Bài giảng Digital Signal Processing - Chapter 9: FIR/IIR Digital Filter Design - Võ Trung Dũng

erval by
Its impulse response
The case k = 0
DSP-Chapter 9-Dr. Dung Trung Vo
Window Method
 Highpass, bandpass, and bandstop filters:
 Complementary properties: their impulse responses add up to a unit impulse 
δ(k) and their frequency responses add up to unity
DSP-Chapter 9-Dr. Dung Trung Vo
Window Method
 Ideal differentiator filter:
 Frequency response:
 Impulse response:
 Hilbert transformer:
 Frequency response:
 Impulse response:
 Symmetric characteristic:
 Symmetric classes: frequency response D(ω) is real and even in ω
 Antisymmetric classes: D(ω) which is imaginary and odd in ω
DSP-Chapter 9-Dr. Dung Trung Vo
Window Method
 Rectangular Window: consists of truncating, or rectangularly windowing, the 
doublesided d(k) to a finite length
 Total number of coefficients: will be odd
 FIR impulse response:
 Causal FIR impulse response: time-delaying the double-sided sequence d(k), 
−M ≤ k ≤ M, by M time units to make it causal
or
DSP-Chapter 9-Dr. Dung Trung Vo
Window Method
 Steps of the rectangular window method: 
 Pick an odd length N = 2M + 1, and let M = (N − 1)/2.
 Calculate the N coefficients d(k), and
 Make them causal by the delay.
DSP-Chapter 9-Dr. Dung Trung Vo
Window Method
 Example: Determine the length-11, rectangularly windowed impulse response 
that approximates:
(a) an ideal lowpass filter of cutoff frequency ωc = π/4, 
(b) the ideal differentiator filter, and 
(c) the ideal Hilbert transformer filter
 Solution: with N = 11, we have M = (N − 1)/2 = 5. 
(a) For the lowpass filter:
(b) For differentiator filter:
(c) For the Hilbert transformer
DSP-Chapter 9-Dr. Dung Trung Vo
Window Method
 DTFT Fourier series expansion:
Its z-transform:
 Final length-N filter:
Frequency response
DSP-Chapter 9-Dr. Dung Trung Vo
Window Method
 Linear phase property - symmetric case: the truncated has the same 
symmetry/antisymmetry properties as D(ω). In the symmetric case, will be 
real and even in ω
where
 Final length-N filter:
its magnitude and phase responses
making the phase response piece-wise linear in ω with 180o jumps at those ω where 
changes sign
DSP-Chapter 9-Dr. Dung Trung Vo
Window Method
 Approximation accuracy example: consider the design of an 
ideal lowpass filter of cutoff frequency ωc = 0.3π, approximated by a rectangularly
windowed response of length N = 41 and then by another one of length N = 121. 
For the case N = 41, we have M = (N − 1)/2 = 20. The designed impulse response 
is given by
For the second design has N = 121 and M = 60
DSP-Chapter 9-Dr. Dung Trung Vo
Window Method
 Approximation accuracy example:
DSP-Chapter 9-Dr. Dung Trung Vo
Window Method
 Gibbs phenomenon: The ripples in the frequency response H(ω), observed in 
arise from the (integrated) ripples of the rectangular window spectrum W(ω). As N 
increases, there are three effects:
 For ω’s that lie well within the passband or stopband (i.e., points of 
continuity), the ripple size decreases as N increases, resulting in flatter 
passband and stopband. For such ω, we have as N→∞.
 The transition width decreases with increasing N. Note also that for any N, 
the windowed response H(ω) is always equal to 0.5 at the cutoff frequency ω
= ωc. (This is a standard property of Fourier series.)
 The largest ripples tend to cluster near the passband-to-stopband
discontinuity (from both sides) and do not get smaller with N. Instead, their 
size remains approximately constant, about 8.9 percent, independent of N. 
Eventually, as N →∞, these ripples get squeezed onto the discontinuity at ω = 
ωc, occupying a set of measure zero.
DSP-Chapter 9-Dr. Dung Trung Vo
Window Method
 Rectangular Window: To eliminate the 8.9% passband and stopband ripples, 
h(n) can be multiplied by the rectangular window to taper off gradually at its 
endpoints, thus reducing the ripple effect. 
 Hamming window: There exist dozens of windows. Hamming window is a 
popular choice:
DSP-Chapter 9-Dr. Dung Trung Vo
Window Method
 Hamming window: consider the design of a length N = 81 lowpass filter with 
cutoff frequency ωc = 0.3π. 
Note how the Hamming impulse response tapers off to zero more gradually
DSP-Chapter 9-Dr. Dung Trung Vo
Window Method
 Hamming window: consider the design of a length N = 81 lowpass filter with 
cutoff frequency ωc = 0.3π. 
Hamming impulse response tapers off to zero more gradually. Ripples are virtually 
eliminated with maximum overshoot of about 0.2% but with wider transition width.
DSP-Chapter 9-Dr. Dung Trung Vo
Kaiser Window
 Kaiser Window: provide better control over the filter design specifications than 
the rectangular and Hamming window designs. A flexible set of specifications in 
which the designer can arbitrarily specify the amount of passband and stopband
overshoot δpass, δstop, as well as the transition width ∆f
DSP-Chapter 9-Dr. Dung Trung Vo
Kaiser Window
 passband/stopband frequencies {fpass, fstop}:
 Digital frequencies:
 passband and stopband overshoots: are usually expressed in dB
 Pass back and forth between the specification sets:
 Equal passband and stopband ripples
DSP-Chapter 9-Dr. Dung Trung Vo
Kaiser Window
 Kaiser window:
where I0(x) is the modified Bessel function of the first kind and 0th order
 Specifications for rectangular, Hamming, and Kaiser windows:
 Other FIR Design Methods: Parks-McClellan method based on the so-called 
optimum equiripple Chebyshev approximation generally results in shorter filters
DSP-Chapter 9-Dr. Dung Trung Vo
Frequency Sampling Method
 Frequency sampling method: For arbitrary frequency responses D(ω) while 
The window method is very convenient for designing ideally shaped filters
 Impulse response:
where
 Frequencies ωi: spanning equally the interval [−π,π], instead of the standard 
DFT interval [0, 2π]
 Final designed filter: will be the delayed and windowed version of # d(k)
DSP-Chapter 9-Dr. Dung Trung Vo
Fast Fourier transform (FFT)
 Homework: 10.1, 10.3-10.4
DSP-Chapter 9-Dr. Dung Trung Vo
IIR Digital Filter Design
 Bilinear Transformation: Instead of designing the digital filter directly, the 
method maps the digital filter into an equivalent analog filter, which can be 
designed by one of the well-developed analog filter design methods, such as 
Butterworth, Chebyshev, or elliptic filter designs. The designed analog filter is then
mapped back into the desired digital filter
DSP-Chapter 9-Dr. Dung Trung Vo
IIR Digital Filter Design
 Mapping between the s and z planes: The z-plane design of the digital filter is 
replaced by an s-plane design of the equivalent analog filter
 Mapping between the physical digital frequency and equivalent analog 
frequency: due to and , then or
 Bilinear transformation: is a particular case of f
 Mapping of frequencies (frequency prewarping transformation):
or
DSP-Chapter 9-Dr. Dung Trung Vo
IIR Digital Filter Design
 Other versions of the bilinear transformation:
 Corresponding frequency maps:
DSP-Chapter 9-Dr. Dung Trung Vo
IIR Digital Filter Design
 Overall design method: 
 Determine magnitude response specifications for the digital filter 
 Transform the specification by the appropriate prewarping transformation 
into the specifications of an equivalent analog filter. 
 Design the equivalent analog filter using an analog filter design technique, 
say Ha(s). 
 Map back the analog filter using the bilinear transformation into the desired 
digital filter H(z)
Or corresponding frequency responses
DSP-Chapter 9-Dr. Dung Trung Vo
IIR Digital Filter Design
 Property of the bilinear transformation: it maps the left-hand s-plane into 
the inside of the unit circle on the z-plane. Because all analog filter design methods 
give rise to stable and causal transfer functions Ha(s), so the digital filter H(z) will 
also be stable and causal
From bilinear transformation
DSP-Chapter 9-Dr. Dung Trung Vo
First-Order Lowpass IIR Design
 First-Order Lowpass: that has a prescribed cutoff frequency, say fc, and 
operates at a given sampling rate fs.
 Filter form: the design problem is to determine the filter coefficients {b0, b1, a1} 
in terms of the cutoff frequency fc and rate fs
Digital cutoff frequency: to be the frequency at which |H(ω)|2 drops by a 
factor of G2c < 1, or a drop in dB
DSP-Chapter 9-Dr. Dung Trung Vo
First-Order Lowpass IIR Design
 Cutoff frequency specifications for lowpass digital filter:
Popular choice
DSP-Chapter 9-Dr. Dung Trung Vo
First-Order Lowpass IIR Design
 Prewarp the cutoff frequency: to get the cutoff frequency of the equivalent 
analog filter:
 Design a first-order analog filter: with cutoff frequency is Ωc
DSP-Chapter 9-Dr. Dung Trung Vo
First-Order Lowpass IIR Design
 Frequency and magnitude responses:
 Cutoff condition:
Lead to 
 Transform the filter to the z-domain by the bilinear transformation
where
DSP-Chapter 9-Dr. Dung Trung Vo
First-Order Lowpass IIR Design
 Frequency and magnitude responses:
 Cutoff condition:
Lead to 
 Transform the filter to the z-domain by the bilinear transformation
where
DSP-Chapter 9-Dr. Dung Trung Vo
First-Order Lowpass IIR Design
 Example: Design a lowpass digital filter operating at a rate of 10 kHz, whose 3-
dB frequency is 1 kHz. 
Solution: digital cutoff frequency is
Prewarped analog version
Filter parameters: with
Digital filter transfer function
DSP-Chapter 9-Dr. Dung Trung Vo
First-Order Lowpass IIR Design
 Example: Design a lowpass digital filter operating at a rate of 10 kHz, whose 3-
dB frequency is 1 kHz. 
Solution:
DSP-Chapter 9-Dr. Dung Trung Vo
First-Order Highpass IIR Design
 Highpass analog first-order filter form:
 Highpass digital filter:
where
 Frequency and magnitude responses:
DSP-Chapter 9-Dr. Dung Trung Vo
First-Order Highpass IIR Design
 Cutoff condition:
lead to