Digital Signal Processing Reference
In-Depth Information
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Frequency (Hz)
FIGURE 7.4
Magnitude frequency response in Example 7.2.
Next, we see that the 3-tap FIR filter does not give an acceptable magnitude frequency response.
To explore this response further,
Figure 7.5
displays the magnitude and phase responses of 3-tap
(
M ¼
1) and 17-tap (
M ¼
8) FIR lowpass filters with a normalized cutoff frequency
U
c
¼
0
:
2
p
We can make the following observations at this point:
1.
The oscillations (ripples) exhibited in the passband (main lobe) and stopband (side lobes) of the
magnitude frequency response constitute the
Gibbs effect.
The Gibbs oscillatory behavior
remedy this problem, window functions will be used and will be discussed in the next section.
2.
Using a larger number of the filter coefficients will produce the sharp roll-off characteristic of the
transition band but may cause increased time delay and increase computational complexity for
implementing the designed FIR filter.
3.
The phase response is linear in the passband. This is consistent with Equation
(7.13)
,
which means
that all frequency components of the filter input within the passband are subjected to the same time
delay at the filter output. This is a requirement for applications in audio and speech filtering, where
phase distortion needs to be avoided. Note that we impose a linear phase requirement, that is, the
FIR coefficients are symmetric about the middle coefficient, and the FIR filter order is an odd
number. If the design methods cannot produce the symmetric coefficients or generate anti-
symmetric coefficients (Proakis and Manolakis, 1996), the resultant FIR filter does not have the
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