SS_03-4

• Harmonically related signal sets

2π )n jk(

{ φk[n] = e N, k = 0, ±1, ±2,……

all with period N, ω0 = N 2π

}

φk+rN[n]= φk[n], only N distinct signals in the set

2π n jk( ) n= Σe N = N , k = 0, ±N, ±2N,……

= 0 , else

• Fourier Series x[n] =Σ ak e

k=

jkωon

=k=Σ ak e

2π )n jk ( N

2π )n1 –jkωon 1 -jk( ak = N Σ x[n]e= Σx[n]e N Nk= k=ak+rN = ak , repeat with period N Note: both x[n] and ak are discrete, and

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periodic with period N, therefore summed over a period of N

Fourier Series Representation • Vector Space Interpretation { x[n] , x[n] is periodic with period N } is a vector space

( x1[n] )．( x2[n] ) =Σx1[k] x2*

[k]

k=( φi[n] )．( φj[n] ) =N , i = j+rN

= 0 , else

{( )

1 ½N

φk [n] = φk'[n] , k = < N > } a set of orthonormal bases x[n] = ∑ ak φk [n]

k= a

k = ( ( )1 ½N x[n] )．( ( )1 ½

N

φk [n] ) =

1

∑ x[n] e –jkωon

N k=2

Fourier Series Representation • No Convergence/ Completeness Issue, No Gibbs Phenomenon - x[n] has only N parameters, represented by N coefficients

sum of N terms gives the exact value - N odd x[n] M M = ∑ a

2k= - M k e

jk( ) π Nn x[n]M = x[n] , if M = (N-1) /2 - N even M x[n] M = ∑ a

k e

jk( 2 πN ) n

k= - M+1 x[n]M = x[n] if M = N / 2

See Fig. 3.18 , P.220 of text

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Properties • Primarily Parallel with those for continuous-time x[n] ak • Multiplication

FS x[n] ak , y[n] bk

FS FS

x[n] y[n] dk = ∑ ajbk-j

j=FS

periodic convolution • First Difference x[n]-x[n-1] (1-e• Parseval’s Relation

1 2 2 ∑ | x[n] |= ∑ | ak |N k= k=

FS

2π -jk( ) N) ak

average power

in a period average power in a period for each harmonic component

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