3.Classification of sequences 3.Classification of sequences 3.Classification of sequences .A length-Nsequence x(n)is called a periodic x(m) x) Folding A sequence (n)satisfying conjugate-symmetric sequence if x(《-》 (n)=n+k)for all n x(m=x'(《-mw)=x'(N-m)0≤n≤N-1 x(N-n) is called a periodic sequence with a period N. (0SnsN-1) —RM where Nis a positive integer and k is any and is called a periodic conjugate-anti- integer symmetric sequence if Smallest value of N satisfying (n)=(n+) xm)=-x'(《-)w)=-x(N-n)0≤n≤N-1 is called the fundamental period Q:How to getx(-m)in the interval0≤n≤N-1 A sequence not satisfying the periodicity 31 condition is called an aperiodic sequence 33 3.Classification of sequences 3.Classification of sequences 3.Classification of sequences Total energy of a sequence x(n)is defined by The average power of a periodic sequence with A sequence x(n)is said to be bounded if B,=2f a period N is given by x(nsB<∞ A sequence x(n)is said to be absolutely An infinite length sequence with finite sample 2r 0 summable if values may or may not have finite energy The average power of an aperiodic sequence is defined by 立olk“ A finite length sequence with finite sample 1 ◆A sequence x(m）is said to be square values has finite energy P=lim summable if 品 立of<“31 3. Classification of sequences A length-N sequence x(n) is called a periodic periodic conjugate conjugate-symmetric sequence symmetric sequence if and is called a periodic conjugate periodic conjugate-anti￾symmetric sequence symmetric sequence if Q: How to get x(ˉn) in the interval 0İnİ Nˉ1 * * () ( ) ( ) 0 1 N xn x n x N n n N  * * () ( ) ( ) 0 1 N xn x n x N n n N  32 3. Classification of sequences * ( ) N x n  * x ( ) N n x(n) Conjugating x*(n) Folding x*(ˉ n) Periodical Extension h RN(n) (0 1) n N x*(n) n x*(—n) n n * ( ) N x n  33 3. Classification of sequences A sequence satisfying is called a periodic sequence periodic sequence with a period N, where N is a positive integer and k is any integer Smallest value of N satisfying is called the fundamental period A sequence not satisfying the periodicity condition is called an aperiodic aperiodic sequence sequence x ( ) ( ) for all n x n kN n  x ( ) n x () ( ) n x n kN  34 3. Classification of sequences Total energy of a sequence x(n) is defined by An infinite length sequence with finite sample values may or may not have finite energy A finite length sequence with finite sample values has finite energy 2 ( ) x n E xn   35 3. Classification of sequences The average power average power of a periodic sequence of a periodic sequence with a period N is given by The average power of an average power of an aperiodic aperiodic sequence sequence is defined by 1 2 0 1 ( ) N x n P xn N 1 2 lim ( ) 2 1 K x K n K P xn  K  36 3. Classification of sequences A sequence x(n) is said to be bounded if A sequence x(n) is said to be absolutely summable summable if A sequence x(n) is said to be square summable summable if ( ) x xn B  2 ( ) n x n     ( ) n x n