signal space analysis of optimum detection , SIGNAL SPACE ANALYSIS in digital communication pdf ?

**INTRODUCTION TO SIGNAL SPACE ANALYSIS**

__Inside this Chapter__

- Introduction
- Concept of Additive White Gaussain Noise (AWGN) Channel
- Concept of Optimum Receiver
- Geometric Representation of Signals
- Schwarz Inequality
- Gram-Schmidt Orthogonalization Procedure

**7.1 INTRODUCTION**

Figure 7.1 shows the most basic form of a digital communication system.

Here, a message source emits one symbol every *T* seconds. The emitted symbols belong to an alphabet of *M* symbols represented by m_{1}, m_{2}, ……m_{M}.

To make things more clear, let us consider two examples as under:

(i) In first example, let us consider the remote connection of two digital computers with one computer acting as an information source which calculates digital outputs based upon observation and inputs fed into it. The resulting computer output is expressed as a sequence of 0s and 1s, which are transmitted to a second computer over a communication channel. In this case, the alphabet consists simply of two binary symbols i.e., 0 and 1.

(ii) In second example, let us consider a quaternary PCM encoder with an alphabet consisting of four possible symbols, i.e., 00, 01 10, and 11.

In case of an event, let probabilities p_{1}, p_{2}, …. p_{M} denote the message source output. For simplicity, let us assume that the *M* symbols of the alphabet are equally likely.

Then, the probability that symbol mi is emitted by the information source, is expressed as

P_{i} = P(m_{i})

or p_{i} = for *i* = 1, 2, …. M …(7.1)

Now, the transmitter takes the message source output m_{i} and codes it into a distinct signal s_{i}(t) which is suitable for transmission over the channel.

**DIAGRAM**

**FIGURE 7.1.*** A most basic form of digital communication system.*

This signal s_{i}(t) occupies the full duration _{T} alloted to symbol m_{i}. Further, s_{i}(t) is a real-valued energy signal as given by

for *i* = 1, 2, …M …(7.2)

**7.2 CONCEPT OF ADDITIVE WHITE GAUSSIAN NOISE (AWGN) CHANNEL **

Now, the channel is assumed to have following two characteristics:

(i) the channel is linear with a bandwidth that is wide enough to accommodate the transmission of signal s_{i}(t) with negligible or no distortion.

(ii) the channel noise, w(t), is the sample function of a zero-mean white Gaussian noise process.

At this stage, it may be noted that the reasons for second assumption are that it makes receiver calculations very easy and also, it is a reasonable description of the type of noise present in Several practical communication systems. Such a channel is popularly known as an additive Mute Gaussian noise (AWGN) channel.

Hence, in view of above discussion, the received signal x(t) is expressed as

**EQUATION**

and thus, we can model an additive white Gaussian noise (AWGN) channel in figure 7.2.

**DIAGRAM**

**FIGURE 7.2** *Additive white Gaussian noise (AWGN) model of a channel.*

**7.3 CONCEPT OF OPTIMUM RECEIVER**

** **As a matter of fact, the receiver observes the received signal x(t) for a duration of *T* seconds and makes a best estimate of the transmitted signal s_{i}(t) or equivalently the estimate of symbols m_{i}. But, due to the presence of channel noise, this decision-making process is statistical in nature. As a result of this, the receiver is likely to make occasional errors.

Therefore, the requirement is to design the receiver so as to minimize the average probability of symbol error.

This average probability of symbol error may be defined as

…(7.4)

where m_{i} = transmitted symbol,

= estimate produced by the receiver, and

P = the conditional error probability given that the ith symbol was sent.

The resulting receiver is called to be optimum in the minimum probability of error sense. The above model provides a basis for the design of the optimum receiver, for which, a geometric representation of the known set of transmitted signals, {s_{i}(t)}, will be used.

**7.4 GEOMETRIC REPRESENTATION OF SIGNALS (Very Important)**

In geometric representation of signals, we represent any set of M energy signals {s

_{i}(t)} as linear combinations of

*N*orthonormal basis functions, where N £ M.

This means that given a set of real-valued signals s

_{i}(t), s

_{2}(t), …, s

_{M}(t), each of duration

*T*seconds, we may write s

_{i}(t) as under:

**EQUATION**

where, the coefficients of the expansion can be defined as

**EQUATION**

Now, the real-valued basis functions , , …, are orthonormal. Here, the wort `orthonormar implies that

**EQUATION**

where is the Kronecker delta.

From equation (7.7), we note the following two points:

(i) the first condition in equation (7.7) states that each basis function is normalized to have unit energy.

(ii) the second condition states that the basis functions , , …, are, orthogonal with respect to each other over the internal 0 £ t £ T.

In fact, the set of coefficients may be viewed as an

*N*-dimensional vector, represented by s

_{i}*

With respect to vector s

_{i}and transmitted signal s

_{i}(t), let us note the following two points:

(i) Given the

*N*elements of the vectors s

_{i}(i.e., s

_{i1}, s

_{i2}, …, s

_{iN}) operating as input, the scheme shown in figure 7.3 may used to generate the signal s

_{i}(t). It consists of a group of

*N*multipliers. Here, each multiplier has its own basis function, followed by a summer. In fact, this scheme may be called as synthesizer.

(ii) Conversely, given the signals s

_{i}(t), i = 1, 2, …, M, operating as input, the scheme shown in figure 7.3(b) may be used to calculate the coefficient s

_{i1}s

_{i2},…s

_{iN}, which follows directly from equations (7.6).

This second scheme consists of a group of

*N*product-integrators or correlators with a common input. Each correlator is supplied with its own basis function.

In fact, the scheme in figure 7.3 (b) may be called as an

**analyzer.**

**DIAGRAM**

**FIGURE 7.3**

*(a) Synthesizer for generating the signal s*

_{i}(t),*(b) Analyzer for generating the set of signal vectors (s*

_{i}).Hence, based upon above two points, we can say that each signal in the set {s

_{i}(t)} is completely determined by the vector of its coefficients

s

_{i}= , i = 1, 2, …,M …(7.8)

Here, the vector s

_{i}is called a signal vector. Also, if we extend the conventional notion of two and three dimensional Euclidean spaces to an

*N-*dimensional Euclidean space, the set of signal vectors {s

_{i}|

*i*= 1, 2, …M} may be viewed as defining a corresponding set of M points in an N-dimensional Euclidean space, with mutually perpendicular axes labeled

* The vector s

_{i}bears a one-to-one relationship with the transmitted signal s

_{i}(t).

In fact, this

*N*-dimensional Euclidean space is called the signal space.

**NOTE It may be noted that the idea of visualizing a set of energy signals geometrically is of utmost importance. In fact, it provides the mathematical basis for the geometric representation of energy signals and hence giving the way for the noise analysis of digital communication systems in a much satisfied manner. Figure 7.4 illustrates this form of representation for the case of a two-dimensional signal space with three signals i.e., N = 2 and M = 3.**

In an

*N*-dimensional Euclidean space, we may define lengths of vectors and angles between vectors. It is customary to denote the length (also called the absolute value or norm) of a signal vector s

_{i}by the symbol

**| |**s

_{i}

**| |**. The squared-length of any signal vector s

_{i}is defined to be the inner product or dot product if s

_{i}with itself, as shown by

**EQUATION**

where s

_{ij}is the

*j*th element of s

_{i}, and the superscript

*T*denotes matrix transposition.

There is an interesting relationship between the energy content of a signal and its representation as a vector. By definition, the energy of a signal s

_{i}(t) of duration

*T*seconds is

Ei = (t) dt …(7.10)

Therefore, substituting equation (7.5) into equation (7.10), we get

**equation**

Interchanging the order of summation and integration, and the rearranging terms, we get

**equation**

But, since the from and orthonormal set, in accordance with the two conditions of equation (7.7), we find that equation (7.11) reduces simply to

**equation**

**diagram**

**FIGURE 7.4**

*Illustrating the geome-tric representation of signals for the case when N = 2 and M=3.*

Thus equations (7.9) and (7.12) show that the energy of a signal s

_{i}(t) is equal to the squared length of the signal vector s

_{i}(t) representing it.

In the case of a pair of signals s

_{i}(t) and s

_{k}(t), represented by the signal vectors s

_{i}and s

_{k}, respectively, we may also show that

Equation (7.13) states that the inner product of the signal s

_{i}(t) and s

_{k}(t) over the interval [0, 7], using their time-domain representations, is equal to the inner product of their respective vector representations s

_{i}and s

_{k}. It may be noted that the inner product of s

_{i}(t) and s

_{k}(t) is invariant to the choice of basis functions in that it only depends upon the components of the signals s

_{i}(t) and s

_{k}(t) projected onto each of the basis functions.

Yet another useful relation involving the vector representations of the signals s

_{i}(t) and s

_{k}(t) is described by

**equation**

or

**equation**

where | | s

_{i}– s

_{k}| | is the Euclidean distance, d

_{ik}, between the points represented by the signal vectors s

_{i}and s

_{k}.

To complete the geometric representation of energy signals, we need to have a representation for the angle

_{ik}subtended between two signal vectors s

_{i}and s

_{k}. By definition, the cosine of the angle

_{ik}is equal to the inner product of these two vectors divided by the product of their individual norms, as shown by

equation …(7.15)

The two vectors s

_{i}and s

_{k}are thus orthogonal or perpendicular to each other if their inner product is zero, in which case

_{ik}, = 90 degrees; this condition is intuitively satisfying.