First Order Complex Adaptive FIR Notch Filter
First Order Complex Adaptive FIR Notch Filter
First Order Complex Adaptive FIR Notch Filter
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to extract the frequency ω0 of an incoming complex IV. RESULTS AND DISCUSSIONS
sinusoidal signal will be discussed. In this section, the computer simulations have been
conducted to demonstrate the effectiveness of the proposed
III. GRADIENT ALGORITHM FIR-ANF.
Now let us introduce the gradient based adaptive algorithm.
The error function of the proposed filter is given by A. Fixed Frequency Estimation
(6) In the first example, the proposed filter is used to
1 π
π ∫0
J (ωˆ0 ) = E [e(k )e * (k )] = H (e j ω )H * (e − j ω )S x (ω)d ω, estimate the frequency of a stationary complex sinusoidal
signal. In order to examine the convergence property of the
proposed technique and the compared algorithm, the initial
where "*" means complex conjugate and S x (ω) is the power values ωˆ0 (0) of the both algorithms are varied to be
spectral density (PSD) of x (k ) as given by ωˆ0 (0) = 0.2π and ωˆ0 (0) = 0.6π . The remaining parameters
used for this simulation are shown in the caption of Fig.1. It is
S x (ω) = A2 δ (ω − ω0 ) + σv2 , (7)
evident from Fig.1 that the convergence property of the
proposed system is better than that of the IIR-ANF, especially
where δ (⋅) is the delta function. By substituting (7) into (6) when the initial value is far from the optimum solution.
and integrating the result, it yields
B. Tracking of Random Frequency Complex Sinusoid
2 j ω0 2 2
J (ωˆ0 ) = A H (e ) + 2σ ,
v (8) A complex sinusoid with random frequency is generated
by using the system shown by Fig.2. The center frequency ω0
is set to be 0.1π. n(k ) is random signal with Gaussian
j ω0 2
where H (e ) = 2(1 − cos(ωˆ0 − ω0 )) . It is obvious that (8) distribution, zero-mean, and variance σn2 = 1 . The cutoff
has only one global minimum point because the noise term frequency of 5th order lowpass filter (LPF) is 0.05 rad/sample.
2σv2 is not a function of filter parameter ω̂0 . As a result, the y (k ) is a complex sinusoid with fixed amplitude A = 2
optimum solution ωopt of (8) can be obtained by and random frequency. Since y (k ) is natural complex, only
differentiating (8) with respect to ω̂0 and setting the result to the real component is considered. The results of tracking the
be zero, which is signal y (k ) using the FIR-ANF and IIR-ANF are shown in
Fig.3. As can be seen, the ability of tracking a complex
ωopt = ω0 . random frequency sinusoidal signal of the proposed FIR-ANF
(9)
is better than that of the IIR-ANF.
It is seen that the optimum solution of (1) is only a function of
an input frequency ω0 , implying that the gradient based 1
FIR-ANF
adaptive algorithm does not provide biased estimate of ω̂0 . 0.9 IIR-ANF
Frequency estimate (× π rad/sample)
0.6 ω(0)=0.6π
⎧ ∂e (k )e * (k )⎪
⎪ ⎫
ωˆ0 (k + 1) = ωˆ0 (k ) − μ Re ⎪
⎨
⎪
⎬
0.5
⎪
⎪ ∂ ω
ˆ (k ) ⎪
⎪
⎩ 0 ⎭ 0.4
⎧
⎪ * ∂e (k ) ⎫⎪ ω(0)=0.2 π
= ωˆ0 (k ) − μ Re ⎪
⎨e (k ) ⎪
⎬ 0.3
⎪
⎪
⎩ ∂ωˆ0 (k )⎪⎪
⎭ 0.2
{
= ωˆ0 (k ) − μ Re je * (k )x (k − 1)e j ωˆ0 (k ) , } (10) 0.1
0 2000 4000 6000 8000 10000
Iteration
⋅ stands for real part. The derivation of (10) is
where R{}
Fig. 1. Trajectories of ωˆ0 (k ) obtained by the FIR-ANF and the IIR-ANF for
obtained by assuming that e * (k ) is not a function of ωˆ0 (k ) .
A= 2 , ω0 = 0.8π , θ = π/3 , ωˆ0 (0) ∈ [0.2π, 0.6π ] , μ = 0.001 ,
In the next section, the performance of the proposed FIR-ANF
ρIIR-ANF = 0.95 , σv2 = 0.1 .
will be demonstrated and compared to the IIR-ANF [3].
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Fig. 2. Complex random frequency sinusoid generator.
3
Input
2.5
FIR-ANF
2 IIR-ANF
1.5
Re[y(k)]
1
0.5
0
-0.5
-1
-1.5
0 5 10 15 20 25 30
Iteration
V. CONCLUSION
The first order complex adaptive FIR notch filter is present
in this paper. The proposed filter provides good convergence
property and very simple. However, the distorted output
signal may be obtained due to the wide bandwidth. In some
applications that require the filter with fast convergence speed,
the proposed FIR-ANF is recommended. The theoretical
analysis of the proposed filter is the interesting topic and the
authors will propose in the future work.
REFERENCES
[1] Soo-Chang Pei, "Complex adaptive IIR notch filter algorithm and its
applications," IEEE Trans. Circuit and System-II: Analog and Digital
Signal Processing, vol. 41, no.2, Feb. 1994, pp. 158-163.
[2] H. Y. Jiang, S. Nishimura, and T. Hinamoto, "Steady-state analysis of
complex adaptive IIR notch filter and its application to QPSK
communication systems." IEEE proceeding ICSP 2000, pp. 551-554.
[3] S. Nishimura and H. Y. Jiang, "Gradient-Based comlex adaptive IIR
notch filters for frequency estimation," IEEE proceeding APCCAS, Nov.
1996, pp. 235-238..
[4] B. Widrow and S. D. Stearns, Adaptive signal processing, Englewood
Cliffs, NJ: Prentice-Hall, 1985.
[5] L. J. Griffiths, "Rapid measurement of digital instantaneous frequency."
IEEE Trans. Acoust. Speech and Signal Process., vol. ASSP-23, no. 2
Apr. 1975, pp. 207-222.
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