数字信号处理教学课件：2-3 Classification of Sequences

1、xn is a real sequence, if the n-th sample xn is real for all values of nOtherwise, xn is a complex sequenceA complex sequence xn can be written as xn=xren+jximn where xre and xim are the real and imaginary parts of xnThe complex conjugate sequence of xn is given by x*n=xren - jxim n2 Discrete-Time S

2、ignals and Systems2.3.3 Classification of Sequences1. Classification based on Symmetry Conjugate-symmetry sequence x*n=x-nConjugate-antisymmetry sequence x*n=-x-nIf xn is a real sequence, what is xn called?2 Discrete-Time Signals and Systemscomplex sequence xn Evenxn=x-n Oddxn=-x-n real xnConjugate-

3、symmetry part Conjugate-antisymmetry part xn is complex sequence, 2 Discrete-Time Signals and Systemsgn=0, 1-j4, -2-j3, 4+j2, -5+j6, j2, 3 g-n=3, j2, -5+j6, 4+j2, -2-j3, 1-j4, 0 2 Discrete-Time Signals and Systemsgn= 0, 1+j4, -2+j3, 4-j2, -5-j6, -j2, 3 Example 2.8 A length-7 sequence(n:-3,3):gcs n=1

4、.5,0.5+j3,-3.5+j4.5,4,-3.5-j4.5,0.5-j3,1.5 gca -n=-1.5,0.5+j,1.5-j1.5,-j2,-1.5-j1.5,-0.5+j,1.5 2.Classification of Sequences based on periodicity Example 2 Discrete-Time Signals and SystemsA sequence satisfying the periodicity condition is called an periodic sequence,for all nAdding two periodic seq

5、uences with different fundamental periodsThe fundamental periodGCD( )-greatest common divider2 Discrete-Time Signals and SystemsLCM( )-least common multiplefundamental periodsExampleIts fundamental period is 48 ,because the GCD of 16 and 12 is 4.2 Discrete-Time Signals and Systems3. Energy and Power

6、 SignalsTotal energy of a sequence xn is defined by2 Discrete-Time Signals and SystemsThe average power of a sequence is defined byExample - Consider the causal sequence defined by Note: xn has infinite energy Its average power is given by2 Discrete-Time Signals and Systems4. Other Types of Classifi

7、cationA sequence xn is said to be bounded if Example - The sequence xn=cos(0.3n) is a bounded sequence as2 Discrete-Time Signals and SystemsA sequence xn is said to be absolutely summable if Example - The sequenceis an absolutely summable sequence as2 Discrete-Time Signals and SystemsA sequence xn i

8、s said to be square-summable if Example - The sequenceis square-summable but not absolutely summable2 Discrete-Time Signals and SystemsIt is neither absolutely summable nor square-summable .2 Discrete-Time Signals and Systems Example - The sequence obtained by adding an absolutely summable sequence

9、with its replicas shifted by integer multiples of N,is called an N-periodic extension of .Unit sample sequence Unit step sequence 2.4 Typical Sequences and Representation2.4.1 Basic Sequences2 Discrete-Time Signals and Systems(2.4.3 )Representation of an Arbitrary Sequence by impulsesAn arbitrary sequence can be represented in the time-domain as a weighted sum of some basic sequence and its delayed (advanced) versions2 Discrete-Time Signals and