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Discrete-Time Signals: Time-Domain Representation

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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 3 Copyright © 2005, S. K. Mitra 4 Copyright © 2005, S. K. Mitra 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 11 Copyright © 2005, S. K. Mitra 12 Copyright © 2005, S. K. Mitra Discrete-Time Signals: Discrete-Time Signals: Time-Domain Representation Time-Domain Representation 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 13 Copyright © 2005, S. K. Mitra 14 Copyright © 2005, S. K. Mitra 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 • If N 1 ≥ 0 , a right-sided sequence is called a causal sequence 15 Copyright © 2005, S. K. Mitra 16 Copyright © 2005, S. K. Mitra 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 17 Copyright © 2005, S. K. Mitra 18 Copyright © 2005, S. K. Mitra Discrete-Time Signals: Discrete-Time Signals: Time-Domain Representation Time-Domain Representation • 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 19 Copyright © 2005, S. K. Mitra 20 Copyright © 2005, S. K. Mitra 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 25 Copyright © 2005, S. K. Mitra 26 Copyright © 2005, S. K. Mitra 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] 27 Copyright © 2005, S. K. Mitra 28 Copyright © 2005, S. K. Mitra 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} 31 Copyright © 2005, S.
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