dt = a k e jωokt e jωolt dt T T k= = a k e jωo(k l)t dt k= T = a k T δ[k l] = a l T k= Solvig for a l provides a explicit formula for the coefficiets: a k = 1 x(t)e jωokt dt = 1 T T T T j 2π x(t)e T kt dt where ω o = 2π T.

5 Cotiuous-Time Fourier Series Represetig a periodic sigal as a sum of harmoic siusoids. a k = 1 x(t)e jkωot dt T T x(t)= x(t + T ) = a k e jkωot k= aalysis equatio sythesis equatio where ω o = 2π T

6 Today: Discrete Time Fourier Series Same high-level idea. Express a periodic sigal as a sum of harmoically related siusoids ( x[] = ck cos kω o + d k si kω o ) k=0 where Ω o represets the fudametal frequecy (radias/sample). Basis fuctios: cos 0Ωo 2π Ω o si 0Ωo 2π Ω o cos Ωo cos 2Ωo. 2π Ωo 2π Ωo si Ωo si 2Ωo. 2π Ωo 2π Ωo

7 Discrete-Time Fourier Series Separatig harmoic compoets relies o the same two key poits. 1. Multiplyig two DT harmoics produces a ew DT harmoic with the same fudametal frequecy: e jkωo e jlωo = e j(k+l)ωo. 2. The sum of a harmoic over ay time iterval with legth equal to the period N is zero uless the harmoic is at DC: 0 +N e jkωo { N if k = 0 e jkωo = = 0 0 if k 0 = Nδ[k]. DT Fourier compoets are orthogoal.

8 Discrete-Time Fourier Series Assume that x[] is periodic i N ad is composed of a weighted sum of harmoics of Ω o = 2π/N. x[] = x[ + N] = a k e jωok k The sift out oe compoet: x[]e jωol = a k e jωok e jωol k = a k e jωo(k l) k = a k Nδ[k l] = a l N k Solvig for a l provides a explicit formula for the coefficiets: a k = 1 N x[]e jωok = 1 N x[]e j 2π N k where Ω o = 2π N.

9 Discrete-Time Fourier Series Represetig a DT periodic sigal as a sum of harmoic siusoids. a k = 1 N x[]e jkωo aalysis equatio x[]= x[ + N] = k a k e jkωo sythesis equatio where Ω o = 2π N

10 Discrete-Time Fourier Series Represetig a DT periodic sigal as a sum of harmoic siusoids. a k = 1 N x[]e jkωo aalysis equatio x[]= x[ + N] = k a k e jkωo sythesis equatio where Ω o = 2π N DT Fourier series are similar to CT Fourier series... but DT siusoids differ from CT siusoids i importat ways! How do CT/DT differeces affect Fourier series?

11 Check Yourself What is the fudametal (shortest) period of each of the followig sigals? 1. x 1 [] = cos π x 2 [] = cos π π + 3 cos x 3 [] = cos + cos 2 + cos 3

12 Check Yourself x 1 [] = cos π ( π ) 12 = cos π π( + 24) = cos = x 1 [ + 24] 12 x 2 [] = cos π 12 = cos π ( π ) ( π ) + 3 cos 15 = cos π + 3 cos π π( + 120) cos π( + 120) 15 x 3 [] = cos + cos 2 + cos 3 is ot periodic. = x 2 [ + 120] cos cos 2 cos 3

13 Check Yourself What is the fudametal (shortest) period of each of the followig sigals? 1. x 1 [] = cos π x 2 [] = cos π cos π x 3 [] = cos + cos 2 + cos 3

14 Check Yourself What is the fudametal (shortest) period of each of the followig sigals? 1. x 1 [] = cos π x 2 [] = cos π cos π x 3 [] = cos + cos 2 + cos 3 The period of a periodic DT sigal must be a iteger. Therefore the fudametal frequecy of a periodic DT sigal must be a iteger submultiple of 2π. No such costraits o fudametal frequecies i CT.

15 Discrete-Time Siusoids The frequecies of discrete-time siusoids alias. Ω = 0.25 x[] = cos(0.25)

16 Discrete-Time Siusoids The frequecies of discrete-time siusoids alias. Ω = 0.5 x[] = cos(0.5)

17 Discrete-Time Siusoids The frequecies of discrete-time siusoids alias. Ω = 1 x[] = cos()

18 Discrete-Time Siusoids The frequecies of discrete-time siusoids alias. Ω = 2 x[] = cos(2)

19 Discrete-Time Siusoids The frequecies of discrete-time siusoids alias. Ω = 3 x[] = cos(3)

20 Discrete-Time Siusoids The frequecies of discrete-time siusoids alias. Ω = 4 x[] = cos(4) = cos(2π 4) cos(2.283)

21 Discrete-Time Siusoids The frequecies of discrete-time siusoids alias. Ω = 5 x[] = cos(5) = cos(2π 5) cos(1.283)

22 Discrete-Time Siusoids The frequecies of discrete-time siusoids alias. Ω = 6 x[] = cos(6) = cos(2π 6) cos(0.283) There are multiple frequecies associated with every DT siusoid. The frequecies of discrete-time siusoids alias.

23 Discrete-Time Siusoids If Ω o is a submultiple of 2π, ad the harmoic frequecies alias, the there are (oly) N distict complex expoetials with period N. (There were a ifiite umber i CT!) If y[] = e jω is periodic i N the y[] = e jω = y[ + N] = e jω(+n) = e jω e jωn ad e jωn must be 1 e jω must be oe of the N th roots of 1. Example: N = 8 Im Re

24 Discrete-Time Siusoids There are N distict complex expoetials with period N. If a DT sigal is periodic with period N, the its Fourier series cotais just N terms. Example: periodic i N=3 3 samples repeated i time 3 complex expoetials Example: periodic i N=4 4 samples repeated i time 4 complex expoetials

25 Discrete-Time Fourier Series DT Fourier series comprise a weighted sum of just N harmoics. x[] = x[ + N] = k=<n> a k e jωok

26 Discrete-Time Fourier Series DT Fourier series comprise a weighted sum of just N harmoics. x[] = x[ + N] = k=<n> a k e jωok The sift out oe compoet: x[]e jωol = a k e jωok e jωol k=<n> = a k e jωo(k l) k=<n> = a k Nδ[k l] = a l N k=<n> Solvig for a l provides a explicit formula for the coefficiets: a k = 1 N x[]e jωok = 1 N x[]e j 2π N k where Ω o = 2π N. Both x[] ad e j 2π N k are periodic i N, a k is periodic i N.

27 Discrete-Time Fourier Series A periodic DT sigal with N samples produces a periodic sequece of N Fourier series coefficiets. a k = a k+n = 1 x[]e jkωo N x[]= x[ + N] = a k e jkωo k=<n> aalysis equatio sythesis equatio where Ω o = 2π N

28 Fourier Series Summary CT ad DT Fourier Series are similar, but DT Fourier Series have just N coefficiets while CT Fourier Series have a ifiite umber. Cotiuous-Time Fourier Series a k = 1 x(t)e jkωot dt T T x(t)= x(t + T ) = a k e jkωot k= Discrete-Time Fourier Series a k = a k+n = 1 x[]e jkωo N = N x[]= x[ + N] = a k e jkωo k= N aalysis equatio sythesis equatio where ω o = 2π T aalysis equatio sythesis equatio where Ω o = 2π N

29 Properties of Discrete-Time Fourier Series Operatios o the time represetatio of a sigal ca ofte be iterpreted as equivalet operatios o the series coefficiets. Example: Fourier series of a liear combiatio of sigals Proof: Let x[] = ax 1 []+bx 2 [] where x 1 [] = x 1 [+N] ad x 2 [] = x 2 [+N] the the Fourier series coefficets for x[] are give by X[k] = 1 x[]e jk 2π N = 1 ( ax1 [] + bx 2 [] ) e jk 2π N N N = a 1 x 1 []e jk 2π N + b 1 x 2 []e jk 2π N N N = ax 1 [k] + bx 2 [k] where X 1 [k] ad X 2 [k] are Fourier series coefficiets for x 1 ad x 2.

30 Properties of Discrete-Time Fourier Series Operatios o the time represetatio of a sigal ca ofte be iterpreted as equivalet operatios o the series coefficiets. Example: Shiftig time chages the phases of Fourier compoets. Proof: Let y[] = x[ 0 ] where x[] = x[ + N] If X[k] = 1 x[]e jk 2π N N the Y [k] = 1 y[]e jk 2π N = 1 x[ 0 ]e jk 2π N N N = 1 x[m]e jk 2π N (m+ 0) where m = 0 N m=<n> = e jk 2π N 0 1 x[m]e jk 2π N m = e jk 2π N 0 X[k] N m=<n>

31 Properties of Discrete-Time Fourier Series Operatios o the time represetatio of a sigal ca ofte be iterpreted as equivalet operatios o the series coefficiets. Example: Flippig time flips frequecy. Proof: Let y[] = x[ ] where x[] = x[ + N] If X[k] = 1 x[]e jk 2π N N the Y [k] = 1 y[]e jk 2π N = 1 x[ ]e jk 2π N N N = 1 x[m]e jk 2π N m where m = N m=<n> = X[ k]

32 Properties of Discrete-Time Fourier Series Operatios o the time represetatio of a sigal ca ofte be iterpreted as equivalet operatios o the series coefficiets. Example: If x[] is a real-valued sequece, the X[ k] = X[k]. X[k] = 1 N X[ k] = 1 N x[]e jk 2π N x[]e jk 2π N = X[k]

33 Properties of Discrete-Time Fourier Series Operatios o the time represetatio of a sigal ca ofte be iterpreted as equivalet operatios o the series coefficiets. Example: Eve ad Odd Decompositio Fid the Fourier series for the eve ad odd parts of a sigal. x[] fs X[k] x[ ] fs X[ k] Eve ( x[] ) = 1 ( ) 2 x[] + x[ ] fs 1 ( ) 2 X[k] + X[ k] If x[] is real, the X[ k] = X[k]. Eve ( x[] ) = 1 ( ) 2 x[] + x[ ] fs 1 2( X[k] + X[k] ) = Re (X[k]) Eve ( x[] ) fs Re ( X[k] ) Similarly Odd ( x[] ) fs jim ( X[k] )

34 Summary Discrete-Time Fourier Series orthogoality of harmoically related DT siusoids DT Fourier series relatios differeces betwee CT ad DT Fourier series properties of DT Fourier series Today s Lab: Use DT Fourier Series to Uderstad a Auditory Perceptio. For today oly: stay here istead. Studets who ormally have lab i should

Signal Processing in Mechatronics. Lecture 3, Convolution, Fourier Series and Fourier Transform

Signal Processing in Mechatronics. Lecture 3, Convolution, Fourier Series and Fourier Transform Sigal Processig i Mechatroics Summer semester, 1 Lecture 3, Covolutio, Fourier Series ad Fourier rasform Dr. Zhu K.P. AIS, UM 1 1. Covolutio Covolutio Descriptio of LI Systems he mai premise is that the