Coding teaser: source coding

Make sure to read the Wiki page about the basics of a channel. Several discussions and interpretations are important for this chapter 😊.

Complete Channel Model

Previously, we showed a simple channel model that consists of the source, channel, and receiver. In modern communication systems, we also add an encoder after the source and a decoder before the receiver. Figure 1 shows the complete channel model. The encoder is used to transform the source symbols into a code that suitable for the channel. The receiver decodes the received message coming out of the channel.

Figure 1: Complete channel model with the encoder added after the source and the decoder added before the receiver.

Let's bring back Bob and Alice into the scene again. This time let's use the complete channel model. Since most of our communication systems use binary representations, Bob needs to encode his data into binary digits before he sends it over the channel.

Let's look at a few examples. Suppose our source alphabet $S$ has the symbols of all combinations of flipping three fair coins. Then our symbols would be $\{TTT,TTH,THT,THH,HTT,HTH,HHT,HHH\}$. Bob needs to map each symbol into a binary representation such that $\{TTT,TTH,THT,THH,HTT,HTH,HHT,HHH\}\rightarrow \{000,001,010,011,100,101,110,111\}$. This is the encoding process. Let's add new definitions:

• The binary set $\{0,1\}$ are the code symbols.
• The encoded patterns $\{000,001,010,011,100,101,110,111\}$ are called the codewords. For example, $000$ is the codeword for the symbol $TTT$. $101$ is the codeword for $HTH$. In practice, we let some random variable $X$ whose outcomes are codewords $\{x_{1},x_{2},x_{3},...,x_{n}\}$ associated with probabilities $\{P(x_{1}),P(x_{2}),P(x_{3}),...,P(x_{n})\}$. For example, $x_{1}=000$ is the codeword for $TTT$. Moreover, from our example, the source symbols are equiprobable and we have a direct mapping from source symbol to codeword (i.e., $s_{i}\rightarrow x_{i}$ ) then it follows that the codewords are equiprobable too: $P(x_{1})=P(x_{2})=P(x_{3})=P(x_{4})=P(x_{5})=P(x_{6})=P(x_{7})=P(x_{8})={\frac {1}{8}}$. The probability distribution of $X$ can change depending on the encoding scheme.
• We usually tabulate the encoding or mapping. We call this table the codebook. It's a simple look-up-table (LUT). Let's call the table below as codebook-1.

Source Symbol	Codeword
TTT	000
TTH	001
THT	010
THH	011
HTT	100
HTH	101
HHT	110
HHH	111

The encoder looks at this table and endcodes message before it gets into the channel. For example, suppose we want to send the message $TTTHHHTHTHTH$. The encoder uses codebook-1 to translate the source message into an encoded message: $000111010101$. Receivers should also know the same codebook-1 so that they can decode the message. For example, if the encoded message is $101011001110$, then we can decode the message as $HTHTHHTTHHHT$. Choosing the encoding where the $T\rightarrow 0$ and $H\rightarrow 1$ is simple. This encoding is arbitrary and it seems convenient for our needs. However, nothing stops us from encoding the source symbols with a different representation. For example, we can choose an encoding where $T\rightarrow 1$ and $H\rightarrow 0$. We can go for more complex ones like codebook-2 as shown below:

Source Symbol	Codeword
TTT	01
TTH	001
THT	0001
THH	00001
HTT	000001
HTH	0000001
HHT	00000001
HHH	000000001

Where a source message $TTTHHHTHTHTH$ translates into $0100000000100010000001$. Codebook-2 can also decode an encoded message. For example, when we receive $00000010000100100000001$ we get $HTHTHHTTHHHT$. The choice of the encoding process is purely arbitrary. However, we are interested if the encoding efficient. One metric is the average code length defined as:

$L(X)=\sum _{i=1}^{n}P(x_{i})l_{i}$

(1)

Where $l_{i}$ is the code length of the codeword $x_{i}$. For example, the average code length of codebook-1 is:

$L(X)=8\cdot {\frac {1}{8}}\cdot 3=3$

The average code length of codebook-2 is:

$L(X)={\frac {1}{8}}\cdot 2+{\frac {1}{8}}\cdot 3+{\frac {1}{8}}\cdot 4+{\frac {1}{8}}\cdot 5+{\frac {1}{8}}\cdot 6+{\frac {1}{8}}\cdot 7+{\frac {1}{8}}\cdot 8+{\frac {1}{8}}\cdot 9=5.5$

Clearly, codebook-1 has a shorter codelength than codebook-2. Let's introduce another metric called the coding efficiency. Coding efficiency is defined as:

$\eta ={\frac {H(S)}{L(X)}}$

(2)

Where $H(S)$ is the source entropy. The coding efficiency $\eta$ has a range of $0\leq \eta \leq 1$. If you recall our "go-to" interpretation of entropy, given some entropy $H(S)$ means we can represent $m=2^{H(S)}$ outcomes. Essentially, $H(S)$ is the minimum number of bits we can use to represent all outcomes of a source regardless of its distribution. Of course, a smaller coding length is desirable if we want to compress representations. For our simple coin flip example, $H(S)=3$. The efficiency of codebook-1 is $\eta ={\frac {3}{3}}=1$. Codebook-1 is said to be maximally efficient because it matches the minimum number of bits to represent the source symbols. The efficiency of codebook-2 is $\eta ={\frac {3}{5.5}}=0.5455$.

$L(X)$ should never be lower than $H(S)$. Otherwise, that doesn't make sense. How can you represent $H(S)=3$ with $L(X)=2$? Something's wrong with your codebook if this happens.

Channel Rates and Channel Capacity

Let's bring back Bob and Alice into the scene again. When Bob sends his file to Alice, it takes a while before she gets the entire message because the channel has limited bandwidth. Bandwidth is the data rate that the channel supports. Let's define $C$ as the maximum channel capacity in units of bits per second (bits/s). We'll provide a quantitative approach later. Channels have different rates because of their physical limitations and noise. For example, an ethernet bandwidth can achieve a bandwidth of 10 Gbps (gigabits per second), while Wifi can only achieve at a maximum of 6.9 Gbps. It really is slower to communicate wirelessly. It is possible to transmit data to the channel at a much slower rate such that we are not utilizing the full capacity of the channel. We'd like to encode our source symbols into some codewords that utilizes the full capacity of a channel. Shannon's source coding theorem says:

"Let a source have entropy $H$ (bits per symbol) and a channel have a capacity $C$ (bits per second). Then it is possible to encode the output of the source in such a way as to transmit at the average rate $C/H-\epsilon$ symbols per second over the channel where $\epsilon$ is arbitrarily small. It is not possible to transmit at an average rate greater than $C/H$."

Take note that, $\epsilon$ in Shannon's statement is not the noise probability that we know. It's just a term that is used to indicate a small margin of error. In a nutshell, Shannon's source coding theorem says that we can calculate a maximum symbol rate (symbols/s) that utilizes the maximum channel rate $C$ while knowing the source entropy $H(S)$. This maximum symbol rate is given by:

$R_{\textrm {max}}={\frac {C}{H(S)}}$

(3)

Let's look at a few examples to appreciate this.

Channel Rate Example 1: Three Coin Flips

Let's re-use the three coin flips example again. Suppose we want to send the three coin flip symbols over some channel that has a channel capacity $C=3$ bits/s. We also know (from the above example) that we need a minimum of $H(S)=3$ bits per symbol to represent each outcome. This translates to:

$R_{\textrm {max}}={\frac {3\ {\textrm {bits/s}}}{3\ {\textrm {bits/symbols}}}}=1\ {\textrm {symbols/s}}$

If we are to utilize the maximum capacity of the channel, we need to encode the source symbols $\{TTT,TTH,THT,THH,HTT,HTH,HHT,HHH\}$ to its binary representation such that each binary digit in each codeword $x_{i}$ provides 1 bit of information . Luckily, codebook-1 does this because we have a coding efficiency of $\eta =1$ which means that every binary digit of the codeword provides at least 1-bit of information. We are maximally efficient with this encoding. If you think about it, if we send 1 symbol/s to the channel and our encoding has 1 symbol (e.g., TTT) is 3 bits long then essentially we are able to send 3 bits/s to the channel. Since $C=3$ bits/s, then we utilized the entire maximum channel capacity.

From the above example, can we actually go beyond $R_{\textrm {max}}={\frac {C}{H(S)}}$ symbols/s? NO. Shannon explicitly said that "It is not possible to transmit at an average rate greater than $C/H$." This is the maximum symbol rate already.

Channel Rate Example 2: A 6-sided Die

This time, suppose we want to send the symbols of a fair 6-sided die on a channel that only has $C=1$ bits/s. Such that $S$ has outcomes $\{1,2,3,4,5,6\}$. Of course, all outcomes are equiprobable with $P(s_{i})={\frac {1}{6}}$. From here, we can determined $H(S)=\log _{2}(6)=2.58$ bits/symbol. Therefore:

$R_{\textrm {max}}={\frac {1\ {\textrm {bits/s}}}{2.58\ {\textrm {bits/symbols}}}}=0.387\ {\textrm {symbols/s}}$

Again, Shannon says that with the given $C$ and $H(S)$ we can only transmit at a maximum symbol rate of $R_{\textrm {max}}$ symbols/s. Now, let's think of an encoding scheme that suits our needs. For convenience, to represent 6 unique outcomes, we can use 3 bits to represent each outcome. Let's say we have codebook-3:

Source Symbol	Codeword
1	000
2	001
3	010
4	011
5	100
6	101

The coding efficiency for codebook-3 and for the 6-die example is:

$\eta ={\frac {H(S)}{L(X)}}={\frac {2.58}{3}}=0.86$

This is not that efficient but it will suffice for now. Here, let's introduce the effective symbol rate $R_{X}$ due to the new encoding:

$R_{X}={\frac {\eta C}{H(S)}}$

(4)

Our encoding scheme affects how much data we're putting into the channel. If our coding efficiency isn't optimal $\eta \neq 1$ then we are only utilizing a fraction of the maximum capacity $C$. Hence we needed the effective channel transmission rate $\eta C$ incorporated into our effective symbol rate $R_{X}$. Calculating $R_{X}$ for the encoding in codebook-3 would give:

$R_{X}={\frac {0.86\cdot 1}{2.58}}=0.333$ symbols/s.

Observe that $R_{X}=0.333<R_{\textrm {max}}=0.387$. We are not even close to the maximum rate. Can we find a better encoding? How about this, suppose we create a composite symbol that consists of the three 6-sided die symbols. For example, we can let $111$ be one symbol, or we can let $555$ be another symbol. If we do so, then we have around $6\times 6\times 6=216$ unique possible outcomes for the new set of symbols. To represent 216 outcomes we need a minimum of 8 bits of codeword length (because $2^{8}=256$). So we can have codebook-4 as:

Source Symbol	Codeword
111	0000_0000
112	0000_0001
113	0000_0010
...	...
665	1101_0110
666	1101_0111

The underscores are just there to separate the 4 bits for clarity. Codebook-4 looks really long but in practice, that's still small. So what changes? Since we have 8 bits per 3 symbols, we effectively have $L(X)={\frac {8}{3}}$. Take note that all outcomes are equiprobable so dividing the 8 bits by 3 symbols is enough. This results in a coding efficiency of:

$\eta ={\frac {H(S)}{L(X)}}={\frac {2.58}{2.66}}=0.97$

Cool! We're much closer to being maximally efficient. Now, let's compute the new $R_{X}$ for codebook-4:

$R_{X}={\frac {\eta C}{H(S)}}={\frac {0.97}{2.58}}=0.376$ symbols/s.

Superb! The new $R_{X}=0.376$ is close to the maximum rate of $R_{\textrm {max}}=0.387$. The key trick was to create a composite symbols that consists of 3 of the original source symbols, then represent them with 8 binary digits of codewords.

In all of the discussions, we've only used different measures of information and channel rates. None of the discussions talked about a methodology of finding the best encoding scheme. This is because information theory does not tell us how to find the best encoder. It's just a measure. Keep that in mind!

Maximum Channel Capacity

Figure 2: $I(R,S)$ of a BSC with varying $p$ and parametrizing $\epsilon$.

This time, let's look into how we can quantitatively determine the maximum channel capacity. In the last module, we comprehensively investigated the measures of $H(R)$, $H(R|S)$, and $I(R,S)$ on a BSC. Recall that:

$I(R,S)=H(R)-H(R|S)$

If you think about it, $I(R,S)$ is a measure of channel quality. It tells us how much we can estimate the input from an observed output. $I(R,S)$ is a measure of the received information contained in the source information. Ideally if there is no noise, then $H(R|S)=0$ resulting in $I(R,S)=H(S)=H(R)$. We want this because this means all the information from the source gets to the receiver! When noise is present $H(R|S)\neq 0$, it degrades the quality of our $I(R,S)$ and the information about the receiver contained in the source is now being shared with noise. This only means one thing: we want $I(R,S)$ to be at maximum. From the previous discussion, we presented how $I(R,S)$ varies with $p$ while parametrizing the noise probability $\epsilon$. Figure 2 shows this plot. We can visualize $I(R,S)$ like a pipe while the noise parameter $\epsilon$ limits the flow of information. Figure 3 GIF shows the pipe visualization.

Figure 3: Pipe visualization for $I(R,S)$

In communication theory, $I(R,S)$ is a metric for transmission rate. So the units are in bits/second. Interchanging between bits or bits/second might be confusing but for now, when we talk about channels (and also $I(R,S)$), let's use bits/s. We can define the maximum channel capacity $C$ as:

$C=\max(I(S,R))$

(5)

We can re-write this as:

$C=\max(H(R)-H(R|S))$

(5)

Here, $C$ is in units of bits/s. It tells us the maximum number of data that we can transmit per unit of time. Both $H(R)$ and $H(R|S)$ are affected by noise parameter $\epsilon$. In a way, noise limits $C$. For example, a certain $\epsilon$ can only provide a certain maximum $I(R,S)$. Looking at figure 2, when $\epsilon =0$ (or $\epsilon =1$) we can get a maximum $I(R,S)=1$ bits/s provided that the probability distribution of the symbols that get into the channel is $p=0.5$. When $\epsilon =0.25$ (or $\epsilon =0.75$) we can also get a maximum $I(R,S)=0.189$ bits/s when $p=0.5$. Let's determine $C$ of a BSC given some $p$ and $\epsilon$. Since we know $H(R)$ follows the Bernoulli entropy $H_{b}(q)$ where $q=p+\epsilon -2\epsilon p$. Then we can write:

$H(R)=H_{b}(q)=-q\log _{2}(q)-(1-q)\log _{2}(1-q)$

$\max(H(R))=1$ bits when $q=0.5$. Since $\epsilon$ is fixed depending on a channel, we can manipulate the input probability $p$ of the symbols that get into the channel such that we can force $q=0.5$ and achieve the maximum $H(R)$.

$p={\frac {0.5-\epsilon }{1-2\epsilon }}$

Interestingly, to achieve $q=0.5$ we simply need to set $p=0.5$ for any $\epsilon \neq 0.5$. The exception is when $\epsilon =0.5$ because that results in an absolute $I(R,S)=0$ and $p$ will be undefined (i.e., ${\frac {0}{0}}$). This result shows that for any $\epsilon \neq 0.5$ we can achieve maximum entropy for that given channel as long as we encode our source symbols to have $p=0.5$ before it gets into the channel! We can continue our derivation by writing $C$ as:

$C=\max(H_{b}(q)-H_{b}(\epsilon ))$

Leading to:

$C=1-H_{b}(\epsilon )$

(6)

In equation 6, we can assume $H_{b}(q)=1$ because we know we can achieve maximum entropy when we set $q=p=0.5$. Equation 6 shows how the maximum capacity is limited by the noise $\epsilon$. It describes the pipe visualization in figure 3. When $\epsilon =0$, $H_{b}(\epsilon )=0$ and our BSC has $C=1.0$ bits/s. When $\epsilon =0.5$, $H_{b}(\epsilon )=1$, the noise entropy is at maximum and our channel has $C=0$ bits/s. Clearly, we can't transmit data this way. Lastly, when $\epsilon =1$, then $H_{b}(\epsilon )=0$ again and we're back to having $C=1$ bits/s.

The most important concept in determining maximum channel capacity is to use equation 5 for any channel model. Moreover, from our simple BSC derivation, we can see that in order to maximize the effective channel capacity, we need to encode our source symbols into codewords such that $p\rightarrow 0.5$ before it gets sent over the channel. This is the essence of the concept source coding where we encode the source to achieve maximum entropy. This is related to equation 4 where $\eta C$ tells us the effective channel rate assuming we have not encoded our source symbols to achieve $p=0.5$. If we have successfully achieved $p=0.5$ for the encoded codewords, then on average we utilize the maximum capacity $C$ (i.e., $\eta =1$).

Don't get mad but despite having rigorous discussions about channels and noise, data compression we assumes a noiseless channel. Phew! So much for going through the hard efforts.

Huffman Coding

As a nice sneak peak into data compression, let's look into Huffman coding.