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.CMD DIMENSIONS_ANALYSIS 0 0
.TXT 2 24 1 0
Cg a30.250000,48.000000,44
{\rtf1\ansi \deff0
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}
{\plain {Small Kernels with Big Transfer
 Functions}}
}
.TXT 2 -5 261 0
Cg a40.625000,61.500000,52
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{\plain Centered DFT, pseudoinverse MMSE,
 and truncated FIR}
}
.TXT 3 43 2 0
Cg a17.250000,24.000000,26
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{\plain {F_16-14}{.MCD, 12/19/94}}
}
.EQN 1 -46 6 0
{0:N}NAME:14
.EQN 0 9 7 0
{0:i}NAME:0;{0:N}NAME-1
.EQN 0 11 8 0
{0:k}NAME:0;{0:N}NAME-1
.EQN 1 -31 5 0
{0:j}NAME:\(-1)
.EQN 2 11 9 0
{0:M}NAME:7
.EQN 0 9 10 0
{0:m}NAME:0;{0:M}NAME-1
.EQN 0 11 11 0
{0:n}NAME:0;{0:M}NAME-1
.TXT 3 -35 13 0
Cg a78.500000,85.625000,119
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{\plain {The }{1-D Discrete Fourier Transform
 computes an M-point}{ transfer function}{
 from an M-point}{ impulse response}{:}}
}
.EQN 6 0 14 0
({0:F}NAME)[({0:n}NAME):((1)/(\({0:M}NAME)))*((0,{0:M}NAME-1,{0:m}NAME,({0:f}NAME)[({0:m}NAME)*{0:exp}NAME(-{0:j}NAME*2*{0:\p}NAME*{0:n}NAME*({0:m}NAME)/({0:M}NAME))){64})
.TXT 0 30 15 0
Cg a1.375000,44.000000,5
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{\plain {or}}
}
.EQN 0 4 16 0
{0:F}NAME:{0:W}NAME*{0:f}NAME
.TXT 0 10 17 0
Cg a4.250000,29.000000,8
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{\plain {where}}
}
.EQN 0 8 18 0
({0:W}NAME)[({0:n}NAME,{0:m}NAME):(1)/(\({0:M}NAME))*{0:exp}NAME(-{0:j}NAME*2*{0:\p}NAME*{0:n}NAME*({0:m}NAME)/({0:M}NAME))
.TXT 6 -52 136 0
Cg a69.500000,86.375000,103
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{\plain {is an M by M kernel matrix. Both
 [f] and [F] have the central element at
 the left end of the vector.}}
}
.EQN 4 0 147 0
{0:f}NAME:(({1,7}ö.5ö0ö0ö0ö0ö.5ö1)){51}
.EQN 0 31 148 0
{0:F}NAME:{0:W}NAME*{0:f}NAME
.EQN 0 17 149 0
({0:F}NAME){51}={0}?_n_u_l_l_
.TXT 4 -48 19 0
Cg a56.000000,73.000000,86
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{\plain {Now l}{et f(m) be an M - point 
FIR, rotated (a places) to the center of
 the vector.}}
}
.EQN 5 0 165 0
{0:f}NAME:(({1,7}ö0ö0ö.5ö1ö.5ö0ö0)){51}
.EQN 0 28 178 0
{0:a}NAME:{0:floor}NAME(({0:M}NAME)/(2))
.EQN 0 12 179 0
{0:a}NAME={0}?_n_u_l_l_
.EQN 0 18 250 0
{0:b}NAME:{0:floor}NAME(({0:N}NAME)/(2))
.EQN 0 13 251 0
{0:b}NAME={0}?_n_u_l_l_
.TXT 4 -71 258 0
Cg a16.625000,86.375000,27
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{\plain {Its M-point transform is}}
}
.EQN 6 0 174 0
({0:F}NAME)[({0:n}NAME):((1)/(\({0:M}NAME)))*((0,{0:M}NAME-1,{0:m}NAME,({0:f}NAME)[({0:m}NAME)*{0:exp}NAME(-{0:j}NAME*2*{0:\p}NAME*{0:n}NAME*({0:m}NAME-{0:a}NAME)/({0:M}NAME))){64})
.EQN 0 41 168 0
({0:F}NAME){51}={0}?_n_u_l_l_
.TXT 6 -41 183 0
Cg a63.500000,86.375000,95
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{\plain {For the M-point transform, this
 works for any rotation, a, of the vector,
 for example, zero:}}
}
.EQN 0 67 180 0
{0:f}NAME:(({1,7}ö.5ö0ö0ö0ö0ö.5ö1)){51}
.EQN 6 -67 181 0
({0:F}NAME)[({0:n}NAME):((1)/(\({0:M}NAME)))*((0,{0:M}NAME-1,{0:m}NAME,({0:f}NAME)[({0:m}NAME)*{0:exp}NAME(-{0:j}NAME*2*{0:\p}NAME*{0:n}NAME*({0:m}NAME-0)/({0:M}NAME))){64})
.EQN 1 41 182 0
({0:F}NAME){51}={0}?_n_u_l_l_
.TXT 6 -41 176 0
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{\plain {Now we can compute N points on 
its transform:}}
}
.EQN 6 5 232 0
{0:f}NAME:(({1,7}ö0ö0ö.5ö1ö.5ö0ö0)){51}
.EQN 0 37 163 0
({0:G}NAME)[({0:k}NAME):((1)/(\({0:M}NAME)))*((0,{0:M}NAME-1,{0:m}NAME,({0:f}NAME)[({0:m}NAME)*{0:exp}NAME(-{0:j}NAME*2*{0:\p}NAME*{0:k}NAME*({0:m}NAME-{0:a}NAME)/({0:N}NAME))){64})
.EQN 7 -42 171 0
({0:G}NAME){51}={0}?_n_u_l_l_
.TXT 4 0 184 0
Cg a59.500000,86.375000,91
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{\plain {But for}{ its N-point transform,
 the vector length is wrong and it doesn't
 wrap properly}}
}
.EQN 6 0 185 0
({0:G}NAME)[({0:k}NAME):((1)/(\({0:M}NAME)))*((0,{0:M}NAME-1,{0:m}NAME,({0:f}NAME)[({0:m}NAME)*{0:exp}NAME(-{0:j}NAME*2*{0:\p}NAME*{0:k}NAME*({0:m}NAME-0)/({0:N}NAME))){64})
.EQN 7 0 187 0
({0:G}NAME){51}={0}?_n_u_l_l_
.TXT 3 0 191 0
Cg a54.625000,87.750000,128
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{\plain {If the M-point function is centered
 on element a, \par and we want its N-point
 spectrum to be centered on element b, we
 can use:}}
}
.EQN 6 0 190 0
{0:f}NAME:(({1,7}ö0ö0ö.5ö1ö.5ö0ö0)){51}
.EQN 0 33 194 0
{0:M}NAME={0}?_n_u_l_l_
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{0:a}NAME={0}?_n_u_l_l_
.EQN 0 25 195 0
{0:N}NAME={0}?_n_u_l_l_
.EQN 0 10 252 0
{0:b}NAME={0}?_n_u_l_l_
.TXT 4 -76 259 0
Cg a13.500000,86.500000,22
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{\plain {so the transform is}}
}
.EQN 6 0 188 0
({0:G}NAME)[({0:k}NAME):((1)/(\({0:M}NAME)))*((0,{0:M}NAME-1,{0:m}NAME,({0:f}NAME)[({0:m}NAME)*{0:exp}NAME(-{0:j}NAME*2*{0:\p}NAME*({0:k}NAME-{0:b}NAME)*({0:m}NAME-{0:a}NAME)/({0:N}NAME))){64})
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{\plain {or}}
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{0:G}NAME:{0:T}NAME*{0:f}NAME
.TXT 0 9 199 0
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{\plain {where}}
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{0:G}NAME:{0:T}NAME*{0:f}NAME
.TXT 0 9 260 0
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{\plain {or}}
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.EQN 0 7 189 0
\(({0:N}NAME)/({0:M}NAME))*({0:G}NAME){51}={0}?_n_u_l_l_
.TXT 4 -16 255 0
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{\plain {This scaling preserves the amplitude
 of the spectrum as it expands to N points.}}
}
.TXT 3 0 249 0
Cg a41.500000,85.375000,61
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{\plain {This is then the general M-point
 to N-point centered DFT. }}
}
.TXT 4 29 233 0
Cg a22.750000,69.375000,33
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{\plain {\b Small kernel from large MTF}}
}
.TXT 3 -30 113 0
Cg a62.875000,86.375000,88
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{\plain {Suppose the desired TF, given as
 an N-point spectrum vector centered on 
element b, is}}
}
.EQN 0 74 213 0
{0:\s}NAME:1.0
.EQN 3 0 246 0
{0:M}NAME={0}?_n_u_l_l_
.EQN 3 -74 212 0
({0:G}NAME)[({0:i}NAME):{0:exp}NAME(-(((({0:i}NAME-{0:b}NAME))^(2))/(2*{0:\s}NAME*{0:\s}NAME)))
.EQN 0 16 202 0
({0:G}NAME){51}={0}?_n_u_l_l_
.EQN 0 58 245 0
{0:N}NAME={0}?_n_u_l_l_
.TXT 6 -74 114 0
Cg a15.500000,86.375000,24
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{\plain {whose N-point FIR is:}}
}
.EQN 5 0 204 0
({0:W}NAME)[({0:i}NAME,{0:k}NAME):(1)/(\({0:N}NAME))*{0:exp}NAME(-{0:j}NAME*2*{0:\p}NAME*({0:i}NAME-{0:b}NAME)*({0:k}NAME-{0:b}NAME)/({0:N}NAME))
.EQN 1 29 203 0
{0:g}NAME:(({0:W}NAME)^(-1)*{0:G}NAME)
.EQN 4 -15 205 0
({0:g}NAME){51}={0}?_n_u_l_l_
.TXT 4 -14 26 0
Cg a68.000000,77.125000,95
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{\plain {We}{ solve G = Tf (by the pseudoinverse
 method) for an M-point MMSE approximation
 FIR, f(m):}}
}
.EQN 6 0 99 0
{0:K}NAME:(((((({0:T}NAME)])){51}*{0:T}NAME))^(-1)*((({0:T}NAME)])){51})
.EQN 0 26 103 0
{0:f}NAME:{0:K}NAME*{0:G}NAME
.EQN 0 11 29 0
({0:f}NAME){51}={148578}?_n_u_l_l_
.TXT 4 -37 239 0
Cg a78.625000,86.375000,112
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{\plain {which, we observe, is simply the
 truncated FIR. The constant rescales T 
to preserve the amplitude of the FIR.}}
}
.TXT 3 0 30 0
Cg a58.750000,72.000000,88
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{\plain {We can}{ transform this back to
 an N-point spectrum, to see}{ the approximation
 TF}{:}}
}
.EQN 4 1 209 0
{0:F}NAME:{0:T}NAME*{0:f}NAME
.EQN 0 15 210 0
({0:F}NAME){51}={0}?_n_u_l_l_
.EQN 4 6 88 0
({0:G}NAME){51}={0}?_n_u_l_l_
.TXT 3 -22 240 0
Cg a44.250000,85.375000,63
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{\plain {Using the N-point formulation as
 a check on the computation,}}
}
.EQN 4 0 235 0
{0:h}NAME:({0:augment}NAME(({1,4}ö0ö0ö0ö0),{0:augment}NAME(({0:f}NAME){51},({1,3}ö0ö0ö0)))){51}
.EQN 4 23 236 0
({0:h}NAME){51}={0}?_n_u_l_l_
.EQN 3 -22 237 0
{0:H}NAME:{0:W}NAME*{0:h}NAME
.EQN 0 15 238 0
({0:H}NAME){51}={0}?_n_u_l_l_
.TXT 5 -15 211 0
Cg a34.375000,85.375000,46
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{\plain {We can compute the MSE of the approximation}}
}
.EQN 0 44 133 0
({0:f}NAME){51}={0}?_n_u_l_l_
.EQN 4 -42 134 0
({0:g}NAME){51}={0}?_n_u_l_l_
.EQN 2 38 110 0
&-.2&(_n_u_l_l_&_n_u_l_l_)&({0:G}NAME)[({0:k}NAME),({0:F}NAME)[({0:k}NAME),0@({0:N}NAME)/(1)&&(_n_u_l_l_&_n_u_l_l_)&{0:k}NAME
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0 0 1 1 0 0 1 
0 0 6 5 NO-TRACE-STRING
0 2 6 0 NO-TRACE-STRING
0 1 6 0 NO-TRACE-STRING
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0 1 4 0 NO-TRACE-STRING
0 2 5 0 NO-TRACE-STRING
0 3 6 0 NO-TRACE-STRING
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0 1 1 0 NO-TRACE-STRING
0 2 2 0 NO-TRACE-STRING
0 3 3 0 NO-TRACE-STRING
0 4 4 0 NO-TRACE-STRING
0 1 5 0 NO-TRACE-STRING
0 2 6 0 NO-TRACE-STRING
0 3 0 0 NO-TRACE-STRING
0 4 1 0 NO-TRACE-STRING
0 1 37 21 0 3 
.EQN 7 -40 139 0
{0:RMSE}NAME:\((1)/({0:N}NAME)*((0,{0:N}NAME-1,{0:i}NAME,((({0:F}NAME)[({0:i}NAME)-({0:G}NAME)[({0:i}NAME)))^(2)){64}))
.TXT 8 30 257 0
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{\plain {Figure 16-14}}
}
.EQN 2 -26 140 0
{0:RMSE}NAME={148583}?_n_u_l_l_