Copyright Feb 2002 Page 1NEARFIELD SYSTEMS, INC.
AMTA EDUCATIONAL SEMINAR 2002
Near-field Antenna Measurement Theory -Planar
Copyright Feb 2002 Page 2NEARFIELD SYSTEMS, INC.
Overview
l Development Of Plane Wave Theoryl Development Using Measurement Approachl Understanding Working Equationsl Planar Transmission Equationsl Solution Using FFTl Sampling And Data Point Spacingl Planar Probe Correction
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Notation Summary
Time Convention Phase increases with +z
Consistent with published papers on Near-field Theory
Time convention in instrumentation and NSI software is Phase decreases with +z.
i te ω−
j te ω+
Only affects complex quantities such as conversion from linear to circular components
-iωt convention +jωt convention
2x y
R
E i EE
−=
2x y
R
E i EE
+=
Copyright Feb 2002 Page 4NEARFIELD SYSTEMS, INC.
Notation Summary Vectors
Vectors = arrow over symbol or underline
Unit vectors = “hat” or lower case e
Vector components shown as subscript
,K t or Kr r
,ˆ ˆˆ , , x Ax A or e eθ
, ,x Ak t sφ
Copyright Feb 2002 Page 5NEARFIELD SYSTEMS, INC.
Theoretical Basis for Planar Near-field Measurements
l Scattering Matrix Theory Developed by Dr. D. M. Kerns in 1960’s
l Does Not Require:l Ideal Probe, measuring one component of field at a pointl Reciprocal antennasl AUT have ideal polarization, field separability or symmetry l Separation between AUT and probe within a special range
l Only Two Approximations Necessaryl Multiple reflections small enough to neglectl Measurements made over a finite plane
l Fast and Efficient Data Processing (Primarily FFT)
Copyright Feb 2002 Page 6NEARFIELD SYSTEMS, INC.
Angular Spectrum Of Plane Waves
Each vector component is a complex number
( , , )
( , , )
( , )
Electric Field E r
E r Superposition of Plane Waves
b Angular Spectrum Vectors
θ φ
θ φ
θ φ
=
=
=
r
r
r
Copyright Feb 2002 Page 7NEARFIELD SYSTEMS, INC.
Propagation Vector
Propagation vectorˆ ˆ ˆ
= Defines direction of plane wave
x y x
kk x k y k z
== + +
r
Copyright Feb 2002 Page 8NEARFIELD SYSTEMS, INC.
Propagation Vector
ˆ ˆ ˆPropagation vector
= Defines direction of plane wave
b(k) = Angular Spectrum Vector, Function of k
b(K) = Angular Spectrum Vector, Function of K
ˆ ˆ Transverse (x-y) Part
x y x
x y
k k x k y k z
K k x k y
= = + +
= + =
r
r r r
r r r
r of k
r
Copyright Feb 2002 Page 9NEARFIELD SYSTEMS, INC.
Propagation Vector
22 2 2 2
2 2 2
2
z
ˆ ˆ ˆ Propagation vector
ˆ
Since For losless medium
Any two components define the third.
= k
Therefore and both define directions.
x y z
z
x y z
y z
k k x k y k z
K k z
k k k k k k
k k k
k K
πλ
γ
= + + =
= +
≡ ⋅ = + + =
± = − −
r
r
r r
r r
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Single Plane Wave
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Two Plane Waves Different Directions
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Evanescent Plane Waves
Propagating in the x-y plane
Exponentially Attenuated in the z-direction
2 2 2x yk k k+ >
γ is Pure Imaginary
Sometimes called Imaginary Space,
θ and φ not defined
Copyright Feb 2002 Page 13NEARFIELD SYSTEMS, INC.
Scattering Matrix for Waveguide
1 11 1 12 2
2 21 1 22 2
b S a S a
b S a S a
= +
= +
1 2
a1
b1
a2
b2
Copyright Feb 2002 Page 14NEARFIELD SYSTEMS, INC.
Planar Scattering Matrix Schematic
0S′
1S 2S′Y
Z
AUTProbe
0S
0b
0a
0b′
0a′
( )b Kr r
( )a Krr
( )b K′r r
( )a K′rr
Copyright Feb 2002 Page 15NEARFIELD SYSTEMS, INC.
Transmission Equation Notation
O = AUT coordinate system originO’ = Probe coordinate system at (x0,y0,z0) in O
S0 = AUT portS’
0 = Probe porta0 = Complex Amplitude incident on AUT Transmission Line portb0 = Complex Amplitude emerging from AUT Transmission Line porta’
0 = Complex Amplitude incident on Probe Transmission Line portb’
0 = Complex Amplitude emerging from Probe Transmission Line porta(K)= Complex Amplitude incident on AUT Space side apertureb(K)= Complex Amplitude emerging from AUT Space side aperturea’(K)= Complex Amplitude incident on Probe Space side apertureb’(K)= Complex Amplitude emerging from Probe Space side aperture
Copyright Feb 2002 Page 16NEARFIELD SYSTEMS, INC.
Single, Linearly Polarized Plane Wave
Equations for AUT
0 10( 0) ( 0)y yb K a t K= = =r r
Equations for Probe
0 02
( 0) ( 0)
( 0) ( 0)
ikdy y
y y
a K b K e
b a K s K
′ = = =
′ ′= = =
r r
r r
0 0 10 02( 0) ( 0)ikdy yb F a t K e s K′ ′ ′= = =
r r
Copyright Feb 2002 Page 17NEARFIELD SYSTEMS, INC.
Single Plane Wave With Two Polarizations
Equations for AUT
0 10(0) (0)b a t=r r
Equations for Probe
0 02
(0) (0)
(0) (0)
ikda b e
b a s
′ =
′ ′ ′= •
rr
r r
0 0 10 02 10 02
0 10 02
(0) (0) (0) (0)
(0) (0)
ikdx x y y
ikd
b F a t s t s e
F a t s e
′ ′ ′ ′ = + ′ ′ = •
r r
Copyright Feb 2002 Page 18NEARFIELD SYSTEMS, INC.
Angular Spectrum Of Plane Waves With Arbitrary Polarization
Equations for AUT
0 10( ) ( )b K a t K=r r rr
Equations for Probe
0 02
( ) ( )
( ) ( )
i da K b K e
b a K s K
γ′ =
′ ′ ′= •
rr rr
r rr r
0 0 10 02( ) ( ) ( ) i dx yb P F a t K s K e dk dkγ′ ′ ′= •∫∫
r r rr r
Copyright Feb 2002 Page 19NEARFIELD SYSTEMS, INC.
Translation Of Probe In Plane
Equations for AUT
0 10( ) ( )b K a t K=r r rr
Equations for Probe
0 02
( ) ( )
( ) ( ) ( )
i d iK Pa K b K e e
b P a K s K
γ •′ =
′ ′ ′= •
r rrr rr
r rr r
0 0 10 02( ) ( ) ( ) i d iK Px yb P F a t K s K e e dk dkγ •′ ′ ′= •∫∫
r rr r rr r
Copyright Feb 2002 Page 20NEARFIELD SYSTEMS, INC.
( )0 0 10 02( , , ) ( ) ( ) x yi k x k yi d
x yb x y d F a t K s K e e dk dkγ +′ ′ ′= •∫∫r rr r
Planar Transmission Equation
Probe Amp and Phase Output
Probe Position
AUT Plane Wave Transmitting Function
Probe Plane Wave Receiving Function
Factors due to Probe Translation in Plane
AUT Input Amp and Phase
Copyright Feb 2002 Page 21NEARFIELD SYSTEMS, INC.
Planar Transmission Equation
( )0 0 10 02( , , ) ( ) ( ) x yi k x k yi d
x yb x y d F a t K s K e e dk dk
multiple reflection term
γ +′ ′ ′= •
+∫∫
r rr r
Neglecting multiple reflection term is only approximation in derivation of Planar Transmission Equation
Copyright Feb 2002 Page 22NEARFIELD SYSTEMS, INC.
Plane Wave Transmitting Spectrum And Electric Field
The Electric Field at a large distance from the AUT in the direction specified by K
r
010( , ) ( )cos
ikri k a eE r K t K
rθ=
r r rr
-80 -60 -40 -20 0 20 40 60 80
Azimuth in Degrees
-80
-60
-40
-20
0
Rel
ativ
e A
mpl
itude
in d
B
Transmitting Spectrum and E-Field ComparisonMain Comp.Transmitting SpectrumMain Comp. E-Field
Copyright Feb 2002 Page 23NEARFIELD SYSTEMS, INC.
Plane Wave Receiving Spectrum And Electric Field
Receiving Spectrum pattern and Far Electric Field pattern are identical
01
01
( , ) ( , )(0,0)(0,0)
E A E t A EtE
=r rr r
-80 -60 -40 -20 0 20 40 60 80
Azimuth in Degrees
-80
-60
-40
-20
0
Rel
ativ
e A
mpl
itude
in d
B
Receiving Spectrum and E-Field ComparisonMain Comp. Receiving SpectrumMain Comp. E-Field
Copyright Feb 2002 Page 24NEARFIELD SYSTEMS, INC.
Gain Equation
( )22
0 10
20
0
0
4 ( )( )
1
Characteristic Admittance of Free Space
Characteristic Admittance of Transmission Line
= Antenna Reflection Coefficient
a
a
Y t KG K
Y
πγ
η
η
=− Γ
==
Γ
rrr
Copyright Feb 2002 Page 25NEARFIELD SYSTEMS, INC.
Planar Near-field Measurement
MEASUREMENT AND DATA PROCESSING
tA
tE
s'A
s'E
Probe
Receiver
NF Data
Computer
-100 0 100Azimuthal Angle in Degrees
-80
-40
0
Re
lativ
e A
mpl
itude
in d
B
AUT Spectrum Patten AUTSpectrum -1 .0 -0.8 -0 .6 -0.4 -0.2 0.0 0.2 0.4 0 . 6 0.8 1 . 0
Relative x-component propagation vector kx/k
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
Re
lativ
e y
-com
pon
ent
pro
pa
gat
ion
vec
tor
ky/
k
Open Ended Waveguide Probe PatternAs Measured on Pattern RangeX-Polarized, AZ-Component, Amplitude F=12.0 GHz
Probe Data
Copyright Feb 2002 Page 26NEARFIELD SYSTEMS, INC.
Sample Planar Near-field Data
-50
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
-7.5 -5.0 -2.5 0.0 2.5 5.0 7.5
Sample Near-Field for Slot Array AntennaNear-Field Amplitude along y=0 centerline
Am
plit
ud
e (
dB
)
X (in)
-55-50-45-40-35-30-25-20-15-10-5-303510Sample Near-Field Data
Main Component Amplitude
Copyright Feb 2002 Page 27NEARFIELD SYSTEMS, INC.
Sample Planar Near-field Data
-150
-100
-50
0
50
100
150
-7.5 -5.0 -2.5 0.0 2.5 5.0 7.5
Sample Near-Field for Slot Array AntennaNear-field Phase along y=0 centerline
Ph
ase
(d
eg
)
X (in)
-180
-150
-120
-90
-60
-30
0
30
60
90
120
150
180
Sample Near-Field DataMain Component Phase
Copyright Feb 2002 Page 28NEARFIELD SYSTEMS, INC.
Sample Planar Near-field Data
-50
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
-7.5 -5.0 -2.5 0.0 2.5 5.0 7.5
Sample Near-Field for Slot Array AntennaCross Pol near-field Amplitude along y=0 centerline
Am
plitu
de (
dB)
X (in)
-60-55-50-45-40-35-30-25-20-15-10-5-303510
Sample Near-Field DataCross Component Amplitude
Copyright Feb 2002 Page 29NEARFIELD SYSTEMS, INC.
Schematic of Waveguide Probe
Y
X
A (main)
E (Cross)
X-Polarized OpenEnded Waveguide
Copyright Feb 2002 Page 30NEARFIELD SYSTEMS, INC.
X-Polarized Probe Az-Component
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
Relative x-component propagation vector kx/k
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
Rel
ativ
e y-
com
pone
nt p
ropa
gatio
n ve
ctor
ky/
k
Open Ended Waveguide Probe PatternAs Measured on Pattern RangeX-Polarized, AZ-Component, Amplitude F=12.0 GHz
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
Relative x-component propagation vector kx/k
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
Rel
ativ
e y-
com
pone
nt p
ropa
gatio
n ve
ctor
ky/
k
Open Ended Waveguide Probe PatternAs Measured on Pattern RangeX-Polarized, AZ-Component, Phase F=12.0 GHz
Copyright Feb 2002 Page 31NEARFIELD SYSTEMS, INC.
X-Polarized Probe El-Component
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
Relative x-component propagation vector kx/k
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
Rel
ativ
e y-
com
pone
nt p
ropa
gatio
n ve
ctor
ky/
k
Open Ended Waveguide Probe PatternAs Measured on Pattern RangeX-Polarized, EL-Component, Amplitude F=12.0 GHz
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
Relative x-component propagation vector kx/k
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
Rel
ativ
e y-
com
pone
nt p
ropa
gatio
n ve
ctor
ky/
k
Open Ended Waveguide Probe PatternAs Measured on Pattern RangeX-Polarized, EL-Component, Phase F=12.0 GHz
Copyright Feb 2002 Page 32NEARFIELD SYSTEMS, INC.
Simplified Notation For Probe Correction Discussion
( )0 0( ) i d i K Pb P F a t s e e dKγ •′ ′ ′= •∫∫
r rr r r
Delete subscripts on
Remember that t is Transmitting and s is Receiving in following equations
10 02t and s′r r
Delete Explicit notation showing t and s as functions of K.
Remember that t and s are not constants but are functions of direction of propagation.
Copyright Feb 2002 Page 33NEARFIELD SYSTEMS, INC.
Planar Near-field Mathematics
( ) ( ) ( ) ( ) ( )' '
' '
[ ]
[ ]
i d i K PA A A E E
i d i K PA A A E E
B P t K s K t K s K e e dK
B t s t s e e dK
γ
γ
•
•
= +
= +
∫
∫
r v
r v
v v v v v v
v
MAIN COMPONENT TRANSMISSION EQUATION
Copyright Feb 2002 Page 34NEARFIELD SYSTEMS, INC.
Planar Near-field Mathematics
( ) ( ) ( ) ( ) ( )" "
" "
[ ]
[ ]
i d i K PE A A E E
i d i K PE A A E E
B P t K s K t K s K e e dK
B t s t s e e dK
γ
γ
•
•
= +
= +
∫
∫
r v
r v
v v v v v v
v
CROSS COMPONENT TRANSMISSION EQUATION
Copyright Feb 2002 Page 35NEARFIELD SYSTEMS, INC.
Solving Transmission Equation
( )' '2( )
4
i di K P
A A A E E A
eD K t s t s B P e dP
γ
π
−− •= + = ∫
r vv v v
( )" "2( )
4
i di K P
E A A E E EeD K t s t s B P e dP
γ
π
−− •= + = ∫
r vv v v
MAIN COMPONENT PLANE WAVE SPECTRUM OF MEASURED DATA
CROSS COMPONENT PLANE WAVE SPECTRUM OF MEASURED DATA
Invert equations using Fourier transform (FFT)
Copyright Feb 2002 Page 36NEARFIELD SYSTEMS, INC.
Planar Near-field Measurement
Receiver
AUT Spectrum
Computer
-100 0 100Azimuthal Angle in Degrees
-80
-40
0
Rel
ativ
e A
mpl
itud
e in
dB
AUT Spectrum Patten
Synthesized Array
Synthesized Beam
PLANE WAVE SPECTRUM ANALYZER
D Ke
B P e dPA
i d
Ai K P( )
v v vr v
=−
− •zγ
π4 2 c h
Copyright Feb 2002 Page 37NEARFIELD SYSTEMS, INC.
Sampling Theorem And Fourier Transform
If the Fourier Transform of the measured data is band limited:
1-Data point spacings are defined .
2-Integration replaced by summation without approximation.
x yandδ δ
2
( )
2
( ) ( , )4
( , )4
x r y s
i di K P
A A
i di k x k yx y
A r sr s
eD K B x y e dx dy
eB x y e
γ
γ
π
δ δπ
−− •
−− +
=
=
∫ ∫
∑∑
r vv
max maxx y
x yk kπ π
δ δ= =
Copyright Feb 2002 Page 38NEARFIELD SYSTEMS, INC.
Band Limited Spectrum
-80
-70
-60
-50
-40
-30
-20
-10
0
-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00
Example of Band Limited Spectrum
Am
plitu
de (
dB)
Kx (cycles/L)
2 2xπλ λ
δπ
= =
Copyright Feb 2002 Page 39NEARFIELD SYSTEMS, INC.
Band Limit Greater than k
0.331.5(2 )y
πλδ λ
π= =
-80
-70
-60
-50
-40
-30
-20
-10
0
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0
Far-field Spectrum, No Probe Correction, No Cos(Theta) Factor
Am
plitu
de d
B
Ky/k Cycles/Lambda
Copyright Feb 2002 Page 40NEARFIELD SYSTEMS, INC.
Span And Spacing Relations
For near-field data with the number and spacing of data in the
x- and y-directions given by
and the FFT size given by ,
, , ,x y x yN N δ δ
,x yF F
x x x y y yNear Field Span S = (N -1) S =(N -1)
Far-Field Spacing
( 1)( 1)Far-Field Span
yx
x x y y
yxkx ky
x x y y
kkk F k F
FFS S
F F
δ δ
λ λδ δ
λλδ δ
∆∆= =
−−= =
Copyright Feb 2002 Page 41NEARFIELD SYSTEMS, INC.
Comparison Of Data Spectrum And Far-field
-80 -60 -40 -20 0 20 40 60 80Azimuth in Degrees
-100
-80
-60
-40
-20
0
Rel
ativ
e A
mpl
itude
in d
B
Spectrum of Meas. Data and E-Field ComparisonSpectrum of Measured DataMain Comp. E-FieldDifference due to
probe correction and Cos(θ) factor
Copyright Feb 2002 Page 42NEARFIELD SYSTEMS, INC.
Probe Correction Equations
E' " '
E"
'
Ds
Main Component Spectrum for A-Polarized AUT1
A
A sA
s
s
Ds
tρ
ρρ
−= =
−
"A" '
A"
'
Ds
Cross Component Spectrum for A-Polarized AUT1
Es
EE
s
s
Ds
tρ
ρρ
−= =
−
Copyright Feb 2002 Page 43NEARFIELD SYSTEMS, INC.
Definition Of Polarization Ratios
'' '
' 1 For A Polarized ProbeAs s
E
sand
sρ ρ= > −
"" "
" 1 For E Polarized ProbeAs s
E
sand
sρ ρ= < −
Copyright Feb 2002 Page 44NEARFIELD SYSTEMS, INC.
Polarization Ratio For OEWG Probe
-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
Relative x-component propogation vector, kx/k
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
Re
lativ
e y-
com
pon
ent p
ropo
gatio
n ve
ctor
. ky/
k
Amplitude, Polarization Ratio, A/E, forOpen Ended WaveguideProbe Main Component E-Polarized( )20*log ( )s Kρ′′
r
Copyright Feb 2002 Page 45NEARFIELD SYSTEMS, INC.
Two Parts Of Probe Correction
l Pattern correction l Due to probe’s main component pattern
l Polarization correctionl Due to probe’s cross component pattern
Copyright Feb 2002 Page 46NEARFIELD SYSTEMS, INC.
Approximate Probe Correction
tDsA
A
A
≅ '
MAIN COMPONENT, FOR CASE OF X-MAIN COMPONENT
PATTERN CORRECTION ONLY
Copyright Feb 2002 Page 47NEARFIELD SYSTEMS, INC.
Probe Pattern Correction Only
-80 -60 -40 -20 0 20 40 60 80Azimuthal Angle in Degrees
-100
-80
-60
-40
-20
0
Relativ
e A
mplitude
in d
B
Main Component Probe Pattern Correction
Probe Main Comp. Pattern
AUT Probe Corrected Main Comp.
Total Spectrum of "Main" N-F DatatDsA
A
A
≅ '
Polarization correction has almost no effect on main component results
Copyright Feb 2002 Page 48NEARFIELD SYSTEMS, INC.
Effect Of Probe Pattern Errors
-80 -60 -40 -20 0 20 40 60 80Azimuthal Angle in Degrees
-100
-80
-60
-40
-20
0
Relative Amplitu
de in
dB
Probe Pattern Errors
Probe Main Comp. Pattern
AUT Probe Corrected Main Comp.
Total Spectrum of "Main" N-F Data
Probe Cross Component Pattern
Probe pattern errors affect single direction
Copyright Feb 2002 Page 49NEARFIELD SYSTEMS, INC.
Theoretical Probe Pattern Error
-20.0
-17.5
-15.0
-12.5
-10.0
-7.5
-5.0
-2.5
0.0
2.5
-75 -50 -25 0 25 50 75
Comparison of Measured and Calculated Probe PatternOEWG, F = 10 GHz, H-Plane Cut
Am
plitu
de (
dB)
Azimuth (deg)
Plot 1 Plot 2 Plot 1 - Plot 2
Copyright Feb 2002 Page 50NEARFIELD SYSTEMS, INC.
Approximate Probe Correction
tDs
DsE
E
E
A
As≅ −" '"ρ
CROSS COMPONENT
PATTERN CORRECTION POLARIZATION CORRECTION
Copyright Feb 2002 Page 51NEARFIELD SYSTEMS, INC.
Probe Pattern Correction Only
-80 -60 -40 -20 0 20 40 60 80
Azimuthal Angle in Degrees
-100
-80
-60
-40
-20
0
Rela
tive
Am
plitud
e in d
B
Cross Component Probe Pattern Correction
Probe Main Comp. Pattern
AUT Cross Component (Probe Corrected)
Total Spectrum of "Cross" N-F DatatDsE
E
E
≅ "
Pattern correction has same effect on cross pol as on main pol
Copyright Feb 2002 Page 52NEARFIELD SYSTEMS, INC.
Polarization Ratios
-80 -60 -40 -20 0 20 40 60 80
Azimuthal Angle in Degrees
-50
-40
-30
-20
-10
0
10
Relat
ive Am
plitu
de in
dB
Polarization Ratios of AUT and Probe
X-Polarized AUT, Reciprocal of Polarization Ratio Y/X
x-Polarized Probe, Reciprocal of Polarization Ratio, Y/X
Copyright Feb 2002 Page 53NEARFIELD SYSTEMS, INC.
Probe Polarization Correction
-80 -60 -40 -20 0 20 40 60 80
Azimuthal Angle in Degrees
-120
-100
-80
-60
-40
-20
Relativ
e A
mpl
itude
in d
B
Cross Component Polarization Correction
Polarization Correction
After Polarization Correction
Spectrum of Data Before Correction
Copyright Feb 2002 Page 54NEARFIELD SYSTEMS, INC.
General Conclusions
l Probe correction performed on spectrum not near-field data
l Probe polarization correction rarely if ever needed for main component probe
l Need for accurate cross polarization data on cross component probe depends on
l Relative polarization ratio of AUT and probel Required accuracy