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 nN2 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 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, 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
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-antisymmetric 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
4.Typical Sequences 4.Typical Sequences 4.Typical Sequences 「1，n20 ·Unit Sample Sequence ∫1，n=0 。Unit Step Sequence Real sinusoidal sequence- 4(m)= 0,n≠0 10.is0 xr(n=Acos(@on+中)】 6侧s.0 where 4 is the amplitude,o is the angular 5(m)vs.u(m) frequency,and is the phase of x(n) Their energies are both equal to 1.5(n) M(m)=6(n-k)=5(k) Complex exponential sequence- is of engineering r(n)=Aa",-0≤n≤o value,but 6(r)only 6(a)=4(a-m-1) has meaning in theory where A and a are real or complex numbers. In general,a=e)and 4=Ale 37 39 4.Typical Sequences 5.The Sampling Process 5.The Sampling Process .An arbitrary sequence can be represented in Often,a discrete-time sequence x(n)is .Time variable t ofx (r)is related to the time the time-domain as a weighted sum of some developed by uniformly sampling a basic sequence and its delayed(advanced) continuous-time signalx(n)as indicated variable n ofx(n)only at discrete-time t versions below instants given by (m)(o(-k) n。2xn 1.=nT=- F Another interpretation is that the above with F,=1/T denoting the sampling frequency equation can be viewed as a convolution of x(n)and (n) The relation between the two signals is and =2F denoting the sampling angular xn卢xt0l-nFx(nT刀，=.，-2，-1,0,1,2. frequency 名 41 42
37 4. Typical Sequences Unit Sample Sequence Unit Sample Sequence ¥(n) vs. ¥(t) Their energies are both equal to 1. ¥(n) is of engineering value, but¥(t) only has meaning in theory 1, 0 ( ) 0, 0 n n n -8 -6 -4 -2 0 2 4 6 8 0 0.5 1 Time (n) 38 4. Typical Sequences Unit Step Sequence Unit Step Sequence 1, 0 ( ) 0, 0 n n n ¥(n) vs. u(n) 0 () ( ) () n k k n nk k ( ) ( ) ( 1) n nn -8 -6 -4 -2 0 2 4 6 8 0 0.5 1 Time u(n) 39 4. Typical Sequences Real sinusoidal sequence Real sinusoidal sequencex(n)=Acos(¹0 n+ ¶) where A is the amplitude amplitude,¹0 is the angular angular frequency, and ¶ is the phase of x(n) Complex exponential sequence Complex exponential sequencewhere A and are real or complex numbers. In general, and () , n xn A n 0 0 ( ) j e j A Ae 40 4. Typical Sequences An arbitrary sequence can be represented in the time-domain as a weighted sum of some basic sequence and its delayed (advanced) versions Another interpretation is that the above equation can be viewed as a convolution of x(n) and ¥(n) () ()( ) k x n xk n k 41 5. The Sampling Process Often, a discrete-time sequence x(n) is developed by uniformly sampling a continuous-time signal xa(n) as indicated below The relation between the two signals is x(n)=xa(t)|t=nT=xa(nT), n=…, ˉ2, ˉ1,0,1,2,… -3 -2 -1 0 1 2 3 T n 42 5. The Sampling Process Time variable t of xa(t) is related to the time variable n of x(n) only at discrete-time tn instants given by with FT=1/T denoting the sampling frequency and denoting the sampling angular sampling angular frequency 2 n T T n n t nT F 2 T T F
5.The Sampling Process 5.The Sampling Process 5.The Sampling Process Consider the continuous-time signal 22 where 4= An Example 2 x.)=Ac0s(2πf1+p)=Ac0s2,1+) is the normalized digital angular frequency of Consider the three continuous-time signals The corresponding discrete-time signal is x(n) g1(0)=c0s(6T0,g2(0=c0s(14T),8(0=c0s(26T0 x(n)=Acos(SonT+)=Acos on If the unit of sampling period Tis in seconds, of frequencies 3 Hz,7 Hz,and 13 Hz,are sampled at the unit of normalized digital angular a sampling rate of 10 Hz,i.e.with T=0.1 sec. frequency o,is radians/sample while the unit generating the three sequences 2 n+ 4cos(,+p of normalized analog angular frequency is g1(n)=cos(0.6 )g(n)=cos(14 n).g;(n)=cos(2.6n) radians/second 43 44 45 5.The Sampling Process 5.The Sampling Process 5.The Sampling Process Plots of these sequences and their parent time functions are shown below: This fact can also be verified by observing that The above phenomenon of a continuous-time signal of higher frequency acquiring the identity of a sinusoidal sequence of lower frequency after g2(n)=cos(1.4mn)=cos((2-0.6r)n)=cos(0.6mn) sampling is called aliasing 8(m)=cos(2.6n)=c0s(2x+0.6x)m）)=c0s(0.6xm) Therefore,additional conditions need to imposed so that the sequence xn)can uniquely represent the As a result,all three sequences are identical and it is parent continuous-time signal x (r) difficult to associate a unique continuous-time function with each of these sequences Note that each sequence has exactly the same sample value for any given n 46
43 5. The Sampling Process Consider the continuous-time signal The corresponding discrete-time signal is 0 0 ( ) cos(2 ) cos( ) a x t A ft A t 0 0 0 0 1 ( ) cos( ) cos 2 cos cos( ) T T x n A nT A n F A n An 44 5. The Sampling Process where is the normalized digital angular frequency of x(n) If the unit of sampling period T is in seconds, the unit of normalized digital angular frequency is radians/sample radians/sample while the unit of normalized analog angular frequency is radians/second 0 0 2 T o 0 45 5. The Sampling Process An Example An Example Consider the three continuous-time signals g1(t)=cos(6±t), g2(t)=cos(14±t), g3(t)=cos(26±t) of frequencies 3 Hz, 7 Hz, and 13 Hz, are sampled at a sampling rate of 10 Hz, i.e. with T = 0.1 sec. generating the three sequences g1(n)=cos(0.6±n), g2(n)=cos(1.4±n),g3(n)=cos(2.6±n) 46 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 Time Amplitude g1(t) g1(n) g2(t) g2(n) g3(t) g3(n) Plots of these sequences and their parent time functions are shown below: Note that each sequence has exactly the same sample value for any given n 5. The Sampling Process 47 This fact can also be verified by observing that As a result, all three sequences are identical and it is difficult to associate a unique continuous-time function with each of these sequences 5. The Sampling Process 2 gn n n n ( ) cos(1.4 ) cos((2 0.6 ) ) cos(0.6 ) 3 gn n n n ( ) cos(2.6 ) cos((2 0.6 ) ) cos(0.6 ) 48 5. The Sampling Process The above phenomenon of a continuous-time signal of higher frequency acquiring the identity of a sinusoidal sequence of lower frequency after sampling is called aliasing Therefore, additional conditions need to imposed so that the sequence x(n) can uniquely represent the parent continuous-time signal xa(t)
5.The Sampling Process 5.The Sampling Process 5.The Sampling Process Another Example The sampling period is T=1/200=0.005sec .Note:v(n)is composed of 3 discrete-time sinusoidal Determine the discrete-time signal wn)obtained by The generated discrete-time signal wn)is thus given signals of normalized angular frequencies:0.3, uniformly sampling at a sampling rate of 200 Hz the by 0.5x.and 0.7 continuous-time signal )=6cs(0.3mn)+3sin(1.5r .Note:An identical discrete-time signal is also v(f)=6cos(60 )+3sin(300)+2cos(3401) +2c0s1.7rm+4c0s2.5xn+10sin(3.3x generated by uniformly sampling at a 200-Hz =6c0s(0.3x)+3sin(2-0.5)xm)2cos(2-0.3)Fm) +4cos(500πf)+10sin(660rt1) sampling rate the following continuous-time signals: +4cos(2+0.5)x+10sin(4-0.7)zn) Note:It is composed of 5 sinusoidal signals of ,t)=8cos(60a)+5c0s100xf+0.6435)-10sin(140f) frequencies 30 Hz,150 Hz,170Hz,250 Hz and 330 =6c0s(0.3x-3sin(0.5xn)+2cos(0.3rm (t)=2cos(60mt)+4cos(100mt)+10sin(260mr) Hz +4c0s(0.5xn)-10sin(0.7πm) +6cos(460r)+3sin(700at) 49 =8cos(0.3rn)+5c0s0.5rn+0.6435)-10sin(0.7xnsm 51 5.The Sampling Process 5.The Sampling Process Sampling Theorem 。Recall 22 .On the other hand,if 2,the Consider an arbitrary continuous-time signal -04 normalized digital angular frequency will foldoyer into a lower digital frequency x(r)composed of a weighted sum of a number of sinusoidal signals Thus if 22 then the corresponding =(22。/2r〉2.in the range-r<w<x because of aliasing .x(r)can be represented uniquely by its normalized digital angular frequency of the Hence,to prevent aliasing,the sampling sampled version (x(n)if the sampling discrete-time signal obtained by sampling the frequency should be greater than 2 times frequency is chosen to be greater than 2 parent continuous-time sinusoidal signal will the frequency o of the sinusoidal signal times the highest frequency contained inx(r) be in the range being sampled This theorem can be proofed via Fourier 。No aliasing From the above analysis,we state the Sample Transform in chapter 5 52 Theorem as follows 53
49 5. The Sampling Process Another Example Determine the discrete-time signal v(n) obtained by uniformly sampling at a sampling rate of 200 Hz the continuous-time signal va(t)=6cos(60±t)+3sin(300±t)+2cos(340±t) +4cos(500±t)+10sin(660±t) Note: It is composed of 5 sinusoidal signals of frequencies 30 Hz, 150 Hz, 170Hz, 250 Hz and 330 Hz 50 5. The Sampling Process The sampling period is T=1/200=0.005sec The generated discrete-time signal v(n) is thus given by ( ) 6cos(0.3 ) 3sin(1.5 ) 2cos(1.7 ) 4cos(2.5 ) 10sin(3.3 ) 6cos(0.3 ) 3sin((2 0.5) )2cos((2 0.3) ) 4cos((2 0.5) ) 10sin((4 0.7) ) 6cos(0.3 ) 3sin(0.5 ) 2cos(0.3 ) 4c vn n n nn n n nn n n nn n os(0.5 ) 10sin(0.7 ) 8cos(0.3 ) 5cos(0.5 0.6435) 10sin(0.7 ) n n nn n 51 5. The Sampling Process Note: v(n) is composed of 3 discrete-time sinusoidal signals of normalized angular frequencies: 0.3±, 0.5±, and 0.7± Note: An identical discrete-time signal is also generated by uniformly sampling at a 200-Hz sampling rate the following continuous-time signals: ( ) 8cos(60 ) 5cos(100 0.6435) 10sin(140 ) ( ) 2cos(60 ) 4cos(100 ) 10sin(260 ) 6cos(460 ) 3sin(700 ) a a wt t t t ut t t t t t 52 5. The Sampling Process Recall Thus if ¡T> 2¡0, then the corresponding normalized digital angular frequency of the discrete-time signal obtained by sampling the parent continuous-time sinusoidal signal will be in the range ˉ±<¹<± No aliasing 0 0 2 T 53 5. The Sampling Process On the other hand, if ¡T< 2¡0, the normalized digital angular frequency will foldover into a lower digital frequency ¹0={2±¡T /¡T} 2in the range ˉ±<¹<± because of aliasing Hence, to prevent aliasing, the sampling frequency ¡T should be greater than 2 times the frequency ¡0 of the sinusoidal signal being sampled From the above analysis, we state the Sample Theorem as follows 0 0 2 2 / T 54 Sampling Theorem Consider an arbitrary continuous-time signal xa(t) composed of a weighted sum of a number of sinusoidal signals xa(t) can be represented uniquely by its sampled version {x(n)} if the sampling frequency ¡T is chosen to be greater than 2 times the highest frequency contained in xa(t) This theorem can be proofed via Fourier Transform in chapter 5
