Implementing IIR/FIR Filters

Z-Domain Transfer Function
Hz ( )
Difference Equation (Direct Form)
Coefficients
α 1 μz 1 –
σz 2 –
( + + )
1/2 γz 1 –
– βz 2 –
= ------------------------------------------------
+
yn ( ) = 2{ α[ x(
n)
+ μxn ( – 1)
+ σxn ( – 2)]
+ γyn ( – 1)
– βyn ( – 2)
}
1
β --
2
1 1/2dsinθ –
0
2tan( θ
= -----------------------------------
0 /2Q)
γ = ( 1/2 + β)
cosθ
1 + 1/2dsinθ d = --------------------------------
0
0
sinθ0
where 0 < β < 1/2 and θ0 2 π f 0 /fs f 0
Q = ----- = -------------------------------- = --------------
Δ0 2π( f2 – f1)/fs f 2 – f 1
where f0 is the center frequency of the bandpass or bandstop filter,
f1 and f2 are the half-power points (where gain is equal to 1/ 2),
and
fs is the sample frequency. Note the f0 is replaced with fc in the lowpass
and highpass cases.
Type α μ σ Unity Gain at
Lowpass
Highpass
Bandpass
Bandstop
NOTE: θ 0 = 2πf 0 /f s
Numerator Coefficients
(1/2 + β−γ)/4
(1/2 + β+γ)/4
(1/2 - β)/2
(1/2 + β)/2
2
-2 0
-2cosθ 0
Figure 2-22 Summary of Digital Coefficients for the Four Basic Filter Types
MOTOROLA 2-27
1
1
-1 1
f = 0
f = f s /2
f = f 0
f = 0 and f = f s/2