Filter Coefs: 0.0, 0.01899487526049, 0.0, -0.07293139167538, 0.0, 0.30596630545168, 0.5, 0.30596630545168, 0.0, -0.07293139167538, 0.0, 0.01899487526049, 0.0 Filter design: h[q,n] = g[n] * cos(2pi/2 * q * (n-6) ); n = 0..12, q = 0,1; -> h[0,n] = g[n] * 1; -> h[1,n] = g[n] * pow(-1,n);
prototype filter coefficients: 0.00746082949812 0.02270420949825 0.04546865930473 0.07266113929591 0.09885108575264 0.11793710567217 0.125 0.11793710567217 0.09885108575264 0.07266113929591 0.04546865930473 0.02270420949825 0.00746082949812 Filter design: N = 13; Q = 8; h[q,n] = g[n] * exp(j * 2 * pi / Q * (q + .5) * (n - 6)); n = 0..(N-1), q = 0..(Q-1); Time Signal: x[t]; Filter Bank Output y[q,t] = conv(x[t],h[q,t]) = conv(h[q,t],x[t]) = sum(x[k] * h[q, t - k] ) = sum(h[q, k] * x[t - k] ); k = 0..(N-1); y[q,t] = x[t - 12]*h[q, 12] + x[t - 11]*h[q, 11] + x[t - 10]*h[q, 10] + x[t - 9]*h[q, 9] + x[t - 8]*h[q, 8] + x[t - 7]*h[q, 7] + x[t - 6]*h[q, 6] + x[t - 5]*h[q, 5] + x[t - 4]*h[q, 4] + x[t - 3]*h[q, 3] + x[t - 2]*h[q, 2] + x[t - 1]*h[q, 1] + x[t - 0]*h[q, 0]; h'[q, n] = h[q,(N-1)-n] = g[n] * exp(j * 2 * pi / Q * (q + .5) * (6 - n)); n = 0..(N-1), q = 0..(Q-1); y[q,t] = x[t - 12]*h'[q, 0] + x[t - 11]*h'[q, 1] + x[t - 10]*h'[q, 2] + x[t - 9]*h'[q, 3] + x[t - 8]*h'[q, 4] + x[t - 7]*h'[q, 5] + x[t - 6]*h'[q, 6] + x[t - 5]*h'[q, 7] + x[t - 4]*h'[q, 8] + x[t - 3]*h'[q, 9] + x[t - 2]*h'[q, 10] + x[t - 1]*h'[q, 11] + x[t - 0]*h'[q, 12]; Try to split off FFT Modulation Term: FFT(x[t], q) = sum(x[t+k]*exp(-j*2*pi/N *q * k)) c m Step 1: h'[q,n] = g[n] * ( exp(j * 2 * pi / 8 * .5 * (6 - n)) ) * ( exp (j * 2 * pi / 8 * q * (6 - n)) ); h'[q,n] = g[n] *c[n] * m[q,n]; (see above) c[n] = exp( j * 2 * pi / 8 * .5 * (6 - n) ); m[q,n] = exp( j * 2 * pi / 8 * q * (6 - n) ); y[q,t] = x[t - 0]*g[0]*c[0]*m[q,0] + x[t - 1]*g[1]*c[ 1]*m[q, 1] + ... ... + x[t - 12]*g[2]*c[12]*m[q,12]; | n m *exp(-j*2*pi) | n' fft ------------------------------------------------------------------------------------------------------------------------- 0 exp( j * 2 * pi / 8 * q * 6) -> exp(-j * 2 * pi / 8 * q * 2) | 2 exp(-j * 2 * pi / 8 * q * 0) 1 exp( j * 2 * pi / 8 * q * 5) -> exp(-j * 2 * pi / 8 * q * 3) | 3 exp(-j * 2 * pi / 8 * q * 1) 2 exp( j * 2 * pi / 8 * q * 4) -> exp(-j * 2 * pi / 8 * q * 4) | 4 exp(-j * 2 * pi / 8 * q * 2) 3 exp( j * 2 * pi / 8 * q * 3) -> exp(-j * 2 * pi / 8 * q * 5) | 5 exp(-j * 2 * pi / 8 * q * 3) 4 exp( j * 2 * pi / 8 * q * 2) -> exp(-j * 2 * pi / 8 * q * 6) | 6 exp(-j * 2 * pi / 8 * q * 4) 5 exp( j * 2 * pi / 8 * q * 1) -> exp(-j * 2 * pi / 8 * q * 7) | 7 exp(-j * 2 * pi / 8 * q * 5) 6 exp( j * 2 * pi / 8 * q * 0) | 0 exp(-j * 2 * pi / 8 * q * 6) 7 exp(-j * 2 * pi / 8 * q * 1) | 1 exp(-j * 2 * pi / 8 * q * 7) 8 exp(-j * 2 * pi / 8 * q * 2) | 2 9 exp(-j * 2 * pi / 8 * q * 3) | 3 10 exp(-j * 2 * pi / 8 * q * 4) | 4 11 exp(-j * 2 * pi / 8 * q * 5) | 5 12 exp(-j * 2 * pi / 8 * q * 6) | 6 now use fft modulation coefficients m[6] = = fft[0] m[7] = = fft[1] m[8] = m[ 0] = fft[2] m[9] = m[ 1] = fft[3] m[10] = m[ 2] = fft[4] m[11] = m[ 3] = fft[5] m[12] = m[ 4] = fft[6] m[ 5] = fft[7] y[q,t] = ( x[t- 6]*g[ 6]*c[ 6] ) * fft[q,0] + ( x[t- 7]*g[ 7]*c[ 7] ) * fft[q,1] + ( x[t- 0]*g[ 0]*c[ 0] + x[t- 8]*g[ 8]*c[ 8] ) * fft[q,2] + ( x[t- 1]*g[ 1]*c[ 1] + x[t- 9]*g[ 9]*c[ 9] ) * fft[q,3] + ( x[t- 2]*g[ 2]*c[ 2] + x[t-10]*g[10]*c[10] ) * fft[q,4] + ( x[t- 3]*g[ 3]*c[ 3] + x[t-11]*g[11]*c[11] ) * fft[q,5] + ( x[t- 4]*g[ 4]*c[ 4] + x[t-12]*g[12]*c[12] ) * fft[q,6] + ( x[t- 5]*g[ 5]*c[ 5] ) * fft[q,7]; pre twiddle factors c[n] = exp(j * 2 * pi / 8 * .5 * (6 - n)); n c] | n c[n] | n c[n] --------------------------------------------------------------------------------------------------- 0 exp( j * 6 * pi / 8) | 1 exp( j * 5 * pi / 8) | 2 exp( j * 4 * pi / 8) 3 exp( j * 3 * pi / 8) | 4 exp( j * 2 * pi / 8) | 5 exp( j * 1 * pi / 8) 6 exp( j * 0 * pi / 8) | 7 exp(-j * 1 * pi / 8) | 8 exp(-j * 2 * pi / 8) 9 exp(-j * 3 * pi / 8) | 10 exp(-j * 4 * pi / 8) | 11 exp(-j * 5 * pi / 8) 12 exp(-j * 6 * pi / 8) | |
l,r0(n) ---\ l,r1(n) ---- + --\ l,r2(n) ---/ \ + --> F0(w) l,r3(n) ---\ / l,r4(n) ---- + --/ l,r5(n) ---/ l,r6(n) ---\ + ---------> F1(w) l,r7(n) ---/ l,r8(n) ---\ + ---------> F2(w) l,r9(n) ---/
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