Package: MATH_COMPLEX

File: math_complex.vhdl

Constants

Name	Type	Value	Description
CopyRightNotice	STRING	"Copyright 2008 IEEE. All rights reserved."
MATH_CBASE_1	COMPLEX	COMPLEX'(1.0, 0.0)	Constant Definitions
MATH_CBASE_J	COMPLEX	COMPLEX'(0.0, 1.0)
MATH_CZERO	COMPLEX	COMPLEX'(0.0, 0.0)

Types

Name	Type	Description
COMPLEX		Type Definitions
COMPLEX_POLAR

Functions

CMPLX (X : in REAL;
Y : in REAL := 0.0) return COMPLEX

Description
Purpose:
Returns TRUE if L is not equal to R and returns FALSE
otherwise
Special values:
COMPLEX_POLAR'(0.0, X) /= COMPLEX_POLAR'(0.0, Y) returns
FALSE regardless of the value of X and Y.
Domain:
L in COMPLEX_POLAR and L.ARG /= -MATH_PI
R in COMPLEX_POLAR and R.ARG /= -MATH_PI
Error conditions:
Error if L.ARG = -MATH_PI
Error if R.ARG = -MATH_PI
Range:
"/="(L,R) is either TRUE or FALSE
Notes:
None

Function Declarations

GET_PRINCIPAL_VALUE (X : in REAL) return PRINCIPAL_VALUE

Description
Purpose:
Returns COMPLEX number X + iY
Special values:
None
Domain:
X in REAL
Y in REAL
Error conditions:
None
Range:
CMPLX(X,Y) is mathematically unbounded
Notes:
None

COMPLEX_TO_POLAR (Z : in COMPLEX) return COMPLEX_POLAR

Description
Purpose:
Returns principal value of angle X; X in radians
Special values:
None
Domain:
X in REAL
Error conditions:
None
Range:
-MATH_PI < GET_PRINCIPAL_VALUE(X) <= MATH_PI
Notes:
None

POLAR_TO_COMPLEX (Z : in COMPLEX_POLAR) return COMPLEX

Description
Purpose:
Returns principal value COMPLEX_POLAR of Z
Special values:
COMPLEX_TO_POLAR(MATH_CZERO) = COMPLEX_POLAR'(0.0, 0.0)
COMPLEX_TO_POLAR(Z) = COMPLEX_POLAR'(ABS(Z.IM),
SIGN(Z.IM)*MATH_PI_OVER_2) if Z.RE = 0.0
Domain:
Z in COMPLEX
Error conditions:
None
Range:
result.MAG >= 0.0
-MATH_PI < result.ARG <= MATH_PI
Notes:
None

ARG (Z : in COMPLEX) return PRINCIPAL_VALUE

Description
Purpose:
Returns absolute value (magnitude) of Z
Special values:
None
Domain:
Z in COMPLEX_POLAR and Z.ARG /= -MATH_PI
Error conditions:
Error if Z.ARG = -MATH_PI
Range:
ABS(Z) >= 0.0
Notes:
ABS(Z) = Z.MAG

ARG (Z : in COMPLEX_POLAR) return PRINCIPAL_VALUE

Description
Purpose:
Returns argument (angle) in radians of the principal
value of Z
Special values:
ARG(Z) = 0.0 if Z.RE >= 0.0 and Z.IM = 0.0
ARG(Z) = SIGN(Z.IM)*MATH_PI_OVER_2 if Z.RE = 0.0
ARG(Z) = MATH_PI if Z.RE < 0.0 and Z.IM = 0.0
Domain:
Z in COMPLEX
Error conditions:
None
Range:
-MATH_PI < ARG(Z) <= MATH_PI
Notes:
ARG(Z) = ARCTAN(Z.IM, Z.RE)

CONJ (Z : in COMPLEX) return COMPLEX

Description
Purpose:
Returns principal value of unary minus of Z
Special values:
"-"(Z) = COMPLEX_POLAR'(Z.MAG, MATH_PI) if Z.ARG = 0.0
Domain:
Z in COMPLEX_POLAR and Z.ARG /= -MATH_PI
Error conditions:
Error if Z.ARG = -MATH_PI
Range:
result.MAG >= 0.0
-MATH_PI < result.ARG <= MATH_PI
Notes:
Returns COMPLEX_POLAR'(Z.MAG, Z.ARG - SIGN(Z.ARG)*MATH_PI) if
Z.ARG /= 0.0

CONJ (Z : in COMPLEX_POLAR) return COMPLEX_POLAR

Description
Purpose:
Returns complex conjugate of Z
Special values:
None
Domain:
Z in COMPLEX
Error conditions:
None
Range:
CONJ(Z) is mathematically unbounded
Notes:
Returns x -jy for Z= x + jy

SQRT (Z : in COMPLEX) return COMPLEX

Description
Purpose:
Returns principal value of complex conjugate of Z
Special values:
CONJ(Z) = COMPLEX_POLAR'(Z.MAG, MATH_PI) if Z.ARG = MATH_PI
Domain:
Z in COMPLEX_POLAR and Z.ARG /= -MATH_PI
Error conditions:
Error if Z.ARG = -MATH_PI
Range:
result.MAG >= 0.0
-MATH_PI < result.ARG <= MATH_PI
Notes:
Returns COMPLEX_POLAR'(Z.MAG, -Z.ARG) if Z.ARG /= MATH_PI

SQRT (Z : in COMPLEX_POLAR) return COMPLEX_POLAR

Description
Purpose:
Returns square root of Z with positive real part
or, if the real part is zero, the one with nonnegative
imaginary part
Special values:
SQRT(MATH_CZERO) = MATH_CZERO
Domain:
Z in COMPLEX
Error conditions:
None
Range:
SQRT(Z) is mathematically unbounded
Notes:
None

EXP (Z : in COMPLEX) return COMPLEX

Description
Purpose:
Returns square root of Z with positive real part
or, if the real part is zero, the one with nonnegative
imaginary part
Special values:
SQRT(Z) = COMPLEX_POLAR'(0.0, 0.0) if Z.MAG = 0.0
Domain:
Z in COMPLEX_POLAR and Z.ARG /= -MATH_PI
Error conditions:
Error if Z.ARG = -MATH_PI
Range:
result.MAG >= 0.0
-MATH_PI < result.ARG <= MATH_PI
Notes:
None

EXP (Z : in COMPLEX_POLAR) return COMPLEX_POLAR

Description
Purpose:
Returns exponential of Z
Special values:
EXP(MATH_CZERO) = MATH_CBASE_1
EXP(Z) = -MATH_CBASE_1 if Z.RE = 0.0 and ABS(Z.IM) = MATH_PI
EXP(Z) = SIGN(Z.IM)*MATH_CBASE_J if Z.RE = 0.0 and
ABS(Z.IM) = MATH_PI_OVER_2
Domain:
Z in COMPLEX
Error conditions:
None
Range:
EXP(Z) is mathematically unbounded
Notes:
None

LOG (Z : in COMPLEX) return COMPLEX

Description
Purpose:
Returns principal value of exponential of Z
Special values:
EXP(Z) = COMPLEX_POLAR'(1.0, 0.0) if Z.MAG =0.0 and
Z.ARG = 0.0
EXP(Z) = COMPLEX_POLAR'(1.0, MATH_PI) if Z.MAG = MATH_PI and
ABS(Z.ARG) = MATH_PI_OVER_2
EXP(Z) = COMPLEX_POLAR'(1.0, MATH_PI_OVER_2) if
Z.MAG = MATH_PI_OVER_2 and
Z.ARG = MATH_PI_OVER_2
EXP(Z) = COMPLEX_POLAR'(1.0, -MATH_PI_OVER_2) if
Z.MAG = MATH_PI_OVER_2 and
Z.ARG = -MATH_PI_OVER_2
Domain:
Z in COMPLEX_POLAR and Z.ARG /= -MATH_PI
Error conditions:
Error if Z.ARG = -MATH_PI
Range:
result.MAG >= 0.0
-MATH_PI < result.ARG <= MATH_PI
Notes:
None

LOG2 (Z : in COMPLEX) return COMPLEX

Description
Purpose:
Returns natural logarithm of Z
Special values:
LOG(MATH_CBASE_1) = MATH_CZERO
LOG(-MATH_CBASE_1) = COMPLEX'(0.0, MATH_PI)
LOG(MATH_CBASE_J) = COMPLEX'(0.0, MATH_PI_OVER_2)
LOG(-MATH_CBASE_J) = COMPLEX'(0.0, -MATH_PI_OVER_2)
LOG(Z) = MATH_CBASE_1 if Z = COMPLEX'(MATH_E, 0.0)
Domain:
Z in COMPLEX and ABS(Z) /= 0.0
Error conditions:
Error if ABS(Z) = 0.0
Range:
LOG(Z) is mathematically unbounded
Notes:
None

LOG10 (Z : in COMPLEX) return COMPLEX

Description
Purpose:
Returns logarithm base 2 of Z
Special values:
LOG2(MATH_CBASE_1) = MATH_CZERO
LOG2(Z) = MATH_CBASE_1 if Z = COMPLEX'(2.0, 0.0)
Domain:
Z in COMPLEX and ABS(Z) /= 0.0
Error conditions:
Error if ABS(Z) = 0.0
Range:
LOG2(Z) is mathematically unbounded
Notes:
None

LOG (Z : in COMPLEX_POLAR) return COMPLEX_POLAR

Description
Purpose:
Returns logarithm base 10 of Z
Special values:
LOG10(MATH_CBASE_1) = MATH_CZERO
LOG10(Z) = MATH_CBASE_1 if Z = COMPLEX'(10.0, 0.0)
Domain:
Z in COMPLEX and ABS(Z) /= 0.0
Error conditions:
Error if ABS(Z) = 0.0
Range:
LOG10(Z) is mathematically unbounded
Notes:
None

LOG2 (Z : in COMPLEX_POLAR) return COMPLEX_POLAR

Description
Purpose:
Returns principal value of natural logarithm of Z
Special values:
LOG(Z) = COMPLEX_POLAR'(0.0, 0.0) if Z.MAG = 1.0 and
Z.ARG = 0.0
LOG(Z) = COMPLEX_POLAR'(MATH_PI, MATH_PI_OVER_2) if
Z.MAG = 1.0 and Z.ARG = MATH_PI
LOG(Z) = COMPLEX_POLAR'(MATH_PI_OVER_2, MATH_PI_OVER_2) if
Z.MAG = 1.0 and Z.ARG = MATH_PI_OVER_2
LOG(Z) = COMPLEX_POLAR'(MATH_PI_OVER_2, -MATH_PI_OVER_2) if
Z.MAG = 1.0 and Z.ARG = -MATH_PI_OVER_2
LOG(Z) = COMPLEX_POLAR'(1.0, 0.0) if Z.MAG = MATH_E and
Z.ARG = 0.0
Domain:
Z in COMPLEX_POLAR and Z.ARG /= -MATH_PI
Z.MAG /= 0.0
Error conditions:
Error if Z.ARG = -MATH_PI
Error if Z.MAG = 0.0
Range:
result.MAG >= 0.0
-MATH_PI < result.ARG <= MATH_PI
Notes:
None

LOG10 (Z : in COMPLEX_POLAR) return COMPLEX_POLAR

Description
Purpose:
Returns principal value of logarithm base 2 of Z
Special values:
LOG2(Z) = COMPLEX_POLAR'(0.0, 0.0) if Z.MAG = 1.0 and
Z.ARG = 0.0
LOG2(Z) = COMPLEX_POLAR'(1.0, 0.0) if Z.MAG = 2.0 and
Z.ARG = 0.0
Domain:
Z in COMPLEX_POLAR and Z.ARG /= -MATH_PI
Z.MAG /= 0.0
Error conditions:
Error if Z.ARG = -MATH_PI
Error if Z.MAG = 0.0
Range:
result.MAG >= 0.0
-MATH_PI < result.ARG <= MATH_PI
Notes:
None

LOG (Z : in COMPLEX;
BASE : in REAL) return COMPLEX

Description
Purpose:
Returns principal value of logarithm base 10 of Z
Special values:
LOG10(Z) = COMPLEX_POLAR'(0.0, 0.0) if Z.MAG = 1.0 and
Z.ARG = 0.0
LOG10(Z) = COMPLEX_POLAR'(1.0, 0.0) if Z.MAG = 10.0 and
Z.ARG = 0.0
Domain:
Z in COMPLEX_POLAR and Z.ARG /= -MATH_PI
Z.MAG /= 0.0
Error conditions:
Error if Z.ARG = -MATH_PI
Error if Z.MAG = 0.0
Range:
result.MAG >= 0.0
-MATH_PI < result.ARG <= MATH_PI
Notes:
None

LOG (Z : in COMPLEX_POLAR;
BASE : in REAL) return COMPLEX_POLAR

Description
Purpose:
Returns logarithm base BASE of Z
Special values:
LOG(MATH_CBASE_1, BASE) = MATH_CZERO
LOG(Z,BASE) = MATH_CBASE_1 if Z = COMPLEX'(BASE, 0.0)
Domain:
Z in COMPLEX and ABS(Z) /= 0.0
BASE > 0.0
BASE /= 1.0
Error conditions:
Error if ABS(Z) = 0.0
Error if BASE <= 0.0
Error if BASE = 1.0
Range:
LOG(Z,BASE) is mathematically unbounded
Notes:
None

SIN (Z : in COMPLEX) return COMPLEX

Description
Purpose:
Returns principal value of logarithm base BASE of Z
Special values:
LOG(Z, BASE) = COMPLEX_POLAR'(0.0, 0.0) if Z.MAG = 1.0 and
Z.ARG = 0.0
LOG(Z, BASE) = COMPLEX_POLAR'(1.0, 0.0) if Z.MAG = BASE and
Z.ARG = 0.0
Domain:
Z in COMPLEX_POLAR and Z.ARG /= -MATH_PI
Z.MAG /= 0.0
BASE > 0.0
BASE /= 1.0
Error conditions:
Error if Z.ARG = -MATH_PI
Error if Z.MAG = 0.0
Error if BASE <= 0.0 Error if BASE = 1.0 Range: result.MAG >= 0.0
-MATH_PI < result.ARG <= MATH_PI
Notes:
None

SIN (Z : in COMPLEX_POLAR) return COMPLEX_POLAR

Description
Purpose:
Returns sine of Z
Special values:
SIN(MATH_CZERO) = MATH_CZERO
SIN(Z) = MATH_CZERO if Z = COMPLEX'(MATH_PI, 0.0)
Domain:
Z in COMPLEX
Error conditions:
None
Range:
ABS(SIN(Z)) <= SQRT(SIN(Z.RE)SIN(Z.RE) + SINH(Z.IM)SINH(Z.IM))
Notes:
None

COS (Z : in COMPLEX) return COMPLEX

Description
Purpose:
Returns principal value of sine of Z
Special values:
SIN(Z) = COMPLEX_POLAR'(0.0, 0.0) if Z.MAG = 0.0 and
Z.ARG = 0.0
SIN(Z) = COMPLEX_POLAR'(0.0, 0.0) if Z.MAG = MATH_PI and
Z.ARG = 0.0
Domain:
Z in COMPLEX_POLAR and Z.ARG /= -MATH_PI
Error conditions:
Error if Z.ARG = -MATH_PI
Range:
result.MAG >= 0.0
-MATH_PI < result.ARG <= MATH_PI
Notes:
None

COS (Z : in COMPLEX_POLAR) return COMPLEX_POLAR

Description
Purpose:
Returns cosine of Z
Special values:
COS(Z) = MATH_CZERO if Z = COMPLEX'(MATH_PI_OVER_2, 0.0)
COS(Z) = MATH_CZERO if Z = COMPLEX'(-MATH_PI_OVER_2, 0.0)
Domain:
Z in COMPLEX
Error conditions:
None
Range:
ABS(COS(Z)) <= SQRT(COS(Z.RE)COS(Z.RE) + SINH(Z.IM)SINH(Z.IM))
Notes:
None

SINH (Z : in COMPLEX) return COMPLEX

Description
Purpose:
Returns principal value of cosine of Z
Special values:
COS(Z) = COMPLEX_POLAR'(0.0, 0.0) if Z.MAG = MATH_PI_OVER_2
and Z.ARG = 0.0
COS(Z) = COMPLEX_POLAR'(0.0, 0.0) if Z.MAG = MATH_PI_OVER_2
and Z.ARG = MATH_PI
Domain:
Z in COMPLEX_POLAR and Z.ARG /= -MATH_PI
Error conditions:
Error if Z.ARG = -MATH_PI
Range:
result.MAG >= 0.0
-MATH_PI < result.ARG <= MATH_PI
Notes:
None

SINH (Z : in COMPLEX_POLAR) return COMPLEX_POLAR

Description
Purpose:
Returns hyperbolic sine of Z
Special values:
SINH(MATH_CZERO) = MATH_CZERO
SINH(Z) = MATH_CZERO if Z.RE = 0.0 and Z.IM = MATH_PI
SINH(Z) = MATH_CBASE_J if Z.RE = 0.0 and
Z.IM = MATH_PI_OVER_2
SINH(Z) = -MATH_CBASE_J if Z.RE = 0.0 and
Z.IM = -MATH_PI_OVER_2
Domain:
Z in COMPLEX
Error conditions:
None
Range:
ABS(SINH(Z)) <= SQRT(SINH(Z.RE)SINH(Z.RE) + SIN(Z.IM)SIN(Z.IM))
Notes:
None

COSH (Z : in COMPLEX) return COMPLEX

Description
Purpose:
Returns principal value of hyperbolic sine of Z
Special values:
SINH(Z) = COMPLEX_POLAR'(0.0, 0.0) if Z.MAG = 0.0 and
Z.ARG = 0.0
SINH(Z) = COMPLEX_POLAR'(0.0, 0.0) if Z.MAG = MATH_PI and
Z.ARG = MATH_PI_OVER_2
SINH(Z) = COMPLEX_POLAR'(1.0, MATH_PI_OVER_2) if Z.MAG =
MATH_PI_OVER_2 and Z.ARG = MATH_PI_OVER_2
SINH(Z) = COMPLEX_POLAR'(1.0, -MATH_PI_OVER_2) if Z.MAG =
MATH_PI_OVER_2 and Z.ARG = -MATH_PI_OVER_2
Domain:
Z in COMPLEX_POLAR and Z.ARG /= -MATH_PI
Error conditions:
Error if Z.ARG = -MATH_PI
Range:
result.MAG >= 0.0
-MATH_PI < result.ARG <= MATH_PI
Notes:
None

COSH (Z : in COMPLEX_POLAR) return COMPLEX_POLAR

Description
Purpose:
Returns hyperbolic cosine of Z
Special values:
COSH(MATH_CZERO) = MATH_CBASE_1
COSH(Z) = -MATH_CBASE_1 if Z.RE = 0.0 and Z.IM = MATH_PI
COSH(Z) = MATH_CZERO if Z.RE = 0.0 and Z.IM = MATH_PI_OVER_2
COSH(Z) = MATH_CZERO if Z.RE = 0.0 and Z.IM = -MATH_PI_OVER_2
Domain:
Z in COMPLEX
Error conditions:
None
Range:
ABS(COSH(Z)) <= SQRT(SINH(Z.RE)SINH(Z.RE) + COS(Z.IM)COS(Z.IM))
Notes:
None