Channel Spectral Characteristics
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Channel Spectral Characteristics
Some Polyphase Filter Options
FDM-to TDM Standard Channelizer: Collection of Downconverters
FDM-to TDM Polyphase Channelizer: Input Commutator, M-Path Filter, and FFT
Conventional Channelizer Application Channelize and Downsample to Nyquist Rate, Interpolate to Two Times Symbol Rate
Enhanced Channelizer Solution Replace Interpolator Function With Buffer Addressing
Standard Downconverter: Heterodyne, Filter, Resample
Equivalency Theorem Heterodyne and Low Pass Filter Same as Band Pass Filter and Heterodyne
y (n, k ) = [ x (n )e =
− jθ k n
] ∗ h (n )
N −1
− j ( n − r )θ k x ( n − r ) e h (r ) ∑ r =0
=
N −1
jr θ k − jn θ k x ( n − r ) e h ( r ) e ∑ r =0
N −1
= e − jn θ k ∑ x ( n − r ) h ( r ) e jrθ k r =0
Slide Heterodyne Through Filter: Convert Low-Pass to Band-Pass
Slide Heterodyne Through Downsampler: Alias Frequency of Heterodyne
Select Input Center Frequency That Aliases to DC Under Resampling
Discard Trivial Heterodyne: Now a Scalar Multiply by Unity
Noble Identity Interchange Order of Filter and Resample
M-Path Partition of FIR Filter N −1
H ( Z ) = ∑ h( n) Z − n n=0
= h(0) + h(1) Z −1 + h(2) Z − 2 + h(3) Z − 3 + L + h( N − 1) Z −( N −1)
H(Z) = h(0)
+
h(M + 0) Z −M + h(2M + 0)Z −( 2M +0) +L
h(1)Z −1
+ h(M + 1)Z −( M +1) + h(2M + 1)Z −( 2M +1) +L
h(2)Z −2
+ h(M + 2)Z −(M +2) + h(2M + 2)Z −( 2M +2) +L
h(3)Z −3 M
+ h(M + 3)Z −( M +3) + h(2M + 3)Z −( 2M +3) +L M M M M M
h(M −1)Z −( M −1) + h(2M −1)Z −(2M −1) + h(3M −1)Z −(3M −1) +L H ( Z ) = H 0 ( Z M ) + Z −1 H1 ( Z M ) + Z −2 H 2 ( Z M ) + LL + Z −( M −1) H ( M −1) ( Z M ) M −1
H (Z ) = ∑ Z H r (Z ) r =0
−r
M
=
M −1
∑Z r =0
−r
( N / M ) −1
∑ h(r + nM )Z n =0
−M n
Example of Polyphase (M-Path) Partition
Block Diagram of M-Path Filter
Pull Resamplers Through Filters in Each path
Delays and Synchronous Rersamplers are Equivalent to Input Commutator
Input Series in Each Path of M-Path Filter have Offset Time Origins
Path Filters are Interpolators: Computing Output Samples with Common Time Origin
Phase Response of Paths in Ten Stage Polyphase Filter S pectral P has e Res pons e of Ten P olyphas e F ilters 0
-1
-2
Normalized P has e (φ /2π)
-3
-4
-5
-6
-7
-8
-9
0
0.05
0.1
0.15
0.2 0.25 0.3 Normaled Frequency (f/fs )
0.35
0.4
0.45
0.5
Group Delay of Paths in Ten-Stage Polyphase Filter S pectral Group Delay Res pons e of Ten P olyphas e Filters -7
-7.2
-7.4
s
Normalized Delay (∆ T/T )
-7.6
-7.8
-8
-8.2
-8.4
-8.6
-8.8
-9
0
0.05
0.1
0.15
0.2 0.25 0.3 Normaled Frequency (f/fs )
0.35
0.4
0.45
0.5
Modulation Property for Z-Transform if H ( Z ) = h( 0) + h(1) Z −1 + h( 2) Z − 2 + L + n( N − 1) Z −( N −1) N −1
= ∑ h( n) Z − n n =0
and G ( Z ) = h( 0) + h(1)e jθ Z −1 + h( 2)e j 2θ Z − 2 + L + h( N − 1)e j ( N −1)θ Z −( N −1) = h(0) + h(1)[e − jθ Z ]−1 + h( 2)[e − j 2θ Z ]− 2 + L + h( N − 1)[e − j ( N −1)θ Z ]−( N −1) N −1
= ∑ h(n)[e − jθ Z ]− n n =0
then G(Z) = H(Z) Z =e - jθ Z = H (e − jθ Z )
Polyphase Partition of Band-Pass Filter
Spectral Response of Signal in Resampled Pass Band Filter
Performance Specifications for 50-Channel Polyphase Channelizer
Required Spectral Response of Channel Filter at Output and at Input Rates
Time and Frequency Response of Remez Filter Design Impuls e Res pons e , P rototype F ilte r 0.025
10
x 10
Impuls e Res pons e , Detail
-4
0.02 0.015
5
0.01 0.005
0
0 -0.005
0
100
200
300
400
500
-5 -10
0
10
20
30
40
50
60
Frequency Res pons e , P rototype F ilte r 10 0 log magnitude (dB)
-10 -20 -30 -40 -50 -60 -70 -80
-30
-20
-10
0 Normalize d Frequenc y f/f
Channel
10
20
30
Time and Frequency response of Remez Filter Design with Modified End Points Impuls e Res pons e, P rototype Filter 0.025
10
x 10
Impuls e Res pons e, Detail
-4
0.02 0.015
5
0.01 0.005
0
0 -0.005
0
100
200
300
400
500
-5 -10
0
10
20
30
40
50
60
Frequency Res pons e, P rototype Filte r 10 0 log magnitude (dB)
-10 -20 -30 -40 -50 -60 -70 -80
-30
-20
-10
0 Normalized Frequency f/f
Channel
10
20
30
Time and Frequency Response of Prototype Filter at Output Sample Rate Impuls e Res pons e , P rototype F ilter
x 10 -3 20 15 10 5 0 -5
0
50
100
150
200
250
300
350
400
450
500
Frequency Res pons e, P rototype and Replicates at Output Ra te 10 0 log magnitude (dB)
-10 -20 -30 -40 -50 -60 -70 -80 -2
-1.5
-1
-0.5 0 Norma lized Frequency f/f
0.5
Channel
1
1.5
2
Commutators for Standard Input Buffer and for Circular Input Buffer
Standard Polyphase Channelizer and Modified Channelizer with Circular Buffers
Content of 64-Point Circular Input Buffer for Two Successive 48 Point Input Blocks
Shifting Time Origin for Input Data of Polyphase Filter and of Resetting FFT
Contents of Transfer Circular Buffer Aligning Origins for Successive Input Blocks
Input Time and Spectrum to 30 Channel, 40 Point FFT Channelizer real part of compos ite time s ignal 100
50
0
-50
-100 300
350
400
450
500
550
600
650
700
750
800
S pec trum: compos ite time s ignal 10 0 -10 -20 -30 -40 -50 -60 -20
-15
-10
-5
0
5
10
15
20
Time Series from 30 Channel Channelizer 10
10
10
10
10
10
0
0
0
0
0
0
-10
0
100
200
-10
0
100
200
-10
0
100
200
-10
0
100
200
-10
0
100
200
-10
10
10
10
10
10
10
0
0
0
0
0
0
-10
0
100
200
-10
0
100
200
-10
0
100
200
-10
0
100
200
-10
0
100
200
-10
10
10
10
10
10
10
0
0
0
0
0
0
-10
0
100
200
-10
0
100
200
-10
0
100
200
-10
0
100
200
-10
0
100
200
-10
10
10
10
10
10
10
0
0
0
0
0
0
-10
0
100
200
-10
0
100
200
-10
0
100
200
-10
0
100
200
-10
0
100
200
-10
10
10
10
10
10
10
0
0
0
0
0
0
-10
0
100
200
-10
0
100
200
-10
0
100
200
-10
0
100
200
-10
0
100
200
-10
0
100
200
0
100
200
0
100
200
0
100
200
0
100
200
Spectra for 30 Channel Channelizer 0
0
0
0
0
0
-20
-20
-20
-20
-20
-20
-40
-40
-40
-40
-40
-40
-60 -1
0
1
-60 -1
0
1
-60 -1
0
1
-60 -1
0
1
-60 -1
0
1
-60 -1
0
0
0
0
0
0
-20
-20
-20
-20
-20
-20
-40
-40
-40
-40
-40
-40
-60 -1
0
1
-60 -1
0
1
-60 -1
0
1
-60 -1
0
1
-60 -1
0
1
-60 -1
0
0
0
0
0
0
-20
-20
-20
-20
-20
-20
-40
-40
-40
-40
-40
-40
-60 -1
0
1
-60 -1
0
1
-60 -1
0
1
-60 -1
0
1
-60 -1
0
1
-60 -1
0
0
0
0
0
0
-20
-20
-20
-20
-20
-20
-40
-40
-40
-40
-40
-40
-60 -1
0
1
-60 -1
0
1
-60 -1
0
1
-60 -1
0
1
-60 -1
0
1
-60 -1
0
0
0
0
0
0
-20
-20
-20
-20
-20
-20
-40
-40
-40
-40
-40
-40
-60 -1
0
1
-60 -1
0
1
-60 -1
0
1
-60 -1
0
1
-60 -1
0
1
-60 -1
0
1
0
1
0
1
0
1
0
1
Input Time and Spectrum to 50 Channel, 64 Point FFT Channelizer Compos ite Time S igna l 1
Amplitude
0.5
0
-0.5
-1 600
650
700
750
800 S ample Index
850
900
950
1000
S pec trum 10
Log Magnitude (dB)
0 -10 -20 -30 -40 -50 -60 -6000
-4000
-2000
0 Fre que nc y (kHz)
2000
4000
6000
Time Series from 50 Channel Matched Filter Channelizer 1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
-1 1
0
100
200
0 -1 1
1
0
100
200
1
0
100
200
1
0
100
200
-1 1
-1 1
-1 1
0
100
200
-1 1
0
100
200
100
200
-1
1
0
100
200
-1 1
0
100
200
-1 1
0
100
200
100
200
-1 1
0
100
200
100
200
-1 1
0
100
200
100
200
-1
-1 1
-1 1
-1 1
0
100
200
-1 1
0
100 200
100
200
-1
1
0
100
200
-1 1
0
100 200
-1 1
0
100
200
100 200
-1 1
0
100
200
100 200
-1 1
0
100
200
100 200
-1
200
-1 1
-1 1
-1 1
0
100
200
-1 1
0
100
200
100
200
-1
1
0
100 200
-1 1
0
100
200
-1 1
0
100 200
100
200
-1 1
0
100 200
100
200
-1 1
0
100 200
100
200
-1
100
200
-1 1
0
100
200
-1 1
0
100
200
-1 1
0
100
200
0
100
200
0
100
200
0 0
100 200
0 0
0
0
0 0
1
0
0 0
-1
0
0
0 0
-1
0
0
0 0
100
0
0 0
1
0
0
0 0
-1
0
0
0 0
-1
0
0
0 0
100 200
0
0 0
1
0
0
0 0
-1
0
0
0 0
-1
0
0
0 -1
200
0
0 -1
100
0
0 -1
1
0
0
0 -1
-1
-1 1 0
0
100 200
-1
Spectra from 50 Channel Matched Filter Channelizer 0
0
0
0
0
0
0
0
-50
-50
-50
-50
-50
-50
-50
-50
-0.5 0
0
0.5 -0.5 0
0
0.5 -0.5 0
-50 -0.5 0
0
0.5 -0.5 0
0
0.5 -0.5 0
0
0.5 -0.5 0
0.5 -0.5
0.5 -0.5 0
0
0.5 -0.5 0
0
0.5 -0.5 0
0.5 -0.5 0
0
0.5 -0.5 0
0
0.5 -0.5 0
0
0.5 -0.5 0
0
0.5 -0.5 0
0.5 -0.5 0
0.5 -0.5
0.5 -0.5 0
0
0.5 -0.5 0
0
0.5 -0.5 0
0.5 -0.5 0
0
0.5 -0.5 0
0
0.5 -0.5 0
0
0.5 -0.5 0
0
0.5 -0.5 0
0.5 -0.5 0
0.5 -0.5
0.5 -0.5 0
0
0.5 -0.5 0
0
0.5 -0.5 0
0.5 -0.5 0
0
0.5 -0.5 0
0
0.5 -0.5 0
0
0.5 -0.5 0
0
0.5 -0.5 0
0.5
0
0.5
0
0.5
0
0.5
0
0.5
-50
-50 0.5 -0.5
0
-50
-50
0
0.5
-50
-50 0
0
-50
-50
-50 0
0
-50
-50
-50 0.5 -0.5
0
-50
-50
0
0.5 -0.5 0
-50
-50 0
0
-50
-50
-50 0
0
-50
-50
-50 0.5 -0.5
0
-50
-50
0
0.5 -0.5 0
-50
-50 0
0
-50
-50
-50 0
0
-50
-50
-50 -0.5
0.5 -0.5 0
-50
-50 -0.5 0
0
-50
-50 -0.5 0
0.5 -0.5 0
-50
-50 -0.5 0
0
-50 0
0.5 -0.5
Some Polyphase Filter Options
FDM-to TDM Standard Channelizer: Collection of Downconverters
FDM-to TDM Polyphase Channelizer: Input Commutator, M-Path Filter, and FFT
Conventional Channelizer Application Channelize and Downsample to Nyquist Rate, Interpolate to Two Times Symbol Rate
Enhanced Channelizer Solution Replace Interpolator Function With Buffer Addressing
Standard Downconverter: Heterodyne, Filter, Resample
Equivalency Theorem Heterodyne and Low Pass Filter Same as Band Pass Filter and Heterodyne
y (n, k ) = [ x (n )e =
− jθ k n
] ∗ h (n )
N −1
− j ( n − r )θ k x ( n − r ) e h (r ) ∑ r =0
=
N −1
jr θ k − jn θ k x ( n − r ) e h ( r ) e ∑ r =0
N −1
= e − jn θ k ∑ x ( n − r ) h ( r ) e jrθ k r =0
Slide Heterodyne Through Filter: Convert Low-Pass to Band-Pass
Slide Heterodyne Through Downsampler: Alias Frequency of Heterodyne
Select Input Center Frequency That Aliases to DC Under Resampling
Discard Trivial Heterodyne: Now a Scalar Multiply by Unity
Noble Identity Interchange Order of Filter and Resample
M-Path Partition of FIR Filter N −1
H ( Z ) = ∑ h( n) Z − n n=0
= h(0) + h(1) Z −1 + h(2) Z − 2 + h(3) Z − 3 + L + h( N − 1) Z −( N −1)
H(Z) = h(0)
+
h(M + 0) Z −M + h(2M + 0)Z −( 2M +0) +L
h(1)Z −1
+ h(M + 1)Z −( M +1) + h(2M + 1)Z −( 2M +1) +L
h(2)Z −2
+ h(M + 2)Z −(M +2) + h(2M + 2)Z −( 2M +2) +L
h(3)Z −3 M
+ h(M + 3)Z −( M +3) + h(2M + 3)Z −( 2M +3) +L M M M M M
h(M −1)Z −( M −1) + h(2M −1)Z −(2M −1) + h(3M −1)Z −(3M −1) +L H ( Z ) = H 0 ( Z M ) + Z −1 H1 ( Z M ) + Z −2 H 2 ( Z M ) + LL + Z −( M −1) H ( M −1) ( Z M ) M −1
H (Z ) = ∑ Z H r (Z ) r =0
−r
M
=
M −1
∑Z r =0
−r
( N / M ) −1
∑ h(r + nM )Z n =0
−M n
Example of Polyphase (M-Path) Partition
Block Diagram of M-Path Filter
Pull Resamplers Through Filters in Each path
Delays and Synchronous Rersamplers are Equivalent to Input Commutator
Input Series in Each Path of M-Path Filter have Offset Time Origins
Path Filters are Interpolators: Computing Output Samples with Common Time Origin
Phase Response of Paths in Ten Stage Polyphase Filter S pectral P has e Res pons e of Ten P olyphas e F ilters 0
-1
-2
Normalized P has e (φ /2π)
-3
-4
-5
-6
-7
-8
-9
0
0.05
0.1
0.15
0.2 0.25 0.3 Normaled Frequency (f/fs )
0.35
0.4
0.45
0.5
Group Delay of Paths in Ten-Stage Polyphase Filter S pectral Group Delay Res pons e of Ten P olyphas e Filters -7
-7.2
-7.4
s
Normalized Delay (∆ T/T )
-7.6
-7.8
-8
-8.2
-8.4
-8.6
-8.8
-9
0
0.05
0.1
0.15
0.2 0.25 0.3 Normaled Frequency (f/fs )
0.35
0.4
0.45
0.5
Modulation Property for Z-Transform if H ( Z ) = h( 0) + h(1) Z −1 + h( 2) Z − 2 + L + n( N − 1) Z −( N −1) N −1
= ∑ h( n) Z − n n =0
and G ( Z ) = h( 0) + h(1)e jθ Z −1 + h( 2)e j 2θ Z − 2 + L + h( N − 1)e j ( N −1)θ Z −( N −1) = h(0) + h(1)[e − jθ Z ]−1 + h( 2)[e − j 2θ Z ]− 2 + L + h( N − 1)[e − j ( N −1)θ Z ]−( N −1) N −1
= ∑ h(n)[e − jθ Z ]− n n =0
then G(Z) = H(Z) Z =e - jθ Z = H (e − jθ Z )
Polyphase Partition of Band-Pass Filter
Spectral Response of Signal in Resampled Pass Band Filter
Performance Specifications for 50-Channel Polyphase Channelizer
Required Spectral Response of Channel Filter at Output and at Input Rates
Time and Frequency Response of Remez Filter Design Impuls e Res pons e , P rototype F ilte r 0.025
10
x 10
Impuls e Res pons e , Detail
-4
0.02 0.015
5
0.01 0.005
0
0 -0.005
0
100
200
300
400
500
-5 -10
0
10
20
30
40
50
60
Frequency Res pons e , P rototype F ilte r 10 0 log magnitude (dB)
-10 -20 -30 -40 -50 -60 -70 -80
-30
-20
-10
0 Normalize d Frequenc y f/f
Channel
10
20
30
Time and Frequency response of Remez Filter Design with Modified End Points Impuls e Res pons e, P rototype Filter 0.025
10
x 10
Impuls e Res pons e, Detail
-4
0.02 0.015
5
0.01 0.005
0
0 -0.005
0
100
200
300
400
500
-5 -10
0
10
20
30
40
50
60
Frequency Res pons e, P rototype Filte r 10 0 log magnitude (dB)
-10 -20 -30 -40 -50 -60 -70 -80
-30
-20
-10
0 Normalized Frequency f/f
Channel
10
20
30
Time and Frequency Response of Prototype Filter at Output Sample Rate Impuls e Res pons e , P rototype F ilter
x 10 -3 20 15 10 5 0 -5
0
50
100
150
200
250
300
350
400
450
500
Frequency Res pons e, P rototype and Replicates at Output Ra te 10 0 log magnitude (dB)
-10 -20 -30 -40 -50 -60 -70 -80 -2
-1.5
-1
-0.5 0 Norma lized Frequency f/f
0.5
Channel
1
1.5
2
Commutators for Standard Input Buffer and for Circular Input Buffer
Standard Polyphase Channelizer and Modified Channelizer with Circular Buffers
Content of 64-Point Circular Input Buffer for Two Successive 48 Point Input Blocks
Shifting Time Origin for Input Data of Polyphase Filter and of Resetting FFT
Contents of Transfer Circular Buffer Aligning Origins for Successive Input Blocks
Input Time and Spectrum to 30 Channel, 40 Point FFT Channelizer real part of compos ite time s ignal 100
50
0
-50
-100 300
350
400
450
500
550
600
650
700
750
800
S pec trum: compos ite time s ignal 10 0 -10 -20 -30 -40 -50 -60 -20
-15
-10
-5
0
5
10
15
20
Time Series from 30 Channel Channelizer 10
10
10
10
10
10
0
0
0
0
0
0
-10
0
100
200
-10
0
100
200
-10
0
100
200
-10
0
100
200
-10
0
100
200
-10
10
10
10
10
10
10
0
0
0
0
0
0
-10
0
100
200
-10
0
100
200
-10
0
100
200
-10
0
100
200
-10
0
100
200
-10
10
10
10
10
10
10
0
0
0
0
0
0
-10
0
100
200
-10
0
100
200
-10
0
100
200
-10
0
100
200
-10
0
100
200
-10
10
10
10
10
10
10
0
0
0
0
0
0
-10
0
100
200
-10
0
100
200
-10
0
100
200
-10
0
100
200
-10
0
100
200
-10
10
10
10
10
10
10
0
0
0
0
0
0
-10
0
100
200
-10
0
100
200
-10
0
100
200
-10
0
100
200
-10
0
100
200
-10
0
100
200
0
100
200
0
100
200
0
100
200
0
100
200
Spectra for 30 Channel Channelizer 0
0
0
0
0
0
-20
-20
-20
-20
-20
-20
-40
-40
-40
-40
-40
-40
-60 -1
0
1
-60 -1
0
1
-60 -1
0
1
-60 -1
0
1
-60 -1
0
1
-60 -1
0
0
0
0
0
0
-20
-20
-20
-20
-20
-20
-40
-40
-40
-40
-40
-40
-60 -1
0
1
-60 -1
0
1
-60 -1
0
1
-60 -1
0
1
-60 -1
0
1
-60 -1
0
0
0
0
0
0
-20
-20
-20
-20
-20
-20
-40
-40
-40
-40
-40
-40
-60 -1
0
1
-60 -1
0
1
-60 -1
0
1
-60 -1
0
1
-60 -1
0
1
-60 -1
0
0
0
0
0
0
-20
-20
-20
-20
-20
-20
-40
-40
-40
-40
-40
-40
-60 -1
0
1
-60 -1
0
1
-60 -1
0
1
-60 -1
0
1
-60 -1
0
1
-60 -1
0
0
0
0
0
0
-20
-20
-20
-20
-20
-20
-40
-40
-40
-40
-40
-40
-60 -1
0
1
-60 -1
0
1
-60 -1
0
1
-60 -1
0
1
-60 -1
0
1
-60 -1
0
1
0
1
0
1
0
1
0
1
Input Time and Spectrum to 50 Channel, 64 Point FFT Channelizer Compos ite Time S igna l 1
Amplitude
0.5
0
-0.5
-1 600
650
700
750
800 S ample Index
850
900
950
1000
S pec trum 10
Log Magnitude (dB)
0 -10 -20 -30 -40 -50 -60 -6000
-4000
-2000
0 Fre que nc y (kHz)
2000
4000
6000
Time Series from 50 Channel Matched Filter Channelizer 1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
-1 1
0
100
200
0 -1 1
1
0
100
200
1
0
100
200
1
0
100
200
-1 1
-1 1
-1 1
0
100
200
-1 1
0
100
200
100
200
-1
1
0
100
200
-1 1
0
100
200
-1 1
0
100
200
100
200
-1 1
0
100
200
100
200
-1 1
0
100
200
100
200
-1
-1 1
-1 1
-1 1
0
100
200
-1 1
0
100 200
100
200
-1
1
0
100
200
-1 1
0
100 200
-1 1
0
100
200
100 200
-1 1
0
100
200
100 200
-1 1
0
100
200
100 200
-1
200
-1 1
-1 1
-1 1
0
100
200
-1 1
0
100
200
100
200
-1
1
0
100 200
-1 1
0
100
200
-1 1
0
100 200
100
200
-1 1
0
100 200
100
200
-1 1
0
100 200
100
200
-1
100
200
-1 1
0
100
200
-1 1
0
100
200
-1 1
0
100
200
0
100
200
0
100
200
0 0
100 200
0 0
0
0
0 0
1
0
0 0
-1
0
0
0 0
-1
0
0
0 0
100
0
0 0
1
0
0
0 0
-1
0
0
0 0
-1
0
0
0 0
100 200
0
0 0
1
0
0
0 0
-1
0
0
0 0
-1
0
0
0 -1
200
0
0 -1
100
0
0 -1
1
0
0
0 -1
-1
-1 1 0
0
100 200
-1
Spectra from 50 Channel Matched Filter Channelizer 0
0
0
0
0
0
0
0
-50
-50
-50
-50
-50
-50
-50
-50
-0.5 0
0
0.5 -0.5 0
0
0.5 -0.5 0
-50 -0.5 0
0
0.5 -0.5 0
0
0.5 -0.5 0
0
0.5 -0.5 0
0.5 -0.5
0.5 -0.5 0
0
0.5 -0.5 0
0
0.5 -0.5 0
0.5 -0.5 0
0
0.5 -0.5 0
0
0.5 -0.5 0
0
0.5 -0.5 0
0
0.5 -0.5 0
0.5 -0.5 0
0.5 -0.5
0.5 -0.5 0
0
0.5 -0.5 0
0
0.5 -0.5 0
0.5 -0.5 0
0
0.5 -0.5 0
0
0.5 -0.5 0
0
0.5 -0.5 0
0
0.5 -0.5 0
0.5 -0.5 0
0.5 -0.5
0.5 -0.5 0
0
0.5 -0.5 0
0
0.5 -0.5 0
0.5 -0.5 0
0
0.5 -0.5 0
0
0.5 -0.5 0
0
0.5 -0.5 0
0
0.5 -0.5 0
0.5
0
0.5
0
0.5
0
0.5
0
0.5
-50
-50 0.5 -0.5
0
-50
-50
0
0.5
-50
-50 0
0
-50
-50
-50 0
0
-50
-50
-50 0.5 -0.5
0
-50
-50
0
0.5 -0.5 0
-50
-50 0
0
-50
-50
-50 0
0
-50
-50
-50 0.5 -0.5
0
-50
-50
0
0.5 -0.5 0
-50
-50 0
0
-50
-50
-50 0
0
-50
-50
-50 -0.5
0.5 -0.5 0
-50
-50 -0.5 0
0
-50
-50 -0.5 0
0.5 -0.5 0
-50
-50 -0.5 0
0
-50 0
0.5 -0.5
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