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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

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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:

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0 0

N M

k mk m

a y n k b x n m

Linear Constant-Coefficient DifferenceEquations

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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

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Linear Constant-Coefficient DifferenceEquations

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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

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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)

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Linear Constant-Coefficient DifferenceEquations

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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]

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Linear Constant-Coefficient Difference Equations

• 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.

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LCCDE

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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

Linear Constant-Coefficient DifferenceEquations

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• We need an initial value for y

• If the system start with y[-1]=0

0

1,0, 1, 2,1n

x n

Linear Constant-Coefficient DifferenceEquations

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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

Linear Constant-Coefficient DifferenceEquations

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• Block diagram of the accumulator:

Linear Constant-Coefficient DifferenceEquations

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• Example: Moving-average system

Linear Constant-Coefficient DifferenceEquations

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• Block diagram of the moving-average system:

Linear Constant-Coefficient DifferenceEquations

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• Effect of moving-average system

Linear Constant-Coefficient DifferenceEquations

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• 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

Linear Constant-Coefficient Difference Equations

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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).

Linear Constant-Coefficient DifferenceEquations

• For the shifted input

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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.

Linear Constant-Coefficient DifferenceEquations

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• 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

Linear Constant-Coefficient DifferenceEquations

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• 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]

Linear Constant-Coefficient DifferenceEquations

• 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”

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zsy n

Linear Constant-Coefficient DifferenceEquations

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0

, 0n

kzs

k

y n a x n k n

nh n a u n (IIR, causal)

Linear Constant-Coefficient DifferenceEquations

• 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”

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ziy n

1 1nziy n a y

Linear Constant-Coefficient DifferenceEquations

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1

0

1 , 0n

n k

k

y n a y a x n k n

zi zsy n y n y n