Digital Two Pair is defined with an analogy to analog two ports. In analog two ports, we have an input applied between two terminals constituting a port and an output between two other terminals, constituting the second port. In a digital device since we are only talking about numbers so we have an input x(n) and an output y(n) as shown;

If a Digital Signal Processing has one input and one output, then it is called a single input single output deice (SISO). For the example (chip), it had a single input and multiple outputs. So, it is a Single Input Multiple Output devices (SIMO).** A device that has two inputs X1(z) and X2(z) and two outputs Y1(z) and Y2(z) respectively are called a** **digital two pairs** (represented as Digital Two-Port or for short D2P). The schematic representation is shown below;

As we can see from the diagram above that the inputs are drawn on the two sides of the Digital Signal Processing block. Although this will be made clear later, it should be understood that in Digital Signal Processing there is no port concept. Meanwhile, we just feed numbers and extract numbers from the Digital Signal Processor. Let us consider a Digital Two Pair having the defined input X1(z) and X2(z) and outputs Y1(z) and Y2(z) respectively as shown below;

Moreover, the Digital Two Pair can be characterized (like in the analog 2-port, where we have Z-parameters and Y-parameters and ABCD-parameters) in two different useful ways (there are many other ways) which are;

**1. The transmission parameter.**

**2. The chain parameter.**

Table of Contents

## Transmission Parameter

Hence, if we express the output in terms of the combination of the two inputs then the parameters that define their relationship are called **transmission parameters**.

Meanwhile, It is represented in the matrix form where the transmission parameters are. The matrix that contains these parameters is called the **“T” matrix (Tee matrix)**. Well, If we write the complete equations we get:

{Y}_{1}{(z)} = {t}_{11}{X}_{1}{(z)} + {t}_{12}{X}_{2}{(z)}

{Y}_{2}{(z)} = {t}_{21}{X}_{1}{(z)} + {t}_{22}{X}_{2}{(z)}

Just like the analog 2-port, we can define the parameters t11, t12, t21 and t22. Likewise, it can be understood if we check the schematic diagram or the equations. Certainly, there is no concept of impedance or admittance or anything like we have in analog 2-port. Instead, we have a ration of numbers.

## Chain Parameter

If we express the variables on the Left-Hand Side of the Digital Two Pair network in terms of the variables on the Right Hand Side. Then, we have the parameters that we define. That is a relationship (just like in analog 2-port where we relate {V}_{1}, {I}_{1}, with {V}_{2}, {−I}_{2}

The difference between the chain parameter in Digital Two-Port and ABCD parameters in analog two-port is that there is no negative sign attached to X2. Again, there is no concept of voltage and currents but purely numbers.

**Why are these parameters ABC and D called chain parameters?**

It is because if there is a cascade of two such Digital Two Pair networks as shown below. Meanwhile, the overall ABCD parameters of a cascade shall be a multiplication of the individual ABCD parameters of the two Digital Two Ports.

This is why they are called chain parameter, then similarly, if we write the equation i.e

{Y}_{1}{(z)} = {AY}_{2}{(z)} + {BX}_{2}{(z)}

{Y}_{1}{(z)} = {CY}_{2}{(z)} + {DX}_{2}{(z)}{Y}_{1}{(z)}

Well, the symbol given for the matrix containing the chain parameters is the **Γ (Gamma) symbol. **In summary, Γ is the transmission matrix and is the chain matrix. Obviously, since the variables are the same is X1(z), X2(z). Therefore there should be a relationship between the transmission parameters and the chain parameters.

The inter relationship are as follows; A=\frac{1}{{t}_{12}}, B=\frac{{-t}_{22}}{{t}_{21}}, C=\frac{{t}_{11}}{{t}_{12}}, and D=\frac{{t}_{12}{t}_{21}-{t}_{11}{t}_{22}}{{t}_{21}}

Meanwhile, this can be easily shown if we take the set of equations for the chain parameter of Digital Two Pair and express it in terms of the set of equations for the transmission parameters Digital Two Pair. Similarly, Also, the chain parameters can be expressed in terms of the transmission parameters by comparing the equations in reverse.

The first observation in all these relationships is that their denominators are the same. Meanwhile, this is true in any conversion including analog conversions. Now, there is a concept of reciprocity as in analog two-port. In analog two-port, *the concept of reciprocity says that if the input and output are interchanged, then the ratio should remain the same if the network is reciprocal*.

## Digital Two Pair: Cascaded Digital Two Pair

Also, in terms of the parameters, t12 and t21 relate input and output and are input and output with interchanged. So a Digital Two Pair is said to be reciprocal if t12 = t21. Exactly like analog two-port, we can *cascade* Digital Two Pair in two ways; Let us consider the chain parameters Digital Two-Port cascade.

As we said earlier, the overall chain parameter matrix W is the product if the individual chain matrix and W1and W2 respectively. i.e. W = W1*W2. But there is nothing sacred about the directions of the arrow or the position of the input or output variables so the cascade can also be drawn as shown:

Now, what digital two-port can we use here? Obviously it is the transmission parameter because the outputs are expressed in terms of the inputs, therefore the overall transmission matrix is the product of the individual transmission matrix. Therefore it does not matter what form of cascading is used in D2P. However, it matters in analog two-ports.

The first order of cascading is called the W cascade and the second order is also called W- cascade. So unlike analog two-port, there are two manners of cascading in digital. Meanwhile, we shall see how these two concepts are useful, however, we have already seen this concept in action in the realization of a magnitude complementary filter. Finally, we discuss the concept of a terminating Digital Two-Port.

## Digital Two Pair: Terminating D2P

So the concept of a terminating D2P is when the variables Y2(z) and X2(z) are connected through transfer function. Also, it behaves like a Single Input Single Output system and the transfer function \frac{{Y}_{1}}{{X}_{1}} can be expressed in terms of the chain parameters or transmission parameters.

The concept of D2P is being utilized to derive the conical realization of All-Pass Filter. Canonical realization is a realization that has the minimum number of delays as well as multipliers. Moreover, the example we earlier cited which is a first-order filter realized with one delay and one multiplier is a canonical realization.

Finally, in this very article, we revisit the discussion of stability. Stability can be examined in two contexts. One is that a Linear Time Invariance system is stable if and only if its impulse response is absolutely summable. While the other context we have seen is that the poles should be inside the unit circle i.e. These examinations are okay but if we are given a very high Infinite Impulse Response transfer function. Then when we want to test for stability then we need to find and finding means we have to inert the Z transform from the transfer function.

## Root finding

The inversion of Z transform to transfer function is a laborious process particularly if we have multiple poles, or ultimately we can have the transfer function H(z) and factorize its denominator. Factorizing the denominator involves root-finding programs. Root finding programs have their own disadvantages. For example, numerical inaccuracy because these programs cannot find roots up to an infinite number of decimals hence they are truncated.

Besides these errors, the step of finding the roots is a laborious process i.e. To check if the roots or poles are bounded by the unit circle or not. However, testing for stability becomes an easy process through an algorithm in terms of All-Pass filters. Given an H(z) to be tested, we first construct an All-Pass filter. Then for stability using a simple algorithm (please the derivation of this algorithm is not easy).

The algorithm simply implements a recursive relation. Let us consider the simplest possible Infinite Impulse Response filter which is a first-order Infinite N(z) ImpulseResponsefilter.In general, stability can be easily tested in this example because the pole is at. Suppose we consider a second-order Infinite Impulse Response filter

*Root finding*

Obviously, this is not a problem because we can find P1 and P2 as

**{ P }_{1,2}=\frac { { -d }_{ 1 }\pm \sqrt { { { d }^{ 2 } }_{ 1 }-{ 4d }_{ 2 } } }{ 2 }**

**Given d1 and d2 can we say whether the filter is stable or not?** It will be possible to state immediately whether a filter is stable or not from the values of d1 of d2. Well, if we have a plot (a parameter plot) of d1 versus d2 that shows the region of stability. The plot is shown below;

Meanwhile, the region of stability is a triangle region (shaded region) for d1 of d2. So if d1and d2 points lie within that region, the Infinite Impulse Response filter is stable otherwise it is not. Also, we will show that if d1d2 points lie within the crossed region. Then the poles will be complex on the other end, if it lies outside the crossed region the poles will be real.

Now an important question, can we test for the stability if a Finite Impulse Response filter is located in the region i.e. They are deep inside the unit circle. A Finite Impulse Response filter is, therefore, unconditionally stable which is a great advantage of finite Impulse Response.

Another great advantage is that it is a linear phase. These are the two virtues for which the Finite Impulse Response is given great respect. But at the same time, there is a disadvantage which is that when designing an FIR filter. Well, we have to increase its order or complexity to achieve what can be achieved by a second-order Infinite Impulse Response filter with this we end this lecture-discussion.

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