Optimum Probability Distribution for Minimum Redundancy of Source Coding ()
Keywords:Mean Codeword Length; Uniquely Decipherable Code; Kraft’s Inequality; Entropy; Optimum Probability Distribution; Escort Distribution; Source Coding
1. Introduction
Any message that brings a specification in a problem which involves a certain degree of uncertainty is called information and it was Shannon [1] who named this measure of information as entropy. In coding theory, the operational role of entropy comes from the source coding theorem which states that if 
is the entropy of the source letters for a discrete memoryless source, then the sequence of source outputs cannot be represented by a binary sequence using fewer than 
binary digits per source digit on the average, but it can be represented by a binary sequence using as close to 
binary digits per source digit on the average as desired. To be clearer, let us consider the discrete source
that emits symbols 
with probability distribution
where
. The aim of source coding is to encode the source using an alphabet of size
, that is, to map each symbol 
to a codeword 
of length 
expressed using the 
letters of the alphabet. It is known that if the set of lengths 
satisfies Kraft’s [2] inequality
(1.1)
then there exists a uniquely decodable code with these lengths, which means that any sequence 
can be decoded unambiguously into a sequence of symbols
. In this respect, Shannon [1] proved the first noiseless coding theorem for the uniquely decipherable code in the form of following inequality
(1.2)
where 
is a Shannon’s entropy and 
is the mean codeword length.
Later, Campbell [3] and Kapur [4] proved the source coding theorems for their own exponentiated mean codeword length in the form of following inequalities
(1.3)
and
(1.4)
respectively, where
is Campbell’s [3] mean codeword length,
is Kapur’s [4] mean codeword length and
is Renyi’s [5] measure of entropy.
Recently, Parkash and Kakkar [6] introduced two mean codeword lengths given by
(1.5)
and
(1.6)
Further, the authors provided two source coding theorems which show that for all uniquely decipherable codes, the mean codeword lengths 
and 
satisfy the relation:
(1.7)
and
(1.8)
respectively where
is a Kapur’s [4] two parameter additive measure of entropy and
is measure of entropy developed by Parkash and Kakkar [6].
This is to emphasize that in the entire literature of source coding theorems, one can observe that the mean codeword length is lower bounded by the entropy of the source and it can never be less than the entropy of the source but can be made closer to it. This phenomenon provides the idea of absolute redundancy which is the number of bits used to transmit a message minus the number of bits of actual information in the message, that is, the mean codeword length minus the entropy of the source. The objective of the present communication is to minimize this redundancy in order to increase the efficiency of the source encoding. For this purpose we have made use of the concept of escort distribution as follows:
If 
is the original distribution, then its escort distribution is given by 
where 
for some parameter
. Many researchers including Harte [7], Bercher [8,9], Beck and Schloegl [10] etc. used this distribution in their respective findings.
The aim of the present paper is to obtain the optimum probability distribution with which the source should deliver messages in order to minimize the absolute redundancy. To obtain our goal, we have taken into consideration the above mentioned generalized mean codeword lengths. Moreover, the upper bound to these codeword lengths has been found for Huffman [4] encoding.
2. Optimum Probability Distribution to Minimize Absolute Redundancy
Let us assume that for discrete source 
that emits symbols 
with probability distribution
, the codewords 
having lengths
, have been obtained using some encoding procedure on noiseless channel. Further, we assume that entropy of the source is 
and average codeword length is
. Since from (1.7), we have
, therefore, the average redundancy of the source code is given by
(2.1)
where 
and
.
In order to minimize the average redundancy, we resort to the following theorem:
Theorem 1: The optimum probability distribution that minimizes the absolute redundancy 
of the source with entropy 
and the mean codeword length 
is the escort distribution, given by
(2.2)
Proof: To minimize the redundancy, we need to minimize
(2.3)
subject to the constraint
(2.4)
To prove this, we first of all, find the extremum of 
which is equivalent to extremizing
and then use the fact that 
is minimum or maximum will depend upon the value of parameter
.
So, in order to extremize
, we consider the Lagrangian given by
where 
is Lagrange’s multiplier.
Now
(2.5)
Letting
, we get
(2.6)
Substituting (2.6) in (2.4), we get
(2.7)
Substituting (2.7) in (2.6), we get the result (2.2).
Now,
(2.8)
We see that 
for 
and 
for
.
Also,
So, 
has minimum value for 
and maximum for
.
Thus, 
has minimum value for 
and maximum for 
and consequently, observing the function
, we see that it has minimum value for
,
.
Thus, the minimum value is given by
. (2.9)
Again, the necessary condition for the construction of uniquely decipherable codes is given by
(2.10)
Therefore, from (2.9), we have
.
NOTE: It is to be noted that 
if the source is Huffman [11] encoded since for the Huffman encoding, we have
. (2.11)
Therefore, for this case, (2.2) becomes
(2.12)
Similarly, if we consider the codeword length 
which satisfies the relation
, then the absolute redundancy of the source code in this case is given by
where 
and
.
Theorem 2. The optimum probability distribution that minimizes the absolute redundancy 
of the source with entropy 
and mean codeword length 
is the escort distribution, given by
(2.13)
Proof: We will find the extremum of 
which is equivalent to extremizing 
subject to constraint
(2.14)
Let us consider the Lagrangian given by
(2.15)
where 
is a Lagrange’s multiplier.
For an extremum, let
, that is,
(2.16)
Using (2.14), we get
(2.17)
Substituting (2.17) in (2.16), we get (2.13).
Also,
and
So, 
reaches its minimum value when 
and is given by
that is,
Note: Again in this case also, if the source is Huffman [11] encoded, then the probabilities are given by 
Next, we will find the upper bound on the codeword lengths 
and 
when the source is Huffman encoded.
Theorem 3. The exponentiated codeword length 
satisfies the following inequality
(2.18)
if the source is encoded using Huffman procedure.
Proof: The exponentiated codeword length 
can be written in the following form
(2.19)
where
.
Considering (2.12), (2.19) becomes
(2.20)
where
.
We need to find the extremum of 
subject to constraint 
(as the source is encoded using Huffman Procedure).
For this purpose, we first of all, find the extremum of 
which is equivalent to extremizing 
and then use the fact that 
is minimum or maximum depending upon the value of parameter
.
So, we consider the Lagrangian given by
(2.21)
where 
is a Lagrange’s multiplier Put
, (2.21) becomes
Letting
, we get
(2.22)
Now, 
gives
(2.23)
Using (2.23) in (2.22), we get
that is, 
Now,
We see that 
for 
and 
for
.
Also,
So, 
has minimum value for 
and maximum for
.
Therefore, 
has minimum value for 
and maximum for 
and consequently, observing the exponentiated mean codeword length
, we see that it has maximum value for
,
.
Thus, the maximum value is given by
.
Theorem 4. The mean codeword length 
is upper bounded by 
, that is,
(2.24)
if the source is encoded using Huffman procedure.
Proof: The exponentiated codeword length 
can be written in the following form
(2.25)
We need to find the extremum of 
subject to constraint 
(as the source is encoded using Huffman Procedure).
So, we consider the Lagrangian given by
(2.26)
where 
is a Lagrange’s multiplier .
Letting
, we get
(2.27)
Since 
, we have
(2.28)
Substitute (2.28) in (2.27), we get
Now,
Also,
So, the mean codeword length 
has maximum value when 
, and is given by
.
Note-I: For the case of Campbell’s codeword length
, we have from (1.3),
. So, the average redundancy of the source code in this case is given by
where 
and 
The absolute redundancy in the case of Campbell’s [3] mean codeword length is the same as in case of exponentiated mean codeword length 
developed by Parkash and Kakkar [6] as given in (2.1). Thus, we see that similar results as proved in theorem (2.1) and theorem (2.3) hold for Campbell’s case also.
Note-II: Absolute redundancy when we use Kapur’s[4] mean codeword length is given by
where 
Theorem 5: The optimum probability distribution that minimizes the absolute redundancy of the source with entropy 
and mean codeword length 
is given by
. (2.29)
Proof: To minimize the redundancy, we need to minimize
(2.30)
subject to the constraint
(2.31)
To prove this, we first of all find the extremum of 
which is equivalent to extremizing 
and then using the fact that 
is minimum or maximum depending upon the value of parameter
.
So, in order to extremize
, we consider the Lagrangian given by
where 
is Lagrange’s multiplier.
(2.32)
Letting
, we get
(2.33)
Substituting (2.33) in (2.31), we get
(2.34)
Substituting (2.34) in (2.33), we get the result (2.29).
Now, 
We see that 
for 
and 
for
.
Also, 
So, 
has maximum value for 
and minimum value for
.
Therefore, 
has maximum value for 
and minimum value for 
and consequently observing the function
, we see that it has minimum value for
,
.
The minimum value is given by
.
Theorem 6. The Kapur’s [8] mean codeword length 
satisfies the following inequality
(2.35)
if the source is encoded using Huffman procedure.
Proof: Proceeding as in Theorem 2.3, we can prove the Theorem 6.
Acknowledgements
The authors are thankful to Council of Scientific and Industrial Research, New Delhi, for providing the financial assistance for the preparation of the manuscript.