Discrete-Time Signals: Time-Domain Representation

Discrete-Time Signals: Discrete-Time Signals: Time-Domain Representation Time-Domain Representation • Signals represented as sequences of • Discrete-time signal may also be written as numbers, called samples a sequence of numbers inside braces: • Sample value of a typical signal or sequence {x[n]} = {K,− 0.2,2.2,1.1,0.2,− 3.7,2.9,K} denoted as x[n] with n being an integer in ↑ the range − ∞ ≤ n ≤ ∞ • In the above, x[−1] = −0.2, x[0] = 2.2, x[1] =1.1, • x[n] defined only for integer values of n and etc. undefined for noninteger values of n • The arrow is placed under the sample at • Discrete-time signal represented by {x[n]} time index n = 0 1 Copyright © 2005, S. K. Mitra 2 Copyright © 2005, S. K. Mitra

Discrete-Time Signals: Discrete-Time Signals: Time-Domain Representation Time-Domain Representation • In some applications, a discrete-time • Graphical representation of a discrete-time sequence {x[n]} may be generated by signal with real-valued samples is as shown periodically sampling a continuous-time below: signal x a ( t ) at uniform intervals of time

Discrete-Time Signals: Discrete-Time Signals: Time-Domain Representation Time-Domain Representation • Here, n-th sample is given by • Unit of sampling frequency is cycles per second, or hertz (Hz), if T is in seconds x[n] = xa (t) t=nT = xa (nT ), n = K,− 2,−1,0,1,K • The spacing T between two consecutive • Whether or not the sequence {x[n]} has samples is called the sampling interval or been obtained by sampling, the quantity sampling period x[n] is called the n-th sample of the • Reciprocal of sampling interval T, denoted sequence •{x[n]} is a real sequence, if the n-th sample as F T , is called the sampling frequency: 1 x[n] is real for all values of n F = T T •Otherwise, {x[n]} is a complex sequence 5 Copyright © 2005, S. K. Mitra 6 Copyright © 2005, S. K. Mitra Discrete-Time Signals: Discrete-Time Signals: Time-Domain Representation Time-Domain Representation • A complex sequence {x[n]} can be written •Example- {x[n]} = {cos0.25n} is a real as { x [ n ]} = { x re [ n ]} + j { x im [ n ]} where sequence j0.3n xre[n] and x im [ n ] are the real and imaginary • {y[n]} = {e } is a complex sequence parts of x[n] • We can write • The complex conjugate sequence of {x[n]} {y[n]}= {cos0.3n + jsin0.3n} {x*[n]} = {x [n]}− j{x [n]} is given by re im = {cos0.3n}+ j{sin0.3n} • Often the braces are ignored to denote a where {y [n]}= {cos0.3n} sequence if there is no ambiguity re {yim[n]}= {sin0.3n} 7 Copyright © 2005, S. K. Mitra 8 Copyright © 2005, S. K. Mitra

Discrete-Time Signals: Discrete-Time Signals: Time-Domain Representation Time-Domain Representation • Two types of discrete-time signals: •Example- - Sampled-data signals in which samples − j0.3n {w[n]} = {cos0.3n}− j{sin0.3n} = {e } are continuous-valued is the complex conjugate sequence of {y[n]} - Digital signals in which samples are • That is, discrete-valued {w[n]} = {y *[n]} • Signals in a practical digital signal processing system are digital signals obtained by quantizing the sample values either by rounding or truncation 9 Copyright © 2005, S. K. Mitra 10 Copyright © 2005, S. K. Mitra

Discrete-Time Signals: Discrete-Time Signals: Time-Domain Representation Time-Domain Representation •Example- • A discrete-time signal may be a finite- length or an infinite-length sequence • Finite-length (also called finite-duration or Amplitude Amplitude finite-extent) sequence is defined only for a time, t time, t finite time interval: N1 ≤ n ≤ N2 where − ∞ < N 1 and N 2 < ∞ with N1 ≤ N2 Boxedcar signal Digital signal • Length or duration of the above finite- length sequence is N = N2 − N1 +1

2 • A length-N sequence is often referred to as •Example- x[n] = n , − 3 ≤ n ≤ 4 is a finite- an N-point sequence length sequence of length 4 − (−3) +1= 8

• The length of a finite-length sequence can y[n] = cos0.4n is an infinite-length sequence be increased by zero-padding, i.e., by appending it with zeros

Discrete-Time Signals: Discrete-Time Signals: Time-Domain Representation Time-Domain Representation •Example- •A right-sided sequence x[n] has zero- 2 ⎧n , − 3 ≤ n ≤ 4 valued samples for n < N1 xe[n] = ⎨ ⎩ 0, 5 ≤ n ≤ 8

is a finite-length sequence of length 12 n N1 obtained by zero-padding x[n] = n2, − 3 ≤ n ≤ 4 with 4 zero-valued samples A right-sided sequence

Discrete-Time Signals: Discrete-Time Signals: Time-Domain Representation Time-Domain Representation •A left-sided sequence x[n] has zero-valued • Size of a Signal samples for n > N 2 Given by the norm of the signal

Lp-norm

N2 1/ p n ⎛ ∞ p ⎞ x p = ⎜ ∑ x[n] ⎟ A left-sided sequence ⎝ n=−∞ ⎠

• If N 2 ≤ 0 , a left-sided sequence is called a where p is a positive integer anti-causal sequence

• The value of p is typically 1 or 2 or ∞ L1-norm x 1 is the mean absolute value of {x[n]}

L2 -norm L -norm x x ∞ ∞ 2 is the peak absolute value of {x[n]}, i.e. is the root-mean-squared (rms) value of {x[n]} x ∞ = x max

Discrete-Time Signals: Time-Domain Representation Operations on Sequences Example • A single-input, single-output discrete-time •Let{y[n]},0 ≤ n ≤ N −1 , be an approximation of system operates on a sequence, called the {x[n]}, 0 ≤ n ≤ N −1 input sequence, according some prescribed • An estimate of the relative error is given by the rules and develops another sequence, called ratio of the L 2-norm of the difference signal and the L2 -norm of {x[n]}: the output sequence, with more desirable 1/ p ⎛ N −1 2 ⎞ properties ⎜ ∑ y[n]− x[n] ⎟ ⎜ n=0 ⎟ Discrete-time Erel = x[n] y[n] ⎜ N −1 2 ⎟ system ⎜ ∑ x[n] ⎟ Input sequence Output sequence ⎝ n=0 ⎠ 21 Copyright © 2005, S. K. Mitra 22 Copyright © 2005, S. K. Mitra

Basic Operations Operations on Sequences • Product (modulation) operation:

• For example, the input may be a signal x[n] × y[n] corrupted with additive noise – Modulator y[n] = x[n]⋅ w[n] • Discrete-time system is designed to w[n] generate an output by removing the noise • An application is in forming a finite-length component from the input sequence from an infinite-length sequence • In most cases, the operation defining a by multiplying the latter with a finite-length particular discrete-time system is composed sequence called an window sequence of some basic operations • Process called windowing

23 Copyright © 2005, S. K. Mitra 24 Copyright © 2005, S. K. Mitra Basic Operations Basic Operations Basic Operations • Time-shifting operation: y[n] = x[n − N] • Addition operation: where N is an integer x[n] + y[n] • If N > 0, it is delaying operation – Adder y[n] = x[n]+ w[n] w[n] – Unit delay x[n] z−1 y[n] y[n] = x[n −1] • Multiplication operation • If N < 0, it is an advance operation A – Multiplier x[n] y[n] y[n] = A⋅ x[n] x[n] z y[n] y[n] = x[n +1] – Unit advance

Basic Operations Basic Operations

• Time-reversal (folding) operation: •Example- Consider the two following y[n] = x[−n] sequences of length 5 defined for 0 ≤ n ≤ 4 : {a[n]}= {3 4 6 − 9 0} • Branching operation: Used to provide {b[n]}= {2 −1 4 5 − 3} multiple copies of a sequence • New sequences generated from the above

x[n] x[n] two sequences by applying the basic operations are as follows: x[n]

Basic Operations Basic Operations {c[n]}= {a[n]⋅b[n]}= {6 − 4 24 − 45 0} {d[n]}= {a[n]+ b[n]}= {5 3 10 − 4 − 3} • However if the sequences are not of same length, in some situations, this problem can {e[n]}= 3{a[n]} = {4.5 6 9 −13.5 0} 2 be circumvented by appending zero-valued samples to the sequence(s) of smaller • As pointed out by the above example, lengths to make all sequences have the same operations on two or more sequences can be range of the time index carried out if all sequences involved are of same length and defined for the same range •Example- Consider the sequence of length {f [n]}= {− 2 1 − 3} of the time index n 3 defined for 0 ≤ n ≤ 2 :

29 Copyright © 2005, S. K. Mitra 30 Copyright © 2005, S. K. Mitra Basic Operations Basic Operations Ensemble Averaging • We cannot add the length-3 sequence { f [ n ]} • A very simple application of the addition to the length-5 sequence {a[n]} defined operation in improving the quality of earlier measured data corrupted by an additive • We therefore first append { f [ n ]} with 2 random noise zero-valued samples resulting in a length-5 • In some cases, actual uncorrupted data sequence{f [n]} = {− 2 1 − 3 0 0} vector s remains essentially the same from e one measurement to next •Then

{g[n]} ={a[n]}+{ fe[n]} ={1 5 3 − 9 0}

Basic Operations Basic Operations

• While the additive noise vector is random • The average data vector, called the and not reproducible ensemble average, obtained after K measurements is given by • Let d i denote the noise vector corrupting the i-th measurement of the uncorrupted K K K x = 1 ∑x = 1 ∑(s + d ) = s + 1 ∑d ave K i K i K i data vector s: i=1 i=1 i=1 x = s + d i i • For large values of K, x ave is usually a reasonable replica of the desired data vector s

Basic Operations Basic Operations • Example

Original uncorrupted data Noise 8 0.5 • We cannot add the length-3 sequence { f [ n ]} 6 e d to the length-5 sequence {a[n]} defined tu

4 li 0 mp Amplitude A 2 earlier

0 -0.5 0 10 20 30 40 50 0 10 20 30 40 50 • We therefore first append { f [ n ]} with 2 Time index n Time index n

Noise corrupted data Ensemble average 8 8 zero-valued samples resulting in a length-5 6 6 sequence{fe[n]} = {− 2 1 − 3 0 0} 4 4

2 2

Amplitude Amplitude •Then 0 0

-2 -2 {g[n]} ={a[n]}+{ f [n]} ={1 5 3 − 9 0} 0 10 20 30 40 50 0 10 20 30 40 50 e Time index n Time index n 35 Copyright © 2005, S. K. Mitra 36 Copyright © 2005, S. K. Mitra Combinations of Basic Sampling Rate Alteration Operations • Employed to generate a new sequence y[n] ' with a sampling rate F T higher or lower •Example- than that of the sampling rate F T of a given sequence x[n] F ' • Sampling rate alteration ratio is R = T FT • If R > 1, the process called interpolation

y[n] = α1x[n]+α2x[n −1]+α3x[n − 2]+α4x[n − 3] • If R < 1, the process called decimation

Sampling Rate Alteration Sampling Rate Alteration •In up-sampling by an integer factor L > 1, L −1equidistant zero-valued samples are • An example of the up-sampling operation

inserted by the up-sampler between each Input Sequence Output sequence up-sampled by 3 1 1 two consecutive samples of the input sequence x[n]: 0.5 0.5 0 0 Amplitude ⎧x[n / L], n = 0,± L,± 2L,L Amplitude xu[n] = ⎨ -0.5 -0.5 ⎩ 0, otherwise -1 -1 0 10 20 30 40 50 0 10 20 30 40 50 Time index n Time index n x[n] L xu[n]

Sampling Rate Alteration Sampling Rate Alteration

•In down-sampling by an integer factor • An example of the down-sampling M > 1, every M-th samples of the input operation

Input Sequence Output sequence down-sampled by 3 sequence are kept and M − 1 in-between 1 1 samples are removed: 0.5 0.5 y[n] = x[nM ] 0 0 Amplitude Amplitude

-0.5 -0.5 x[n] M y[n] -1 -1 0 10 20 30 40 50 0 10 20 30 40 50 Time index n Time index n

41 Copyright © 2005, S. K. Mitra 42 Copyright © 2005, S. K. Mitra Classification of Sequences Classification of Sequences Based on Symmetry Based on Symmetry • Conjugate-symmetric sequence: • Conjugate-antisymmetric sequence: x[n] = x*[−n] x[n] = −x*[−n] If x[n] is real, then it is an even sequence If x[n] is real, then it is an odd sequence

An even sequence An odd sequence

Classification of Sequences Classification of Sequences Based on Symmetry Based on Symmetry • It follows from the definition that for a • Any complex sequence can be expressed as conjugate-symmetric sequence {x[n]}, x[0] a sum of its conjugate-symmetric part and must be a real number its conjugate-antisymmetric part: • Likewise, it follows from the definition that x[n] = xcs[n]+ xca[n] for a conjugate anti-symmetric sequence where {y[n]}, y[0] must be an imaginary number 1 xcs[n] = ()x[n]+ x*[−n] • From the above, it also follows that for an 2 x [n] = 1 x[n]− x*[−n] odd sequence {w[n]}, w[0] = 0 ca 2 ()

Classification of Sequences Classification of Sequences Based on Symmetry Based on Symmetry •Example- Consider the length-7 sequence • Therefore {g [n]} = 1{g[n]+ g *[−n]} cs 2 defined for − 3 ≤ n ≤ 3 : = {1.5, 0.5+ j3, −3.5+ j4.5, 4, −3.5− j4.5, 0.5− j3, 1.5} {g[n]} = {0, 1+ j4, −2+ j3, 4− j2, −5− j6, − j2, 3} ↑ ↑ • Its conjugate sequence is then given by • Likewise {g [n]}= 1{g[n]− g *[−n]} ca 2 {g * [n]} = {0, 1− j4, −2− j3, 4+ j2, −5+ j6, j2, 3} ↑ = {−1.5, 0.5+ j, 1.5− j1.5, − j2, −1.5− j1.5, −0.5− j, 1.5} • The time-reversed version of the above is ↑ * {g * [−n]} = {3, j2, −5+ j6, 4+ j2, −2− j3, 1− j4, 0} • It can be easily verified that g cs [ n ] = g cs [ − n] * ↑ and gca[n] = −gca[−n] 47 Copyright © 2005, S. K. Mitra 48 Copyright © 2005, S. K. Mitra Classification of Sequences Classification of Sequences Based on Symmetry Based on Symmetry • A length-N sequence x[n], 0 ≤ n ≤ N −1, • Any real sequence can be expressed as a can be expressed as x[n] = x [n]+ x [n] sum of its even part and its odd part: pcs pca where x[n] = xev[n]+ xod [n] 1 xpcs[n] = (x[n]+ x *[〈−n〉 N ]), 0 ≤ n ≤ N −1, where 2 is the periodic conjugate-symmetric part x [n] = 1 x[n]+ x[−n] ev 2 () and 1 1 xpca[n] = (x[n]− x *[〈−n〉 N ]), 0 ≤ n ≤ N −1, xod [n] = ()x[n]− x[−n] 2 2 is the periodic conjugate-antisymmetric part 49 Copyright © 2005, S. K. Mitra 50 Copyright © 2005, S. K. Mitra

Classification of Sequences Classification of Sequences Based on Symmetry Based on Symmetry • For a real sequence, the periodic conjugate- • A length-N sequence x[n] is called a symmetric part, is a real sequence and is periodic conjugate-symmetric sequence if called the periodic even part x [n] pe x[n] = x *[〈−n〉 N ] = x *[N − n] • For a real sequence, the periodic conjugate- and is called a periodic conjugate- antisymmetric part, is a real sequence and is antisymmetric sequence if called the periodic odd part xpo[n] x[n] = −x *[〈−n〉 N ] = −x *[N − n]

Classification of Sequences Classification of Sequences Based on Symmetry Based on Symmetry • A finite-length real periodic conjugate- •Example- Consider the length-4 sequence symmetric sequence is called a symmetric defined for 0 ≤ n ≤ 3 : sequence {u[n]} = {1+ j4, − 2 + j3, 4 − j2, − 5 − j6} • A finite-length real periodic conjugate- • Its conjugate sequence is given by antisymmetric sequence is called a {u *[n]}= {1− j4, − 2 − j3, 4 + j2, − 5 + j6} antisymmetric sequence • To determine the modulo-4 time-reversed version { u * [ 〈 − n 〉 4 ]} observe the following:

u *[〈−0〉4 ] = u *[0] =1− j4 • Therefore 1 u *[〈−1〉4 ] = u *[3] = −5 + j6 {u [n]} = {u[n]+ u *[〈−n〉 ]} pcs 2 4 u *[〈−2〉 ] = u *[2] = 4 + j2 4 = {1, − 3.5 + j4.5, 4, − 3.5 − j4.5} u *[〈−3〉4] = u *[1] = −2 − j3 • Hence • Likewise {u *[〈−n〉 ]} = {1− j4, − 5 + j6, 4 + j2, − 2 − j3} {u [n]}= 1{u[n]− u *[〈−n〉 ]} 4 pca 2 4 = {j4, 1.5 − j1.5, − 2, −1.5 − j1.5}

Classification of Sequences Classification of Sequences Based on Periodicity Based on Periodicity • A sequence ~x [ n ] satisfying x~[n] = ~x[n + kN] •Example- 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 • A sequence not satisfying the periodicity condition is called an aperiodic sequence

Classification of Sequences: Classification of Sequences: Energy and Power Signals Energy and Power Signals • Total energy of a sequence x[n] is defined by •The average power of an aperiodic ∞ 2 sequence is defined by K εx = ∑ x[n] 1 2 n=−∞ Px = lim 2K +1 ∑ x[n] K →∞ n=−K • An infinite length sequence with finite sample values may or may not have finite energy • Define the energy of a sequence x[n] over a • A finite length sequence with finite sample finite interval − K ≤ n ≤ K as values has finite energy K 2 ε x,K = ∑ x[n] n=−K

59 Copyright © 2005, S. K. Mitra 60 Copyright © 2005, S. K. Mitra Classification of Sequences: Classification of Sequences: Energy and Power Signals Energy and Power Signals •Then Energy and Power Signals 1 Px = lim ε x.K •Example- Consider the causal sequence K→∞ 2K +1 defined by •The average power of a periodic sequence ~ ⎧3(−1)n, n ≥ 0 x[n] with a period N is given by x[n] = ⎨ 0, n < 0 N −1 ⎩ P = 1 x~[n]2 x N ∑ •Note: x[n] has infinite energy n=0 • Its average power is given by • The average power of an infinite-length sequence may be finite or infinite 1 ⎛ K ⎞ 9(K +1) Px = lim ⎜9 ∑1⎟ = lim = 4.5 K→∞ 2K +1⎝ n=0 ⎠ K →∞ 2K +1 61 Copyright © 2005, S. K. Mitra 62 Copyright © 2005, S. K. Mitra

Classification of Sequences: Other Types of Classifications Energy and Power Signals • A sequence x[n] is said to be bounded if • An infinite energy signal with finite average x[n] ≤ B < ∞ power is called a power signal x Example - A periodic sequence which has a finite average power but infinite energy •Example- The sequence x [ n ] = cos 0 . 3 π n is a bounded sequence as • A finite energy signal with zero average power is called an energy signal x[n] = cos0.3πn ≤1 Example - A finite-length sequence which has finite energy but zero average power

Other Types of Classifications Other Types of Classifications • A sequence x[n] is said to be absolutely • A sequence x[n] is said to be square- summable if ∞ ∑ x[n] < ∞ summable if ∞ 2 n=−∞ ∑ x[n] < ∞ •Example- The sequence n=−∞ •Example- The sequence ⎧0.3n, n ≥ 0 y[n] = ⎨ sin 0.4n ⎩ 0, n < 0 h[n] = π n is an absolutely summable sequence as is square-summable but not absolutely ∞ 1 summable ∑ 0.3n = =1.42857 < ∞ n=0 1− 0.3 65 Copyright © 2005, S. K. Mitra 66 Copyright © 2005, S. K. Mitra Basic Sequences Basic Sequences ⎧1, n = 0 • Real sinusoidal sequence - • Unit sample sequence - δ[n] = ⎨ x[n] = Acos(ω n + φ) ⎩0, n ≠ 0 o 1 where A is the amplitude, ω o is the angular frequency, and φ is the phase of x[n] n –4 –3 –2 –1 0 123456 Example - ω = 0.1 o • Unit step sequence - ⎧1, n ≥ 0 2 µ[n] = ⎨ 1 ⎩0, n < 0 0 1 Amplitude -1

Basic Sequences Basic Sequences

• Exponential sequence - • xre[n] andxim[n] of a complex exponential x[n] = Aαn, − ∞ < n < ∞ sequence are real sinusoidal sequences with where A and α are real or complex numbers constant ( σ o = 0 ) , growing () σ o > 0 , and decaying ( σ < 0 ) amplitudes for n > 0 • If we write α = e(σo + jωo ), A = A e jφ, o Real part Imaginary part then we can express 1 1 0.5 0.5 jφ (σo + jωo )n x[n] = A e e = xre[n]+ j xim[n], 0 0 Amplitude Amplitude where -0.5 -0.5 σ n o -1 -1 xre[n] = Ae cos(ωon + φ), 0 10 20 30 40 0 10 20 30 40 Time index n Time index n σon 1 π xim[n] = Ae sin(ωon + φ) x[n] = exp(− + j )n 69 Copyright © 2005, S. K. Mitra 70 12 6 Copyright © 2005, S. K. Mitra

Basic Sequences Basic Sequences

• Real exponential sequence - • Sinusoidal sequence A cos( ω o n + φ ) and n x[n] = Aα , − ∞ < n < ∞ complex exponential sequence B exp( j ωon) where A and α are real numbers are periodic sequences of period N if ωoN = 2πr where N and r are positive integers α = 1.2 α = 0.9 50 20

40 • Smallest value of N satisfying ω N = 2πr 15 o 30 is the fundamental period of the sequence 10 20 Amplitude Amplitude • To verify the above fact, consider 10 5

0 0 x [n] = cos(ω n + φ) 0 5 10 15 20 25 30 0 5 10 15 20 25 30 1 o Time index n Time index n x2[n] = cos(ωo (n + N) + φ)

•Now x2[n] = cos(ωo (n + N) + φ)

= cos(ωon + φ)cosωoN − sin(ωon + φ)sin ωoN • If 2 π / ω o is a noninteger rational number, then which will be equal to cos( ω o n + φ ) = x 1 [ n ] the period will be a multiple of 2π/ωo only if • Otherwise, the sequence is aperiodic

sin ωoN = 0 and cosωoN =1 •Example- x[n] = sin( 3n + φ) is an aperiodic • These two conditions are met if and only if sequence 2π N ωoN = 2π r or = ωo r

Basic Sequences Basic Sequences ω = 0.1π 0 ω = 0 2 0 2 1 1.5 0

1 Amplitude -1 Amplitude 0.5 -2 0 10 20 30 40 0 0 10 20 30 40 Time index n Time index n •Hereωo = 0.1π •Hereωo = 0 2πr • Hence for r = 1 2πr N = = 20 • Hence periodN = =1 for r = 0 0.1π 0 75 Copyright © 2005, S. K. Mitra 76 Copyright © 2005, S. K. Mitra

Basic Sequences Basic Sequences

• Property 1 - Consider x [ n ] = exp( j ω 1 n ) and • Property 2 - The frequency of oscillation of Acos(ω n) increases as ω increases from 0 y[n] = exp( jω2n) with 0 ≤ ω 1 < π and o o to π, and then decreases as ω o increases from 2πk ≤ ω2 < 2π(k +1) where k is any positive integer π to 2π • Thus, frequencies in the neighborhood of • If ω 2 = ω 1 + 2 π k , then x[n] = y[n] ω = 0 are called low frequencies, whereas, frequencies in the neighborhood of ω = π are • Thus, x[n] and y[n] are indistinguishable called high frequencies

77 Copyright © 2005, S. K. Mitra 78 Copyright © 2005, S. K. Mitra Basic Sequences Basic Sequences • Frequencies in the neighborhood of ω = 2π k • Because of Property 1, a frequency ω o in are usually called low frequencies the neighborhood of ω = 2π k is • Frequencies in the neighborhood of indistinguishable from a frequency ω o − 2 π k ω = π (2k+1) are usually called high in the neighborhood of ω = 0 frequencies

and a frequency ω o in the neighborhood of • v1[n] = cos(0.1πn) = cos(1.9πn) is a low- ω = π(2k +1) is indistinguishable from a frequency signal frequency ω − π ( 2 k + 1 ) in the o • v2[n] = cos(0.8πn) = cos(1.2πn) is a high- neighborhood of ω = π frequency signal

Basic Sequences The Sampling Process • An arbitrary sequence can be represented in • Often, a discrete-time sequence x[n] is the time-domain as a weighted sum of some developed by uniformly sampling a basic sequence and its delayed (advanced) continuous-time signal x a ( t ) as indicated versions below

• The relation between the two signals is x[n] = 0.5δ[n + 2]+1.5δ[n −1]−δ[n − 2] x[n] = x (t) = x (nT ), n = ,− 2,−1,0,1,2, +δ[n − 4]+ 0.75δ[n − 6] a t=nT a K K

The Sampling Process The Sampling Process • Consider the continuous-time signal • Time variable t of x a ( t ) is related to the time variable n of x[n] only at discrete-time x(t) = Acos(2πfot + φ) = Acos(Ωot + φ)

instants t n given by • The corresponding discrete-time signal is 2πΩo n 2πn x[n] = Acos(ΩonT + φ) = Acos( n + φ) tn = nT = = Ω FT ΩT T = Acos(ωon + φ) with F T = 1 / T denoting the sampling frequency and where ωo = 2πΩo / ΩT = ΩoT ΩT = 2πFT denoting the sampling angular is the normalized digital angular frequency frequency of x[n]

83 Copyright © 2005, S. K. Mitra 84 Copyright © 2005, S. K. Mitra The Sampling Process The Sampling Process • The three continuous-time signals g (t) = cos(6πt) • If the unit of sampling period T is in 1 seconds g2(t) = cos(14πt) g (t) = cos(26πt) • The unit of normalized digital angular 3 frequency ω o is radians/sample of frequencies 3 Hz, 7 Hz, and 13 Hz, are • The unit of normalized analog angular sampled at a sampling rate of 10 Hz, i.e. with T = 0.1 sec. generating the three frequency Ω o is radians/second • The unit of analog frequency f is hertz sequences o g [n] = cos(0.6πn) g [n] = cos(1.4πn) (Hz) 1 2 g3[n] = cos(2.6πn) 85 Copyright © 2005, S. K. Mitra 86 Copyright © 2005, S. K. Mitra

The Sampling Process • Plots of these sequences (shown with circles) The Sampling Process and their parent time functions are shown • This fact can also be verified by observing that below: 1 g2[n] = cos(1.4πn) = cos()(2π − 0.6π)n = cos(0.6πn) 0.5

0 g3[n] = cos(2.6πn) = cos()(2π + 0.6π)n = cos(0.6πn) Amplitude -0.5 • As a result, all three sequences are identical

-1 and it is difficult to associate a unique 0 0.2 0.4 0.6 0.8 1 time continuous-time function with each of these • Note that each sequence has exactly the same sequences sample value for any given n 87 Copyright © 2005, S. K. Mitra 88 Copyright © 2005, S. K. Mitra

The Sampling Process The Sampling Process • Since there are an infinite number of • The above phenomenon of a continuous- continuous-time signals that can lead to the time signal of higher frequency acquiring same sequence when sampled periodically, the identity of a sinusoidal sequence of additional conditions need to imposed so lower frequency after sampling is called that the sequence { x [ n ]} = { x a ( n T )} can aliasing uniquely represent the parent continuous- time signal xa (t)

• In this case, x a ( t ) can be fully recovered from {x[n]}

89 Copyright © 2005, S. K. Mitra 90 Copyright © 2005, S. K. Mitra The Sampling Process The Sampling Process •Example- Determine the discrete-time signal v[n] obtained by uniformly sampling • The sampling period is T = 1 = 0.005 sec 200 at a sampling rate of 200 Hz the continuous- • The generated discrete-time signal v[n] is time signal thus given by va (t) = 6cos(60πt) + 3sin(300πt) + 2cos(340πt) v[n] = 6 cos(0.3πn) + 3sin(1.5πn) + 2 cos(1.7πn) + 4cos(500πt) +10sin(660πt) + 4 cos(2.5πn) + 10sin(3.3πn) = 6 cos(0.3πn) + 3sin((2π − 0.5π)n) + 2 cos((2π − 0.3π)n) •Note: v a ( t ) is composed of 5 sinusoidal signals of frequencies 30 Hz, 150 Hz, 170 + 4 cos((2π + 0.5π)n) + 10sin((4π − 0.7π)n) Hz, 250 Hz and 330 Hz

The Sampling Process The Sampling Process

= 6 cos(0.3πn) − 3sin(0.5πn) + 2 cos(0.3πn) + 4 cos(0.5πn) •Note: An identical discrete-time signal is − 10 sin(0.7πn) also generated by uniformly sampling at a = 8cos(0.3πn) + 5cos(0.5πn + 0.6435) − 10sin(0.7πn) 200-Hz sampling rate the following continuous-time signals: •Note:v[n] is composed of 3 discrete-time sinusoidal signals of normalized angular wa (t) = 8cos(60πt) + 5cos(100πt + 0.6435) − 10sin(140πt) frequencies: 0.3π, 0.5π,and0.7π ga (t) = 2 cos(60πt) + 4 cos(100πt) + 10sin(260πt) + 6 cos(460πt) + 3sin(700πt)

The Sampling Process The Sampling Process 2πΩ • On the other hand, if Ω T < 2 Ω o , the • Recall ω = o normalized digital angular frequency will o Ω T foldover into a lower digital frequency • Thus if Ω > 2 Ω , then the corresponding ωo = 〈2πΩo / ΩT 〉2π in the range − π < ω < π T o because of aliasing normalized digital angular frequency ω o of the discrete-time signal obtained by • Hence, to prevent aliasing, the sampling sampling the parent continuous-time frequency Ω T should be greater than 2 sinusoidal signal will be in the range − π < ω < π times the frequency Ω o of the sinusoidal • No aliasing signal being sampled

• Generalization: Consider an arbitrary • The condition to be satisfied by the continuous-time signal x a ( t ) composed of a sampling frequency to prevent aliasing is weighted sum of a number of sinusoidal called the sampling theorem signals • A formal proof of this theorem will be • xa (t) can be represented uniquely by its presented later sampled version {x[n]} if the sampling frequency Ω T is chosen to be greater than 2 times the highest frequency contained in xa (t) 97 Copyright © 2005, S. K. Mitra 98 Copyright © 2005, S. K. Mitra