CORDIC Unit Design using Verilog HDL

Introduction to CORDIC Algorithm

CORDIC (COordinate Rotation DIgital Computer) is an iterative algorithm which is used to calculate trignometric functions, vector rotations, complex multiplication and division, hyperbolic functions and many more in digital hardware with efficiency. One of the common applications can be found in DSP(Digital Signal Processing) hardwares.

Iterative Equations of CORDIC

These iterative equations rotates the vector (x,y) by the given angle angle. 'σ' is the sign of z[i+1]. The angle to be rotated is initialized into z[0] and after N number of iterations, the rotated vector is obtained which is scaled by a factor of K = 0.6072 .

Reference: (https://www.allaboutcircuits.com/technical-articles/an-introduction-to-the-cordic-algorithm/)

Choosing the fixed-point representation format

As most of the DSP processors are 32-bit, 32-bit representation is used here. To represent integers(decimal + fraction) in binary, Qm.n fixed-point representation is employed in most of the processors. Where 'm' represents the number of bits used to indicate decimal part(including sign bit) and 'n' represents the number of bits used to represent fractional part. In the design Q3.29 representation is chosen. 29-bits for fractional part to provide high resolution to the fractional part.

Module Design CORDIC_UNIT.v

The module can work in two modes

1. Trigonometric mode
2. Vector rotation mode

In the Trigonometric mode (trig_rot = 1) the sin and cos of the given angle is computed. In the Vector rotation mode.(trig_rot = 0) , inputs Xi and Yi are rotated by the angle provided by the input angle

CORDIC Architecture

A parallel architecture is employed in the design. add_sub.v is instantiated inside the Verilog's generate block. Parameter N tells the data-width and I-controls the number of iterations to be performed. The architecture of the single stage is given below:

The incrementat/decrement angle( arctan(2^(-i)) ) is taken from the look-up table given below:

i	2^(-i)	Angle (Degrees)	Binary Value
0	1.000000000	45	32'b0000_1100100100001111110110101010
1	0.500000000	26.565051	32'b0000_0111011010110001100111000001
2	0.250000000	14.036243	32'b0000_0011111010110110111010111111
3	0.125000000	7.125016	32'b0000_0001111111010101101110101001
4	0.062500000	3.576334	32'b0000_0000111111111010101011011101
5	0.031250000	1.789911	32'b0000_0000011111111111010101010110
6	0.015625000	0.895174	32'b0000_0000001111111111111010101010
7	0.007812500	0.447614	32'b0000_0000000111111111111111010101
8	0.003906250	0.223811	32'b0000_0000000011111111111111111010
9	0.001953125	0.111906	32'b0000_0000000001111111111111111111
10	0.000976563	0.055953	32'b0000_0000000000111111111111111111
11	0.000488281	0.027976	32'b0000_0000000000011111111111111111
12	0.000244141	0.013988	32'b0000_0000000000001111111111111111
13	0.000122070	0.006994	32'b0000_0000000000000111111111111111
14	0.000061035	0.003497	32'b0000_0000000000000011111111111111
15	0.000030518	0.001749	32'b0000_0000000000000001111111111111
16	0.000015259	0.000874	32'b0000_0000000000000000111111111111
17	0.000007629	0.000437	32'b0000_0000000000000000011111111111
18	0.000003815	0.000219	32'b0000_0000000000000000001111111111
19	0.000001907	0.000109	32'b0000_0000000000000000000111111111
20	0.000000954	0.000055	32'b0000_0000000000000000000011111111
21	0.000000477	0.000027	32'b0000_0000000000000000000001111111
22	0.000000238	0.000014	32'b0000_0000000000000000000000111111
23	0.000000119	0.000007	32'b0000_0000000000000000000000011111
24	0.000000060	0.000004	32'b0000_0000000000000000000000001111
25	0.000000030	0.000002	32'b0000_0000000000000000000000000111
26	0.000000015	0.000001	32'b0000_0000000000000000000000000011
27	0.000000008	0.000001	32'b0000_0000000000000000000000000001
28	0.000000004	0.000000	32'b0000_0000000000000000000000000000

Test Bench and Results

To test the module in both modes following inputs are provided:

    trig_rot = 0;
    
    #10 
    Xi = 32'b0000_1011010100000100111100110011; // 1/sqrt(2)
    Yi = 32'b0000_1011010100000100111100110011; // 1/sqrt(2)
    angle = 32'b0000_1000011000001010100001001011; // pi/6
    #10
    angle = 32'b1111_0111100111110101011110110101; // -pi/6
    #10
    angle = 32'b0010_0001100000101010010001110000; // 2pi/3
    #10
    angle = 32'b1101_1110011111010101101110010000; // -2pi/3
    #10
    angle = 32'b0011_1010101001001001111111000100; // 7pi/6
    #10
    angle = 32'b1100_0101010110110110000000111100; // -7pi/6
    #10
    angle = 32'b0101_0011110001101001101100001001; // 5pi/3
    #10
    angle = 32'b1010_1100001110010110010011110111; // -5pi/3
    #10
    trig_rot = 1;
    angle = 32'b0000_1000011000001010100001001011; // pi/6
    #10
    angle = 32'b1111_0111100111110101011110110101; // -pi/6
    #10
    angle = 32'b0010_0001100000101010010001110000; // 2pi/3
    #10
    angle = 32'b1101_1110011111010101101110010000; // -2pi/3
    #10
    angle = 32'b0011_1010101001001001111111000100; // 7pi/6
    #10
    angle = 32'b1100_0101010110110110000000111100; // -7pi/6
    #10
    angle = 32'b0101_0011110001101001101100001001; // 5pi/3
    #10
    angle = 32'b1010_1100001110010110010011110111; // -5pi/3
    #10
    $finish;

Both rotation and trigonometric modes with both positive and negative angles in all the quadrants are tested here in this test bench.

The output is verified by converting the Q4.28 format to signed decimal using

YOSYS Synthesis

Netlist dot file (very large to show in image)

Netlist simulation

Matching with the pre-synthesis result