數位信號處理器_林顯易_第六單元 數位系統實現_Part1 直接架構

Hi everyone, welcome to the digital signal processor (DSP) course, Unit 6 – Realization of Digital Systems. In this unit, we will teach everyone the means of creating a digital system such as direct forms, or series forms, or frequency sampling as well as lattice forms in creating our digital system. So what are the contents? Let's go and take a look. The learning objectives and outline of this unit will start off with the direct structure, serial connection structure, frequency sampling structure, and finally the lattice structure. Now basically, we can think of this problem: if a linear time-invariant system can be written as this difference equation, where this part is the output, the past output. and this part is the input, and the past input. Now how do we create this digital system? Let's take a look at this. Firstly, this is a formula, an algorithm. We can re-write it as a Z-transform method. The system's Z-transform can be equated to this rational, can be expressed as a rational function. This is what we've previously mentioned. So there are some equations that can be written in this way. But we can add some block diagrams or use some delay components, for example, z to the -k power, to represent these delay components and create the system. How do we create this system? To create this system, we basically have a few considerations. First, we will consider the complexity of the calculations. For example, the system and the mathematical operations it uses such as multipliers, adders, and division, and its quantities. Because we know when we are trying to create early hardware, the multiplication usually takes a significant amount of time. So if we want to make the system faster, we must consider the frequency of using the multiplier. Secondly, when creating the system, we must consider the memory requirements. Memory requirements mean that when we perform this algorithm, we may be using past outputs or inputs, or we must store system coefficients or other temporary data of the system. These data must be stored in the memory. The quantity of the variables needed is strongly correlated to the size of the memory required. So when we look at the hardware, the memory would represent our cost. So how can we reduce this cost? This of course involves making the memory required smaller. Finally we must consider the effects of limited character length. This is also what we refer to as the quantization error. In a digital system, we know all data are represented by numbers. When a number is used to represent a data, it must be quantified. You can use many levels of precision such as 16 bits, or 32 bits to represent a number. Different bit sizes will lead to different levels of precision. Different levels of precision can help us generate different final outputs. This would be what we define as quantization errors. So these 3 points must be considered when we create a system. Next, we will talk about in the following structure on our considerations when creating the system form and if there are better forms including parallel processing which can help accelerate our computing speed. OK, first, we will introduce the direct form. For direct form, we will first introduce FIR. We have mentioned FIR for the first time here. I believe that the course on Filter designs and later units will mention this. So here we have mentioned it first so that you will know how to create an FIR form. Later, when we get to filter design, we will provide a more detailed explanation. FIR is the finite impulse response filter. which is abbreviated as FIR. Mathematically, it is expressed as this equation. So its output is equal to its past input multiplied by the filter coefficient. This is its length, the length of the filter. When we look at this formula, we know that to achieve this output, we need M -1 an input of M-1, as well as M multipliers. Because we have 1 multiplier here and M-1 adders. Given the large numbers involved, we would need M-1 adders. So if you use a direct form to express this type of output, we can see that our inputs will undergo different delays because it’s K will change. So after passing through different delays, different delays and multiplied by this coefficient, we would add them up, finally add them up to get the final output of the system. So from this direct form, we can see that an FIR filter and its basic expression. It is created via direct form which looks very simple. Next we will consider if we were to design an FIR an FIR filter, the benefit is that the filter has linear phases. Linear phase means that the system, its phase, which is Theta. We know that when differentiating Theta and Omega, we would obtain a constant. This property is the linear phase. The benefit of this is that for a system output, if your system, this system, has a linear phase, it means that your input signal will have a fixed delay time. After a Fourier transform, we would have an additional term. If we differentiate this by Omega we will get a constant. This means that in the time domain, all frequency components would be subject to the same time delay, otherwise the filtered signal would be distorted. So linear phases have this benefit. Now let's look at FIR with linear phase. The property is that its filter coefficients are symmetric. They are symmetrical. So if the formula is symmetrical, if it has this property, we can correct the direct form and change this single way from here to there and break it in half. Why half? because we know that its coefficients are symmetric. Coefficients at the beginning and at the end can be shared. For example, h0 can be shared with h M-1 So when we share these coefficients and bring x(n) in after delay, and allow this to go out. Once this go out, both will use the same coefficient. Let's look at an example to make it clearer for everyone. For example, when M = 5 so in this example, M = 5, we have a formula where M = 5 we can expand this formula into this. When M = 5, we would find out that for linear phase, it will satisfy this formula. This means that h0 and h4 are basically the same. My h3 and h1 are the same as well. So, I can combine these pairs and bring out the coefficient. So we represent this by h0, and combining these two, we would represent this with h1, and combine the x terms behind the h together. What is the benefit of this? We can cut down on the number of coefficients to give us this. In other words, once my equation becomes this I can take h0 and h1 and multiply them with this term and this term. Adding these two together, and these two together, this route and this route together, sharing the same coefficient we basically reduced the number of multiplier needed by half. So linear phase and its symmetrical property help us to significantly reduce the multipliers we need, and the time needed by the multipliers. So we have IIR, standing for infinite impulse response happens to be another type of mathematical formula depicted in this way. So it uses the past output summed with the current input. So this formula, if we intend to rewrite it as a Z-transform, then we have previously talked about this before. OK, so now we can see the means of creating the direct form. This is the way it looks. Here is the input. So after the delay, it is multiplied by the coefficient to generate the output here. After the delay, we take this back and multiply it by the coefficient add them all up, and add these together and use the direct form to create an IIR. The reason is simple. We need M + N delays and M + N + 1 multipliers. We have mentioned this before. If we use the Canonical form, Direct Form II, what happens? We transpose the left and right sides, which is to say, I would transpose these two things from the previous page so that we can share a single delay. Hence, we can change this into a Direct Form II with the delay at the middle the delay it needs will become the maximum value of M and N. Hence, you can reduce the number of delays. Since it needs so many multipliers, so we would use Direct Form II for the IIR realization. So in this part, we have introduced the direct form for digital system realization. We have learned that to create an actual digital system, we must consider its requirements such as its computational load and memory requirements. So in the direct form, we achieve less significant reductions in computational load. But we have introduced linear phase filters that can help us reduce the number of multipliers needed. In the next unit, we will introduce the series form, which helps achieve a further reduction in computational load. Let's go on and take a look.