cordiccexp

CORDIC-based approximation of complex exponential

Syntax

y = cordiccexp(theta)

y = cordiccexp(theta,niters)

Description

y = cordiccexp(theta) computes cos(theta) + j*sin(theta) using a CORDIC algorithm approximation and returns the approximated complex result in y.

y = cordiccexp(theta,niters) performs niters iterations of the algorithm.

Examples

Effect of Number of Iterations on CORDIC Approximation

See the effect of the number of iterations on the result of the cordiccexp approximation.

wl = 8;
theta = fi(pi/2,1,wl);

output_type = fi([], 1,wl,wl - 2);
results_array = zeros(wl - 1,1,'like',output_type)';

for niters = 1:(wl - 1)
  cis    = cordiccexp(theta,niters);
  fl     = cis.FractionLength;
  x      = real(cis);
  y      = imag(cis);

  x_dbl  = double(x);
  y_dbl  = double(y);

  x_err  = abs(x_dbl - cos(double(theta)));
  y_err  = abs(y_dbl - sin(double(theta)));

  result = [niters,y_dbl,y_err,(y_err*pow2(fl)),...
      x_dbl,x_err,(x_err*pow2(fl))];
  results_array = [results_array; result];
end

results_table = array2table(results_array,'VariableNames',{'NITERS',...
    'Y (SIN)','Y ERROR','Y LSBs','X (COS)','X ERROR','X LSBs'})

results_table=8×7 table
    NITERS    Y (SIN)    Y ERROR     Y LSBs    X (COS)     X ERROR     X LSBs 
    ______    _______    ________    ______    ________    ________    _______

         0          0           0         0           0           0          0
         1    0.70312     0.29688    1.9844    -0.70312     0.70312     1.9844
    1.9844     0.9375      0.0625    1.9844     -0.3125      0.3125     1.9844
    1.9844    0.96875     0.03125    1.9844     -0.0625      0.0625     1.9844
    1.9844    0.96875     0.03125    1.9844      0.0625      0.0625     1.9844
    1.9844    0.98438    0.015625         1           0           0    0.46875
    1.9844    0.98438    0.015625         1     0.03125     0.03125     1.9844
    1.9844          1           0         0    0.015625    0.015625     1.4688

Input Arguments

theta — Input array

scalar | vector | matrix | multidimensional array

Input array, specified as a signed or unsigned scalar, vector, matrix, or multidimensional array. All values of theta must be real and in the range [-2π 2π).

If the input is a fi object, it must use binary-point scaling.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | fi

niters — Number of iterations CORDIC algorithm performs

positive scalar integer

Number of iterations the CORDIC algorithm performs, specified as a positive scalar integer. Increasing the number of iterations can produce more accurate results but also increases the expense of the computation and adds latency.

If you do not specify niters, or if you specify a value that is too large, the algorithm uses a maximum value based on the data type of the inputs:

• Fixed-point inputs — The maximum number of iterations is one less than the word length of theta.

• Floating-point inputs — The maximum value is 52 for double or 23 for single.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | fi

Output Arguments

y — Approximated complex exponential

scalar | vector | matrix | multidimensional array

Approximated complex exponential e^(j*theta), returned as a scalar, vector, matrix, or multidimensional array. The data type of the output depends on the input:

• When the input theta is floating point, the output data type is the same as the input type.

• When the input theta is fixed point, the output has the same word length as the input and a fraction length equal to the word length minus 2.

Algorithms

CORDIC

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

The precision of the CORDIC algorithm is a function of the data type used and the maximum shift value or number of iterations of the CORDIC kernel. Using a data type with a larger word length and performing more iterations of the CORDIC algorithm can reduce the numeric error of the result. However, doing so also increases the latency of the computation and the utilizes more hardware resources. For more information, see How to Set CORDIC Input Word Length and Maximum Shift Value to Achieve Desired Precision.

Signal Flow Diagrams

CORDIC Rotation Kernel

X represents the real part, Y represents the imaginary part, and Z represents theta. The accuracy of the CORDIC rotation kernel depends on the choice of initial values for X, Y, and Z. This algorithm uses these initial values:

• z₀ is initialized to the θ input argument value.

• x₀ is initialized to 1/A_N.

• y₀ is initialized to 0.

fimath Propagation Rules

CORDIC functions discard any local fimath attached to the input.

The CORDIC functions use their own internal fimath when performing calculations:

• OverflowAction — Wrap

• RoundingMethod — Floor

The output has no attached fimath.

References

[1] Volder, Jack E. “The CORDIC Trigonometric Computing Technique.” IRE Transactions on Electronic Computers. EC-8, no. 3 (Sept. 1959): 330–334.

[2] Andraka, Ray. “A Survey of CORDIC Algorithm for FPGA Based Computers.” In Proceedings of the 1998 ACM/SIGDA Sixth International Symposium on Field Programmable Gate Arrays, 191–200. https://dl.acm.org/doi/10.1145/275107.275139.

[3] Walther, J.S. “A Unified Algorithm for Elementary Functions.” In Proceedings of the May 18-20, 1971 Spring Joint Computer Conference, 379–386. https://dl.acm.org/doi/10.1145/1478786.1478840.

[4] Schelin, Charles W. “Calculator Function Approximation.” The American Mathematical Monthly, no. 5 (May 1983): 317–325. https://doi.org/10.2307/2975781.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• Variable-size signals are not supported.

• The number of iterations the CORDIC algorithm performs, niters, must be a constant.

HDL Code Generation

Generate VHDL, Verilog and SystemVerilog code for FPGA and ASIC designs using HDL Coder™.

The cordiccexp function also supports MATLAB^® to High-Level Synthesis (HLS) code generation.

Version History

Introduced in R2010a