ECE 383
Date: March 14/15
Lecture: 27
File: LUT.xlsx
Status: Complete
Handout: hand24.docx
Lesson SlidesECE_383_Lec24.pdf

LUT

Let's say that you wanted to compute the square root of an value using the microBlaze. If you were lucky enough to have a compiler which provided this function you could just use the math library functions. If on the other hand, you did not have the use of such a library, you would have to figure out a way to compute the square root.

There are a lot of ways that you could compute the SQRT function. You could crack open a math book and find an analytic expression which would produce a value for the SQRT function. Unfortunately, this approach often leads to timely computations. Another approach would be to enumerate every possible value of x and its square root. We could then look-up a SQRT in the table, by going to the row corresponding to x and retrieving its value. This approach seems silly because it would use a lot of space. If we reduced the size of the table by eliminating some of the entries then we would save space at the expense of introducing error - if you wanted the SQRT for x and its entry was not in the table then you would have to use the closest x in the table. A good compromise among all three of these design constraints (space, time, and error) is to use interpolation in a partial look-up table (LUT).

"Interpolation is a mathematical method of creating missing data. ... There are many methods of interpolation, but one simple method would be to generate a new value by using the average of the value of the two values on either side of the one to be created." This average is also referred to as linear interpolation. For example if you have a value of x which is 1/2 way between 0 and 4 then you assume that the SQRT is 1/2 between 0 and 2.

We know that if f(x)=y and f(x+4)=z then we estimate the intermediate values of f(x+1), f(x+2), and f(x+3) by drawing a straight line between y and z and using the points on this line to estimate the function between x and x+4. For example, let: F(8)=2.8284 and F(12)=3.4641 then We understand that this is an approximation and consequently we will have error, but sometimes close is better than exact in embedded computing especially when time is of the essence.

//----------------------------------------------
//	A code chunk to perform linear
//	interpolation of some unknown fnc at
//	x+i where i is between 0 and 4 
//	inclusive.  You are given that
//	f(x)=y		f(x+4)=z
//	You are given i, please return f(x+i)
//----------------------------------------------
	delta = (z-y)>>2;
	f = y + delta*i;
It would be better to do the division by 4 (shift right by 2-bits) after the multiplication of delta*i because the difference (z-y) might be small and the division may result in a 0 value.

Now let's examine how to determine a square root. The purple curve in the graph below shows the true value of the SQRT function for the integers 0-15. If we have a table with 16 rows we could store these values and generate really good values for the SQRT function. This approach becomes infeasible as we seek to make the table larger. The yellow curve in the graph below represents the values of the SQRT function for 4 equally spaced values. These values are given in the table to the left of the graph.
x sqrt(x) 2.6 fixed point
0 0 0x00
4 2 0x80
8 2.828427125 0xB5
12 3.464101615 0xDE
16 4 0xFF


If we stopped here then our values for the SQRT would be very coarse. For example if you wanted the SQRT(1), the closest value in our table would be 0. Now that's a really bad approximation, but an approximation none-the-less.

Let's consider how to use interpolation to find a better value for the SQRT(9). Clearly, the SQRT will be between SQRT(8) and SQRT(12). How much between? Well 1/4 of the way because 9 is a 1/4 of the way between 9 and 12. Have the class write an equation describing SQRT(9) in terms of SQRT(12) and SQRT(8). 
    SQRT(9) = SQRT(8) + 1/4*(SQRT(12)-SQRT(8))
Now write the equation replacing the 1/4 by a statement using 9,8, and 12. 
    SQRT(9) = SQRT(8) + (9-8)/(12-8)*(SQRT(12)-SQRT(8))
Now divide the class into 16 sections and have them compute all the values of SQRT for 0-15. Compare these to the excel spreadsheet (4pt SQRT tab). In this spreadsheet, I've broken the computation down so that you can see what the individual parts are doing. Now we have to write a function to do this. 
//----------------------------------------------
// Fnc	SQRT
// In	A 4-bit integer 
// Out	An approximate SQRT
// Pur	This function computes a linearly
//	interpolated value for the SQRT
//	function.  There are some significant
//	data type issues that will have to
//	be resolved - note the use of "type"
//	in the function is a place-holder.
//----------------------------------------------
fixed SQRT(int4 x) {

    fixed lut[5] = {0, 2, 2.828427125, 3.464101615, 4};
    int8 index;
    fixed base;

    index = x >> 2;		// We are looking at sets of 4 points
    base = lut[index];		// Get the base value to start interpolation
    offset = (x & 0x03)>>2;	// The proportion into the interval - PROBLEM
    delta = lut[x+1] - lut[x];	// The difference between consecutive SQRTs
    return(base + offset*delta);
} // end SQRT

Data Type

Let's ponder the data type of the SQRT type. The input argument could be a 4-bit value. The output is a number between 0.0 and 4.0 and include some fractional part. Since 8-bit values are the norm with many micro-controllers, it makes sense to make the output an 8-bit value, but where to stick the decimal point? It makes sense to place the decimal at the 6th bit. Now it remains to write the SQRT code for this representation. 
//----------------------------------------------
// Fnc	SQRT
// In	A 4-bit integer 
// Out	An approximate SQRT
//	8-bit fixed point with decimal at 6th bit.
// Pur	This function computes a linearly
//	interpolated value for the SQRT.  The
//	5th entry in the lut is an approximation
//	to 4.
//----------------------------------------------
int8 SQRT(int8 x) {

    int8 lut[5] = {0x00, 0x80, 0xB5, 0xDE, 0xFF};
    int8 index;
    fixed base;

    index = x >> 2;	
    base = lut[index];	
    offset = (x & 0x03)	
    delta = lut[index+1] - lut[index];
    return(base + offset*delta)>>2;
} // end SQRT