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Design Specification
1. Purpose2. Design enviroment
3. Principle
3-1. CORDIC method
3-2. Extension to −π~π
3-3. Algorithm
4. Design
4-1. Design of the updated
value αi and δi
4-2. Judgement of the angle z
by the angle θ
4-3. Determination of the
initialvalue x0
4-5. Output example of the
value
5. Example to Level2
6. Challenge
7. Unit of measurement of the
circuit scale and speed
18th LSI Design Contests・in Okinawa Design Specification - 3
3-1. CORDIC method([1])
The CORDIC, it is one way to compute the value of the elementary functions, the calculation is performed while rotating the (or coordinates) vector plane.
θ as the angle to be obtained finally, shows a schematic diagram of a method of determining the sinθ and cosθ in the CORDIC method in Figure 1. As shown in Figure 1, CORDIC method by repeating the process angle z is to approach the angle θ, cosθ as the coordinates x, is calculated as sinθ the coordinates y. Also from Figure 1, there is a relationship, such as the following angle the z.

To approaches the angle θ, (1.1) shows that it will add to the angle Z angle A. At this time, the angle between each of the relationship, such as the following.

As shown in (1.10), the angle z is closer to θ eventually. To determine sinθ,cosθ in this way. The following will describe the main principle of the CORDIC method.
Shows the how the angle z whether the angle θ closer to the CORDIC method in Figure 2. I to R the length of the line segment connecting the origin (xi, yi) coordinates first. I and z the angle at that time. Coordinates (xi, yi) and angle z_i+1 is expressed by the following equation.



Consider the update value α to approximate to the angle θ the angle z here.

When defined by the equation (1.6), sinαi and cosαi can be expressed as follows.


By using (1.7), (1.8) equation, it can be expressed as follows: coordinate transformation (x(i+1),y(i+1)) to the point from (xi,yi) point.




Similarly,

Then, how sinθ and cosθ or go is calculated as an example, or try to follow the process.
As shown in Figure 3,It is assumed that the x-axis on the initial position of the vector will rotate,Initial coordinates point is (x0,y0(=0)).Assuming that zn=θ,





In determining the initial value of x0 from (1.13) and (1.11), by obtaining the xn and yn coordinates after the rotation of n times, sinθ and cosθ of the angle θ given can be determined as follows.


Attention must be paid to, It is assumed that the angle zn approaches the angle θ by repeating n times the rotation, Assuming that the error θ,

As (1.16), there is an error occurs even after repeated n times.
Reference
[1] 青木由直,『BASIC計算法』,コロナ社,1984