Discrete-Time Signals: Time-Domain Representation

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1 Discrete-Time Signals: Time-Domain Representation 1 Signals represented as sequences of numbers, called samples Sample value of a typical signal or sequence denoted as x[n] with n being an integer in the range n x[n] defined only for integer values of n and undefined for noninteger values of n Discrete-time signal represented by {x[n]}

2 Discrete-Time Signals: Time-Domain Representation 2 Discrete-time signal may also be written as a sequence of numbers inside braces: { x[ n]} = { K, 0.2, 2.2,1.1,0.2, 3.7, 2.9, K} In the above, x [ 1] = 0.2, x [ 0] = 2.2, x[ 1] = 1.1, etc. The arrow is placed under the sample at time index n = 0

3 Discrete-Time Signals: Time-Domain Representation Graphical representation of a discrete-time signal with real-valued samples is as shown below: 3

4 Discrete-Time Signals: Time-Domain Representation In some applications, a discrete-time sequence {x[n]} may be generated by periodically sampling a continuous-time signal x a (t) at uniform intervals of time 4

5 Discrete-Time Signals: Time-Domain Representation 5 Here, n-th sample is given by x[ n] = xa ( t) t= nt = xa ( nt ), n = K, 2, 1,0,1,K The spacing T between two consecutive samples is called the sampling interval or sampling period Reciprocal of sampling interval T, denoted as, is called the sampling frequency: F T 1 F T = T

6 Discrete-Time Signals: Time-Domain Representation Unit of sampling frequency is cycles per second, or hertz (Hz), if T is in seconds 6 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, if the n-th sample x[n] is real for all values of n Otherwise,{x[n]} is a complex sequence

7 Discrete-Time Signals: Time-Domain Representation 7 A complex sequence {x[n]} can be written as { x[ n]} = { xre [ n]} + j{ xim[ n]} where x re [n] and x im [n] are the real and imaginary parts of x[n] The complex conjugate sequence of {x[n]} is given by { x*[ n]} = { xre[ n]} j{ xim[ n]} Often the braces are ignored to denote a sequence if there is no ambiguity

8 Discrete-Time Signals: Time-Domain Representation 8 Example-{ x[ n]} = {cos0. 25n} is a real sequence j0. 3n { y[ n]} = { e } is a complex sequence We can write { y[ n]} = {cos0. 3n + jsin0. 3n} = {cos0. 3n} + j{sin0. 3n} where { y re [ n]} = {cos0. 3n} { y im [ n]} = {sin0. 3n}

9 Discrete-Time Signals: Time-Domain Representation Example - j0. 3n { w[ n]} = {cos0. 3n} j{sin0. 3n} = { e } is the complex conjugate sequence of {y[n]} That is, { w [ n]} = { y *[ n]} 9

10 10 Discrete-Time Signals: Time-Domain Representation Two types of discrete-time signals: - Sampled-data signals in which samples are continuous-valued - Digital signals in which samples are discrete-valued Signals in a practical digital signal processing system are digital signals obtained by quantizing the sample values either by rounding or truncation

11 Discrete-Time Signals: Time-Domain Representation Example - Amplitude Amplitude time, t time, t Boxedcar signal Digital signal 11

12 12 Discrete-Time Signals: Time-Domain Representation A discrete-time signal may be a finitelength or an infinite-length sequence Finite-length (also called finite-duration or finite-extent) sequence is defined only for a finite time interval: N1 n N 2 where < N 1 and N2 < with N1 N2 Length or duration of the above finitelength sequence is N = N N

13 Discrete-Time Signals: Time-Domain Representation 2 Example- x [ n] = n, 3 n 4 is a finitelength sequence of length 4 ( 3) + 1 = 8 y[ n] = cos0. 4n is an infinite-length sequence 13

14 Discrete-Time Signals: Time-Domain Representation A length-n sequence is often referred to as an N-point sequence The length of a finite-length sequence can be increased by zero-padding, i.e., by appending it with zeros 14

15 Discrete-Time Signals: Time-Domain Representation Example- 2 x = n 3 n 4 e [ n] 0,, 5 n 8 is a finite-length sequence of length 12 2 obtained by zero-padding x [ n] = n, 3 n with 4 zero-valued samples 4 15

16 Discrete-Time Signals: Time-Domain Representation A right-sided sequence x[n] has zerovalued samples for n < N 1 N 1 n 16 A right-sided sequence If N 1 0, a right-sided sequence is called a causal sequence

17 Discrete-Time Signals: Time-Domain Representation A left-sided sequence x[n] has zero-valued samples for n > N 2 N 2 n A left-sided sequence If N 2 0, a left-sided sequence is called a anti-causal sequence 17

18 Operations on Sequences A single-input, single-output discrete-time system operates on a sequence, called the input sequence, according some prescribed rules and develops another sequence, called the output sequence, with more desirable properties 18 x[n] Input sequence Discrete-time system y[n] Output sequence

19 Operations on Sequences For example, the input may be a signal corrupted with additive noise Discrete-time system is designed to generate an output by removing the noise component from the input In most cases, the operation defining a particular discrete-time system is composed of some basic operations 19

20 Basic Operations Product (modulation) operation: Modulator x[n] w[n] y[n] y[ n] = x[ n] w[ n] 20 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 Process called windowing

21 Basic Operations Addition operation: Adder x[n] + w[n] Multiplication operation y[n] y [ n] = x[ n] + w[ n] Multiplier A x[n] y[n] y[ n] = A x[ n] 21

22 Basic Operations Time-shifting operation: y[ n] = x[ n N] where N is an integer If N > 0, it is delaying operation Unit delay x[n] 1 z y[n] If N < 0, it is an advance operation y[ n] = x[ n 1] Unit advance x[n] z y[n] y[ n] = x[ n + 1] 22

23 Basic Operations Time-reversal (folding) operation: y[ n] = x[ n] Branching operation: Used to provide multiple copies of a sequence x[n] x[n] x[n] 23

24 Basic Operations Example- Consider the two following sequences of length 5 defined for 0 n 4: { a[ n]} = { } { b[ n]} = { } New sequences generated from the above two sequences by applying the basic operations are as follows: 24

25 25 Basic Operations { c[ n]} = { a[ n] b[ n]} = { } { d[ n]} = { a[ n] + b[ n]} = { } 3 2 { e[ n]} = { a[ n]} = { } As pointed out by the above example, 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

26 Basic Operations 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 Example- Consider the sequence of length 3 defined for 0 n 2: { f [ n]} = { 2 1 3} 26

27 Basic Operations 27 We cannot add the length-3 sequence { f [ n]} to the length-5 sequence {a[n]} defined earlier We therefore first append { f [ n]} with 2 zero-valued samples resulting in a length-5 sequence{ f e [ n]} = { } Then { g[ n]} = { a[ n]} + { f [ n]} = {1 5 3 e 9 0}

28 Combinations of Basic Operations Example - y n] = α1x[ n] + α2x[ n 1] + α3x[ n 2] + α x[ n [ 4 3] 28

29 Sampling Rate Alteration Employed to generate a new sequence y[n] ' with a sampling rate F T higher or lower than that of the sampling rate F T of a given sequence x[n] ' FT Sampling rate alteration ratio is R = F If R > 1, the process called interpolation If R < 1, the process called decimation T 29

30 Sampling Rate Alteration In up-sampling by an integer factor L > 1, L 1equidistant zero-valued samples are inserted by the up-sampler between each two consecutive samples of the input sequence x[n]: x[ n / L], n = 0, ± L, ± 2L, L x u [ n] = 0, otherwise x L x u [n] [n] 30

31 Sampling Rate Alteration An example of the up-sampling operation 1 Input Sequence 1 Output sequence up-sampled by Amplitude 0 Amplitude Time index n Time index n 31

32 Sampling Rate Alteration In down-sampling by an integer factor M > 1, every M-th samples of the input sequence are kept and M 1 in-between samples are removed: y [ n] = x[ nm ] x [n] y[n] M 32

33 Sampling Rate Alteration An example of the down-sampling operation 1 Input Sequence 1 Output sequence down-sampled by Amplitude 0 Amplitude Time index n Time index n

34 Classification of Sequences Based on Symmetry Conjugate-symmetric sequence: x[ n] = x*[ n] If x[n] is real, then it is an even sequence 34 An even sequence

35 Classification of Sequences Based on Symmetry Conjugate-antisymmetric sequence: x[ n] = x*[ n] If x[n] is real, then it is an odd sequence 35 An odd sequence

36 Classification of Sequences Based on Symmetry It follows from the definition that for a conjugate-symmetric sequence {x[n]}, x[0] must be a real number Likewise, it follows from the definition that for a conjugate anti-symmetric sequence {y[n]}, y[0] must be an imaginary number From the above, it also follows that for an odd sequence {w[n]}, w[0] = 0 36

37 37 Classification of Sequences Based on Symmetry Any complex sequence can be expressed as a sum of its conjugate-symmetric part and its conjugate-antisymmetric part: where x[ n] = x [ n] x [ n] x cs x ca [ n] = [ n] = cs + ca 1 x[ n] + x*[ n ] 2 ( ) 1 x[ n] x*[ n ] 2 ( )

38 38 Classification of Sequences Classification of Sequences Based on Symmetry Based on Symmetry Example - Consider the length-7 sequence defined for : Its conjugate sequence is then given by The time-reversed version of the above is },,,,,, { ]} [ { j j j j j n g + + = },,,,,, { ]} * [ { j j j j j n g + + = },,,,,, { ]} * [ { j j j j j n g + + = 3 3 n

39 Classification of Sequences Based on Symmetry Therefore { [ n]} { g[ n] + g *[ n]} g cs = = { 1. 5, j3, j4. 5, 4, 3. 5 j4. 5, 0. 5 j3, 1. 5} Likewise { [ n]} { g[ n] g *[ n]} g ca = 2 1 = { 1. 5, j, 1. 5 j1. 5, j2, 1. 5 j1. 5, 0. 5 j, 1. 5} It can be easily verified that and g * ca [ n] = g ca [ n] g * cs [ n] = g cs [ n]

40 Classification of Sequences Based on Symmetry Any real sequence can be expressed as a sum of its even part and its odd part: x[ n] = xev [ n] + xod[ n] where [ n] = 1 ( x[ n] + x[ n ]) x ev 2 x od [ n] = 1 x[ n] x[ n ] 2 ( ) 40

41 41 Classification of Sequences Based on Symmetry A length-n sequence x[n], 0 n N 1, can be expressed as x[ n] = xpcs [ n] + xpca[ n] where x ( pcs[ n] = 1 2 x[ n] + x *[ n N ]), 0 n N 1, is the periodic conjugate-symmetric part and x [ ] ( pca n = 1 2 x[ n] x *[ n N ]), 0 n N 1, is the periodic conjugate-antisymmetric part

42 Classification of Sequences Based on Symmetry For a real sequence, the periodic conjugatesymmetric part, is a real sequence and is called the periodic even part x pe [n] For a real sequence, the periodic conjugateantisymmetric part, is a real sequence and is called the periodic odd part x po [n] 42

43 Classification of Sequences Based on Symmetry A length-n sequence x[n] is called a periodic conjugate-symmetric sequence if x[ n] = x *[ n N ] = x *[ N n] and is called a periodic conjugateantisymmetric sequence if x[ n] = x *[ n N ] = x *[ N n] 43

44 Classification of Sequences Based on Symmetry A finite-length real periodic conjugatesymmetric sequence is called a symmetric sequence A finite-length real periodic conjugateantisymmetric sequence is called a antisymmetric sequence 44

45 Classification of Sequences Based on Symmetry Example- Consider the length-4 sequence defined for 0 n 3: { u[ n]} = { 1+ j4, 2 + j3, 4 j2, 5 j6} Its conjugate sequence is given by { u *[ n]} = { 1 j4, 2 j3, 4 + j2, 5 + j6} To determine the modulo-4 time-reversed version u *[ n ]} observe the following: { 4 45

46 Classification of Sequences Based on Symmetry u *[ 0 4 ] = u *[ 0] = 1 j4 u *[ 1 4 ] = u *[ 3] = 5 + j6 u *[ 2 4 ] = u *[ 2] = 4 + j2 u *[ 3 4 ] = u *[ 1] = 2 j3 Hence { u *[ n ]} = { 1 j4, 5 + j6, 4 + j2, 2 3} 4 j 46

47 Classification of Sequences Therefore Based on Symmetry 1 2 { u pcs [ n]} = { u[ n] + u *[ n 4]} = { 1, j4. 5, 4, 3. 5 j4. 5} Likewise [ n]} = { u[ n] u *[ n { u 1 pca 2 4 ]} = { j4, 1. 5 j1. 5, 2, 1. 5 j1. 5} 47

48 Classification of Sequences Based on Periodicity A sequence ~ x [ n ] satisfying ~ x [ n] = ~ x [ n + kn ] is called a periodic sequence with a period N where N is a positive integer and k is any integer Smallest value of N satisfying ~ x [ n] = ~ x [ n + kn ] is called the fundamental period 48

49 Classification of Sequences Example - Based on Periodicity A sequence not satisfying the periodicity condition is called an aperiodic sequence 49

50 Classification of Sequences: Energy and Power Signals Total energy of a sequence x[n] is defined by ε = x x[ n] n= 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 50

51 51 Classification of Sequences: Energy and Power Signals The average power of an aperiodic sequence is defined by K 1 K 2K + 1 n= K Px = lim x[ n] Define the energy of a sequence x[n] over a finite interval K n K as ε x K, = x[ n] K n= K 2 2

52 Classification of Sequences: Energy and Power Signals Then 1 P x = lim K x. K K The average power of a periodic sequence ~ x [ n ] with a period N is given by 1 1 N N n= 0 ε P = ~ x [ n] x The average power of an infinite-length sequence may be finite or infinite 2 52

53 53 Classification of Sequences: Energy and Power Signals Example- Consider the causal sequence defined by n x[ n] = 3( 1), n 0 0, n < 0 Note: x[n] has infinite energy Its average power is given by P x = lim K 1 9 2K + 1 K 1 = 0 n = lim K 9( K 2K + 1) + 1 = 4.5

54 Classification of Sequences: Energy and Power Signals An infinite energy signal with finite average power is called a power signal Example - A periodic sequence which has a finite average power but infinite energy A finite energy signal with zero average power is called an energy signal Example - A finite-length sequence which has finite energy but zero average power 54

55 Other Types of Classifications A sequence x[n] is said to be bounded if x [ n] < B x Example- The sequence x[ n] = cos0. 3πn is a bounded sequence as x[ n] = cos0.3πn 1 55

56 Other Types of Classifications 56 A sequence x[n] is said to be absolutely summable if x [ n] < n= Example - The sequence y[ n] = , is an absolutely summable sequence as n = = < n= 0 n, n n < 0 0

57 Other Types of Classifications A sequence x[n] is said to be squaresummable if 2 x[ n] < n= Example- The sequence sin 0.4n h[ n] = π n is square-summable but not absolutely summable 57

58 Basic Sequences Unit sample sequence - 1 δ[ n] = 1, 0, n n = n Unit step sequence - 1 µ [ n] = 1, 0, n n < n

59 Basic Sequences Real sinusoidal sequence - x[ n] = Acos( ωon + φ) where A is the amplitude, ω o is the angular frequency, and φ is the phase of x[n] Example - 2 ω o = Amplitude Time index n

60 60 Basic Sequences Exponential sequence - n x[ n] = Aα, < n < where A and α are real or complex numbers ( σ If we write o + jωo ) jφ α = e, A = A e, then we can express jφ ( σ ) x[ n] A e e o + jω n = o = xre[ n] + j xim[ n], where x x re im σ n [ n] = Ae o cos( ω n + φ), σ n [ n] = Ae o sin( ω n + φ) o o

61 Basic Sequences x re [n] and x im [n] of a complex exponential sequence are real sinusoidal sequences with constant ( σ o = 0), growing ( σ o > 0), and decaying ( < 0) amplitudes for n > 0 σ o Real part Imaginary part Amplitude Amplitude Time index n 61 1 π [ 12 6 x n] = exp( + j ) n Time index n

62 Basic Sequences Real exponential sequence - n x[ n] = Aα, < n < where A and α are real numbers 50 α = α = Amplitude Amplitude Time index n Time index n 62

63 63 Basic Sequences Sinusoidal sequence Acos( ωon + φ) and complex exponential sequence Bexp( jωon) are periodic sequences of period N if ωon = 2πr where N and r are positive integers Smallest value of N satisfying ωon = 2πr is the fundamental period of the sequence To verify the above fact, consider x1[ n] = cos( ωon + φ) x [ n] = cos( ω ( n + N) + ) 2 o φ

64 Basic Sequences 64 Now x2[ n] = cos( ωo( n + N) + φ) = cos( ωon + φ)cosωon sin( ωon + φ) sin ω which will be equal to cos( ωo n + φ) = x 1[ n ] only if sin ω o N = 0 and cos ω o N = 1 These two conditions are met if and only if ωon = 2π r or N ω 2π = o r o N

65 Basic Sequences If 2π/ω o is a noninteger rational number, then the period will be a multiple of 2π/ω o Otherwise, the sequence is aperiodic Example- x[ n] = sin( 3n + φ) is an aperiodic sequence 65

66 Basic Sequences 2 ω 0 = Amplitude Here ω o Time index n = πr Hence period N = = 1 for r = 0 0

67 Basic Sequences 2 ω 0 = 0.1π 1 Amplitude 0-1 Here Time index n ω = 0. 1π o 2πr Hence N = = 20 for r = 1 0.1π 67

68 Basic Sequences Property 1 - Consider x [ n ] = exp( jω 1 n ) and y [ n ] = exp( jω 2 n ) with 0 ω 1 < π and 2πk ω2 < 2π( k + 1) where k is any positive integer If ω = ω + 2π, then x[n] = y[n] 2 1 k Thus, x[n] and y[n] are indistinguishable 68

69 Basic Sequences Property 2 - The frequency of oscillation of Acos( ωon) increases as ωo increases from 0 to π, and then decreases as ω o increases from π to 2π Thus, frequencies in the neighborhood of ω = 0 are called low frequencies, whereas, frequencies in the neighborhood of ω = π are called high frequencies 69

70 Basic Sequences ω o Because of Property 1, a frequency in the neighborhood of ω = 2π k is indistinguishable from a frequency ωo 2πk in the neighborhood of ω = 0 and a frequency ω o in the neighborhood of ω = π( 2k + 1) is indistinguishable from a frequency ωo π( 2k + 1) in the neighborhood of ω = π 70

71 Basic Sequences Frequencies in the neighborhood of ω = 2π k are usually called low frequencies Frequencies in the neighborhood of ω = π (2k+1) are usually called high frequencies v1[ n] = cos(0.1π n) = cos(1.9π n) is a lowfrequency signal v2[ n] = cos(0.8πn) = cos(1.2π n) is a highfrequency signal 71

72 Basic Sequences An arbitrary sequence can be represented in the time-domain as a weighted sum of some basic sequence and its delayed (advanced) versions 72 x[ n] = 0.5δ [ n + 2] + 1.5δ [ n 1] δ[ n 2] + δ[ n 4] δ [ n 6]

73 The Sampling Process Often, a discrete-time sequence x[n] is developed by uniformly sampling a continuous-time signal x a (t) as indicated below The relation between the two signals is x[ n] = x ( t) x ( nt ), n = K, 2, 1,0,1, 2, K = a = t nt a 73

74 The Sampling Process Time variable t of x a (t) is related to the time variable n of x[n] only at discrete-time instants given by t t n n = nt = n F T 2πn Ω with F T =1/T denoting the sampling frequency and ΩT = 2πF T denoting the sampling angular frequency = T 74

75 The Sampling Process Consider the continuous-time signal x( t) = Acos(2πf t + φ) = Acos( Ω t + φ) o The corresponding discrete-time signal is 2πΩ x[ n] = Acos( Ω nt + φ) = Acos( o o n + φ) ΩT = Acos( ω n + φ) o where ω o = 2πΩo / ΩT = Ωo T is the normalized digital angular frequency of x[n] o 75

76 The Sampling Process If the unit of sampling period T is in seconds The unit of normalized digital angular frequency ω o is radians/sample The unit of normalized analog angular frequency Ω o is radians/second The unit of analog frequency f o is hertz (Hz) 76

77 77 The Sampling Process The three continuous-time signals g1( t) = cos(6πt) g2( t) = cos(14πt) g ( t) = cos(26π ) 3 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(0.6πn) g [ n] = cos(1.4π g [ n] = cos(2.6π ) 2 n 3 n )

78 The Sampling Process Plots of these sequences (shown with circles) and their parent time functions are shown below: Amplitude Note that each sequence has exactly the same sample value for any given n time

79 The Sampling Process This fact can also be verified by observing that g2[ n] = cos(1.4π n) = cos n n ((2π 0.6π) ) = cos(0.6π ) g3[ n] = cos(2.6πn) = cos n ((2π + 0.6π) n) = cos(0.6π ) As a result, all three sequences are identical and it is difficult to associate a unique continuous-time function with each of these sequences 79

80 The Sampling Process The above phenomenon of a continuoustime signal of higher frequency acquiring the identity of a sinusoidal sequence of lower frequency after sampling is called aliasing 80

81 The Sampling Process Since there are an infinite number of continuous-time signals that can lead to the same sequence when sampled periodically, additional conditions need to imposed so that the sequence { x[ n]} = { xa( nt )} can uniquely represent the parent continuoustime signal x a (t) In this case, x a (t) can be fully recovered from {x[n]} 81

82 The Sampling Process 82 Example- Determine the discrete-time signal v[n] obtained by uniformly sampling at a sampling rate of 200 Hz the continuoustime signal v a ( t) = 6cos(60πt) + 3sin(300πt) + 2cos(340πt) + 4cos(500πt) + 10sin(660πt) Note: v a (t) is composed of 5 sinusoidal signals of frequencies 30 Hz, 150 Hz, 170 Hz, 250 Hz and 330 Hz

83 The Sampling Process The sampling period is 1 = sec The generated discrete-time signal v[n] is thus given by v[ n] = 6 cos(0.3πn) + 3sin(1.5πn) + 2 cos(1.7πn) = + T = 4 cos(2.5πn) + 10 sin(3.3πn) ((2π 0.5π) n) + 2 cos( (2π 0.3π) π) n) + 10 sin( (4π 0.7π ) 6 cos(0.3πn) + 3sin n + 4 cos( (2π ) n ) 83

84 The Sampling Process = 6 cos(0.3πn) 3sin(0.5πn) + 2 cos(0.3πn) + 4 cos(0.5πn) 10 sin(0.7πn) = 8 cos(0.3πn) + 5 cos(0.5πn ) 10 sin(0.7πn) Note: v[n] is composed of 3 discrete-time sinusoidal signals of normalized angular frequencies: 0.3π, 0.5π, and 0.7π 84

85 The Sampling Process Note: An identical discrete-time signal is also generated by uniformly sampling at a 200-Hz sampling rate the following continuous-time signals: wa ( t) = 8 cos(60πt) + 5cos(100πt ) 10 sin(140πt) ga ( t) = 2 cos(60πt) + 4 cos(100πt) + 10 sin(260πt) + 6 cos(460πt) + 3sin(700πt) 85

86 The Sampling Process Recall ω o = 2πΩ Ω T o 86 Thus if ΩT > 2Ω o, then the corresponding normalized digital angular frequency ω o of the discrete-time signal obtained by sampling the parent continuous-time sinusoidal signal will be in the range π < ω < π No aliasing

87 The Sampling Process On the other hand, if ΩT < 2Ω o, the normalized digital angular frequency will foldover into a lower digital frequency ωo = 2πΩo / ΩT 2π in the range π < ω < π because of aliasing Hence, to prevent aliasing, the sampling frequency Ω T should be greater than 2 times the frequency Ω o of the sinusoidal signal being sampled 87

88 The Sampling Process 88 Generalization: Consider an arbitrary continuous-time signal x a (t) composed of a weighted sum of a number of sinusoidal signals x a (t) can be represented uniquely by its sampled version {x[n]} if the sampling frequency Ω T is chosen to be greater than 2 times the highest frequency contained in x a (t)

89 The Sampling Process The condition to be satisfied by the sampling frequency to prevent aliasing is called the sampling theorem A formal proof of this theorem will be presented later 89

Discrete-Time Signals: Time-Domain Representation

Discrete-Time Signals: Time-Domain Representation 1 Signals represented as sequences of numbers, called samples Sample value of a typical signal or sequence denoted as x[n] with n being an integer in the