INF5410 Array signal processing.
Ch. 3: Apertures and Arrays
Andreas Austeng
January 2005
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Aperture and Arrays
Study apertures: Examine the effect of sensors that gather
signal energy over finite areas.
Arrays: Group of sensors combined to produce single output.
At m’th sensor position, ~xm :
Fields value: f (~xm , t).
Sensors output: ym (t).
If sensor is perfect (i.e. linear transf., infinite bandwidth):
ym (t) = κ · f (~xm , t), κ ∈ < (or C).
Directional ↔ omni-directional
If sensor has (significant) spatial extent, it will spatially
integrate energy, i.e. focises in particular propagation direction.
Example: Parabolic dish.
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Aperture and Arrays
Study apertures: Examine the effect of sensors that gather
signal energy over finite areas.
Arrays: Group of sensors combined to produce single output.
At m’th sensor position, ~xm :
Fields value: f (~xm , t).
Sensors output: ym (t).
If sensor is perfect (i.e. linear transf., infinite bandwidth):
ym (t) = κ · f (~xm , t), κ ∈ < (or C).
Directional ↔ omni-directional
If sensor has (significant) spatial extent, it will spatially
integrate energy, i.e. focises in particular propagation direction.
Example: Parabolic dish.
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Aperture and Arrays
Study apertures: Examine the effect of sensors that gather
signal energy over finite areas.
Arrays: Group of sensors combined to produce single output.
At m’th sensor position, ~xm :
Fields value: f (~xm , t).
Sensors output: ym (t).
If sensor is perfect (i.e. linear transf., infinite bandwidth):
ym (t) = κ · f (~xm , t), κ ∈ < (or C).
Directional ↔ omni-directional
If sensor has (significant) spatial extent, it will spatially
integrate energy, i.e. focises in particular propagation direction.
Example: Parabolic dish.
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Aperture and Arrays
Study apertures: Examine the effect of sensors that gather
signal energy over finite areas.
Arrays: Group of sensors combined to produce single output.
At m’th sensor position, ~xm :
Fields value: f (~xm , t).
Sensors output: ym (t).
If sensor is perfect (i.e. linear transf., infinite bandwidth):
ym (t) = κ · f (~xm , t), κ ∈ < (or C).
Directional ↔ omni-directional
If sensor has (significant) spatial extent, it will spatially
integrate energy, i.e. focises in particular propagation direction.
Example: Parabolic dish.
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Aperture and Arrays
Study apertures: Examine the effect of sensors that gather
signal energy over finite areas.
Arrays: Group of sensors combined to produce single output.
At m’th sensor position, ~xm :
Fields value: f (~xm , t).
Sensors output: ym (t).
If sensor is perfect (i.e. linear transf., infinite bandwidth):
ym (t) = κ · f (~xm , t), κ ∈ < (or C).
Directional ↔ omni-directional
If sensor has (significant) spatial extent, it will spatially
integrate energy, i.e. focises in particular propagation direction.
Example: Parabolic dish.
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Aperture and Arrays
Study apertures: Examine the effect of sensors that gather
signal energy over finite areas.
Arrays: Group of sensors combined to produce single output.
At m’th sensor position, ~xm :
Fields value: f (~xm , t).
Sensors output: ym (t).
If sensor is perfect (i.e. linear transf., infinite bandwidth):
ym (t) = κ · f (~xm , t), κ ∈ < (or C).
Directional ↔ omni-directional
If sensor has (significant) spatial extent, it will spatially
integrate energy, i.e. focises in particular propagation direction.
Example: Parabolic dish.
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Aperture and Arrays
Study apertures: Examine the effect of sensors that gather
signal energy over finite areas.
Arrays: Group of sensors combined to produce single output.
At m’th sensor position, ~xm :
Fields value: f (~xm , t).
Sensors output: ym (t).
If sensor is perfect (i.e. linear transf., infinite bandwidth):
ym (t) = κ · f (~xm , t), κ ∈ < (or C).
Directional ↔ omni-directional
If sensor has (significant) spatial extent, it will spatially
integrate energy, i.e. focises in particular propagation direction.
Example: Parabolic dish.
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Aperture and Arrays
Study apertures: Examine the effect of sensors that gather
signal energy over finite areas.
Arrays: Group of sensors combined to produce single output.
At m’th sensor position, ~xm :
Fields value: f (~xm , t).
Sensors output: ym (t).
If sensor is perfect (i.e. linear transf., infinite bandwidth):
ym (t) = κ · f (~xm , t), κ ∈ < (or C).
Directional ↔ omni-directional
If sensor has (significant) spatial extent, it will spatially
integrate energy, i.e. focises in particular propagation direction.
Example: Parabolic dish.
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Aperture and Arrays
Study apertures: Examine the effect of sensors that gather
signal energy over finite areas.
Arrays: Group of sensors combined to produce single output.
At m’th sensor position, ~xm :
Fields value: f (~xm , t).
Sensors output: ym (t).
If sensor is perfect (i.e. linear transf., infinite bandwidth):
ym (t) = κ · f (~xm , t), κ ∈ < (or C).
Directional ↔ omni-directional
If sensor has (significant) spatial extent, it will spatially
integrate energy, i.e. focises in particular propagation direction.
Example: Parabolic dish.
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Aperture function and aperture smooting function
Apertures (6= point sources) are described by the aperture
function, w (~x ).
Spatial extend reflects size and shape
Aperture weighting; relative weighting of the field within the
aperture (also known as shading, tapering, apodization).
Aperture smooting function: W (~k) =
R∞
~ · ~x )d~x
x ) exp(k
−∞ w (~
Given a finite aperture, w (~x ), the sensor output, z(~x , t):
z(~x , t) = w (~x )f (~x , t).
Space-time Fourier transform:
R∞
Z (~k, w ) =
W (~k − ~l)F (~l, w )d~l.
−∞
Wavefield spectum, F (~l, w ), smoothed by the kernel W (~k).
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Aperture function and aperture smooting function
Apertures (6= point sources) are described by the aperture
function, w (~x ).
Spatial extend reflects size and shape
Aperture weighting; relative weighting of the field within the
aperture (also known as shading, tapering, apodization).
Aperture smooting function: W (~k) =
R∞
~ · ~x )d~x
x ) exp(k
−∞ w (~
Given a finite aperture, w (~x ), the sensor output, z(~x , t):
z(~x , t) = w (~x )f (~x , t).
Space-time Fourier transform:
R∞
Z (~k, w ) =
W (~k − ~l)F (~l, w )d~l.
−∞
Wavefield spectum, F (~l, w ), smoothed by the kernel W (~k).
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Aperture function and aperture smooting function
Apertures (6= point sources) are described by the aperture
function, w (~x ).
Spatial extend reflects size and shape
Aperture weighting; relative weighting of the field within the
aperture (also known as shading, tapering, apodization).
Aperture smooting function: W (~k) =
R∞
~ · ~x )d~x
x ) exp(k
−∞ w (~
Given a finite aperture, w (~x ), the sensor output, z(~x , t):
z(~x , t) = w (~x )f (~x , t).
Space-time Fourier transform:
R∞
Z (~k, w ) =
W (~k − ~l)F (~l, w )d~l.
−∞
Wavefield spectum, F (~l, w ), smoothed by the kernel W (~k).
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Aperture function and aperture smooting function
Apertures (6= point sources) are described by the aperture
function, w (~x ).
Spatial extend reflects size and shape
Aperture weighting; relative weighting of the field within the
aperture (also known as shading, tapering, apodization).
Aperture smooting function: W (~k) =
R∞
~ · ~x )d~x
x ) exp(k
−∞ w (~
Given a finite aperture, w (~x ), the sensor output, z(~x , t):
z(~x , t) = w (~x )f (~x , t).
Space-time Fourier transform:
R∞
Z (~k, w ) =
W (~k − ~l)F (~l, w )d~l.
−∞
Wavefield spectum, F (~l, w ), smoothed by the kernel W (~k).
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Aperture function and aperture smooting function
Apertures (6= point sources) are described by the aperture
function, w (~x ).
Spatial extend reflects size and shape
Aperture weighting; relative weighting of the field within the
aperture (also known as shading, tapering, apodization).
Aperture smooting function: W (~k) =
R∞
~ · ~x )d~x
x ) exp(k
−∞ w (~
Given a finite aperture, w (~x ), the sensor output, z(~x , t):
z(~x , t) = w (~x )f (~x , t).
Space-time Fourier transform:
R∞
Z (~k, w ) =
W (~k − ~l)F (~l, w )d~l.
−∞
Wavefield spectum, F (~l, w ), smoothed by the kernel W (~k).
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Aperture function and aperture smooting function
Apertures (6= point sources) are described by the aperture
function, w (~x ).
Spatial extend reflects size and shape
Aperture weighting; relative weighting of the field within the
aperture (also known as shading, tapering, apodization).
Aperture smooting function: W (~k) =
R∞
~ · ~x )d~x
x ) exp(k
−∞ w (~
Given a finite aperture, w (~x ), the sensor output, z(~x , t):
z(~x , t) = w (~x )f (~x , t).
Space-time Fourier transform:
R∞
Z (~k, w ) =
W (~k − ~l)F (~l, w )d~l.
−∞
Wavefield spectum, F (~l, w ), smoothed by the kernel W (~k).
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Aperture function and aperture smooting function
Apertures (6= point sources) are described by the aperture
function, w (~x ).
Spatial extend reflects size and shape
Aperture weighting; relative weighting of the field within the
aperture (also known as shading, tapering, apodization).
Aperture smooting function: W (~k) =
R∞
~ · ~x )d~x
x ) exp(k
−∞ w (~
Given a finite aperture, w (~x ), the sensor output, z(~x , t):
z(~x , t) = w (~x )f (~x , t).
Space-time Fourier transform:
R∞
Z (~k, w ) =
W (~k − ~l)F (~l, w )d~l.
−∞
Wavefield spectum, F (~l, w ), smoothed by the kernel W (~k).
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Classical resolution
Spatial extent of w (~x ) determines the resulution with which
two plane waves can be separated.
Ideally, W (~k) = δ(~k), i.e. infinit spatial extent!
Rayleigh criterion:
Two incoherent plane waves, propagating in two slightly different
directions, are resolved if the mainlobe peak of one aperture
smoothing function replica falls on the first zero of the other
aperture smoothing function replica, i.e. half the mainlobe width.
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Classical resolution
Spatial extent of w (~x ) determines the resulution with which
two plane waves can be separated.
Ideally, W (~k) = δ(~k), i.e. infinit spatial extent!
Rayleigh criterion:
Two incoherent plane waves, propagating in two slightly different
directions, are resolved if the mainlobe peak of one aperture
smoothing function replica falls on the first zero of the other
aperture smoothing function replica, i.e. half the mainlobe width.
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Classical resolution
Spatial extent of w (~x ) determines the resulution with which
two plane waves can be separated.
Ideally, W (~k) = δ(~k), i.e. infinit spatial extent!
Rayleigh criterion:
Two incoherent plane waves, propagating in two slightly different
directions, are resolved if the mainlobe peak of one aperture
smoothing function replica falls on the first zero of the other
aperture smoothing function replica, i.e. half the mainlobe width.
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Classical resolution ...
Linear aperture of size D
W (kx ) =
sin(kx D/2)
(=
kx /2
Dsinc(kx D/2)) =
sin(π sin θD/λ)
π sin θ/λ
-3 dB width: θ−3dB ≈ 0.89λ/D
-6 dB width: θ−6dB ≈ 1.21λ/D
Zero-to-zero distance: θ0−0 = 2λ/D
Circular aperture of diameter D
W (kxy ) =
J1 (kxy D/2)
kxy D/2
(??)
-3 dB width: θ−3dB ≈ 1.02λ/D
-6 dB width: θ−6dB ≈ 1.41λ/D
Zero-to-zero distance: θ0−0 ≈ 2.44λ/D
Rule-of-thumb; Angular resolution: θ = λ/D
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Classical resolution ...
Linear aperture of size D
W (kx ) =
sin(kx D/2)
(=
kx /2
Dsinc(kx D/2)) =
sin(π sin θD/λ)
π sin θ/λ
-3 dB width: θ−3dB ≈ 0.89λ/D
-6 dB width: θ−6dB ≈ 1.21λ/D
Zero-to-zero distance: θ0−0 = 2λ/D
Circular aperture of diameter D
W (kxy ) =
J1 (kxy D/2)
kxy D/2
(??)
-3 dB width: θ−3dB ≈ 1.02λ/D
-6 dB width: θ−6dB ≈ 1.41λ/D
Zero-to-zero distance: θ0−0 ≈ 2.44λ/D
Rule-of-thumb; Angular resolution: θ = λ/D
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Classical resolution ...
Linear aperture of size D
W (kx ) =
sin(kx D/2)
(=
kx /2
Dsinc(kx D/2)) =
sin(π sin θD/λ)
π sin θ/λ
-3 dB width: θ−3dB ≈ 0.89λ/D
-6 dB width: θ−6dB ≈ 1.21λ/D
Zero-to-zero distance: θ0−0 = 2λ/D
Circular aperture of diameter D
W (kxy ) =
J1 (kxy D/2)
kxy D/2
(??)
-3 dB width: θ−3dB ≈ 1.02λ/D
-6 dB width: θ−6dB ≈ 1.41λ/D
Zero-to-zero distance: θ0−0 ≈ 2.44λ/D
Rule-of-thumb; Angular resolution: θ = λ/D
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Classical resolution ...
Linear aperture of size D
W (kx ) =
sin(kx D/2)
(=
kx /2
Dsinc(kx D/2)) =
sin(π sin θD/λ)
π sin θ/λ
-3 dB width: θ−3dB ≈ 0.89λ/D
-6 dB width: θ−6dB ≈ 1.21λ/D
Zero-to-zero distance: θ0−0 = 2λ/D
Circular aperture of diameter D
W (kxy ) =
J1 (kxy D/2)
kxy D/2
(??)
-3 dB width: θ−3dB ≈ 1.02λ/D
-6 dB width: θ−6dB ≈ 1.41λ/D
Zero-to-zero distance: θ0−0 ≈ 2.44λ/D
Rule-of-thumb; Angular resolution: θ = λ/D
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Classical resolution ...
Linear aperture of size D
W (kx ) =
sin(kx D/2)
(=
kx /2
Dsinc(kx D/2)) =
sin(π sin θD/λ)
π sin θ/λ
-3 dB width: θ−3dB ≈ 0.89λ/D
-6 dB width: θ−6dB ≈ 1.21λ/D
Zero-to-zero distance: θ0−0 = 2λ/D
Circular aperture of diameter D
W (kxy ) =
J1 (kxy D/2)
kxy D/2
(??)
-3 dB width: θ−3dB ≈ 1.02λ/D
-6 dB width: θ−6dB ≈ 1.41λ/D
Zero-to-zero distance: θ0−0 ≈ 2.44λ/D
Rule-of-thumb; Angular resolution: θ = λ/D
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Classical resolution ...
Linear aperture of size D
W (kx ) =
sin(kx D/2)
(=
kx /2
Dsinc(kx D/2)) =
sin(π sin θD/λ)
π sin θ/λ
-3 dB width: θ−3dB ≈ 0.89λ/D
-6 dB width: θ−6dB ≈ 1.21λ/D
Zero-to-zero distance: θ0−0 = 2λ/D
Circular aperture of diameter D
W (kxy ) =
J1 (kxy D/2)
kxy D/2
(??)
-3 dB width: θ−3dB ≈ 1.02λ/D
-6 dB width: θ−6dB ≈ 1.41λ/D
Zero-to-zero distance: θ0−0 ≈ 2.44λ/D
Rule-of-thumb; Angular resolution: θ = λ/D
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Classical resolution ...
Linear aperture of size D
W (kx ) =
sin(kx D/2)
(=
kx /2
Dsinc(kx D/2)) =
sin(π sin θD/λ)
π sin θ/λ
-3 dB width: θ−3dB ≈ 0.89λ/D
-6 dB width: θ−6dB ≈ 1.21λ/D
Zero-to-zero distance: θ0−0 = 2λ/D
Circular aperture of diameter D
W (kxy ) =
J1 (kxy D/2)
kxy D/2
(??)
-3 dB width: θ−3dB ≈ 1.02λ/D
-6 dB width: θ−6dB ≈ 1.41λ/D
Zero-to-zero distance: θ0−0 ≈ 2.44λ/D
Rule-of-thumb; Angular resolution: θ = λ/D
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Classical resolution ...
Linear aperture of size D
W (kx ) =
sin(kx D/2)
(=
kx /2
Dsinc(kx D/2)) =
sin(π sin θD/λ)
π sin θ/λ
-3 dB width: θ−3dB ≈ 0.89λ/D
-6 dB width: θ−6dB ≈ 1.21λ/D
Zero-to-zero distance: θ0−0 = 2λ/D
Circular aperture of diameter D
W (kxy ) =
J1 (kxy D/2)
kxy D/2
(??)
-3 dB width: θ−3dB ≈ 1.02λ/D
-6 dB width: θ−6dB ≈ 1.41λ/D
Zero-to-zero distance: θ0−0 ≈ 2.44λ/D
Rule-of-thumb; Angular resolution: θ = λ/D
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Classical resolution ...
Linear aperture of size D
W (kx ) =
sin(kx D/2)
(=
kx /2
Dsinc(kx D/2)) =
sin(π sin θD/λ)
π sin θ/λ
-3 dB width: θ−3dB ≈ 0.89λ/D
-6 dB width: θ−6dB ≈ 1.21λ/D
Zero-to-zero distance: θ0−0 = 2λ/D
Circular aperture of diameter D
W (kxy ) =
J1 (kxy D/2)
kxy D/2
(??)
-3 dB width: θ−3dB ≈ 1.02λ/D
-6 dB width: θ−6dB ≈ 1.41λ/D
Zero-to-zero distance: θ0−0 ≈ 2.44λ/D
Rule-of-thumb; Angular resolution: θ = λ/D
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Geometrical Optics
Validity: down to about a wavelength
Nearfield-farfield transition
dR = D 2 /λ for a maximum phase error of λ/8 over aperture
f-number
Ratio of range and aperture: f# = R/D
Resolution
Angular resolution: θ = λ/D
Azimuth resolution: u = Rθ = f# λ
Depth of focus
Aperture is focused at range R. Phase error of λ/8 yields
r = ±f#2 λ or DOF=2f#2 λ (proportional to phase error)
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Geometrical Optics
Validity: down to about a wavelength
Nearfield-farfield transition
dR = D 2 /λ for a maximum phase error of λ/8 over aperture
f-number
Ratio of range and aperture: f# = R/D
Resolution
Angular resolution: θ = λ/D
Azimuth resolution: u = Rθ = f# λ
Depth of focus
Aperture is focused at range R. Phase error of λ/8 yields
r = ±f#2 λ or DOF=2f#2 λ (proportional to phase error)
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Geometrical Optics
Validity: down to about a wavelength
Nearfield-farfield transition
dR = D 2 /λ for a maximum phase error of λ/8 over aperture
f-number
Ratio of range and aperture: f# = R/D
Resolution
Angular resolution: θ = λ/D
Azimuth resolution: u = Rθ = f# λ
Depth of focus
Aperture is focused at range R. Phase error of λ/8 yields
r = ±f#2 λ or DOF=2f#2 λ (proportional to phase error)
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Geometrical Optics
Validity: down to about a wavelength
Nearfield-farfield transition
dR = D 2 /λ for a maximum phase error of λ/8 over aperture
f-number
Ratio of range and aperture: f# = R/D
Resolution
Angular resolution: θ = λ/D
Azimuth resolution: u = Rθ = f# λ
Depth of focus
Aperture is focused at range R. Phase error of λ/8 yields
r = ±f#2 λ or DOF=2f#2 λ (proportional to phase error)
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Geometrical Optics
Validity: down to about a wavelength
Nearfield-farfield transition
dR = D 2 /λ for a maximum phase error of λ/8 over aperture
f-number
Ratio of range and aperture: f# = R/D
Resolution
Angular resolution: θ = λ/D
Azimuth resolution: u = Rθ = f# λ
Depth of focus
Aperture is focused at range R. Phase error of λ/8 yields
r = ±f#2 λ or DOF=2f#2 λ (proportional to phase error)
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Geometrical Optics
Validity: down to about a wavelength
Nearfield-farfield transition
dR = D 2 /λ for a maximum phase error of λ/8 over aperture
f-number
Ratio of range and aperture: f# = R/D
Resolution
Angular resolution: θ = λ/D
Azimuth resolution: u = Rθ = f# λ
Depth of focus
Aperture is focused at range R. Phase error of λ/8 yields
r = ±f#2 λ or DOF=2f#2 λ (proportional to phase error)
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Geometrical Optics
Validity: down to about a wavelength
Nearfield-farfield transition
dR = D 2 /λ for a maximum phase error of λ/8 over aperture
f-number
Ratio of range and aperture: f# = R/D
Resolution
Angular resolution: θ = λ/D
Azimuth resolution: u = Rθ = f# λ
Depth of focus
Aperture is focused at range R. Phase error of λ/8 yields
r = ±f#2 λ or DOF=2f#2 λ (proportional to phase error)
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Geometrical Optics
Validity: down to about a wavelength
Nearfield-farfield transition
dR = D 2 /λ for a maximum phase error of λ/8 over aperture
f-number
Ratio of range and aperture: f# = R/D
Resolution
Angular resolution: θ = λ/D
Azimuth resolution: u = Rθ = f# λ
Depth of focus
Aperture is focused at range R. Phase error of λ/8 yields
r = ±f#2 λ or DOF=2f#2 λ (proportional to phase error)
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Geometrical Optics
Validity: down to about a wavelength
Nearfield-farfield transition
dR = D 2 /λ for a maximum phase error of λ/8 over aperture
f-number
Ratio of range and aperture: f# = R/D
Resolution
Angular resolution: θ = λ/D
Azimuth resolution: u = Rθ = f# λ
Depth of focus
Aperture is focused at range R. Phase error of λ/8 yields
r = ±f#2 λ or DOF=2f#2 λ (proportional to phase error)
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Geometrical Optics
Validity: down to about a wavelength
Nearfield-farfield transition
dR = D 2 /λ for a maximum phase error of λ/8 over aperture
f-number
Ratio of range and aperture: f# = R/D
Resolution
Angular resolution: θ = λ/D
Azimuth resolution: u = Rθ = f# λ
Depth of focus
Aperture is focused at range R. Phase error of λ/8 yields
r = ±f#2 λ or DOF=2f#2 λ (proportional to phase error)
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Ultrasound imaging
Nearfield-farfield transition, D=28mm, f=3.5MHz ⇒
λ = 1540/3.5 · 106 = 0.44mm and dR = D 2 /R = 1782mm
All diagnostic ultrasound imaging occurs in the extreme near
field!
Azimuth resolution, D=28mm, f=7MHz ⇒
λ = 0.22mm and θ = λ/D = 0.45◦ ,
i.e. about 200 lines are required to scan ±45◦
Depth of focus, f# = 2, f=5MHz ⇒
λ = 0.308mm and DOF = 2f#2 λ ≈ 2.5mm.
Ultrasound requires T = 2 · 2.5 · 10−3 /1540 = 3.2µs to travel
the DOF. This is the minimum update rate for the delays in a
dynamically focused system.
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Ultrasound imaging
Nearfield-farfield transition, D=28mm, f=3.5MHz ⇒
λ = 1540/3.5 · 106 = 0.44mm and dR = D 2 /R = 1782mm
All diagnostic ultrasound imaging occurs in the extreme near
field!
Azimuth resolution, D=28mm, f=7MHz ⇒
λ = 0.22mm and θ = λ/D = 0.45◦ ,
i.e. about 200 lines are required to scan ±45◦
Depth of focus, f# = 2, f=5MHz ⇒
λ = 0.308mm and DOF = 2f#2 λ ≈ 2.5mm.
Ultrasound requires T = 2 · 2.5 · 10−3 /1540 = 3.2µs to travel
the DOF. This is the minimum update rate for the delays in a
dynamically focused system.
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Ultrasound imaging
Nearfield-farfield transition, D=28mm, f=3.5MHz ⇒
λ = 1540/3.5 · 106 = 0.44mm and dR = D 2 /R = 1782mm
All diagnostic ultrasound imaging occurs in the extreme near
field!
Azimuth resolution, D=28mm, f=7MHz ⇒
λ = 0.22mm and θ = λ/D = 0.45◦ ,
i.e. about 200 lines are required to scan ±45◦
Depth of focus, f# = 2, f=5MHz ⇒
λ = 0.308mm and DOF = 2f#2 λ ≈ 2.5mm.
Ultrasound requires T = 2 · 2.5 · 10−3 /1540 = 3.2µs to travel
the DOF. This is the minimum update rate for the delays in a
dynamically focused system.
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Ultrasound imaging
Nearfield-farfield transition, D=28mm, f=3.5MHz ⇒
λ = 1540/3.5 · 106 = 0.44mm and dR = D 2 /R = 1782mm
All diagnostic ultrasound imaging occurs in the extreme near
field!
Azimuth resolution, D=28mm, f=7MHz ⇒
λ = 0.22mm and θ = λ/D = 0.45◦ ,
i.e. about 200 lines are required to scan ±45◦
Depth of focus, f# = 2, f=5MHz ⇒
λ = 0.308mm and DOF = 2f#2 λ ≈ 2.5mm.
Ultrasound requires T = 2 · 2.5 · 10−3 /1540 = 3.2µs to travel
the DOF. This is the minimum update rate for the delays in a
dynamically focused system.
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Ultrasound imaging
Nearfield-farfield transition, D=28mm, f=3.5MHz ⇒
λ = 1540/3.5 · 106 = 0.44mm and dR = D 2 /R = 1782mm
All diagnostic ultrasound imaging occurs in the extreme near
field!
Azimuth resolution, D=28mm, f=7MHz ⇒
λ = 0.22mm and θ = λ/D = 0.45◦ ,
i.e. about 200 lines are required to scan ±45◦
Depth of focus, f# = 2, f=5MHz ⇒
λ = 0.308mm and DOF = 2f#2 λ ≈ 2.5mm.
Ultrasound requires T = 2 · 2.5 · 10−3 /1540 = 3.2µs to travel
the DOF. This is the minimum update rate for the delays in a
dynamically focused system.
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Ultrasound imaging
Nearfield-farfield transition, D=28mm, f=3.5MHz ⇒
λ = 1540/3.5 · 106 = 0.44mm and dR = D 2 /R = 1782mm
All diagnostic ultrasound imaging occurs in the extreme near
field!
Azimuth resolution, D=28mm, f=7MHz ⇒
λ = 0.22mm and θ = λ/D = 0.45◦ ,
i.e. about 200 lines are required to scan ±45◦
Depth of focus, f# = 2, f=5MHz ⇒
λ = 0.308mm and DOF = 2f#2 λ ≈ 2.5mm.
Ultrasound requires T = 2 · 2.5 · 10−3 /1540 = 3.2µs to travel
the DOF. This is the minimum update rate for the delays in a
dynamically focused system.
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Ultrasound imaging
Nearfield-farfield transition, D=28mm, f=3.5MHz ⇒
λ = 1540/3.5 · 106 = 0.44mm and dR = D 2 /R = 1782mm
All diagnostic ultrasound imaging occurs in the extreme near
field!
Azimuth resolution, D=28mm, f=7MHz ⇒
λ = 0.22mm and θ = λ/D = 0.45◦ ,
i.e. about 200 lines are required to scan ±45◦
Depth of focus, f# = 2, f=5MHz ⇒
λ = 0.308mm and DOF = 2f#2 λ ≈ 2.5mm.
Ultrasound requires T = 2 · 2.5 · 10−3 /1540 = 3.2µs to travel
the DOF. This is the minimum update rate for the delays in a
dynamically focused system.
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Ultrasound imaging
Nearfield-farfield transition, D=28mm, f=3.5MHz ⇒
λ = 1540/3.5 · 106 = 0.44mm and dR = D 2 /R = 1782mm
All diagnostic ultrasound imaging occurs in the extreme near
field!
Azimuth resolution, D=28mm, f=7MHz ⇒
λ = 0.22mm and θ = λ/D = 0.45◦ ,
i.e. about 200 lines are required to scan ±45◦
Depth of focus, f# = 2, f=5MHz ⇒
λ = 0.308mm and DOF = 2f#2 λ ≈ 2.5mm.
Ultrasound requires T = 2 · 2.5 · 10−3 /1540 = 3.2µs to travel
the DOF. This is the minimum update rate for the delays in a
dynamically focused system.
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Ultrasound imaging
Nearfield-farfield transition, D=28mm, f=3.5MHz ⇒
λ = 1540/3.5 · 106 = 0.44mm and dR = D 2 /R = 1782mm
All diagnostic ultrasound imaging occurs in the extreme near
field!
Azimuth resolution, D=28mm, f=7MHz ⇒
λ = 0.22mm and θ = λ/D = 0.45◦ ,
i.e. about 200 lines are required to scan ±45◦
Depth of focus, f# = 2, f=5MHz ⇒
λ = 0.308mm and DOF = 2f#2 λ ≈ 2.5mm.
Ultrasound requires T = 2 · 2.5 · 10−3 /1540 = 3.2µs to travel
the DOF. This is the minimum update rate for the delays in a
dynamically focused system.
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Aperture ambiguities
Due to symmetries
Aberrations
Deviation in the waveform from its intended form.
In optics; due to deviation of a lense from its ideal shape.
More generally; Turbulence in the medium, inhomogent
medium or position errors in the aperture.
φ −→ sin φ
Co-array for continuius apertures
R
c(~
χ) ≡ w (~x )w (~x + χ
~ )d~x , χ
~ called lag and its domain lag
space.
Fourier transform of c(~
χ)(= |W (~k)|2 ) gives a smoothed
estimate of the power spectum Sf (~k, w ).
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Aperture ambiguities
Due to symmetries
Aberrations
Deviation in the waveform from its intended form.
In optics; due to deviation of a lense from its ideal shape.
More generally; Turbulence in the medium, inhomogent
medium or position errors in the aperture.
φ −→ sin φ
Co-array for continuius apertures
R
c(~
χ) ≡ w (~x )w (~x + χ
~ )d~x , χ
~ called lag and its domain lag
space.
Fourier transform of c(~
χ)(= |W (~k)|2 ) gives a smoothed
estimate of the power spectum Sf (~k, w ).
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Aperture ambiguities
Due to symmetries
Aberrations
Deviation in the waveform from its intended form.
In optics; due to deviation of a lense from its ideal shape.
More generally; Turbulence in the medium, inhomogent
medium or position errors in the aperture.
φ −→ sin φ
Co-array for continuius apertures
R
c(~
χ) ≡ w (~x )w (~x + χ
~ )d~x , χ
~ called lag and its domain lag
space.
Fourier transform of c(~
χ)(= |W (~k)|2 ) gives a smoothed
estimate of the power spectum Sf (~k, w ).
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Aperture ambiguities
Due to symmetries
Aberrations
Deviation in the waveform from its intended form.
In optics; due to deviation of a lense from its ideal shape.
More generally; Turbulence in the medium, inhomogent
medium or position errors in the aperture.
φ −→ sin φ
Co-array for continuius apertures
R
c(~
χ) ≡ w (~x )w (~x + χ
~ )d~x , χ
~ called lag and its domain lag
space.
Fourier transform of c(~
χ)(= |W (~k)|2 ) gives a smoothed
estimate of the power spectum Sf (~k, w ).
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Aperture ambiguities
Due to symmetries
Aberrations
Deviation in the waveform from its intended form.
In optics; due to deviation of a lense from its ideal shape.
More generally; Turbulence in the medium, inhomogent
medium or position errors in the aperture.
φ −→ sin φ
Co-array for continuius apertures
R
c(~
χ) ≡ w (~x )w (~x + χ
~ )d~x , χ
~ called lag and its domain lag
space.
Fourier transform of c(~
χ)(= |W (~k)|2 ) gives a smoothed
estimate of the power spectum Sf (~k, w ).
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Aperture ambiguities
Due to symmetries
Aberrations
Deviation in the waveform from its intended form.
In optics; due to deviation of a lense from its ideal shape.
More generally; Turbulence in the medium, inhomogent
medium or position errors in the aperture.
φ −→ sin φ
Co-array for continuius apertures
R
c(~
χ) ≡ w (~x )w (~x + χ
~ )d~x , χ
~ called lag and its domain lag
space.
Fourier transform of c(~
χ)(= |W (~k)|2 ) gives a smoothed
estimate of the power spectum Sf (~k, w ).
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Aperture ambiguities
Due to symmetries
Aberrations
Deviation in the waveform from its intended form.
In optics; due to deviation of a lense from its ideal shape.
More generally; Turbulence in the medium, inhomogent
medium or position errors in the aperture.
φ −→ sin φ
Co-array for continuius apertures
R
c(~
χ) ≡ w (~x )w (~x + χ
~ )d~x , χ
~ called lag and its domain lag
space.
Fourier transform of c(~
χ)(= |W (~k)|2 ) gives a smoothed
estimate of the power spectum Sf (~k, w ).
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Aperture ambiguities
Due to symmetries
Aberrations
Deviation in the waveform from its intended form.
In optics; due to deviation of a lense from its ideal shape.
More generally; Turbulence in the medium, inhomogent
medium or position errors in the aperture.
φ −→ sin φ
Co-array for continuius apertures
R
c(~
χ) ≡ w (~x )w (~x + χ
~ )d~x , χ
~ called lag and its domain lag
space.
Fourier transform of c(~
χ)(= |W (~k)|2 ) gives a smoothed
estimate of the power spectum Sf (~k, w ).
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Aperture ambiguities
Due to symmetries
Aberrations
Deviation in the waveform from its intended form.
In optics; due to deviation of a lense from its ideal shape.
More generally; Turbulence in the medium, inhomogent
medium or position errors in the aperture.
φ −→ sin φ
Co-array for continuius apertures
R
c(~
χ) ≡ w (~x )w (~x + χ
~ )d~x , χ
~ called lag and its domain lag
space.
Fourier transform of c(~
χ)(= |W (~k)|2 ) gives a smoothed
estimate of the power spectum Sf (~k, w ).
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Aperture and Arrays
w (~
x)andW (~
k)
Classical resolution
Aperture ambiguities
Due to symmetries
Aberrations
Deviation in the waveform from its intended form.
In optics; due to deviation of a lense from its ideal shape.
More generally; Turbulence in the medium, inhomogent
medium or position errors in the aperture.
φ −→ sin φ
Co-array for continuius apertures
R
c(~
χ) ≡ w (~x )w (~x + χ
~ )d~x , χ
~ called lag and its domain lag
space.
Fourier transform of c(~
χ)(= |W (~k)|2 ) gives a smoothed
estimate of the power spectum Sf (~k, w ).
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Sampling in one dimension
Periodic spatial sampling in one dimension
Array:
Consists of individual sensors that sample the environment
spatially
Each sensor coulr be an aperture or omni-directional transducer
Spatial sampling introduces some complications
(Nyquist sampling, folding, . . .)
Question to be asked/answered:
When can f (x, t0 ) be reconstructed by {ym (to )}?
f (x, t) is the continuous signal and
{ym (t)} is a sequence of temporal signals where
ym (t) = f (md, t), d being the spatial sampling interval.
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Sampling in one dimension
Periodic spatial sampling in one dimension
Array:
Consists of individual sensors that sample the environment
spatially
Each sensor coulr be an aperture or omni-directional transducer
Spatial sampling introduces some complications
(Nyquist sampling, folding, . . .)
Question to be asked/answered:
When can f (x, t0 ) be reconstructed by {ym (to )}?
f (x, t) is the continuous signal and
{ym (t)} is a sequence of temporal signals where
ym (t) = f (md, t), d being the spatial sampling interval.
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Sampling in one dimension
Periodic spatial sampling in one dimension
Array:
Consists of individual sensors that sample the environment
spatially
Each sensor coulr be an aperture or omni-directional transducer
Spatial sampling introduces some complications
(Nyquist sampling, folding, . . .)
Question to be asked/answered:
When can f (x, t0 ) be reconstructed by {ym (to )}?
f (x, t) is the continuous signal and
{ym (t)} is a sequence of temporal signals where
ym (t) = f (md, t), d being the spatial sampling interval.
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Sampling in one dimension
Periodic spatial sampling in one dimension
Array:
Consists of individual sensors that sample the environment
spatially
Each sensor coulr be an aperture or omni-directional transducer
Spatial sampling introduces some complications
(Nyquist sampling, folding, . . .)
Question to be asked/answered:
When can f (x, t0 ) be reconstructed by {ym (to )}?
f (x, t) is the continuous signal and
{ym (t)} is a sequence of temporal signals where
ym (t) = f (md, t), d being the spatial sampling interval.
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Sampling in one dimension
Periodic spatial sampling in one dimension
Array:
Consists of individual sensors that sample the environment
spatially
Each sensor coulr be an aperture or omni-directional transducer
Spatial sampling introduces some complications
(Nyquist sampling, folding, . . .)
Question to be asked/answered:
When can f (x, t0 ) be reconstructed by {ym (to )}?
f (x, t) is the continuous signal and
{ym (t)} is a sequence of temporal signals where
ym (t) = f (md, t), d being the spatial sampling interval.
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Sampling in one dimension
Periodic spatial sampling in one dimension
Array:
Consists of individual sensors that sample the environment
spatially
Each sensor coulr be an aperture or omni-directional transducer
Spatial sampling introduces some complications
(Nyquist sampling, folding, . . .)
Question to be asked/answered:
When can f (x, t0 ) be reconstructed by {ym (to )}?
f (x, t) is the continuous signal and
{ym (t)} is a sequence of temporal signals where
ym (t) = f (md, t), d being the spatial sampling interval.
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Sampling in one dimension
Periodic spatial sampling in one dimension
Array:
Consists of individual sensors that sample the environment
spatially
Each sensor coulr be an aperture or omni-directional transducer
Spatial sampling introduces some complications
(Nyquist sampling, folding, . . .)
Question to be asked/answered:
When can f (x, t0 ) be reconstructed by {ym (to )}?
f (x, t) is the continuous signal and
{ym (t)} is a sequence of temporal signals where
ym (t) = f (md, t), d being the spatial sampling interval.
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Sampling in one dimension
Periodic spatial sampling in one dimension ...
Sampling theorem (Nyquist):
If a continuius-variable signal is bandlimited to frequences
below k0 , then it can be periodically sampled without loss of
information so long as the sampling period d ≤ π/k0 = λ0 /2.
Periodic sampling of one-dimensional signals can be
straightforwardly extended to multidimensional signals.
“Rectangular/regular” sampling not nessesary for
multidimensional signals.
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Sampling in one dimension
Periodic spatial sampling in one dimension ...
Sampling theorem (Nyquist):
If a continuius-variable signal is bandlimited to frequences
below k0 , then it can be periodically sampled without loss of
information so long as the sampling period d ≤ π/k0 = λ0 /2.
Periodic sampling of one-dimensional signals can be
straightforwardly extended to multidimensional signals.
“Rectangular/regular” sampling not nessesary for
multidimensional signals.
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Sampling in one dimension
Periodic spatial sampling in one dimension ...
Sampling theorem (Nyquist):
If a continuius-variable signal is bandlimited to frequences
below k0 , then it can be periodically sampled without loss of
information so long as the sampling period d ≤ π/k0 = λ0 /2.
Periodic sampling of one-dimensional signals can be
straightforwardly extended to multidimensional signals.
“Rectangular/regular” sampling not nessesary for
multidimensional signals.
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Regular arrays
Grating lobes
Irregular arrays
Regular arrays
Assume point sources (Wtot = Warray · Wel )).
Easy to analyse and fast algorithms available (FFT).
Consider linear array; M equally spaced ideal sensor with
interelement spacing d along the x direction.
The discrete aperture function, wm .
The discrete aperture smoothing function, W (k):
P
W (k) ≡ m wm e kmd
Spatial aliasing given by d relative to λ.
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Regular arrays
Grating lobes
Irregular arrays
Regular arrays
Assume point sources (Wtot = Warray · Wel )).
Easy to analyse and fast algorithms available (FFT).
Consider linear array; M equally spaced ideal sensor with
interelement spacing d along the x direction.
The discrete aperture function, wm .
The discrete aperture smoothing function, W (k):
P
W (k) ≡ m wm e kmd
Spatial aliasing given by d relative to λ.
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Regular arrays
Grating lobes
Irregular arrays
Regular arrays
Assume point sources (Wtot = Warray · Wel )).
Easy to analyse and fast algorithms available (FFT).
Consider linear array; M equally spaced ideal sensor with
interelement spacing d along the x direction.
The discrete aperture function, wm .
The discrete aperture smoothing function, W (k):
P
W (k) ≡ m wm e kmd
Spatial aliasing given by d relative to λ.
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Regular arrays
Grating lobes
Irregular arrays
Regular arrays
Assume point sources (Wtot = Warray · Wel )).
Easy to analyse and fast algorithms available (FFT).
Consider linear array; M equally spaced ideal sensor with
interelement spacing d along the x direction.
The discrete aperture function, wm .
The discrete aperture smoothing function, W (k):
P
W (k) ≡ m wm e kmd
Spatial aliasing given by d relative to λ.
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Regular arrays
Grating lobes
Irregular arrays
Regular arrays
Assume point sources (Wtot = Warray · Wel )).
Easy to analyse and fast algorithms available (FFT).
Consider linear array; M equally spaced ideal sensor with
interelement spacing d along the x direction.
The discrete aperture function, wm .
The discrete aperture smoothing function, W (k):
P
W (k) ≡ m wm e kmd
Spatial aliasing given by d relative to λ.
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Regular arrays
Grating lobes
Irregular arrays
Regular arrays
Assume point sources (Wtot = Warray · Wel )).
Easy to analyse and fast algorithms available (FFT).
Consider linear array; M equally spaced ideal sensor with
interelement spacing d along the x direction.
The discrete aperture function, wm .
The discrete aperture smoothing function, W (k):
P
W (k) ≡ m wm e kmd
Spatial aliasing given by d relative to λ.
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Regular arrays
Grating lobes
Irregular arrays
Grating lobes
Given an linear array of M sensors with element spacing d.
kMd/2
W (k) = sin
sin kd/2 .
Mainlobe given by D = Md.
Gratinglobes (if any) given by d.
Maximal response for φ = 0. Does it exist other φg with the
same maximal response?
kx = 2π/λ sin φg ? ± 2π/dn ⇒ sin φg = ±λ/dn.
n = 1: No gratinglobes for λ/d > 1, i.e. d < λ.
d = 4λ: sin φg ± n · 1/4 ⇒ φg = ±14.5◦ , ±30◦ , ±48.6◦ , ±90◦ .
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Regular arrays
Grating lobes
Irregular arrays
Grating lobes
Given an linear array of M sensors with element spacing d.
kMd/2
W (k) = sin
sin kd/2 .
Mainlobe given by D = Md.
Gratinglobes (if any) given by d.
Maximal response for φ = 0. Does it exist other φg with the
same maximal response?
kx = 2π/λ sin φg ? ± 2π/dn ⇒ sin φg = ±λ/dn.
n = 1: No gratinglobes for λ/d > 1, i.e. d < λ.
d = 4λ: sin φg ± n · 1/4 ⇒ φg = ±14.5◦ , ±30◦ , ±48.6◦ , ±90◦ .
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Regular arrays
Grating lobes
Irregular arrays
Grating lobes
Given an linear array of M sensors with element spacing d.
kMd/2
W (k) = sin
sin kd/2 .
Mainlobe given by D = Md.
Gratinglobes (if any) given by d.
Maximal response for φ = 0. Does it exist other φg with the
same maximal response?
kx = 2π/λ sin φg ? ± 2π/dn ⇒ sin φg = ±λ/dn.
n = 1: No gratinglobes for λ/d > 1, i.e. d < λ.
d = 4λ: sin φg ± n · 1/4 ⇒ φg = ±14.5◦ , ±30◦ , ±48.6◦ , ±90◦ .
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Regular arrays
Grating lobes
Irregular arrays
Grating lobes
Given an linear array of M sensors with element spacing d.
kMd/2
W (k) = sin
sin kd/2 .
Mainlobe given by D = Md.
Gratinglobes (if any) given by d.
Maximal response for φ = 0. Does it exist other φg with the
same maximal response?
kx = 2π/λ sin φg ? ± 2π/dn ⇒ sin φg = ±λ/dn.
n = 1: No gratinglobes for λ/d > 1, i.e. d < λ.
d = 4λ: sin φg ± n · 1/4 ⇒ φg = ±14.5◦ , ±30◦ , ±48.6◦ , ±90◦ .
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Regular arrays
Grating lobes
Irregular arrays
Grating lobes
Given an linear array of M sensors with element spacing d.
kMd/2
W (k) = sin
sin kd/2 .
Mainlobe given by D = Md.
Gratinglobes (if any) given by d.
Maximal response for φ = 0. Does it exist other φg with the
same maximal response?
kx = 2π/λ sin φg ? ± 2π/dn ⇒ sin φg = ±λ/dn.
n = 1: No gratinglobes for λ/d > 1, i.e. d < λ.
d = 4λ: sin φg ± n · 1/4 ⇒ φg = ±14.5◦ , ±30◦ , ±48.6◦ , ±90◦ .
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Regular arrays
Grating lobes
Irregular arrays
Grating lobes
Given an linear array of M sensors with element spacing d.
kMd/2
W (k) = sin
sin kd/2 .
Mainlobe given by D = Md.
Gratinglobes (if any) given by d.
Maximal response for φ = 0. Does it exist other φg with the
same maximal response?
kx = 2π/λ sin φg ? ± 2π/dn ⇒ sin φg = ±λ/dn.
n = 1: No gratinglobes for λ/d > 1, i.e. d < λ.
d = 4λ: sin φg ± n · 1/4 ⇒ φg = ±14.5◦ , ±30◦ , ±48.6◦ , ±90◦ .
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Regular arrays
Grating lobes
Irregular arrays
Grating lobes
Given an linear array of M sensors with element spacing d.
kMd/2
W (k) = sin
sin kd/2 .
Mainlobe given by D = Md.
Gratinglobes (if any) given by d.
Maximal response for φ = 0. Does it exist other φg with the
same maximal response?
kx = 2π/λ sin φg ? ± 2π/dn ⇒ sin φg = ±λ/dn.
n = 1: No gratinglobes for λ/d > 1, i.e. d < λ.
d = 4λ: sin φg ± n · 1/4 ⇒ φg = ±14.5◦ , ±30◦ , ±48.6◦ , ±90◦ .
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Regular arrays
Grating lobes
Irregular arrays
Irregular arrays
Discrete co-array function:
P
c(~
χ) = (m1 ,m2 )∈ϑ(~χ) wm1 wm∗ 2 , where ϑ(~
χ) denotes the set of
indices (m1 , m2 ) for which ~xm2 − ~xm1 = χ
~.
0 ≤ c(~
χ) ≤ M = c(~0).
Equals the inverse Fourier Transform of |W (~k)|2 .
Redundant lag: The number of distinct baselines of a given
length is grater than one.
Sparse arrays
Underlying regular grid, all position not filled.
Position fills to aquire a given co-array
Non-redundant arrays with minimum number of gaps
Maximal length redundant arrays with no gaps.
Sparse array optimization
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Regular arrays
Grating lobes
Irregular arrays
Irregular arrays
Discrete co-array function:
P
c(~
χ) = (m1 ,m2 )∈ϑ(~χ) wm1 wm∗ 2 , where ϑ(~
χ) denotes the set of
indices (m1 , m2 ) for which ~xm2 − ~xm1 = χ
~.
0 ≤ c(~
χ) ≤ M = c(~0).
Equals the inverse Fourier Transform of |W (~k)|2 .
Redundant lag: The number of distinct baselines of a given
length is grater than one.
Sparse arrays
Underlying regular grid, all position not filled.
Position fills to aquire a given co-array
Non-redundant arrays with minimum number of gaps
Maximal length redundant arrays with no gaps.
Sparse array optimization
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Regular arrays
Grating lobes
Irregular arrays
Irregular arrays
Discrete co-array function:
P
c(~
χ) = (m1 ,m2 )∈ϑ(~χ) wm1 wm∗ 2 , where ϑ(~
χ) denotes the set of
indices (m1 , m2 ) for which ~xm2 − ~xm1 = χ
~.
0 ≤ c(~
χ) ≤ M = c(~0).
Equals the inverse Fourier Transform of |W (~k)|2 .
Redundant lag: The number of distinct baselines of a given
length is grater than one.
Sparse arrays
Underlying regular grid, all position not filled.
Position fills to aquire a given co-array
Non-redundant arrays with minimum number of gaps
Maximal length redundant arrays with no gaps.
Sparse array optimization
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Regular arrays
Grating lobes
Irregular arrays
Irregular arrays
Discrete co-array function:
P
c(~
χ) = (m1 ,m2 )∈ϑ(~χ) wm1 wm∗ 2 , where ϑ(~
χ) denotes the set of
indices (m1 , m2 ) for which ~xm2 − ~xm1 = χ
~.
0 ≤ c(~
χ) ≤ M = c(~0).
Equals the inverse Fourier Transform of |W (~k)|2 .
Redundant lag: The number of distinct baselines of a given
length is grater than one.
Sparse arrays
Underlying regular grid, all position not filled.
Position fills to aquire a given co-array
Non-redundant arrays with minimum number of gaps
Maximal length redundant arrays with no gaps.
Sparse array optimization
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Regular arrays
Grating lobes
Irregular arrays
Irregular arrays
Discrete co-array function:
P
c(~
χ) = (m1 ,m2 )∈ϑ(~χ) wm1 wm∗ 2 , where ϑ(~
χ) denotes the set of
indices (m1 , m2 ) for which ~xm2 − ~xm1 = χ
~.
0 ≤ c(~
χ) ≤ M = c(~0).
Equals the inverse Fourier Transform of |W (~k)|2 .
Redundant lag: The number of distinct baselines of a given
length is grater than one.
Sparse arrays
Underlying regular grid, all position not filled.
Position fills to aquire a given co-array
Non-redundant arrays with minimum number of gaps
Maximal length redundant arrays with no gaps.
Sparse array optimization
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Regular arrays
Grating lobes
Irregular arrays
Irregular arrays
Discrete co-array function:
P
c(~
χ) = (m1 ,m2 )∈ϑ(~χ) wm1 wm∗ 2 , where ϑ(~
χ) denotes the set of
indices (m1 , m2 ) for which ~xm2 − ~xm1 = χ
~.
0 ≤ c(~
χ) ≤ M = c(~0).
Equals the inverse Fourier Transform of |W (~k)|2 .
Redundant lag: The number of distinct baselines of a given
length is grater than one.
Sparse arrays
Underlying regular grid, all position not filled.
Position fills to aquire a given co-array
Non-redundant arrays with minimum number of gaps
Maximal length redundant arrays with no gaps.
Sparse array optimization
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Regular arrays
Grating lobes
Irregular arrays
Irregular arrays
Discrete co-array function:
P
c(~
χ) = (m1 ,m2 )∈ϑ(~χ) wm1 wm∗ 2 , where ϑ(~
χ) denotes the set of
indices (m1 , m2 ) for which ~xm2 − ~xm1 = χ
~.
0 ≤ c(~
χ) ≤ M = c(~0).
Equals the inverse Fourier Transform of |W (~k)|2 .
Redundant lag: The number of distinct baselines of a given
length is grater than one.
Sparse arrays
Underlying regular grid, all position not filled.
Position fills to aquire a given co-array
Non-redundant arrays with minimum number of gaps
Maximal length redundant arrays with no gaps.
Sparse array optimization
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Regular arrays
Grating lobes
Irregular arrays
Irregular arrays
Discrete co-array function:
P
c(~
χ) = (m1 ,m2 )∈ϑ(~χ) wm1 wm∗ 2 , where ϑ(~
χ) denotes the set of
indices (m1 , m2 ) for which ~xm2 − ~xm1 = χ
~.
0 ≤ c(~
χ) ≤ M = c(~0).
Equals the inverse Fourier Transform of |W (~k)|2 .
Redundant lag: The number of distinct baselines of a given
length is grater than one.
Sparse arrays
Underlying regular grid, all position not filled.
Position fills to aquire a given co-array
Non-redundant arrays with minimum number of gaps
Maximal length redundant arrays with no gaps.
Sparse array optimization
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Regular arrays
Grating lobes
Irregular arrays
Irregular arrays
Discrete co-array function:
P
c(~
χ) = (m1 ,m2 )∈ϑ(~χ) wm1 wm∗ 2 , where ϑ(~
χ) denotes the set of
indices (m1 , m2 ) for which ~xm2 − ~xm1 = χ
~.
0 ≤ c(~
χ) ≤ M = c(~0).
Equals the inverse Fourier Transform of |W (~k)|2 .
Redundant lag: The number of distinct baselines of a given
length is grater than one.
Sparse arrays
Underlying regular grid, all position not filled.
Position fills to aquire a given co-array
Non-redundant arrays with minimum number of gaps
Maximal length redundant arrays with no gaps.
Sparse array optimization
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Regular arrays
Grating lobes
Irregular arrays
Irregular arrays
Discrete co-array function:
P
c(~
χ) = (m1 ,m2 )∈ϑ(~χ) wm1 wm∗ 2 , where ϑ(~
χ) denotes the set of
indices (m1 , m2 ) for which ~xm2 − ~xm1 = χ
~.
0 ≤ c(~
χ) ≤ M = c(~0).
Equals the inverse Fourier Transform of |W (~k)|2 .
Redundant lag: The number of distinct baselines of a given
length is grater than one.
Sparse arrays
Underlying regular grid, all position not filled.
Position fills to aquire a given co-array
Non-redundant arrays with minimum number of gaps
Maximal length redundant arrays with no gaps.
Sparse array optimization
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Regular arrays
Grating lobes
Irregular arrays
Irregular arrays
Discrete co-array function:
P
c(~
χ) = (m1 ,m2 )∈ϑ(~χ) wm1 wm∗ 2 , where ϑ(~
χ) denotes the set of
indices (m1 , m2 ) for which ~xm2 − ~xm1 = χ
~.
0 ≤ c(~
χ) ≤ M = c(~0).
Equals the inverse Fourier Transform of |W (~k)|2 .
Redundant lag: The number of distinct baselines of a given
length is grater than one.
Sparse arrays
Underlying regular grid, all position not filled.
Position fills to aquire a given co-array
Non-redundant arrays with minimum number of gaps
Maximal length redundant arrays with no gaps.
Sparse array optimization
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays
Finite Continuous Apetrures
Spatial sampling
Arrays of discrete sensors
Regular arrays
Grating lobes
Irregular arrays
Random arrays
Read Sparse Sampling in Array Processing
Andreas Austeng
INF5410 Array signal processing. Ch. 3: Apertures and Arrays