3. Fourier Correlation
• Fourier
(f ? )(x) :=
Z
R
(x0
x)f(x0
) dx0
=
Z
R
dx0
Z
R
ˆ(k0
)eik0
(x0
x)
dk0
Z
R
ˆf(k)eikx0
dk
=
Z
R
dk0 ˆ(k0
)e ik0
x
Z
R
dk ˆf(k)
Z
R
dx0
ei(k+k0
)x0
= (2⇡)
Z
R
dk0 ˆ(k0
)e ik0
x
Z
R
dk ˆf(k) (k + k0
)
= (2⇡)
Z
R
dk ˆ( k) ˆf(k)eikx
= (2⇡)
Z
R
dk ˆ†
(k) ˆf(k)eikx
<latexit sha1_base64="fVLCyoy4wdkvxqmF3TaWI2o5hG4=">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</latexit><latexit sha1_base64="fVLCyoy4wdkvxqmF3TaWI2o5hG4=">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</latexit><latexit sha1_base64="fVLCyoy4wdkvxqmF3TaWI2o5hG4=">AAAEF3icpVPNi9NAFJ9N/FjrV1ePXgaL2xRpSRZBEYRFLx5XsbsLnW6ZTCbtkMkHMxNpCfkvvPivePGgiFe9+d84maRgmy578EHg8d7v471Hxs84k8p1/+xZ9rXrN27u3+rcvnP33v3uwYNTmeaC0DFJeSrOfSwpZwkdK6Y4Pc8ExbHP6Zkfvan6Zx+pkCxNPqhVRqcxnicsZAQrXZodWEMnhEgqLCDKJBs4ywE8fPkKIpaoGYqxWvh+8b40TWfZh0OoAaHOBhDBQBcQ6hzugNe97eoCq6JSKp1IC9CLgkX9taohRZeQQs1oCMt+jbzE2UjscBpqq2XZRq/BtUOrX21hfJ0IPoVay9gbZ+cIZWwH5f8GCChXeG12ldOm0bDib1/rymk3NC4KFOD5nIp6mJbYrNtzR64J2E68JumBJk5m3d8oSEke00QRjqWceG6mpgUWihFOyw7KJc0wifCcTnSa4JjKaWH+6xI+0ZUAhqnQX6Kgqf7LKHAs5Sr2NbLaSG73quKu3iRX4YtpwZIsVzQhtVGYc6hSWD0SGDBBieIrnWAimJ4VkgUWmCj9lDr6CN72yu3k9GjkuSPv3bPe8evmHPvgEXgMHOCB5+AYvAUnYAyI9cn6Yn2zvtuf7a/2D/tnDbX2Gs5DsBH2r79N6EoP</latexit><latexit sha1_base64="fVLCyoy4wdkvxqmF3TaWI2o5hG4=">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</latexit>
Filter Signal
Translation
!3
4. Fourier Correlation
• Fourier
(f ? )(x) :=
Z
R
(x0
x)f(x0
) dx0
=
Z
R
dx0
Z
R
ˆ(k0
)eik0
(x0
x)
dk0
Z
R
ˆf(k)eikx0
dk
=
Z
R
dk0 ˆ(k0
)e ik0
x
Z
R
dk ˆf(k)
Z
R
dx0
ei(k+k0
)x0
= (2⇡)
Z
R
dk0 ˆ(k0
)e ik0
x
Z
R
dk ˆf(k) (k + k0
)
= (2⇡)
Z
R
dk ˆ( k) ˆf(k)eikx
= (2⇡)
Z
R
dk ˆ†
(k) ˆf(k)eikx
<latexit sha1_base64="fVLCyoy4wdkvxqmF3TaWI2o5hG4=">AAAEF3icpVPNi9NAFJ9N/FjrV1ePXgaL2xRpSRZBEYRFLx5XsbsLnW6ZTCbtkMkHMxNpCfkvvPivePGgiFe9+d84maRgmy578EHg8d7v471Hxs84k8p1/+xZ9rXrN27u3+rcvnP33v3uwYNTmeaC0DFJeSrOfSwpZwkdK6Y4Pc8ExbHP6Zkfvan6Zx+pkCxNPqhVRqcxnicsZAQrXZodWEMnhEgqLCDKJBs4ywE8fPkKIpaoGYqxWvh+8b40TWfZh0OoAaHOBhDBQBcQ6hzugNe97eoCq6JSKp1IC9CLgkX9taohRZeQQs1oCMt+jbzE2UjscBpqq2XZRq/BtUOrX21hfJ0IPoVay9gbZ+cIZWwH5f8GCChXeG12ldOm0bDib1/rymk3NC4KFOD5nIp6mJbYrNtzR64J2E68JumBJk5m3d8oSEke00QRjqWceG6mpgUWihFOyw7KJc0wifCcTnSa4JjKaWH+6xI+0ZUAhqnQX6Kgqf7LKHAs5Sr2NbLaSG73quKu3iRX4YtpwZIsVzQhtVGYc6hSWD0SGDBBieIrnWAimJ4VkgUWmCj9lDr6CN72yu3k9GjkuSPv3bPe8evmHPvgEXgMHOCB5+AYvAUnYAyI9cn6Yn2zvtuf7a/2D/tnDbX2Gs5DsBH2r79N6EoP</latexit><latexit sha1_base64="fVLCyoy4wdkvxqmF3TaWI2o5hG4=">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</latexit><latexit sha1_base64="fVLCyoy4wdkvxqmF3TaWI2o5hG4=">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</latexit><latexit sha1_base64="fVLCyoy4wdkvxqmF3TaWI2o5hG4=">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</latexit>
!4
5. S2-Correlation?
•
• Translation?
•
(f ? )(x) :=
Z
R
(x0
x)f(x0
) dx0
=
Z
R
dx0
Z
R
ˆ(k0
)eik0
(x0
x)
dk0
Z
R
ˆf(k)eikx0
dk
=
Z
R
dk0 ˆ(k0
)e ik0
x
Z
R
dk ˆf(k)
Z
R
dx0
ei(k+k0
)x0
= (2⇡)
Z
R
dk0 ˆ(k0
)e ik0
x
Z
R
dk ˆf(k) (k + k0
)
= (2⇡)
Z
R
dk ˆ( k) ˆf(k)eikx
= (2⇡)
Z
R
dk ˆ†
(k) ˆf(k)eikx
<latexit sha1_base64="fVLCyoy4wdkvxqmF3TaWI2o5hG4=">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</latexit><latexit sha1_base64="fVLCyoy4wdkvxqmF3TaWI2o5hG4=">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</latexit><latexit sha1_base64="fVLCyoy4wdkvxqmF3TaWI2o5hG4=">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</latexit><latexit sha1_base64="fVLCyoy4wdkvxqmF3TaWI2o5hG4=">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</latexit>
?
!5