Digital Signal Processing Reference
In-Depth Information
with coecientsf2, 0, 0.5gandf1,1, 3g, respectively. The eect is the same
as having a single FIR lter with coecientsf2,2, 5.5,0:5, 1.5g. But which
is easier to implement, the rst two lters, or the equivalent one? Notice that the
rst lter can be made without multipliers; all we need is a shifter. For xed-point
numbers, a left-shift is equivalent to multiplying by 2, while a right-shift is equivalent
to dividing by 2. For the second lter, no multiplier is needed for multiplying by
1, and multiplying by1 can be done by ipping the sign bit. The only tricky one
is the multiplication by 3, and this can be done with a shift and an add, that is,
x + 2x = 3x. For a system with just these coecients, the rst two lters can be
implemented without multipliers.
Output
Input
a, b, c, d
f, g, k, m
w[n]
x[n]
y[n]
Input
Output
???
y[n]
x[n]
Figure 8.1: FIR lters in series can be combined.
Let's examine how we can nd the resulting combined coecients, in the fol-
lowing manner. First, we will write the equations describing the output of the two
lters along the top of Figure 8.1, as if they were two separate lters; that is, one
lter with coecientsfa;b;c;dgand an input labeled x and output labeled w, and
another lter of coecientsff;g;k;mgwith input w and output y. All lter coe-
cients are assumed to be constants.
w[n] = ax[n] + bx[n1] + cx[n2] + dx[n3]
y[n] = fw[n] + gw[n1] + kw[n2] + mw[n3]
We use w as a label simply to make this description less confusing, but what we
really need is the nal output y in terms of the original input x. To gure this out,
we must have an expression for w[nk], instead of w[n]. In the next step, we use
the expression for w[n], but replace n with nk throughout.
w[nk] = ax[nk] + bx[nk1] + cx[nk2] + dx[nk3]
