cordiccexp - CORDIC-based approximation of complex exponential

Syntax

y = cordiccexp(theta,niters)

Description

y = cordiccexp(theta,niters) computes cos(theta) + j*sin(theta) using a CORDIC algorithm approximation. y contains the approximated complex result.

Input Arguments

theta

theta can be a signed or unsigned scalar, vector, matrix, or N-dimensional array containing the angle values in radians. All values of theta must be real and in the range [–2π 2π).

niters

niters is the number of iterations the CORDIC algorithm performs. This is an optional argument. When specified, niters must be a positive, integer-valued scalar. If you do not specify niters or if you specify a value that is too large, the algorithm uses a maximum value. For fixed-point operation, the maximum number of iterations is one less than the word length of theta. For floating-point operation, the maximum value is 52 for double or 23 for single. Increasing the number of iterations can produce more accurate results, but it also increases the expense of the computation and adds latency.

Output Arguments

y

y is the approximated complex result of the cordiccexp function. When the input to the function is floating point, the output data type is the same as the input data type. When the input is fixed point, the output has the same word length as the input, and a fraction length equal to the WordLength – 2.

Definitions

CORDIC

CORDIC is an acronym for COordinate Rotation DIgital Computer. The Givens rotation-based CORDIC algorithm is among one of the most hardware-efficient algorithms available because it requires only iterative shift-add operations (see [1], [2]) The CORDIC algorithm eliminates the need for explicit multipliers. Using CORDIC, you can calculate various functions, such as sine, cosine, arc sine, arc cosine, arc tangent, and vector magnitude. You can also use this algorithm for divide, square root, and hyperbolic, and logarithmic functions.

Increasing the number of CORDIC iterations can produce more accurate results, but doing so also increases the expense of the computation and adds latency.

Examples

The following example illustrates the effect of the number of iterations on the result of the cordiccexp approximation.

wrdLn = 8;
theta = fi(pi/2, 1, wrdLn);
fprintf('\n\nNITERS\t\tY (SIN)\t ERROR\t LSBs\t\tX (COS)\t ERROR\t LSBs\n');
fprintf('------\t\t-------\t ------\t ----\t\t-------\t ------\t ----\n');
for niters = 1:(wrdLn - 1)
  cis    = cordiccexp(theta, niters);
  fl     = cis.FractionLength;
  x      = real(cis);
  y      = imag(cis);
  x_dbl  = double(x);
  x_err  = abs(x_dbl - cos(double(theta)));
  y_dbl  = double(y);
  y_err  = abs(y_dbl - sin(double(theta)));
  fprintf('%d\t\t%1.4f\t %1.4f\t %1.1f\t\t%1.4f\t %1.4f\t %1.1f\n', niters, y_dbl, y_err,(y_err * pow2(fl)), x_dbl, x_err,(x_err * pow2(fl)));
end
fprintf('\n');

The output table appears as follows:

NITERS   Y (SIN)  ERROR   LSBs    X (COS)  ERROR   LSBs
------    -------  ------  ----   -------  ------  ----
1         0.7031   0.2968  19.0    0.7031   0.7105  45.5
2         0.9375   0.0625  4.0     0.3125   0.3198  20.5
3         0.9844   0.0156  1.0     0.0938   0.1011  6.5
4         0.9844   0.0156  1.0     -0.0156  0.0083  0.5
5         1.0000   0.0000  0.0     0.0312   0.0386  2.5
6         1.0000   0.0000  0.0     0.0000   0.0073  0.5
7         1.0000   0.0000  0.0     0.0156   0.0230  1.5

References

[1] Volder, J.E. "The CORDIC Trigonometric Computing Technique," IRE Transactions on Electronic Computers. Vol. EC-8, September 1959, pp. 330–334.

[2] Andraka, R. "A survey of CORDIC algorithm for FPGA based computers." Proceedings of the 1998 ACM/SIGDA sixth international symposium on Field programmable gate arrays. Feb. 22–24, 1998, pp. 191–200.