Digital Signal Processing Reference
In-Depth Information
Keeping track of whether index n is even or odd, we can nd y[n] = w
f
[n] + z
f
[n]:
y[n] = dw[n] + bw[n2]az[n]cz[n2]; n is even
y[n] = cw[n1] + aw[n3] + bz[n1] + dz[n3]; n is odd:
These expressions can be expanded, to put the output y[n] in terms of the original
input x[n]:
y[n]
= d(ax[n] + bx[n1] + cx[n2] + dx[n3])
+b(ax[n2] + bx[n3] + cx[n4] + dx[n5])
a(dx[n]cx[n1] + bx[n2]ax[n3])
c(dx[n2]cx[n3] + bx[n4]ax[n5]); n is even
y[n]
= c(ax[n1] + bx[n2] + cx[n3] + dx[n4])
+a(ax[n3] + bx[n4] + cx[n5] + dx[n6])
+b(dx[n1]cx[n2] + bx[n3]ax[n4])
+d(dx[n3]cx[n4] + bx[n5]ax[n6]); n is odd:
Simplifying:
y[n]
= adx[n] + bdx[n1] + cdx[n2] + ddx[n3]
+abx[n2] + bbx[n3] + bcx[n4] + bdx[n5]
adx[n] + acx[n1]abx[n2] + aax[n3]
cdx[n2] + ccx[n3]bcx[n4] + acx[n5]; n is even
y[n]
= acx[n1] + bcx[n2] + ccx[n3] + cdx[n4]
+aax[n3] + abx[n4] + acx[n5] + adx[n6]
+bdx[n1]bcx[n2] + bbx[n3]abx[n4]
+ddx[n3]cdx[n4] + bdx[n5]adx[n6]; n is odd:
Eliminating terms that cancel each other out, and simplifying further:
