Page 202 - Mechanical Engineers' Handbook (Volume 2)

2 Basic Analog Filters 191

FAST FOURIER TRANSFORM 1 Given a vector {x ,..., x } of complex numbers,
n
1
p
where n is some integer and there exists some integer p such that n 2 . This algorithm
outputs the Fourier transform overwriting the input vector x.
k ← 1
For i 1 To n
If k i
swap(x ,x )
i
k
End If
m n/2
While m 1 And k m
k ← k m
m ← m/2
End While
k ← k m
Next i
M max ← 1
While n M max
i ← 2 M max
c
← /M max
2
wp ← 2 sin ( /2) j sin( )
w ← 1
For m 1 To M max
For i mTonByi c
k ← i M max
xtemp ← x w x k
i
x ← x w x k
i
i
x ← xtemp
k
Next i
w ← w (wp 1)
Next m
M max i c
End While

2 BASIC ANALOG FILTERS
Linear filters apply frequency-specific gains to a signal. This is often done to enhance desired
portions of the spectrum while attenuating or eliminating other portions. Four common filters
are low pass, high pass, bandpass, and band reject. The objective of an ideal low-pass filter
is to eliminate a range of undesired high frequencies from a signal and leave the remaining
portion undistorted. To this end, an ideal low-pass filter will have a gain of 1 for all fre-
quencies less than some desired cutoff frequency ƒ and a gain of 0 for all frequencies
c
greater than ƒ , as seen in Fig. 1. There are various rational functions that approximate this
c
ideal. But because of the discontinuity in the ideal low-pass response, all realizations of this
ideal with be an approximation. The various approximation functions generally trade off
between three characteristics: passband ripple, stop-band ripple, and the transition width,
shown in Fig. 2. Four common rational function approximations for low-pass filters are the
Butterworth, the Tchebyshev Types I and II, and the elliptical filter. By convention, we use
H(s) as the transfer function from which we determine the frequency response, where