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17. Numerical Filtering
� = 0 � = �
• Consider a signal �(�), such as velocity, varies in space � (or time �) between 0, � .
For example:
� � = 0.5 × sin (��) − 0.3 × cos (2��) + 0.7 × cos (12��) + 0.5 × sin (48��) − 0.6 × cos (64��)
• The fluctuating signal is decomposed into Fourier components with various wavenumbers:
0.5 × sin (��)
−0.3 × cos (2��)
0.7 × cos (12��)
0.5 × sin (48��)
−0.6 × cos (64��)
� = 0 � = �
Function in Physical Domain versus Spectrum in Wavenumber (or Frequency) Domain
� =2�
�
�� = �� = �2�
�
� � = 0.5 × sin (��) − 0.3 × cos (2��) + 0.7 × cos (12��) + 0.5 × sin (48��) − 0.6 × cos (64��)
0.5 × sin (��)
0.5 × sin (��) − 0.3 × cos (2��)
0.5 × sin (��) − 0.3 × cos (2��) + 0.7 × cos (12��)
0.5 × sin (��) − 0.3 × cos (2��) + 0.7 × cos (12��) + 0.5 × sin (48��)
� = 0 � = �� =2�
�
� = 0 � = �
� � = 0.5 × sin (��) − 0.3 × cos (2��) + 0.7 × cos (12��) + 0.5 × sin (48��) − 0.6 × cos (64��)
• Spectrum of � :
Sample the signal at discrete �� and expressed by discrete Fourier transform:
The largest ⇒ � = 128. Means � must be sampled at least on 128 grids.� = 6 4 =�
2
�� = � ���������
�
��
��
�� ���
= 0 − 0.25� ����� + 0 + 0.25� ������
+ −0.15 + 0� ������ + −0.15 − 0� �������
+ 0.35 + 0� ������� + 0.35 − 0� ��������
+ 0 − 0.25� ������� + 0 + 0.25� ��������
+ −0.3 + 0� ������� + −0.3 − 0� ��������
�� = �� = �2�
�
�� = � ��� ������
�
��
��
�� ���
= � ��� ������
��
�
��
��
�� ���
= � ��� ������
��
��
�� ���
� = 0, 1, 2, ⋯ , ��� =�
��
�
����
What is the physical meaning of spectral density ����
?
�� = � ���������
�
��
��
�� ���
= 0 − 0.25� ����� + 0 + 0.25� ������
+ −0.15 + 0� ������ + −0.15 − 0� �������
+ 0.35 + 0� ������� + 0.35 − 0� ��������
+ 0 − 0.25� ������� + 0 + 0.25� ��������
+ −0.3 + 0� ������� + −0.3 − 0� ��������
� = 0 � = �
0.090.1
0.05
0.0225
0.0625
0.1225
0.0625
0.15
� �� ⋅ ��∗
� ��
�� �
= � � ��� �����
� ��
�� �
⋅ � ���∗ ������
� ��
�� �
� ��
�� �
= � � ���
� ��
�� �
� ���∗ ��(���)��
� ��
�� �
� ��
�� �
= � ��� � ���∗ � ��(���)
���
� � ��
�� �
� ��
�� �
� ��
�� �
�� = � ��� �����
��
��
�� ���
= � ��������
� ��
�� �
��� =1
�� �� ������
� ��
�� �
discrete Fourier transform pair
� ���∗ ������
��
��
�� ���
= � ���∗������
� ��
�� �
= ��∗
1
�� ��
∗ �����
� ��
�� �
= ���∗
discrete Fourier transform pair of conjugate
� ��(���)���
� � ��
�� �
= � � if
� − �
� is an integer
0 otherwise
But,���
�= integer only happens only � = �.
Recall the orthogonality property of ����:
� �� ⋅ ��∗
� ��
�� �
= � � ��� ⋅ ���∗
� ��
�� �
∴ � ���
� ��
�� �
= � � ����
� ��
�� �
or
Parseval’s Theorem of Discrete Fourier Transform0 1 2� = ��−1�
�� =2�
��
• Parseval’s theorem for Fourier transform of continuous function:
� � � ⋅ �∗ � ���
��
= � �� � ⋅ ��∗ � ���
��
� � � � ���
��
= � �� ��
���
��or
� �� ⋅ ��∗
� ��
�� �
= � � ��� ⋅ ���∗
� ��
�� �
� ���
� ��
�� �
= � � ����
� ��
�� �
or
• Parseval’s theorem for discrete Fourier transform:
• If � is the velocity, the left-hand side of the equality represents the total kinetic energy.
Parseval’s theorem can then be interpreted as :
The total energy is the summation of energies contributed by all spectral components ���(or �� � ).
• The contribution of the spectral component of wavenumber � (or �) is proportional to ����
(or �� ��
).
• Therefore, we can interpret ����
(or �� ��
) as the energy per unit wavenumber of the
spectral component.
In other word, ����
(or �� ��
) is the energy spectral density of �.
Averaging and Filtering
�̅ � =1
�� � � ��
����
����
• The averaged function �̅ � depends on the averaging interval � employed.
The larger � will result in smoother �̅ � .
• In analyzing the fluctuating function �(�), it is desirable to decompose � � into a averaged
(smoothed) part �̅ � plus a fluctuation �′(�), i.e., � � = �̅ � + �′(�)
• This corresponds to filtered out the fluctuation �′ from �, or find the smoothed counterpart �.̅
• The most straightforward smoothing scheme is to use the vicinal mean at �:
�
�� ��
original function � averaged function �̅ with �� averaged function �̅ with ��
�
�
• Thus, a more general way which leads to smoothing is to employ a filter function � � − � :
�̅ � = � � � � � − � ���
��
• The vicinal-mean smoothing can be represented in term of a filter function:
�̅ � =1
�� � � ��
����
����
= � � �1
���
����
����
≡ � � � � � − � ���
��
�
� � − � =1
� where � − � <
�
2
� � � ���
��
= 1
� � =6
� 1
� exp −
6
�
�
� �
• A commonly used Gaussian filter function � � :
0 10 20−10−20
� = 8
� = 16
0.2
0.1
0
�
• Discrete form of filtering:
�̅ � = � � � � � − � ���
��
��̅ = � �� ����
� ��
�� �
= �� ∙ �� +�� ∙ ���� + ⋯ + ���� ∙ �� +�� ∙ �� +���� ∙ ��� + ⋯ + �� �� ∙ ���� �� +�� �� ∙ ���� ��
1 2�= 0 ��−1
��
��
���� ����
������� �� �� ���� ���� ������ ��� �� ������� ��
�� ���� �� ��
• Filtering of a continuous function �:
Filtering in Physical versus Wavenumber (or Frequency) Domain
� ∗ � � = � � � � � − � ���
��
= � � � − � � � ���
��
�̅ � = � � � � � − � ���
��
• Filtering carried out in physical domain:
• This suggests that the filtering can be done by:
• Convolution of two functions:
� � ∗ � = 2� � � ⋅ � �
• Fourier transform of a convolution (convolution theorem):
This is the form of filtering.
�̅ � = (� ∗ �)(�) = ��� � � ∗ � = ��� 2� � � ⋅ � �
�� � =1
2�� � � �������
�
��
≡ � �
� � = � �� � �������
��
≡ ��� ��
Fourier transform pair
• Convolution of discrete functions:
• Discrete Fourier transform of a convolution:
��� =1
�� �� �����
���
� ��
�� �
= � ��
�� = � ��� �������
��
��
�� ���
= � ��� �������
� ��
�� �
= ���(���)
discrete Fourier transform pair
• Discrete form of filtering: ��̅ = � �� ����
� ��
�� �
�� ∗ �� ≡ � �� ����
� ��
�� �
= � � ��� �������
� ��
�� �
� ��ℓ ����ℓ ���
�
� ��
ℓ� �
� ��
�� �
= � ��� � ��ℓ
� ��
ℓ� �
� �������
� ��
�� �
����ℓ ���
�
� ��
�� �
= � ��� � ��ℓ
� ��
ℓ� �
����ℓ�� � ����
(��ℓ)��
� ��
�� �
� ��
�� �
= � ��� � ��ℓ
� ��
ℓ� �
����ℓ�� � ∙ �(� − ℓ)
� ��
�� �
= � ∙ � ��� ∙ ��� �������
� ��
�� �
= � ∙ ���(��� ∙ ���)
� �� ∗ �� = � � ���(��� ∙ ���) = � ∙ ��� ∙ ��� = � ∙ � �� ∙ � ��
• Again, this suggests that the filtering can be done by:
��̅ = �� ∗ �� = ��� � �� ∗ �� = ��� � ∙ � �� ⋅ � ��
• Computing discrete filtering using FFT:
��̅ = � �� ����
� ��
�� �
= �� ∗ �� = ��� � �� ∗ �� = ��� � ∙ � �� ⋅ � ��
� �
������ ���� �� ���� �� ��
��̅��̅�� ��̅�� ��̅ ����̅ ��̅ ��̅
• In general, the transformed filtered function
ℑ �� can be derived analytically.
• So only one forward and one backward transform are needed to be computed.
• Since an FFT only requires � log�� operations, while filtering in physical domain requires ��
operations; filtering in wavenumber domain is far more efficient than doing in physical domain.
� � � = ����
� � � = ����
0 0.5 1 1.5 2 2.5 3
�� =2�
��
spec
tral
den
sity
�
� � = �1
� for � <
�
20 otherwise
• Top-Hat Filter Function:
� � =sin ��
σ2
��σ2
0 50 100 150 200
0.4
-0.4
0
�
��
��̅
��̅
�
2= 4
σ
2= 8
0 10 20−10−20�
0.2
0.1
0
�� = �� = �2�
�
��Δ = �2�
�
�
�= 2�
�
�
σ
2= 4
σ
2= 8
0
1
0 1 2 3��Δ
�
σ
2= 4
σ
2= 8
� � = ����
����
�
� � =6
�
1
Δ�
��
�
�
��
• Gaussian Filter Function
0 50 100 150 200
0.4
-0.4
0
�
��
��̅
��̅
0 1 2 3
0
1
� = 8
� = 16
��Δ = 2��
�
0 10 20−10−20
σ = 8
σ = 16
0.2
0.1
0
�
� �(�) = �1 for �� < ��
0 otherwise• Sharp-cutoff Filter Function
�� =�
�� � =
sin(���)
��
0 50 100 150 200
0.4
-0.4
0
�
��
��̅
��̅
�� =2�
16
0
1
0 1 2 3
�� =2�
8
0.2
0
0 10 20−10−20�
�� =2�
16
�� =2�
8
��Δ = 2��
�
cutoff wavenumber
0
1
1
� � =sin ��
σ2
��σ2
Top-hat
0
1
� = 8
� = 16
� � = ����
����
�
Gaussian
�� =2�
16
0
1
0 1 2 3
�� =2�
8
��Δ = 2��
�
� �(�) = �1 for �� < ��
0 otherwise
Sharp-cutoff
0 50 100 150 200
�
�� ��̅��̅
σ = 8
σ = 16
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