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DISCRETE.TIME SIGNALS AND SYSTEM1. Discrete-timesignalsA
discretetime signal is represented as a sequence of
numbers:y=1x[n], -6< n
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is defined as l'O n*Odlnl= I -Ll n=0This sequence is often
referred to as a discrete-time impulse or just impulse. lt plays
the same rolefor discrete-time signals as the Dirac delta function
or 6 function does for continuous-time signals.However, there are
no mathematical complications in its definition.An important aspect
of the impulse sequen@ is that an arbitrary sequence can be
represented as asum of scaled, delayed impulses. For example, the
sequence
can be represented asxlnf=a-o|ln+41+ a-r6ln+31+
a-r6fn+21+a-r|ln+ll+ar6fnl+ar6ln-lf+ar6ln-21+
ar6ln-3]+ ao6ln-4).In general, any sequence can be expressed
as
xfnl= ixfklilfn- kl.t=
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a
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{nl= Aa'll A and dare rcal numbers the s6quenc6 is real. lf
0
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. ln the continuous-time case, sinusoidal and complex
exponential sequences are alwaysperiodic. Discrete-time sequences
are periodic (with period N) ifxlnf = -r[-r + //] for all NThus the
discrete-time sinusoid is only periodic ifAcos(oton+ 0) =
Acos(roon+ aoN + Q),which requires that osN = 2rk tor I an
integer.
The same condition is required for the complex exponential
sequence Cet^o' to beperiodic.The two factors just described can be
combined to reach the conclusion that there are onlyN
distinguishable frequencies for which the corresponding sequences
are periodic with period N .One such set is or=4,
t=0,1,...,.ly'-1.Additionally, for discrete-time sequences the
interpretation of high and low frequencies has to bemodified: the
discrete-time sinusoidal sequene xlnl= Acos(oon+0) oscillates more
rapidly asato increases from 0to tt, but the oscillations become
slower as it increases further from a to2r.
l
1A
-tto-tIG
_:tG
-tIo-!1o
-l
3ta
_t I I I.iI i i?' ''l I-
I
-g -tThe sequence corresponding to @o=0 is indistinguishable
from that with aro:(tr*2td
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2. DISCRETE.TIME SYSTEMSA discrete{ime system is defined as a
transformation or mapping operator that maps an input signalx[n]to
an output signal y[n]. This can be denoted asv Jrr I - 7'l.t [rr
l].
l.lrarrrple : Idcal drlalt'lnl-.r[n - n.1l:
-3 t,, t.ltr3\\\\\'\r'[u]=rln-ll
This operation shifts input sequence later by na
samples.Exantple: inoving average t !b,finl= ) x[n -f]M, + M, +7
rt'---u,For M, = I and = i, the input sequence
ol.:3{L+ fl:I
n-2 tr-[ t, o t. I\\\\\\\\\\\\\\\
yields an output with
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i4z1 = 11,[r]+ ,[z]*,[s]rFI=]t tzl.,trl*,[al)
ln g6n6ral, system can be classified by placing constraints on
the transformatlon 7{.}2.1 Memoryless SystomsA system is memoryless
if the output y[z] oepenos onty on x[z] at th6 same ,2.For example,
yb)= QbD' is memoryless, but the ideal aeny ylnl= tb -" ol is not
unless /,d = 0.2.2 Linear SystemsA system is linear if the
principle of superposition applies, thus if y, [n] is the response
of the system tothe input .r, [n ] ano yrln) w response to ,, [, j,
tfr"n linearity implieso Additivity:
rlx,[nl+x,fnll=r{x,[n]+r{x,ln}=y,Irl*y,blo Scaling:
r k,,bl = ar $,[n] = ay,["|These properties combine to form the
general principle of superposition
rfox,[nl+ bx,fn]= arlx,fn!+ brb,b]-- oy,b)* ty,blln all cases a
and b are arbitrary constants.This property generalizes to many
inputs, so the response of a linear system to
,lrf= lra r* ofn] will oe y[n] = Loouy obl2.3 Time-invariant
systemsA system is time invarianl if a time shift or delay of the
input sequence causes a corresponding shift inthe output sequonce.
That is, it y[z] is tne response to x[z], tnen y[z - zo ] is the
response to,b- rolFor example, lhe accumulator system
yl"l= f,Ik),t =-.is time invariant, but the compressor
system
yful= *[u"]
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for M a positive integer (which selects every il,fih sample from
sequence) is not.2.4 CausalityA system is casual if the output at r
depends only on th6 input at n and earlier inputs.For example, the
bactward difference system
ybl= -b,l-,b-:lis causal, but the forward difference system
is not.2.5 Stability
3.
