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1 Summary of the influence of the parameters of RLS and NLMS algorithms on the behaviour of proposed feedback cancellation design with shaped probe noise
2 Summary of the influence of the parameters of RLS and NLMS algorithms on the behaviour of the linear prediction-based AFC
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system without probe noise 3 Explanation of PESQ values
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Figure 1: Linear prediction-based AFC system with shaped probe noise signal
Figure 2: Linear prediction-based AFC system without probe noise [6]
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Figure 3: Magnitude response of the original feedback path
Figure 4: Frequency Response of the high-pass filter Hp(k) and the low-pass filter Lp(k)
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(a)
(b)
Figure 5: Simulation results for linear prediction-based feedback canceller (a) with shaped probe noise and (b) without probe noise for the frequency bin m = 7
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(a)
(b)
Figure 6: Simulation results for linear prediction-based feedback canceller (a) with shaped probe noise and (b) without probe noise for the frequency bin m = 12
Figure 7: Simulation result for linear prediction-based feedback canceller with shaped probe noise for the desired steady-state error of -0.015 dB per iterative instant n
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(a)
(b)
(c)
Figure 8: Spectrogram for (a) Speech input. (b) Loudspeaker output of the basic AFC with probe noise in [8, 18]. (c) Loudspeaker output of the linear prediction-based feedback canceller with shaped probe noise
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Figure 9: Simulation runs for linear prediction-based feedback canceller with shaped probe signal when speech signal is used as input m = 12
Table 1: Summary of the influence of the parameters of RLS and NLMS algorithms on the behaviour of proposed feedback cancellation design with shaped probe noise
Algorithm/parameters CR SSE TE
RLS Algorithm
λ IP (25) IP (28) DP (28)
Lf - DP (28)
-Sq(ω) - -
-Sp(ω) - IP (28)
Sx(ω) - DP (28)
-S^f (ω) - - DP (28)
|A (ω)|2 - IP (28)
-NLMS Algorithm
˜
µ (n) DP (24, 39) DP (42) IP (42)
Lf
(δ+Lfσ2p) - DP (42)
-Sp(ω)
(δ+Lfσ2p) DP (24, 39) - IP (42)
Sx(ω) - DP (42)
-S^
f (ω) - - DP (42)
|A (ω)|2 DP (24, 39) - IP (42)
(DP: Directly proportional, IP: Inversely proportional)
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Table 2: Summary of the influence of the parameters of RLS and NLMS algorithms on the behaviour of the linear prediction-based AFC system without probe noise
Algorithm/parameters CR SSE TE
RLS Algorithm
λ IP (25) IP (C.11) DP (C.11)
Lf - DP (C.11)
-Sq(ω) - IP (C.11)
-Sx(ω) - DP (C.11)
-S^f (ω) - - DP (C.11)
NLMS Algorithm
˜
µ (n) DP (24, D.6) DP (D.9) IP (D.9)
Lf
(δ+Lfσ2q) - DP (D.9)
-Sq(ω)
(δ+Lfσ2q) DP (24, D.6) - IP (D.9)
Sx(ω) - DP (D.9)
-S^
f (ω) - - DP (D.9)
(DP: Directly proportional, IP: Inversely proportional)
Table 3: Explanation of PESQ values
PESQ value Signal Quality Comments on impairment
1 Poor Very annoying
2 Bad Annoying
3 Fair Slightly annoying
4 Good Perceptible but not annoying
5 Excellent Imperceptible
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Appendix A.
Proof of Theorem 1
Proof. Using the shaped-probe-noise approach for feedback path estimation, the microphone signal in (5) can be rewritten as
y (n) = x (n) + F (k, n− 1) qr(n) . (A.1) Rewriting (6), we have
e (n) = y (n)− ˆF (k, n− 1) qr(n) . (A.2) Combining (10) with (A.1) and substituting into (A.2), we get
e (n) = x (n) + F (k, n− 1) qr(n)− ˆF (k, n− 1) qr(n)
= x (n)−
F (k, nˆ − 1) − F (k, n − 1)
q (n)−
F (k, nˆ − 1) − F (k, n − 1) rs(n) . (A.3)
Rewriting (4) in the frequency domain and substituting into (A.3), we have e (n) = x (n)− ˜F (k, n− 1) q (n) − ˜F (k, n− 1) rs(n) . (A.4) The RLS weight update equation for the feedback path estimation filter ˜F (k) in Fig. 1 is given as [11]
ˆf (n) = ˆf (n− 1) + D (n) rs(n) e (n) , (A.5) where
D (n) = R−1(n− 1)
λ + rsT(n) R−1(n− 1) rs(n), (A.6) and λ denotes the forgetting factor such that 0 < λ≤ 1. In (A.6), R (n) is the Lfˆ× Lfˆinput auto-correlation matrix which can be computed recursively as
R (n) = λR (n− 1) − rs(n) rsT(n) . (A.7) In order to compute the filter coefficients of the adaptive filter, a recursive expression for R−1(n) is required in terms of R−1(n− 1) [11]. Using matrix inversion lemma [10], R−1(n) can be obtained from (A.7) as
R−1(n) = λ−1h
R−1(n− 1) − D (n) rs(n) rTs (n) R−1(n− 1)i
. (A.8)
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In (A.8), R−1(0) = δ I, where I is an identity matrix and the regularization parameter δ is a positive real number such that R (n) does not become singular for a small value of n [11]. Substituting the combination of (3), (4) and (A.4) into (A.5), we get
˜f (n) =
I− D (n) rs(n) qT(n)− D (n) rs(n) rsT(n)˜f (n− 1)
+ D (n) rs(n) x (n)−^f (n) . (A.9) For further analysis, we consider the estimation-error covariance matrix
F (n) = E¯ h
˜f (n) ˜fT(n)i
. (A.10)
In order to compute ¯F (n) from (A.10),^f (n) is assumed to be uncorrelated with all the other terms in (A.9) assumption (v)
. Substituting (A.9) into (A.10), we have
F (n) = E¯ h
˜f (n− 1)˜fT(n− 1) − ˜f(n − 1)˜fT(n− 1) q (n) rTs (n) D (n)
− ˜f(n − 1)˜fT(n− 1) rs(n) rTs (n) D (n) + ˜f (n− 1) x (n) qT(n) D (n)
− D (n) rs(n) qT(n) ˜f (n− 1)˜fT(n− 1)
+ D (n) rs(n) qT(n) ˜f (n− 1)˜fT(n− 1) q (n) rTs (n) D (n)
− D (n) rs(n) qT(n) ˜f (n− 1) x (n) qT(n) D (n)
+ D (n) rs(n) qT(n) ˜f (n− 1)˜fT(n− 1) rs(n) rTs (n) D (n)
− D (n) rs(n) rTs (n) ˜f (n− 1)˜fT(n− 1)
+ D (n) rs(n) rTs (n) ˜f (n− 1)˜fT(n− 1) q (n) rTs (n) D (n)
− D (n) rs(n) rTs (n) ˜f (n− 1) x (n) qT(n) D (n)
+ D (n) rs(n) rTs (n) ˜f (n− 1)˜fT(n− 1) rs(n) rTs (n) D (n)
+ D (n) q (n) x (n) ˜fT(n− 1) − D (n) q (n) x (n)˜fT(n− 1) q (n) rTs (n) D (n) + D (n) q (n) x (n) x (n) qT(n) D (n)
− D (n) q (n) x (n)˜fT(n− 1) rs(n) rTs (n) D (n) +^f (n)^fT(n)i
. (A.11)
To represent (A.11) in a simplified form, let us represent a matrix ∆ (n) such that
∆ (n) = I− D (n) rs(n) qT(n)− D (n) rs(n) rsT(n) . (A.12)
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Collecting terms in (A.11), substituting (A.12) into (A.11) and using assumption (iv), we get
is the feedback path variation vector covariance matrix. It can be seen that (A.13) is a difference equation in terms of ¯F (n).
According to the direct averaging method [14, 24], the term rs(n) rsT(n− m) can be replaced by its sample average as
Rp(m) = lim by its sample average which can be expressed using assumption (i) as
Rrq(m) = lim
Combining (A.14) and (A.15) for m = 0, and substituting into (A.13), we get the approximate feedback estimation-error covariance matrix as
F¯a(n) = ¯Fa(n− 1) − D (n) Rp(0) ¯Fa(n− 1) − ¯Fa(n− 1) Rp(0) D (n) +^F (n)
Further, there is a need to evaluate the terms Eh
D (n) rs(n) x (n) ˜fT(n− 1)i and Eh
˜f (n− 1) x (n) rTs (n) D (n)i
in (A.16) to obtain the final expression for
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F¯a(n). The evaluation of both the terms is presented in Appendix B in detail.
Finally, we have
DFT matrix such that Γ is a complex quantity. Then, we can say that Γ diagonalizes a Toeplitz matrix asymptotically as Lf → ∞ [14, 25]. Therefore, diagonalizing ¯Fa(n) using Γ, we obtain
ˆ
χ (n) = Γ ¯Fa(n) ΓH. (A.18) Here, ˆχ (n) approaches a diagonal matrix as Lf → ∞ and has diagonal elements expressed by ˆζ (ω, n). Substituting (A.17) into (A.18), we have
ˆ having diagonal elements Sp(ω), which is the PSD of rs(n) and is defined as the Fourier transform of the auto-correlation function of rs(n). We can express Sp(ω) as
Sp(ω) =|A (ω)|2Sr(ω) , (A.20) where A (ω) is the frequency response A (k) and Sr(ω) is the PSD of unshaped probe noise r (n). The PTF-approximation ˆζ (ω, n) forms the diagonal elements of the matrix ˆχ (n). Therefore, replacing the terms ˆχ (n), Γ ¯Fa(n− 1) ΓH,
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LfΓ Rp(0) ΓH and ΓF (n) Γ^ H in (A.19) with their respective diagonal ele-ments, the expression for ˆζ (ω, n) can be written as
ζ (ω, n) = (1ˆ − 2 d (ω, n) Sp(ω)) ˆζ (ω, n− 1) + Lf d2(ω, n) Sp(ω) Sx(ω) + S^f(ω)
=
1− 2 d (ω, n) |A (ω)|2Sr(ω)
ζ (ω, nˆ − 1)
+ Lf d2(ω, n)|A (ω)|2Sr(ω) Sx(ω) + S^f(ω) , (A.21) where d (ω, n) is the diagonal entry of L1fΓ D (n) ΓH (see Appendix B) ex-pressed as
d (ω, n) = 1− λ
Sp(ω)= 1− λ
|A (ω)|2Sr(ω) , (A.22) and Sx(ω) is the PSD of the incoming desired acoustic signal x (n) and is defined
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as the Fourier transform of rx(m). Substituting (A.22) into (A.21), the PTF of (15) can be written approximately as (16).
Appendix B.
Evaluation of the term Eh
D (n) rs(n) x (n) ˜fT(n− 1)i
We substitute (A.12) into (A.9) and rewrite the expression for ˜f (n) as
˜f (n) = ∆ (n) ˜f (n− 1) + D (n) rs(n) x (n)−^f (n) . (B.1)
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Simplifying the expanded expression in (B.2), we get
˜f (n) =
In (B.3), it is assumed that the iteration for the adaptation process begins at n = 0 and also
, using assumption (iv) and considering
˜f (0) = ˆf (0)− f (0) to be uncorrelated with rs(n), we have
where t = m−n. Since only the steady-state behaviour of the PTF is influenced by (B.4), we can consider a large value of n for n− 1 ≥ t0and rewrite (B.4) as
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In the above equation, the term nQ−1
l=n+1+t
∆ (l)
!
can be simplified for t =−t0, where the factors of ∆ (l) can be expressed following (A.12) as
nY−1 Expanding (B.6), collecting terms and substituting into (B.5), it can be observed that apart from I, all other terms result in higher-order terms consisting of D (n) DT(n + t) rs(n) rsT(n + t) in (B.5) and can be neglected. Hence, we can rewrite (B.6) as
nY−1 l=n−t0+1
∆ (l)≈ I. (B.7)
Substituting (B.7) into (B.5), we have
Eh Here, it can be assumed that D (n) is varying slowly over time such that the variation in D (n) is slower than the decay of rx(t). Then, we have
D (n) DT(n + t) = D (n) DT(n) . (B.9) Then, (B.8) can be expressed with the help of (B.9) as
Eh
Evaluation of the term Eh
˜f (n− 1) x (n) rsT(n) D (n)i
Substituting (B.3) into (A.16) similarly as above for the term Eh
˜f (n− 1) x (n) rsT(n) D (n)i
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Substituting (B.10) and (B.11) into (A.16), we get the final expression for ¯Fa(n) in (A.17).
Evaluation of diagonal elements of L1fΓ D (n) ΓH
To obtain the expression for ˆζ (ω, n), the time-average shaped probe noise signal correlation matrix R (n) for the RLS algorithm in Fig. 2 can be expressed after the addition of the regularization term in the cost function as
R (n) = Xn i=1
λn−irs(i) rTs (i) + δ λn I (B.12)
where R (n) is a diagonal matrix and I is the Lf × Lf identity matrix [11].
Asymptotically, for a large value of n and as λto1, the matrix Pn i=1
λn−irs(i) rTs (i) tends to have large values [14]. As a result, R−1(n) tends to contain small entries and D (n) in (A.6) can be expressed as
D (n)≈ R−1(n) . (B.13)
In (B.13), it is assumed that R−1(n) has converged2 such that R−1(n) = R−1(n− 1). Thus, R−1(n) can be written by substituting (B.13) into (A.8) as
D (n)≈ λ−1
D (n)− D (n) rs(n) rTs (n) D (n)
. (B.14)
Upon convergence, D (n) in (B.14) becomes a Toeplitz matrix. Following (A.18), D (n) can also be diagonalized as L1fΓ D (n) ΓH. Then, the diagonal entries of
1
LfΓ D (n) ΓH can be written as d (ω, n)≈ λ−1
d (ω, n)− d2(ω, n) Sp(ω)
. (B.15)
The solution to the second-order difference equation of (B.15) can be written as
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(A.22).
2It is not necessarily implied that n→ ∞ for R−1(n− 1) to converge.
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Appendix C.
Proof of Lemma 1
Proof. The PTF expression in (16) is a first-order difference equation in terms of ˆζ (ω, n) and it can be expressed as
η (k) = β
1− α k−1 (C.1)
where α, β are real numbers. The coefficient α determines the convergence rate of the feedback canceller and can be written in terms of the forgetting factor as3
α = 2λ− 1, (C.2)
whereas the coefficient β in (C.1) can be expressed as
β = Lf(1− λ)2
Sp(ω) Sx(ω) + S^f(ω) . (C.3) As defined in (17), the convergence rate in this case is determined as the decay of ˆζ (ω, n), and can be computed in dB per iterative instant n as
CR = d
dn10 log10(β|α|n) , (C.4) where β|α|nis the impulse response function of η (k). Further solving (C.4), we have
CR = 10 d dn
ln (α)n ln 10
= 10
ln α ln 10
d dn(n)
= 10 log10(α) . (C.5)
Substituting (C.2) in (C.5), we have
CR = 10 log10(2λ− 1) . (C.6)
3In (C.2), we consider values of λ such that λ→ 1. This allows for slow convergence of the RLS algorithm due to which a good approximation of ¯F (n) by ¯Fa(n) can be obtained in (A.17).
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It can be observed from the above equation that the convergence rate is neither dependent on the frequency nor on the incoming acoustic signal x (n), but only dependent on λ. In fact, it decreases with increasing λ and the convergence becomes slower. From (C.1), it can be seen that η (k) is stable when the poles lie inside the unit circle, i.e. |α| < 1. Thus, it can be said that the linear prediction-based feedback cancellation system with shaped probe noise (see Fig.
1) is stable for
0 < λ < 1. (C.7)
Using (18), the steady-state behaviour of the aforementioned system can be expressed as
ζ (ω,ˆ ∞) = limn
→∞
ζ (ω, n) .ˆ (C.8)
Substituting (16) into (C.8), we get ζ (ω,ˆ ∞) = Lf In the above equation, the first term denotes the steady-state error, i.e. the minimum possible value of ˆζ (ω, n) in the steady state, and the second term
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denotes the tracking error due to changes in the feedback path. It can be observed in (C.9) that Lf and Sx(ω) are directly proportional to the steady-state error, while the steady-steady-state error decreases with increase in λ and Sp(ω).
The tracking error is independent of Sp(ω) but its value increases with increase in the value of λ for the feedback path variations when S^f (ω) > 0. This implies
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a slower tracking. Infact, the convergence rate in (C.6) is also independent of Sp(ω), but dependent on λ. Hence, it can be said that the value of ˆζ (ω,∞) in (C.9) is a trade-off between the steady-state behaviour for a time-invariant feedback path and the tracking ability for a time-varying feedback path.
Proof of Corollary 1
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Proof. Using (C.6), a required rate of convergence can be obtained by choosing the value of λ as (20), where the convergence rate is in dB per iterative instant
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n. For a time-invariant feedback path, ignoring the tracking error in (C.9) and rearranging the terms, we have the expression in (21) that can be used to select a value of λ for obtaining a required steady-state error.
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Appendix D.
Proof of Corollary 2
Proof. Consider the AFC system of Fig. 1. The weight update equation for NLMS algorithm is given as [11]
ˆf (n) = ˆf (n− 1) + µ (n) rs(n) e (n) , (D.1)
where e (n) is expressed as (A.2) and µ (n) is written as µ (n) = µ (n)˜
δ + rsT(n) rs(n), (D.2) where ˜µ (n) is the normalized step size parameter and δ is a positive real number.
The term rsT(n) rs(n) in (D.2) can be expressed as
rsT(n) rs(n) = Lfσˆp2, (D.3) where ˆσp2is an estimate of the shaped probe noise signal variance ˆσp2. However for ˜µ (n) → 0, ˆσp2 can be replaced by σp2 in (D.3) [17]. This is because, for a small value of step size, the NLMS algorithm can have a low-pass influence on the loudspeaker signal [11]. Thus, we can rewrite (D.2) as
µ (n) = µ (n)˜
δ + Lf σp2, (D.4)
where σp2 is the variance of rs(n). Following the proof of Theorem 1, the ex-pression for approximated feedback estimation-error covariance matrix can be written as
F¯a(n) = ¯Fa(n− 1) − µ (n) Rp(0) ¯Fa(n− 1) − µ (n) ¯Fa(n− 1) Rp(0) +^F (n)
+ µ2(n)
t0
X
t=−t0
Rp(t) rx(t). (D.5)
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Diagonalizing ¯Fa(n) in (D.5) using DFT matrix Γ, we have ˆ
χ (n) = Γ ¯Fa(n− 1) ΓH+ Γ^F (n) ΓH− µ (n) 1 Lf
Γ Rp(0) ΓHΓ ¯Fa(n− 1) ΓH
− µ (n) 1 Lf
Γ ¯Fa(n− 1) ΓHΓ Rp(0) ΓH+ µ2(n)
t0
X
t=−t0
Γ Rp(t) ΓHrx(t).
(D.6) The diagonal elements of ˆχ (n) are defined as the PTF approximation as
ζ (ω, n) = (1ˆ − 2µ (n) Sp(ω)) ˆζ (ω, n− 1) + Lf µ2(n) Sp(ω) Sx(ω) + S^f (ω) . (D.7) Substituting (D.4) into (D.7), we have the approximate PTF expression for the NLMS adaptive algorithm-based feedback cancellation system with shaped probe noise as (22).
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Proof of Lemma 2
Proof. Equation (22) is a first-order difference equation in ˆζ (ω, n) and can be written following (C.1), where α is frequency dependent as
α = 1− 2 µ (n)˜
δ + Lfσp2Sp(ω)
= 1− 2 µ (n)˜
δ + Lfσp2|A (ω)|2Sr(ω) , (D.8) whereas β can be written as
β = Lf µ˜2(n)
δ + Lfσp22Sp(ω) Sx(ω) + S^f(ω) (D.9) From (C.1), it can be seen that the system is stable|α| < 1 and therefore, the range of ˜µ (n) which ensures stability of the system can be given as
0 < ˜µ (n) < δ + Lfσp2
maxωSp(ω). (D.10)
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We can express the steady-state behaviour of the system in Fig. 1 for the NLMS algorithm by substituting (22) into (C.8) as
ζ (ω,ˆ ∞) = limn
→∞Lf µ (n)˜
2 δ + Lfσp2 Sx(ω) + lim
n→∞ δ + Lfσ2p S^f(ω) 2˜µ (n) Sp(ω)
= lim
n→∞Lf µ (n)˜
2 δ + Lfσ2p Sx(ω) + lim
n→∞ δ + Lfσp2 S^f(ω) 2˜µ (n)|A (ω)|2Sr(ω).
(D.11) In the above equation, the first term denotes the steady-state error and the second term denotes the tracking error due to changes in the feedback path.
It can be seen from (D.11) that Lf and ˜µ (n) are directly proportional to the steady-state error, while the steady-state error decreases with increasing σp2.
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The tracking error is inversely proportional to the ratio Sp(ω) / δ + Lf σ2p , but increases with increasing Lf. It can be seen from (C.5) and (D.8) that the rate of convergence is dependent upon the ratio Sp(ω) / δ + Lf σ2p
and also decreases with increasing Lf. Also, the rate of convergence decreases and the tracking error increases owing to a small value of|A (ω)|2.
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Proof of Corollary 3
Proof. Substituting (D.8) into (C.5) and rearranging, a required given rate of convergence can be obtained for the step size ˜µ (n) as (23). Similarly from (D.11), ignoring the tracking-error term and rearranging, the expression for determining the value of ˜µ (n) to achieve a required value of ˆζ (ω,∞) is obtained
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as (24).
Appendix E.
Proof of Corollary 4
Proof. The microphone signal in (5) can be rewritten as
y (n) = x (n) + F (k, n− 1) q (n) . (E.1)
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We can rewrite the expression for e (n) in (6) as e (n) = x (n)−
F (k, nˆ − 1) − F (k, n − 1) q (n)
= x (n)− ˜F (k, n− 1) q (n) . (E.2) The RLS adaptive weight update equation for the feedback path estimation filter ˆF (k) is given as [11] The approximate feedback estimation-error covariance matrix can be written as F¯a(n) = ¯Fa(n− 1) − D (n) Rq(0) ¯Fa(n− 1) − ¯Fa(n− 1) Rq(0) D (n) +F (n)^ where m is the time delay, using direct-averaging method. After further evaluat-ing the terms Eh
D (n) q (n) x (n) ˜fT(n− 1)i and Eh
˜f (n− 1) x (n) qT(n) D (n)i in (E.6) following Appendix B to obtain the final expression for ¯Fa(n), we get F¯a(n) = ¯Fa(n− 1) − D (n) Rq(0) ¯Fa(n− 1) − ¯Fa(n− 1) Rq(0) D (n) +F (n)^
Following the proof of Theorem 1, we can diagonalize (E.8) using an Lf× Lf
DFT matrix Γ, which consists of complex entries, as ζ (ω, n) =ˆ
1− 2 d (ω, n) Sq(ω)
ζ (ω, nˆ − 1) + Lfd2(ω, n) Sq(ω) Sx(ω) + S^f(ω) , (E.9)
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where d (ω, n) is the diagonal entry of L1f Γ D (n) ΓH and is expressed as
d (ω, n) = 1− λ
Sq(ω) (E.10)
and Sx(ω) is the PSD of the incoming desired acoustic signal x (n) and is defined as the Fourier transform of rx(t). Then, the PTF of (15) for the linear
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prediction-based feedback cancellation design without probe noise (shown in Fig. 2) can be written as (28).
Proof of Lemma 3
Proof. Following the proof of Lemma 1, the steady-state behaviour of the system in Fig. 2 can be expressed using (28) as
ζ (ω,ˆ ∞) = Lf 1− λ
2 Sq(ω)Sx(ω) + S^f(ω)
2 (1− λ). (E.11) The expression for ˆζ (ω,∞) in (E.11) is the sum of two terms. The first term denotes the minimum possible value of ˆζ (ω, n) in the steady state and the
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second term denotes the tracking error due to changes in the feedback path. In (E.11), it is seen that Lf and Sx(ω) are directly proportional to the steady-state error, while the steady-steady-state error decreases with increase in Sq(ω) and λ. The tracking error is unaffected by Sq(ω), but increases with increasing λ for the feedback path variations when S^f(ω) > 0 leading to a slower tracking.
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Following the proof of Lemma 1, it can be concluded that the convergence rate is also unaffected by Sq(ω) , but depends on the value of λ as shown in (C.6). So, it can be said that the overall value of ˆζ (ω,∞) is a trade-off between the steady-state behaviour for a time-invariant feedback path and tracking behaviour for a time-varying feedback path.
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Proof of Corollary 5
Proof. The expression for selecting a forgetting factor value for RLS adaptive algorithm to obtain a required given rate of convergence for the AFC system without probe noise can be obtained following Corollary 1. Also, following Corollary 1 in ignoring the tracking-error term in (E.11) and rearranging, we
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obtain the expression for choosing the value of λ to obtain a desired steady-state error ˆζ (ω,∞) as (29).
Appendix F.
Proof of Corollary 6
Proof. The weight update equation for the NLMS algorithm-based adaptive filter in Fig. 2 is given by [11]
ˆf (n) = ˆf (n− 1) + µ (n) q (n) e (n) , (F.1)
where
µ (n) = µ (n)˜
δ + Lfσq2. (F.2)
The expression for ¯Fa(n) can be written as
F¯a(n) = ¯Fa(n− 1) − µ (n) Rq(0) ¯Fa(n− 1) − µ (n) ¯Fa(n− 1) Rq(0) +^F (n)
Diagonalizing ¯Fa(n) in (F.3) using the DFT matrix Γ, we have Ψ (n) = Γ¯ˆ Fa(n− 1) ΓH+ Γ^F (n) ΓH− µ (n) 1 The diagonal elements of ˆΨ (n) are defined as the PTF of (15), and expressed as
ζ (ω, n) =ˆ
1− 2µ (n) Sq(ω)
ζ (ω, nˆ − 1) + Lfµ2(n) Sq(ω) Sx(ω) + S^f (ω) . (F.5) Substituting the expression for µ (n) from (F.2) into (F.5), we have the PTF
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of (15) for the NLMS adaptive-algorithm-based feedback cancellation system without probe noise as (30).
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Proof of Lemma 4
Proof. Following the proof of Lemma 2, we have from Corollary 6 α = 1− 2 µ (n)˜
The steady-state behaviour of the feedback cancellation system in Fig. 2 can be expressed as denotes the minimum possible value of ˆζ (ω, n) in the steady state and the second
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term denotes the tracking error due to changes in the feedback path. It can be seen from (F.9) that both Lf and the step size ˜µ (n) are directly proportional to the steady-state error, while the steady-state error decreases with increase in σq2. Owing to the presence of σ2q, the tracking error is inversely proportional to the ratio Sq(ω) / δ + Lfσq2
instead of only the value of Sq(ω). Also, the
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tracking error is increased with an increase in Lf. Similarly, it can be seen from (C.5) and (F.6) that the convergence rate is dependent upon the ratio Sq(ω) / δ + Lfσq2
and decreases with increase in Lf. Proof of Corollary 7
Proof. Following the proof of Corollary 3, the expression for ˜µ (n) for a given
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rate of convergence can be obtained from (F.9) as (31), where the convergence rate is in dB per iterative instant n. Similarly, using (F.9) and ignoring the tracking error, we have the expression for determining the value of ˜µ (n) for obtaining a desired steady-state error ˆζ (ω,∞) as (32).