Hi, in this lecture, we will think about the concept of information, especially in a digital format. In our life, we always get information from what we sense. For example, the woman can understand the man's thought by listening to him. What do you see? To see this picture delivers the visual information of a beautiful scenery to us. If it is quite impressive, we may describe a visual information by writing a letter to friends. Nowadays, every information including video and audio is stored and delivered in a digital format by using zero and one. I think most of you enjoy to see interesting videos by using your smart phones. So do I. And some you know that for a video, there is a number of unit meta bps. Here, in this example, the number is 20 meta bps, and it is related to the video quality. Originally, a video is what we see if we were there, but how can it be delivered to us in a digital format? Then naturally arriving questions are as follows. First, what is the definition and essence of information? And then, how can we quantify or measure the amount of information? And finally, how can we represent visual information in a digital format? In this lecture, you will get the answers. Suppose that a die is cast, but you cannot see the result due to a wall. In other words, there is an uncertainty to you on the result. Now, suppose that there is a friend of you who can see the result. Then your friend is able to deliver the information on the result, by talking to you. So information is transferred. Once the information of the result is transferred to you, the uncertainty is now cleared. That means that the amount of the original uncertainty is same to the required amount of information for clearing the uncertainty. This shows the essence of information, simply every information is what we don't know. And uncertainty is cleared if we get information. Now, consider how we can quantify information. Here, a man is flipping a fair coin. Suppose that he flips the coin once in a second, and the result during four seconds are head, tail, heads, and tail. Now it's your turn to describe the result in a digital format with zeros and ones. One straightforward way is to map one for head and zero for tail. Then the first result is zero, and the next one is one. And then zero, and the final one is one. So the result is zero, one, zero, one. So we need four binary digit for describing the result during four seconds. So you get how to quantify information. Bit is the unit for the amount of information, and 1 bit means the amount of information that can describe the flipping result of a fair coin. Also, the unit for the information rate is the number of bits per second, bps. In this case, we need 1 bit per second, so that the information rate is 1 bps. So 20 meta bps video has information equivalent to about 20 million coin flipping result in each second. Now think about the case where the coin is not fair. So that the probability of tail is P and that of head is 1- P. Then what happens? Do we need more amount of information to describe it, or less? The answer is simple. We call that the amount of information is the amount of uncertainty. If the tail probability is zero, there is no uncertainty, and consecutive heads will occur. So you can easily see that the uncertainty is maximized when the coin is fair. Again, bit is the unit for the amount of information, and 1 bit means the amount of information that can describe the flipping result of a fair coin. And the unit for the information rate is the number of bits per second, an acronym bps. And one bps means that the information rate, that can describe the results of one fair coin flipping per second. Now you know what the information is and how to quantify the information. So you already get the answers to the first two questions. And let's get how a visual information is represented in a decent format. Voice or images signal is analog. That means it can be represented as a continuous time function having continuous values at each time. Then, how can we represent it with binary digits? Here, we define the term frequency. The unit for the frequency is hertz, and it denotes the number of repetitions in a second. One could think that mathematicians found is that an analog signal can be decomposed according to its continuous frequencies. Here, low frequency means that a signal component is slowly changing. And high frequency means that a signal component is fast changing as time goes. And suppose that the maximum frequency component of the signal is W hertz. First thing we need to do is to sample the signal. After sampling, we get a discrete time sequence having continuous values. Here what we want is that we can recover the original signal from the samples later. Here, in case the signal changes fast, that means W is large, then we need more samples per second to represent the fast changing signal. On the other hand, if the signal changes slowly, that means W is small, then we need less number of samples per second. Here 2 times the maximum frequency W is called the Nyquist rate. And the recovery of the original signal is guaranteed, if the sample rate is above or equal to the Nyquist rate ionized theorem called sampling theorem. After a proper sampling, we get a sequence having continuous values. Then if we can represent the sample with binary digits, then the signal can be represented by a sequence of binary digits. But if representing a continuous value exactly, it requires infinite number of bits because the resolution is infinite. So we need to restrict the resolution and represent each continuous value in V(n) by selecting one of the finite candidates in V hat n, such an operation is called quantization. Unlike the sampling, quantization causes distortion. If we increase the resolution of quantization, the distortion is reduced, but the information rate, the number of bits per sample times the number of samples per second, will increase. So the original video information is converted into digital information by sampling and quantization. In practice, further compression techniques are typically used. Once it is converted into digital format, it can be stored or transmitted and then finally reconstructed. In this way, you can see a video by using your smart phone. And finally, as you can see, there is a trade-off between distortion and information rate. So as we want to improve the video quality, we need larger information rate. Now, you get the answers.