Cascade Form Structure for IIR Systems

5.3.3 Cascade Form Structure for IIR Systems

Consider the rational system function of IIR system,

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The numerator and denominator polynomials of above equation can be expressed as multiplication of second order polynomials. i.e.,

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We know that the system functions H₁(z), H₂(z) etc of equation (14) can be connected in cascade to obtain realization of H(z). This is shown in fig. 5.9.

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Now each H₁(z), H₂(z),...etc can be realized by direct form I or II structures.

We know that H_k(z) = Y_k(z)/X_k(z). This can also be written as,

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This is all pole second order subsystem and,

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We have discussed the procedure for obtaining direct form II in last subsection. Proceeding on the same lines we can obtain the direct form-II structure for H(z), which is splitted into two functions given by equation (16) and equation (17). This direct form-II structure is shown below in fig. 5.10.

 

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This cascade structure is described by following equations.

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This equation represent the second order subsystem of Fig 5.7. And cascading is represented by following equation:

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The above equations represent the cascading of second order subsystems as shown in Fig 5.9.

Problem 3:

Realize the following system function in cascade form.

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Solution :

The given transfer function can be written as the product of two functions.

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The above two equation are in the form of equation (15). They are written as follows:

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Realization of equation (15) is given in fig. 5.11. The above two equations can be realized by using fig. 5.11 shows the cascade realization of H(z) = H₁(z) . H₂(z).

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5.3.4 Parallel Form Structure for IIR Systems

We know that the rational system function of IIR system is given as,

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The above system function can be expanded in partial fractions as follows:

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Here 'C' is constant and each H₁(z), H₂(z),....etc is the second order subsystems which is given as,

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These second order subsystems are formed by combining complex conjugate poles. Because of this, the coefficients b_k0, b_k1, a_kl, a_k2 are real.

We know that addition of system functions results in parallel connection. Then the realization of H(z) of equation (22) becomes as shown in Fig. 5.12.

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Here each H₁(z), H₂(z)... etc can be realized by direct form -I or direct form-II Fig. 5.13 shows the direct form II realization of H_k(z) of equation (23).

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The parallel form structure discussed here can be described by following equations.

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Problem 4:

Realize the following systems function in parallel form.

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Solution:

Let us write the given system function as,

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Calculating the values of A₁, A₂, A₃ and A₄, we get,

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Let us combine the first two terms and last two terms. Because of this, the complex values will be combined into real coefficients. i.e.,

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The above equation can also be written as,

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The above equation has two terms, they can be called as,

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Observe that the above two equations are in the form of equation (28). The realization of equation (28) is shown in fig 5.10. The realization of H₁(z) and H₂(z) in parallel is shown in Fig. 5.14.

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Problem 5:

Consider a causal LTI system whose input x(n) and output y(n) are related through the block diagram representation shown in fig 19.

1)Determine a difference equation relating y(n) and x(n).

2)Is this system stable?

 

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Solutiion:

1) To obtain difference equation :

From fig. 5.15.

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The generalized difference equation is given as,

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Putting values in above equation,

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2) To check stability of this system :

Taking z-transform of the difference equation,

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There are two poles at z = 1/3. Since the poles lie inside the unit circle, the system is stable.

 

Problem 6:

i) Develop a direct form-I realization of the difference equation.

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(ii) Draw the transposed direct forms -I and II structures of third order IIR filter.

Solution:

i) Direct form -I Realization.

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(ii) Transposed direct forms I and II structures of third order IIR filter:

Fig. 5.17 (a) and (b) shows the direct form I and II structures of 3^rd order IIR filter.

 

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The transposed direct form I and II can be obtained by reversing the directions of all branches and positions of inputs and outputs. Fig. 5.18 shows the transposed structures.

Problem 7:

Draw the direct form -II block diagram representation for the system function.

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Solution:

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Putting these values in direct form-II, we get the realization as shown in fig.5.19

 

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Problem 8:

Draw the direct form, cascade form and parallel form block diagram of the following system functions: H(z) = eeeeeeeeee

Solution:

1) Direct form Realization:

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Fig. 5.20 shows the direct form-II realization obtained by putting values in standard realization.

2) Cascade form realization:

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Thus H(z) is product of H₁(z) and H₂(z). They can be realized in cascade by putting values in standard direct form-II as shown in fig. 5.21.

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3) Parallel form Realization:

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Thus H(z) is sum of H₁(z) and H₂(z). They can be realized in parallel by putting values in standard direct form-II as shown in fig. 5.22.

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Problem 9:

Consider the causal linear shift invariant filter with system function eeeeeeeeeeee Draw the following realization structures of the system.

i) Direct form -II

ii) A parallel form connection of first and second order systems realized in direct form -II.

 

Solution :

1) Direct form -II structure

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Here b₀ = 1, b₁ = 0.875, and a₁ = -0.5, a₂ = 0.76, a₃ = -0.63 fig 5.23 shows direct form -II realization structure.

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ii) Parallel form structure

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Fig. 5.24 shows the parallel form realization

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Problem 10:

Obtain the cascade realization of

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Solution:

Taking z-transform of given equation,

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Putting these values in standard direct form -II structures,

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Problem 11:

Obtain the cascade and parallel realizations for the system described by the system functions :

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Solution:

i) Cascade Realization.

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Where

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Fig. 5.26 Shows the cascade of H₁(z) and H₂(z) realized in direct form II.

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ii) Parallel form Realization.

Expressing the given H(z) in partial fractions,

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Above two H₁(z) and H₂(z) are realized in parallel form as shown in fig. 5.27

 

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