Chapter 2 Chapter 2A Part A-Discrete-Time Signals Part A ●●● Part B--Discrete-Time Systems ●●● ●●●● Discrete-Time Signals and Part C--Time-Domain Characterization of Discrete-Time Signals Systems in the Time-Domain LTI Discrete-Time Systems Part A:Discrete-Time Signals 1.Time-Domain Representation 1.Time-Domain Representation Signals represented as sequences of numbers, Discrete-time signal may also be written as a Time-Domain Representation called samples(采样、样本) sequence of numbers inside braces: Operations on Sequences Sample value of a typical signal or sequence denoted as x(n)with n being an integer in the {(m}={.,-0.2.22,1.10.2,-3,7,2.9} Classification of Sequences range-∞≤m≤∞ In the above,x(-1=-0.2,x0=2.2,x3=-3 ·Typical Sequences .x(n)defined only for integer values of n and etc. ·The Sample Process undefined for non-integer values of n The arrow is placed under the sample at time Discrete-time signal represented by {x (n) indexn=0 4 6
Chapter 2A Discrete-Time Signals and Systems in the Time-Domain 2 Chapter 2 Part A -- Discrete Discrete-Time Signals Time Signals Part B -- Discrete Discrete-Time Systems Time Systems Part C -- Time-Domain Characterization of Domain Characterization of LTI Discrete LTI Discrete-Time Systems Time Systems Part A Discrete-Time Signals 4 Part A: Discrete-Time Signals Time-Domain Representation Domain Representation Operations on Sequences Operations on Sequences Classification of Sequences Classification of Sequences Typical Sequences Typical Sequences The Sample Process The Sample Process 5 1. Time-Domain Representation Signals represented as sequences of numbers, called samples samples ˄䟷ṧǃṧᵜ˅ Sample value of a typical signal or sequence denoted as x (n) with n being an integer in the range ˉĞİnİĞ x(n) defined only for integer values of n and undefined for non-integer values of n Discrete-time signal represented by {x (n)} 6 Discrete-time signal may also be written as a sequence of numbers inside braces: {x(n)}={…, ˉ0.2, 2.2, 1.1, 0.2, ˉ3, 7, 2.9, …} In the above, x(ˉ1)= ˉ0.2, x(0)=2.2, x(3)=ˉ3 etc. The arrow is placed under the sample at time index n = 0 1. Time-Domain Representation
1.Time-Domain Representation 1.Time-Domain Representation 1.Time-Domain Representation Graphical representation of a discrete-time .In some applications,a discrete-time sequence Here,n-th sample is given by signal with real-valued samples is as shown (x(n))may be generated by periodically x(n斤x(0l-mFx(nD2n=.,-2,-1,0,-1 below: sampling a continuous-time signal at uniform -5 intervals of timex(r) The spacing T between two consecutive x(n) x-sn samples is called the sampling interval or sampling period Reciprocal of sampling interval T.denoted as, is called the sampling frequency: FFVT 1.Time-Domain Representation 1.Time-Domain Representation 1.Time-Domain Representation Unit of sampling frequency is cycles per A discrete-time signal may be a finite-length second,or hertz(Hz),if Tis in seconds Two types of discrete-time signals: or an infinite-length sequence --Sampled-data signals Whether or not the sequence (x(n))has been -Digital signals Finite-length (also called finite-duration or obtained by sampling.the quantity x(n)is finite-extent)sequence is defined only for a called the n-th sample of the sequence Signals in a practical digital signal processing finite time interval NnN2, system are digital signals obtained by .(x(n))is a real sequence,if the n-th sample x(n) quantizing the sample values either by where-<N and N <o with N<N2 is real for all values ofn ounding(取整)or truncation(舍位) Length or duration of the above sequence is Otherwise,(x(n))is a complex sequence N=N-N+1
7 Graphical representation of a discrete-time signal with real-valued samples is as shown below: 1. Time-Domain Representation x( 5) x( ) n x(3) 8 In some applications, a discrete-time sequence {x(n)} may be generated by periodically sampling a continuous-time signal at uniform intervals of time xa(t) 1. Time-Domain Representation (5) a x T ( ) a x t (3 ) a x T 9 Here, n-th sample is given by x(n)=xa(t)|t=nT=xa(nT), n =…, ˉ2, ˉ1,0, ˉ1,… The spacing T between two consecutive samples is called the sampling interval sampling interval or sampling period Reciprocal of sampling interval T, denoted as , is called the sampling frequency: FT=1/T 1. Time-Domain Representation 10 Unit of sampling frequency is cycles per second, or hertz (Hz), if T is in seconds Whether or not the sequence {x(n)} has been obtained by sampling, the quantity x(n) is called the n-th sample of the sequence {x(n)} is a real sequence real sequence, if the n-th sample x(n) is real for all values of n Otherwise, {x(n)} is a complex sequence complex sequence 1. Time-Domain Representation 11 Two types of discrete-time signals: -- Sampled-data signals data signals -- Digital signals Digital signals Signals in a practical digital signal processing system are digital signals obtained by quantizing the sample values either by rounding ˄ਆᮤ˅ or truncation ˄㠽ս˅ 1. Time-Domain Representation 12 A discrete-time signal may be a finite-length or an infinite infinite-length sequence Finite-length (also called finite-duration or finite-extent) sequence is defined only for a finite time interval N1 İ n İ N2 , where ˉĞ< N1 and N2 <Ğ with N1 < N2 Length or duration of the above sequence is N = N2ˉN1 + 1 1. Time-Domain Representation
1.Time-Domain Representation 2.Operations on Sequences 2.Operations on Sequences The length of a finite-length sequence can be .A single-input,single-output discrete-time Basic Operations increased by zero-padding,i.e.,by appending system operates on a sequence,called the input ·Product it with zeros sequence,according to some prescribed rules and develops another sequence,called the ●Addition Infinite-length sequences can be classified as output sequence,with more desirable ·Multiplication following properties ·Time-Shifting x(n)=0 for n<N right-sided sequence x(n) Discrete-Time v(n) left-sided sequence Time-Reverse (folding) x(n)=0 for n>N2 Input Sequence System Output Sequence x(m)≠0 for oo≤n≤o double-sided sequence ◆Branching 13 h(n) 15 2.Operations on Sequences 2.Operations on Sequences 2.Operations on Sequences Product (modulation)operation: 。Addition operation: + Time-shifting operation:y(n)=x(n-N) 8 If N>0,it is delaying operation --Modulator w(m) --Unit delay --Adder ynx(n+w(n) →n-) An application is in forming a finite-length 回 sequence from an infinite-length sequence by Multiplication operation: If N<0,it is an advance operation multiplying the latter with a finite-length sequence called an window sequence.The --Unit advance process is called windowing(加窗) --Multiplier y(n)=Ax(n) 月 回 18
13 The length of a finite-length sequence can be increased by zero-padding, i.e., by appending it with zeros Infinite-length sequences can be classified as following x(n) =0 for n <N1 right-sided sequence sided sequence x(n) =0 for n >N2 left-sided sequence sided sequence x(n)Į0 for ĞİnİĞ double-sided sequence sided sequence 1. Time-Domain Representation 14 2. Operations on Sequences A single-input, single-output discrete-time system operates on a sequence, called the input sequence sequence, according to some prescribed rules and develops another sequence, called the output sequence output sequence, with more desirable properties Discrete-Time Input Sequence System Output Sequence x(n) y(n) h(n) 15 2. Operations on Sequences Basic Operations Product Product Addition Multiplication Time-Shifting Time-Reverse (folding) Reverse (folding) Branching 16 2. Operations on Sequences Product Product (modulation) operation: (modulation) --Modulator Modulator An application is in forming a finite-length sequence from an infinite-length sequence by multiplying the latter with a finite-length sequence called an window sequence window sequence. The process is called windowing˄࣐デ˅ w(n) x(n) h y(n) 17 2. Operations on Sequences Addition operation: --Adder y(n)=x(n)+w(n) Multiplication operation: --Multiplier Multiplier y(n)=Ax(n) x(n) + w(n) y(n) x(n) A y(n) 18 2. Operations on Sequences Time-shifting operation: y(n)=x(nˉN) If N > 0, it is delaying operation --Unit delay If N < 0, it is an advance operation --Unit advance x(n) z ˉ y(n)=x(nˉ1) 1 x(n) z y(n)=x(n+1)
2.Operations on Sequences 2.Operations on Sequences 2.Operations on Sequences Time-reversal (folding)operation:y(n)=x(-n) However if the sequences are not of same An Example Branching operation:Used to provide length,in some situations,this problem can be multiple copies of a sequence circumvented by appending zero-valued dn) + 回回回 samples to the sequence(s)of smaller lengths to make all sequences have the same range of a the time index Operations on two or more sequences can be carried out if all sequences involved are of The combination of basic operations can same length and defined for the same range realize desirable functions of the time index n 19 20 21 2.Operations on Sequences 2.Operations on Sequences 2.Operations on Sequences Sampling Rate Alteration ·Inup-sampling(升采样)by an integer factorL Employed to generate a new sequence wn) >1,equidistant zero-valued samples are with a sampling rate F higher or lower than inserted by the up-sampler between each two that of the sampling rate F of a given consecutive samples of the input sequence x(n): sequence x(n) Sampling rate alteration ratio is R=F/F x(n/L),n=0,±L,±2L. x(m)= IfR>l,the process called interpolation(内插) 0 otherwise IfR<l,the process called decimation(抽取) 同) t L →x( 22
19 2. Operations on Sequences Time-reversal (folding) reversal (folding) operation: y(n)=x(ˉn) Branching operation: Used to provide multiple copies of a sequence Operations on two or more sequences can be carried out if all sequences involved are of same length and defined for the same range of the time index n x(n) x(n) x(n) 20 2. Operations on Sequences However if the sequences are not of same length, in some situations, this problem can be circumvented by appending zero-valued samples to the sequence(s) of smaller lengths to make all sequences have the same range of the time index The combination of basic operations can realize desirable functions 21 2. Operations on Sequences An Example z-1 a1 z-1 a2 z-1 a3 a4 + y(n) x(n) 22 2. Operations on Sequences Sampling Rate Alteration Employed to generate a new sequence y(n) with a sampling rate higher or lower than that of the sampling rate of a given sequence x(n) Sampling rate alteration ratio is If R > 1, the process called interpolation (ᨂ) If R < 1, the process called decimation (ᣭਆ) ' FT FT ' / RFF T T 23 2. Operations on Sequences In up-sampling (ॷ䟷ṧ) by an integer factor L > 1, equidistant zero-valued samples are inserted by the up-sampler sampler between each two consecutive samples of the input sequence x(n): ( / ), 0, , 2 ,... ( ) 0, u x nL n L L x n otherwise x(n) Ė L xu(n) 24 2. Operations on Sequences
2.Operations on Sequences 2.Operations on Sequences 3.Classification of sequences ·n down-sampling(降采样)by an integer .A discrete-time signal can be classified in factor M>1,every M-th samples of the input various ways,such as length,symmetry, sequence are kept and M-1 in-between summability,energy and power. samples are removed: .Conjugate-symmetric sequence:x(n=x*(-n) y(n)=x(nM) If x(n)is real,then it is an even sequence Conjugate-antisymmetric sequence:x(n)=- xin) I M x*(-n),Ifx(n)is real,then it is an odd sequence 27 3.Classification of sequences 3.Classification of sequences 3.Classification of sequences It follows from the definition that for a Any complex sequence can be expressed as a ●For a length-V sequence defined for 0≤n≤W conjugate-symmetric sequence (x(n),x(0) sum of its conjugate-symmetric part and its -1,it has a different definition as follows must be a real number conjugate anti-symmetric part: m）=xe(m）+x(m)0≤n≤N-1 Likewise,it follows from the definition that for x(m）=m)+xa(m) where a conjugate anti-symmetric sequence n), where 0)must be an imaginary number x(m)=(1/2)xn)H.x*W-m】0≤n≤N-1 From the above,it also follows that for an odd x.(m)=(1/2)r(n)tx*(-nm] is the periodic conjugate-symmetric part,and sequence fw(n),w(0)=0 xem)=(1/2)[x(m)-x*(-n] xe(m=(1/2)x(m)-x*W-n】0≤n≤N-1 is the periodic conjugate-antisymmetric part 28 30
25 2. Operations on Sequences In down-sampling (䱽䟷ṧ) by an integer factor M > 1, every M-th samples of the input sequence are kept and Mˉ1 in-between samples are removed: y n x nM () ( ) x(n) Ę M y(n) 26 2. Operations on Sequences 27 3. Classification of sequences A discrete-time signal can be classified in various ways, such as length, symmetry, summability, energy and power. Conjugate Conjugate-symmetric symmetric sequence: x(n)=x*(ˉn) If x(n) is real, then it is an even sequence Conjugate Conjugate-antisymmetric antisymmetric sequence: x(n)=ˉ x*(ˉn), If x(n) is real, then it is an odd sequence 28 3. Classification of sequences It follows from the definition that for a conjugate-symmetric sequence {x(n)}, x(0) must be a real number real number Likewise, it follows from the definition that for a conjugate anti-symmetric sequence {y(n)}, y(0) must be an imaginary number an imaginary number From the above, it also follows that for an odd sequence {w(n)}, w(0) = 0 29 3. Classification of sequences Any complex sequence can be expressed as a sum of its conjugate conjugate-symmetric part symmetric part and its conjugate anti conjugate anti-symmetric part symmetric part: x(n) =xcs(n) + xca(n) where xcs(n) =(1/2)[x(n)+x*(ˉn)] xca(n) =(1/2)[x(n)ˉx*(ˉn)] 30 3. Classification of sequences For a length-N sequence defined for 0İnİN ˉ1, it has a different definition as follows x(n) =xpcs(n) + xpca(n) 0İnİ Nˉ1 where xpcs(n) =(1/2)[x(n)+x*(Nˉn)] 0İnİ Nˉ1 is the periodic conjugate periodic conjugate-symmetric part symmetric part, and xpca(n) =(1/2)[x(n)ˉx*(Nˉn)] 0İnİ Nˉ1 is the periodic conjugate periodic conjugate-antisymmetric antisymmetric part