presentation-4 discrete time systems-2ehm.kocaeli.edu.tr/upload/duyurular/161018112341dc074.pdf · Title: Microsoft PowerPoint - presentation-4 discrete time systems-2 Author: aysun

MEH329DIGITAL SIGNAL PROCESSING

Dept. Of Electronics & Telecomm. Eng.Kocaeli University

-4-Discrete Time Systems-2

Linear Constant-Coefficient Difference Equations (LCCDE)

• We calculate the output (response) of an LTI system using the input and the system’s impulse response

• However, as n gets larger, the convolution sum results in increased computation time and memory requirement

• Systems for which the output can be represented in terms of the input signal’s present and past values and the output signal’s past values reduce these requirement

MEH329 Digital Signal Processing 2

Linear Constant-Coefficient DifferenceEquations (LCCDE)

• An important class of LTI systems consists ofthose systems for which the input x[n] andoutput y[n] satisfy Nth-order eq. form of:

3MEH329 Digital Signal Processing

0 0

N M

k mk m

a y n k b x n m

Linear Constant-Coefficient DifferenceEquations


01 0

N M

k kk m

a y n a y n k b x n m

0 10

1 M N

k km k

y n b x n m a y n ka

• If we choose a0=1;

0 1

M N

k km k

y n b x n m a y n k

• All systems cannot be represented in terms of an LCCD




0

M

mm

y n b x n m

0

M

mm

h n b n m

If the output is not related with the previous outputvalues:

As a special case, if the output of the system does not depend on past output values:

Linear Constant-Coefficient Difference Equations


Sistemin dürtü yanıtı doğrudan giriş değerlerinin katsayıları olarakbulunduğundan LCCDE gösterimikonvolusyon toplamınadönüşmektedir.

• Sistem çıkışının sadece giriş değerlerine bağlı olduğu bu sistemşerde LCCDE ile konvolusyon toplamının sonucu aynıdır.

• Sistemin dürtü yanıtı sonlu sayıda sıfırdan farklı değerlere sahip olduğu için bu tür sistemler yapı itibarıyle FIR sistemlerdir.

• The length of impulse response is M+1 (FIR)




1y n x n x n • Example:

n < -1 : x[n]=0 and y[n]=0n = -1 : y[-1]=x[-1]-x[-2]=2n = 0 : y[0]=x[0]-x[-1]=-1n = 1 : y[1]=x[1]-x[0]=-1n = 2 : y[2]=x[2]-x[1]=-1n = 3 : y[3]=x[3]-x[2]=-1n = 4 : y[4]=x[4]-x[3]=3n = 5 : y[5]=x[5]-x[4]=-1n > 5 : y[n]=0

0

2, 1,0, 1, 2,1n

x n

• X[n] y[n]



• Özyinesiz(Non-Recursive): Bir sistem LCCDE şeklinde tanımlandığında sistem çıkışı girişin o andaki ve eski değerlerine bağlı ise

• Özyineli (Recursive): Bir sistem LCCDE şeklinde tanımlandığında sistem çıkışı girişin o andaki ve eski değerlerine ek olarak çıkışın da eski değerlerine bağlı ise

• Recursive sistemlerde giriş değerleri yanında çıkışın başlangıç koşulları da verilmelidir.

• Başlangıç koşulları sistemin çalışma biçimine doğrudan etkisi bulunduğu için onemlidir.


LCCDE


n

k

y n x k

• Example: Find the output of the accumulator

system

1

1

n

k

y n x n x k

y n x n y n

1y n y n x n 0

1

0

1

1

1

1

N

a

a

b

LCCDE representation



• We need an initial value for y

• If the system start with y[-1]=0

0

1,0, 1, 2,1n

x n



n < 0 : y[n]=0n = 0 : y[0]=x[0]+y[-1]=1n = 1 : y[1]=x[1]+y[0]=1n = 2 : y[2]=x[2]+y[1]=0n = 3 : y[3]=x[3]+y[2]=-2n = 4 : y[4]=x[4]+y[3]=-1n = 5 : y[5]=x[5]+y[4]=-1n > 5 : y[n]=-1



• Block diagram of the accumulator:



• Example: Moving-average system



• Block diagram of the moving-average system:



• Effect of moving-average system



• Example:

1

, 1 1

y n ay n x n

x n b n y

2

3 2

4 3

0

1

2

3

y a b

y a ab

y a a b

y a a b

1

2

3

4

2

3

4

5

y a

y a

y a

y a

1 , 0n ny n a a b n 1 , 0ny n a n



1 , 0n ny n a a b n 1 , 0ny n a n

1n ny n a a bu n

• If b=0 → x[n]=0, but y[n]=an+1 is not equal tozero.

• Therefore, scaling the input with zero is notgives zero output (the system is not linear).


• For the shifted input


1 d dx n x n n b n n

11

dn nndy n a a bu n n

1 dy n y n n

• the system is time variant.



• If y[-1]=0 is given

• The system is linear and time invariant in thiscase (evaluate with ).

• NOTE: Initial conditions of a LCCDE for systemsaffect the characteristics directly!

• In general, x[n]=0, n<0 and initial conditionsare chosen as zero for the causal LTI systems.

ny n a bu n

1x n b n



• Alternative representation for theaccumulator 1y n ay n x n

2

3 2

1

0

0 1 0

1 0 1 1 0 1 1 0 1

2 1 2 1 0 1 2

1 , 0n

n k

k

y ay x

y ay x a ay x x a y ax x

y ay x a y a x ax x

y n a y a x n k n

Response to the initial conditions Response to the input signal x[n]


• If the system initially relaxed at time n=0y[-1]=0

• Thus a recursive system is relaxed if it starts withzero initial conditions.

• We say that the system is at “zero state” in thiscase.

• The response of the system is called as“ZERO STATE RESPONSE”


zsy n



0

, 0n

kzs

k

y n a x n k n

nh n a u n (IIR, causal)


• Suppose that the system is initially nonrelaxedand x[n]=0 for all n.

• Then the output of the system with zero inputis called the

“ZERO INPUT RESPONSE”


ziy n

1 1nziy n a y



1

0

1 , 0n

n k

k

y n a y a x n k n

zi zsy n y n y n

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presentation-4 discrete time systems-2ehm.kocaeli.edu.tr/upload/duyurular/161018112341dc074.pdf · Title: Microsoft PowerPoint - presentation-4 discrete time systems-2 Author: aysun