TYPES OF DISCRETE - TIME SYSTEMS1.22007/11/05 07:23:55 US/Central2008/07/01 22:01:13.457 GMT-5PhuongHuuNguyennhphuong@hcmuns.edu.vnPhuongHuuNguyennhphuong@hcmuns.edu.vnDicrete-time (digital) systems comprise of serveral basic types with different characteristics. The categorization gives us a deeper understanding of systems and the choice of appropriate analysis method.Memoryless systems, and systems with memoryA memoryless (or static) system does not need memory. It processes the input and output signals taking place at the same instant. For exampley(n)=2x(n) size 12{ matrix {
{} # {} # {}
} y \( n \) =2x \( n \) } {}y(n)=2x(n)−x2(n) size 12{ matrix {
{} # {} # {}
} y \( n \) =2x \( n \) - x rSup { size 8{2} } \( n \) } {}Actually there is a small delay between input and output due to the propagation delay of the system.A system with memory (or dynamic) needs memory to store past and future values needed for the processing. For exampley(n)=x(n)+0.8x(n−1):onememorycelly(n)=13[x(n−1)+x(n)+x(n+1)]:twomemorycelly(n)=∑k=−∞+∞x(n−k):infinitememory
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOabaeqabaGaamyEaiaacIcacaWGUbGaaiykaiabg2da9iaadIhacaGGOaGaamOBaiaacMcacqGHRaWkcaaIWaGaaiOlaiaaiIdacaWG4bGaaiikaiaad6gacqGHsislcaaIXaGaaiykaiaaywW7caaMf8UaaGzbVlaaywW7caaMf8UaaiOoaiaaysW7caWGVbGaamOBaiaadwgacaaMe8UaamyBaiaadwgacaWGTbGaam4BaiaadkhacaWG5bGaaGjbVlaadogacaWGLbGaamiBaiaadYgaaeaacaWG5bGaaiikaiaad6gacaGGPaGaeyypa0ZaaSaaaeaacaaIXaaabaGaaG4maaaacaGGBbGaamiEaiaacIcacaWGUbGaeyOeI0IaaGymaiaacMcacqGHRaWkcaWG4bGaaiikaiaad6gacaGGPaGaey4kaSIaamiEaiaacIcacaWGUbGaey4kaSIaaGymaiaacMcacaGGDbGaaGzbVlaacQdacaaMe8UaamiDaiaadEhacaWGVbGaaGjbVlaad2gacaWGLbGaamyBaiaad+gacaWGYbGaamyEaiaaysW7caWGJbGaamyzaiaadYgacaWGSbaabaGaamyEaiaacIcacaWGUbGaaiykaiabg2da9maaqahabaGaamiEaiaacIcacaWGUbGaeyOeI0Iaam4AaiaacMcaaSqaaiaadUgacqGH9aqpcqGHsislcqGHEisPaeaacqGHRaWkcqGHEisPa0GaeyyeIuoakiaaywW7caaMf8UaaGzbVlaaywW7caaMf8UaaGzbVlaaywW7caaMe8UaaGjbVlaacQdacaaMe8ocbiGaa8xAaiaa=5gacaWFMbGaamyAaiaad6gacaWGPbGaamiDaiaadwgacaaMe8UaamyBaiaadwgacaWGTbGaam4BaiaadkhacaWG5baaaaa@BA41@Causal and noncausal systemsIn causal system the result comes after the cause, or, at the same time (simultaneously). This is to say that the output at index n only depends on the input at n, n – 1, n – 2,…, and not on n + 1, n + 2,… In noncausal systems, on the other hand the output also depends on future inputs. Following is a few examples.(a)y(n)=2x(n)−3x2(n) size 12{ matrix {
{} # {} # {}
} \( a \) matrix {
{} # {} # {}
} y \( n \) =2x \( n \) - 3x rSup { size 8{2} } \( n \) : matrix {
{} # {} # {} # {}
} ital "causal"} {} : noncausal(b)y(n)=13[x(n−1)+x(n)+x(n+1)] size 12{ matrix {
{} # {} # {}
} \( b \) matrix {
{} # {} # {}
} y \( n \) = { {1} over {3} } \[ x \( n - 1 \) +x \( n \) +x \( n+1 \) \] : matrix {
{} # {} # {} # {}
} matrix {
ital "noncausal" {} # {}
} ital "due" matrix {
{}
} ital "to" matrix {
{}
} ital "the" matrix {
{}
} ital "last" matrix {
{}
} ital "term"} {}: noncausal due to the last term(c)yn=∑k=0∞xn−k size 12{ matrix {
{} # {} # {}
} \( c \) matrix {
{} # {} # {}
} y left (n right )= Sum cSub { size 8{k=0} } cSup { size 8{ infinity } } {x left (n - k right )} : matrix {
{} # {} # {} # {}
} ital "causal"} {}
: causal
(d)y(n)=∑n=−∞∞xn size 12{ matrix {
{} # {} # {}
} \( d \) matrix {
{} # {} # {}
} y \( n \) = Sum cSub { size 8{n= - infinity } } cSup { size 8{ infinity } } {x left (n right )} : matrix {
{} # {} # {} # {}
} ital "noncausal"} {} : noncausal(e)y(n)=∑k=−∞∞xn−k size 12{ matrix {
{} # {} # {}
} \( e \) matrix {
{} # {} # {}
} y \( n \) = Sum cSub { size 8{k= - infinity } } cSup { size 8{ infinity } } {x left (n - k right )} : matrix {
{} # {} # {} # {}
} ital "noncausal"} {}: noncausal(f)y(n)=x(−n) size 12{ matrix {
{} # {} # {}
} \( f \) matrix {
{} # {} # {}
} y \( n \) =x \( - n \) : matrix {
{} # {} # {} # {}
} ital "noncausal"} {} : noncausal
(g)y(n)=x2(n) size 12{ matrix {
{} # {} # {}
} \( g \) matrix {
{} # {} # {}
} y \( n \) =x rSup { size 8{2} } \( n \) : matrix {
{} # {} # {} # {}
} ital "noncausal"} {} : noncausal
(h)y(n)=x(n2) size 12{ matrix {
{} # {} # {}
} \( h \) matrix {
{} # {} # {}
} y \( n \) =x \( n rSup { size 8{2} } \) : matrix {
{} # {} # {} # {}
} ital "noncausal"} {} : noncausal
In real-time processing (or on-line processing), systems must be causal, off-line processing (or batch processing or block processing) systems can be noncausal since all samples have been stored in memory, many of those will be future values with respect to the chosen time origin.The concept of causality is also applied to signals but the definition is modified. A signal x(n) can be classified as Causal (or right-sided) if x(n) = 0 for n < 0Anticausal (or left-sided) if x(n) = 0 for n ≥ 0Two-sided (or bilateral) if x(n) exists for all n (<0 and ≥ 0)For example, the unit step u(n) is causal, u(-n-1) is anticausal,
a∣n∣ size 12{a rSup { size 8{ lline n rline } } } {} is two-sided. We can plot out these signals to really see the difference.Time-invariant and time-variant systemsThe characteristics of a system may change with time so that the output depends on the input as well as the instant the input is applied. This is a time-variant system. On the other hand, many systems can be assumed to be time-invariant, i.e. the output does not depend on the time the input is applied. The terms shift-variant and shift- invariant can be used instead of time-variant and time-invariant respectively.The time (shift) invariance is judged as follows.If
x(n)→y(n) size 12{x \( n \) matrix {
{} # {}
} rightarrow y \( n \) } {}then
x(n−k)→y(n−k) size 12{x \( n - k \) matrix {
{} # {}
} rightarrow y \( n - k \) } {}This criterion is illustrated in
Time (shift) invariant system
Are the following systems time-invariant?(a)y(n)=13[x(n−1)+x(n)+x(n+1)] size 12{ matrix {
{} # {} # {}
} \( a \) matrix {
{} # {} # {}
} y \( n \) = { {1} over {3} } \[ x \( n - 1 \) +x \( n \) +x \( n+1 \) \] } {}(b)y(n)=nx(n) size 12{ matrix {
{} # {} # {}
} \( b \) matrix {
{} # {} # {}
} y \( n \) = ital "nx" \( n \) } {}(c)y(n)=x(−n) size 12{ matrix {
{} # {} # {}
} \( c \) matrix {
{} # {} # {}
} y \( n \) =x \( - n \) } {} Solution
(a) For the system y(n)=13[x(n−1)+x(n)+x(n+1)] size 12{ matrix {
{} # {} # {}
} matrix {
{} # {} # {}
} y \( n \) = { {1} over {3} } \[ x \( n - 1 \) +x \( n \) +x \( n+1 \) \] } {}If the present input is delayed by k (i.e. by replacing x(n) by x(n – k)…) then the output isy(n−k)=13[x(n−1−k)+x(n−k)+x(n+1+k)] size 12{ matrix {
{} # {} # {}
} matrix {
{} # {} # {}
} y \( n - k \) = { {1} over {3} } \[ x \( n - 1 - k \) +x \( n - k \) +x \( n+1+k \) \] } {}and if the present output is delayed by k (i.e. by replacing n by n – k)y'(n−k)=13[x(n−1−k)+x(n−k)+x(n+1+k)] size 12{ matrix {
{} # {} # {}
} matrix {
{} # {} # {}
} y rSup { size 8{'} } \( n - k \) = { {1} over {3} } \[ x \( n - 1 - k \) +x \( n - k \) +x \( n+1+k \) \] } {}Sincey'(n−k)=y(n−k) size 12{ matrix {
{} # {} # {}
} matrix {
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} y rSup { size 8{'} } \( n - k \) =y \( n - k \) } {}the system is time-invariant.(b) For the systemy(n)=nx(n) size 12{ matrix {
{} # {} # {}
} matrix {
{} # {} # {}
} y \( n \) = ital "nx" \( n \) } {}if the present input is delayed by k then the output is y(n−k)=nx(n−k) size 12{ matrix {
{} # {} # {}
} matrix {
{} # {} # {}
} y \( n - k \) = ital "nx" \( n - k \) } {}and if the present output is delayed by k then the output isy'(n−k)=(n−k)x(n−k) size 12{ matrix {
{} # {} # {}
} matrix {
{} # {} # {}
} y rSup { size 8{'} } \( n - k \) = \( n - k \) x \( n - k \) } {}Sincey'(n−k)≠y(n−k) size 12{ matrix {
{} # {} # {}
} matrix {
{} # {} # {}
} y rSup { size 8{'} } \( n - k \) <> y \( n - k \) } {}the system is time-variant.(c) For the systemy(n)=x(−n) size 12{ matrix {
{} # {} # {}
} matrix {
{} # {} # {}
} y \( n \) =x \( - n \) } {}we havey(n−k)=x(−n−k) size 12{ matrix {
{} # {} # {}
} matrix {
{} # {} # {}
} y \( n - k \) =x \( - n - k \) } {}y'(n−k)=x[−(n−k)]=x(−n+k)≠y(n−k) size 12{ matrix {
{} # {} # {}
} matrix {
{} # {} # {}
} y rSup { size 8{'} } \( n - k \) =x \[ - \( n - k \) \] =x \( - n+k \) <> y \( n - k \) } {}So the system is time-variant.Linear and nonlinear systemsThe significance of linearity and nonlinearity for discrete-time systems is about the same as for analog systems. Suppose two input signals
x1(n) size 12{x rSub { size 8{R} } \( n \) } {}
and
x2(n) size 12{x rSub { size 8{R} } \( n \) } {}
when applied separately to a system give corresponding outputs
y1(n) size 12{x rSub { size 8{R} } \( n \) } {}
and
y2(n) size 12{x rSub { size 8{R} } \( n \) } {}
. Now if a linear combination of the two inputs give the same linear combination of the outputs then the system is linear, otherwise the system is nonlinear. Thus linearity implies both scalability (proportionality) and superposition. The definition of linearity is illustrated in .
Linear systems
Consider the linearity of the following systems:(a)y(n)=n2x(n) size 12{ matrix {
{} # {} # {}
} \( a \) matrix {
{} # {} # {}
} y \( n \) =n rSup { size 8{2} } x \( n \) } {}(b)y(n)=nx(n2) size 12{ matrix {
{} # {} # {}
} \( b \) matrix {
{} # {} # {}
} y \( n \) = ital "nx" \( n rSup { size 8{2} } \) } {}(c)y(n)=nx2(n) size 12{ matrix {
{} # {} # {}
} \( c \) matrix {
{} # {} # {}
} y \( n \) = ital "nx" rSup { size 8{2} } \( n \) } {}(d)y(n)=Ax(n)+B,A,Bconstants size 12{ matrix {
{} # {} # {}
} \( d \) matrix {
{} # {} # {}
} y \( n \) = ital "Ax" \( n \) +B, matrix {
{} # {} # {}
} A,B matrix {
{} # {}
} ital "cons""tan" ital "ts"} {} Solution (a) The system isy(n)=n2x(n) size 12{ matrix {
{} # {} # {}
} matrix {
{} # {} # {}
} y \( n \) =n rSup { size 8{2} } x \( n \) } {}The two separate inputs and corresponding outputs arey1(n)=n2x1(n) size 12{ matrix {
{} # {} # {}
} matrix {
{} # {} # {}
} y rSub { size 8{1} } \( n \) =n rSup { size 8{2} } x rSub { size 8{1} } \( n \) } {}y2(n)=n2x2(n) size 12{ matrix {
{} # {} # {}
} matrix {
{} # {} # {}
} y rSub { size 8{2} } \( n \) =n rSup { size 8{2} } x rSub { size 8{2} } \( n \) } {}Now for the combined inputx(n)=a1x1(n)+a2x2(n) size 12{ matrix {
{} # {} # {}
} matrix {
{} # {} # {}
} x \( n \) =a rSub { size 8{1} } x rSub { size 8{1} } \( n \) +a rSub { size 8{2} } x rSub { size 8{2} } \( n \) } {}the output isy(n)=n2[a1x1(n)+a2x2(n)]=a1[n21(n)]+a2[n22(n)]=a1y1(n)+a2y2(n) size 12{y \( n \) =n rSup { size 8{2} } \[ a rSub { size 8{1} } x rSub { size 8{1} } \( n \) +a rSub { size 8{2} } x rSub { size 8{2} } \( n \) \] =a rSub { size 8{1} } \[ n rSup { size 8{2} rSub { size 8{1} } } \( n \) \] +a rSub { size 8{2} } \[ n rSup { size 8{2} rSub { size 8{2} } } \( n \) \] =a rSub { size 8{1} } y rSub { size 8{1} } \( n \) +a rSub { size 8{2} } y rSub { size 8{2} } \( n \) } {}So the system is linear.(b) The system isy(n)=nx(n2) size 12{ matrix {
{} # {} # {}
} matrix {
{} # {} # {}
} y \( n \) = ital "nx" \( n rSup { size 8{2} } \) } {}The procedure is summarized as followsx1(n)→y1(n)=nx1(n2) size 12{ matrix {
{} # {} # {}
} x rSub { size 8{1} } \( n \) rightarrow y rSub { size 8{1} } \( n \) = ital "nx" rSub { size 8{1} } \( n rSup { size 8{2} } \) } {}x2(n)→y2(n)=nx2(n2) size 12{ matrix {
{} # {} # {}
} x rSub { size 8{2} } \( n \) rightarrow y rSub { size 8{2} } \( n \) = ital "nx" rSub { size 8{2} } \( n rSup { size 8{2} } \) } {}x(n)=a1x1(n)+a2x2(n) size 12{ matrix {
{} # {} # {}
} x \( n \) =a rSub { size 8{1} } x rSub { size 8{1} } \( n \) +a rSub { size 8{2} } x rSub { size 8{2} } \( n \) } {}then y(n)=a1nx1(n2)+a2nx2(n2)=a1y