Block Diagram Representation

BLOCK DIAGRAM REPRESENTATION

The discrete time system are often represented by block diagram. These are also called structures of discrete time systems. These diagrams indicates the manner in which computations are performed.

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Consider the system function given by equation

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Hence we can write equation (1) as

H(z) = H₁(z) . H₂(z). .............(4)

The overall IIR system can be realized as cascade of two functions H₁(z) and H₂(z). Here observe that H₁(z) represents zero of H(z). It is all zero system. Similarly H₂(z) represents poles of H(z). It is all pole system.

5.3.1 Direct Form I Structure

Let us first prepare direct form structures of H₁(z) and H₂(z). We can write H₁(z) of equation (2) as,

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Since H₁(z) = Y₁(z)/X₁(z) above equation can be written as

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Taking inverse z transform of above equation

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This figure shows direct form realization of above equation.

 

H₂(z) is all pole system. We will prepare the direct form realization of H₂(z).

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We know that H₂(z) = Y₂(z)/H₂(z), hence above equation becomes,

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Expanding the summation of this equation we get

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Taking inverse z transform of above equation,

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Here observe that the feedback terms are also present.

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From equation (4) we know that,

H(z) = H₁(z). H₂(z)

This represents cascading of two systems

H₁(z) and H₂(z). It is shown below

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It was demonstrated and thus they are cascaded in above diagram. The direct form realization for H(z) as per above figure. This connection is shown in Fig 5.3.

Observe that this realization requires (M+N+1) number of multiplications, (M+N) number of additions and (M+N+1) number of memory locations.

 

5.3.2 Direct Form - II Structure

We have seen that the overall system function can be connected as a cascade of H₁(z) and H₂(z). Recall H(z) of equation (1).

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Let H(z) be written as,

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Now let H₁(z) be given as,

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Gross multiply the terms of equation (9). Then we get,

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Taking inverse z - transform of this equation,

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Fig 5.3 Shows the direct form implementation of above equation.

 

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In the realization of Fig. 5.5 observe that x(n) is the input and ω(n) is the output.

Now let us obtain the realization of H₂(z) of equation (10) from this equation, we obtain

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Taking inverse z-transform of this equation we get,

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Fig. 5.6 shows the direct form implementation of above equation.

Observe that H₁(z) of equation (9) is all pole system function. Hence its realization of fig (5) is also all pole system. Similarly H₂(z) of equation (10) is all zero system function. Hence its realization given in fig 5.5 is also all zero system. In this fig a(n) is the input and y(n) is the output.

 

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Since H(z) = H₁(z). H₂(z), the realization for H(z) can be obtained by cascading H₁(z) and H₂(z) as shown in fig 5.6.

Problem 1:

A difference equation of a discrete time system is given below:

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Solution:

The given difference equation can be written as,

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Below fig shows the direct form -I structure of this difference equation.

 

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The direct form -II structure of given difference equation is shown in fig. 5.8.

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Problem 2:

Obtain the difference equation for the block digram shown in fig below.

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Solution :

Let us redraw the given block diagram and indicate the inputs to delay elements as shown.

 

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From the above diagram we can write.

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From equation (1) we can write ω(n-2) as,

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Putting above expression in equation (2)

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This is the required difference equation of the given system.