L. Vandenberghe
EE236A (Fall 2013-14)
Lecture 10
FIR filter design
• linear phase filter design
• magnitude filter design
• equalizer design
10–1
Finite impulse response (FIR) filter
y(t) =
n−1
X
τ =0
hτ u(t − τ )
• u : Z → R is input signal; y : Z → R is output signal
• hi ∈ R are filter coefficients; n is filter order or length
frequency response: a function H : R → C defined as
H(ω) = h0 + h1e
=
n−1
X
t=0
−jω
+ · · · + hn−1e
ht cos tω − j
n−1
X
−j(n−1)ω
(with j =
√
−1)
ht sin tω
t=0
periodic and conjugate symmetric; we only need to consider ω ∈ [0, π]
design problem: choose hi so that H satisfies/optimizes specifications
FIR filter design
10–2
Example: lowpass FIR filter
impulse response (order n = 21)
h(t)
0.2
0.1
0
−0.1
−0.2
0
2
4
6
8
10
12
14
16
18
20
t
frequency response: magnitude |H(ω)| and phase 6 H(ω)
1
3
2
H(ω)
0
10
−1
10
1
0
−1
−2
10
6
|H(ω)|
10
−2
−3
10
0
0.5
1
1.5
ω
FIR filter design
2
2.5
3
−3
0
0.5
1
1.5
2
2.5
3
ω
10–3
Linear-phase filters
suppose n = 2N + 1 is odd and impulse response is symmetric about hN :
ht = hn−1−t,
t = 0, . . . , n − 1
frequency response
H(ω) = h0 + h1e−jω + · · · + hn−1e−j(n−1)ω
= e−jN ω (2h0 cos N ω + 2h1 cos(N −1)ω + · · · + hN )
= e−jN ω G(ω)
• term e−jN ω represents N -sample delay
• G(ω) is real-valued and |H(ω)| = |G(ω)|
• ‘linear phase’:
FIR filter design
6
H(ω) is linear except for jumps of ±π
10–4
Lowpass filter specifications
δ1
1/δ1
δ2
ωp ωs
π
ω
• maximum passband ripple (±20 log10 δ1 in dB):
1/δ1 ≤ |H(ω)| ≤ δ1
for ω ∈ [0, ωp]
• minimum stopband attenuation (−20 log10 δ2 in dB):
|H(ω)| ≤ δ2
FIR filter design
for ω ∈ [ωs, π]
10–5
Linear-phase lowpass filter design
• sample the frequency axis: ωk = kπ/K, k = 0, . . . , K − 1
• assume without loss of generality that G(0) > 0, so ripple spec. is
1/δ1 ≤ G(ωk ) ≤ δ1
maximum stopband attenuation (for given passband ripple δ1)
minimize δ2
subject to 1/δ1 ≤ G(ωk ) ≤ δ1 for ωk ∈ [0, ωp]
−δ2 ≤ G(ωk ) ≤ δ2 for ωk ∈ [ωs, π]
• a linear program in variables hi, δ2
• known and used since 1960’s
• can add other constraints, e.g., |hi| ≤ α
FIR filter design
10–6
Example
• linear-phase filter of order n = 31
• passband [0, 0.12π]; stopband [0.24π, π]
• maximum ripple δ1 = 1.059 (±0.5dB)
0.2
0.15
h(t)
0.1
0.05
0
−0.05
−0.1
0
5
10
15
20
25
30
t
1
10
0
|H(ω)|
10
−1
10
−2
10
−3
10
−4
10
0
0.5
1
1.5
2
2.5
3
ω
FIR filter design
10–7
Variations
minimize passband ripple (variables δ1, h)
minimize δ1
subject to 1/δ1 ≤ G(ωk ) ≤ δ1 for ωk ∈ [0, ωp]
−δ2 ≤ G(ωk ) ≤ δ2 for ωk ∈ [ωs, π]
minimize transition bandwidth (variables ωs, h)
minimize ωs
subject to 1/δ1 ≤ G(ωk ) ≤ δ1 for ωk ∈ [0, ωp]
−δ2 ≤ G(ωk ) ≤ δ2 for ωk ∈ [ωs, π]
minimize filter order (variables N , h)
minimize N
subject to 1/δ1 ≤ G(ωk ) ≤ δ1 for ωk ∈ [0, ωp]
−δ2 ≤ G(ωk ) ≤ δ2 for ωk ∈ [ωs, π]
not LPs, but can be solved by bisection/LP feasibility problems
FIR filter design
10–8
Outline
• linear phase filter design
• magnitude filter design
• equalizer design
Filter magnitude specifications
magnitude specification: a constraint
L(ω) ≤ |H(ω)| ≤ U (ω)
∀ω
L, U : R → R+ are given and
H(ω) =
n−1
X
t=0
ht cos tω − j
n−1
X
ht sin tω
t=0
• arises in many applications, e.g., audio, spectrum shaping
• not equivalent to linear inequalities in hi (linear inequalities can not
express the lower bound on absolute value)
• can change variables and convert to set of linear inequalities
FIR filter design
10–9
Autocorrelation coefficients
definition: autocorrelation coefficients of h = (h0, . . . , hn−1) ∈ Rn
rt =
n−1−t
X
hτ hτ +t
τ =0
(with hk = 0 for k < 0 or k ≥ n)
rt = r−t and rt = 0 for |t| ≥ n; hence suffices to specify r = (r0, . . . , rn−1)
Fourier transform of autocorrelation coefficients:
R(ω) =
X
τ
e−jωτ rτ = r0 +
n−1
X
t=1
2rt cos ωt = |H(ω)|2
magnitude specifications are linear inequalities in coefficients rt:
L(ω)2 ≤ R(ω) ≤ U (ω)2
FIR filter design
for ω ∈ [0, π]
10–10
Spectral factorization
when is r ∈ Rn the vector of autocorrelation coefficients of some h ∈ Rn?
spectral factorization theorem: if and only if R(ω) ≥ 0 for all ω
• condition is an infinite set of linear inequalities in r
• many algorithms for spectral factorization (find h s.t. R(ω) = |H(ω)|2)
consequence: to cast magnitude design problem as an LP,
• use r = (r0, . . . , rn−1) as variable instead of h = (h0, . . . , hn−1)
• add spectral factorization condition as constraint: R(ω) ≥ 0 for all ω
• discretize the frequency axis
• optimize over r and use spectral factorization to recover h
FIR filter design
10–11
Magnitude lowpass filter design
maximum stopband attenuation design (with variables r)
minimize γ2
subject to 1/γ1 ≤ R(ω) ≤ γ1 for ω ∈ [0, ωp]
R(ω) ≤ γ2 for ω ∈ [ωs, π]
R(ω) ≥ 0 for ω ∈ [0, π]
(γi corresponds to δi2 in original problem)
discretization: impose constraints at finite set of frequencies ωk
minimize γ2
subject to 1/γ1 ≤ R(ωk ) ≤ γ1 for ωk ∈ [0, ωp]
R(ωk ) ≤ γ2 for ωk ∈ [ωs, π]
R(ωk ) ≥ 0 for ωk ∈ [0, π]
this is a linear program in r, γ2
FIR filter design
10–12
Outline
• linear phase filter design
• magnitude filter design
• equalizer design
Equalizer design
g(t)
h(t)
(time-domain) equalization
• given g (unequalized impulse response), gdes (desired impulse response)
• design FIR equalizer h so that convolution g̃ = h ∗ g approximates gdes
example
• gdes is pure delay D: gdes(t) =
1 t=D
0 t=
6 D
• find equalizer h by solving
minimize
max |g̃(t)|
t6=D
subject to g̃(D) = 1
this can be cast as an LP in the coefficients hi
FIR filter design
10–13
Example
unequalized system (10th order FIR)
• impulse response
1
0.8
g(t)
0.6
0.4
0.2
0
−0.2
−0.4
0
1
2
3
4
5
6
7
8
9
t
• frequency response magnitude
1
10
3
G(ω)
0
10
1
0
−1
6
|G(ω)|
2
−2
−1
10
0
0.5
1
1.5
ω
FIR filter design
2
2.5
3
−3
0
0.5
1
1.5
2
2.5
3
ω
10–14
time-domain equalization (30th order FIR, D = 10)
minimize max |g̃(t)|
t6=10
• equalized system impulse response
1
g̃(t)
0.8
0.6
0.4
0.2
0
−0.2
0
5
10
15
20
25
30
35
t
• equalized frequency response
1
10
3
e
G(ω)
0
10
1
0
−1
6
e
|G(ω)|
2
−2
−1
10
0
0.5
1
1.5
ω
FIR filter design
2
2.5
3
−3
0
0.5
1
1.5
2
2.5
3
ω
10–15
Magnitude equalizer design
H(ω)
G(ω)
problem
• given system frequency response G : [0, π] → C
• design FIR equalizer H so that |G(ω)H(ω)| ≈ 1:
minimize
2
max |G(ω)H(ω)| − 1 ω∈[0,π]
LP formulation: use autocorrelation coefficients as variables
minimize α
2
subject to
|G(ω)| R(ω) − 1 ≤ α for ω ∈ [0, π]
R(ω) ≥ 0 for ω ∈ [0, π]
after discretizing the frequency axis, we obtain an LP in r and α
FIR filter design
10–16
Multi-system magnitude equalization
problem
• we are given M frequency responses Gk : [0, π] → C
• design FIR equalizer H so that |Gk (ω)H(ω)| ≈ constant:
minimize
max
2
max |Gk (ω)H(ω)| − γk k=1,...,M ω∈[0,π]
subject to γk ≥ 1,
k = 1, . . . , M
LP formulation: use autocorrelation coefficients as variables
minimize α
2
subject to |Gk (ω)| R(ω) − γk ≤ α
R(ω) ≥ 0 for ω ∈ [0, π]
γk ≥ 1, k = 1, . . . , M
for ω ∈ [0, π],
k = 1, . . . , M
after discretizing the frequency axis, we obtain an LP in γk , r, α
FIR filter design
10–17
Example
• M = 2 systems, equalizer of order n = 25
2.5
2.5
2
2
|Gk (ω)H(ω)|2
|Gk (ω)|2
• unequalized and equalized frequency responses
1.5
1
0.5
0
0
0.5
1
1.5
ω
FIR filter design
2
2.5
3
1.5
1
0.5
0
0
0.5
1
1.5
2
2.5
3
ω
10–18