GB2334392A

GB2334392A - Tracking a centre frequency

Info

Publication number: GB2334392A
Application number: GB9827410A
Authority: GB
Inventors: Suk Phil Lee
Original assignee: Daewoo Electronics Co Ltd
Current assignee: WiniaDaewoo Co Ltd
Priority date: 1998-02-13
Filing date: 1998-12-11
Publication date: 1999-08-18
Also published as: KR19990069891A; GB9827410D0

Abstract

A method capable of tracking a centre frequency of a random signal by using an adaptive recursive algorithm comprises the steps of: sampling the random signal to provide a sequence of sample signals [x(n)]; filtering the sequence of sample signals [x(n)] to generate an output signal y(n); calculating a gradient α(n) of the output signal y(n); generating an updated weight vector W(n) based on a previous weight vector, a previous output signal and a previous gradient, storing the updated weight vector in a memory; calculating, responsive to the updated weight vector W(n), the centre frequency f c (n); updating a new weight vector W(n+1) for calculating a next centre frequency f c (n+1); and repeating the steps for the rest of the sequence of sample signals.

Description

METHOD AND APPARATUS CAPABLE OF TRACKING A CENTER FREQUENCY BY USING AN ADAPTIVE RECURSIVE ALGORITHM The present invention relates to a method and apparatus capable of tracking a center frequency of a random signal; and more particularly, to a method and apparatus capable of tracking a center frequency of a random signal by using an adaptive recursive algorithm.

Conventionally, signals are classified into a deterministic and a random signals. The deterministic signal has a finite energy and a periodic characteristic as a function of time, whereas the random signal may be represented as a function of one or more independent variables, e.g., probability distributions. For example, a voice or a noise is a random signal having an irregular characteristic as a function of time. Such random signal may be analyzed by employing, e.g., an adaptive filtering algorithm, wherein the adaptive filtering algorithm starts with a predetermined set of initial conditions to converge to an optimum state.

The adaptive filtering algorithm is well known and its typical example are a Least Mean Square (LMS) and a Recursive Least Square (RLS) algorithm. The LMS algorithm is a linear adaptive filtering algorithm that includes two basic processes: The first is a filtering process that involves (a) computing the output of a transversal filter produced by a set of tap inputs and (b) generating an estimation error by comparing this output to a desired response. The second is an adaptive process which involves the automatic adjustment of the tap weights of the transversal filter according to an estimation error. Thus, the combination of these two process working together constitutes a feedback loop around the LMS algorithm.

The RLS algorithm may be viewed as a special case of the Kalman filter. The RLS algorithm utilizes information contained in input data, extending back to the instant when the algorithm is initiated. The resulting rate of convergence is typically an order of magnitude faster than the simple LMS algorithm. See, e.g., Simon Haykin, "Adaptive filter theory", Prentice Hall, 1996.

Referring to Fig. 1, there is shown a schematic block diagram of a conventional center frequency tracking apparatus.

As shown in Fig. 1, the prior art apparatus comprises an adaptive filtering block 110, an FFT (Fast Fourier Transform) block 120, a spectrum analysis block 130 and a tracking block 140. The adaptive filtering block 110 receives an input signal, e.g., a random signal, and filters the received input signal by using an adaptive filtering algorithm, wherein filter coefficients are recursively updated for the opt mation thereof. Thereafter, the FFT block 12C converts the filtered result from the adaptive filtering block 110 into a power spectrum density in a frequency domain.

The spectrum analysis block 130 analyses the power spectrum density inputted from the FFT block 120 to provide the tracking block 140 with the analysis result of the power spectrum density. And, the tracking block i40 tracks a center frequency fc of the input signal based on the analysis result fed thereto from the spectrum analysis block 130, wherein the center frequency f, fc can be defined as a frequency having a largest power spectrum density.

In other words, the conventional center frequency tracking apparatus updates the filter coefficients of an input signal in order to optimize the filter coefficients by using the adaptive filtering algorithm; converts the input signal in a time domain into a power spectrum density in a frequency domain; and analyses the power spectrum density to track a center frequency fc of the input signal based on the analysis result.

However, the adaptive filtering algorithm of the prior art has a large variation in the filter coefficients as a function of time to thereby cause a degraded stability thereof. It is also difficult to implement a real-time center frequency tracking system due to the computational burden involved in performing the FFT and the spectral analysis for the input signal.

It is, therefore, an object of the present invention to provide a method and apparatus capable of tracking a center frequency of a random signal by using an adaptive recursive algorithm without employing the FFT and the spectral analysis.

In accordance with one aspect of the invention, there is provided a method for tracking a center frequency fc(n) of a random signal by using a recursive algorithm, comprising the steps of: (a) sampling the random signal to provide a sequence of sample signals [x(n)]; (b) filtering the sequence of sample signals [x(n)] to generate an output signal y(n), n representing time sequence; (c) calculating a gradient a(n) of the output signal y(n); (d) generating an updated weight vector W(n) based on a previous weight vector W(n-l), a previous output signal y(n-1) and a previous gradient a(n), wherein W(n-1), y(n-1) and a(n) are stored in a memory; (d) storing the updated weight vector W(n) in the memory; (e) calculating, responsive to the updated weight vector W(n), the center frequency fc(n); (f) updating a new weight vector W(n+1) for calculating a next center frequency fc(n+l); and (g) repeating the steps (a) to (f) for the rest of the sequence of sample signals [x(n)j.

In accordance with another aspect of the invention, there is provided an apparatus for tracking a center frequency fc(n) of a random signal, comprising: a sampling block for sampling the random signal to provide a sequence of sample signals [x(n)] ; a filtering block for filtering the sequence of sample signals [x(n)] to provide an output signal y(n); a weight update block for generating a weight vector W(n+l) based on the output signal y(n), a gradient a(n) thereof and a previous weight vector W(n); a gradient providing block for calculating the gradient a(n) of the output signal y(n); a tracking block for computing, responsive to the previous weight vector W(n), the center frequency fc(n) of the random signal; and a memory for storing the sequence of sample signals [x(n)], the output signal y(n) and the gradient a(n) of the output signal.

The above and other objects and features of the present invention will become apparent from the following description of preferred embodiments given in conjunction with the accompanying drawings, in which: Fig. 1 shows a schematic block diagram of a conventional center frequency tracking apparatus; and Fig. 2 represents a schematic block diagram of a center frequency tracking apparatus in accordance with the present invention.

A preferred embodiment in accordance with the present invention will be described with reference to Fig. 2.

Referring to Fig. 2, there is shown a schematic block diagram of a center frequency tracking apparatus 200 in accordance with the present invention, which comprises a sampling block 201; a first memory 202; an adaptive band-pass filtering circuit 203 including filtering block 204 and a second memory 205; an update circuit 206 including therein a weight update block 207, a third memory 208 and a gradient providing block 209; and a tracking block 210.

The sampling block 201 receives a random input signal and samples same with a time interval T to provide a sequence of sample signals [x(n)] to the first memory 202 and the filtering block 204, n representing an index of a sample signal. The sample signals are delayed at the first memory 202 by a predetermined time interval to provide a set of delayed sample signals including, e.g., x(n-2) and x(n-4) to the filtering block 204.

The adaptive band-pass filtering circuit 203 including the filtering block 204 and the second memory 205 may be implemented by, e.g., an IIR (Infinite Impulse Response) filter having a transfer function defined as: a0+a2z 2+a4z4 H(z) = 1+bWz-1+( b2W+b'2)z-2+b3Wz-3+b4z-4 (Eq. 1) wherein a0=1/(k&+2+1); a2=-2a0; a4=a0; b1=-2k(2k+2)a0; b2=4k2aO; b' 2=2 (k2-l)a0; b3=2k(-2k+21/2)aO; b4=(k2-21/2k+l)aO; k=cot[(f2-f)T; W is a weight vector with its initial value W0=cos[#(f2+f1)T]/cos[#(f2-f1)T]; f1 and f2 are a lower and an upper cutoff frequencies, respectively; and T is the sampling interval.

An output signal at time n, y(n), can be obtained from the transfer function in Eq. 1 as: y(n)=aOx(n)+a2x(n-2)+a4x(n-4) -[b1W(n)y(n-1)+(b2W(n)+b'2)y(n-2) +b3W(n)y(n-3)+b4y(n-4)] (Eq. 2) wherein x(i) represents a sample signal at time i; y(i), the output signal at time i; and aO to b4 are same as defined Eq.

1.

The output signal y(n) is calculated at the filtering block 204 and provided to the second memory 205 and the weight update block 207. The second memory stores output signals from the filtering block 204 by a preset time interval and provides delayed output signals, e.g., y(n-l), y(n-2), y(n-3), y(n-4) to the filtering block 204 and y(n-1) , y(n-2), y(n-3) to the gradient block 209. The weight vector W(n) used in the calculation of the output signal y(n) defined in Eq. 2 is provided from the weight update block 207.

A center frequency at time n, fc(n), can be derived, from the initial weight vector w0=cos[#(fz+f1)T]/cos[#(f2-f1)T] by setting fc=(f1+f2)/2 as: fc(n) = cos1 [W(n)cos(nBT)/2rT] (Eq. 3) wherein B is a bandwidth and B=f2-f1.

Thereafter, the update circuit 206 receives the output signal y(n) from filtering block 204 and delayed output signals, e.g., y(n-l), y(n-2) and y(n-3) from the second memory 205 to thereby update the weight vector W(n) by maximizing a cost function J(n) of y(n) based on a recursive algorithm. The cost function J(n) may be defined by an expectation value of the square of the output signal y(n) as follows: J(n)=E[y(n)] (Eg 4) In order to maximize the cost function J(n), the weight vector is updated as follows: W(n+l) =W(n) -1/2 (-V(J(n))) (Eq. 5) wherein, W(n+l) is a weight vector at time n+l; W(n) is a weight vector at time n, is a positive real-valued constant, e.g., 0.02; the factor 1/2 is merely used for the purpose of canceling the factor 2 that appears in the formula for VJ(n); and #(J(n))=#y(n)/#W(n)=2y(n) (#y(n)/#dW(n)).

VJ(n) is a gradient vector at time n. If the gradient is positive, W(n+1) is greater than w(n); and if otherwise, W(n+l) is smaller than W(n). By applying Eq. 3 in updating a weight vector, the cost function may approach to a maximum value thereof. In contrast to the present invention, a mean squared error function E(n) is minimized in the steepest descent algorithm well known in the art by defining the weight vector as W(n+l)=W(n)+l/2ji(-(E(n))) Eq. 5 can be represented by substituting #y(n)/#W(n) with a(n) as follows: W(n+l) =W(n) n) a (n) Eq. 6) wherein a(n) is a gradient of the output signal y(n).

The weight update block 207 calculates the updated weight vector W(n+l) based on the Eq. 6, wherein W(n) is the previously weight vector calculated therein and y(n) and α(n) are provided from the filtering block 204 and the gradient providing block 209, respectively. The gradient a(n) is provided from the gradient providing block 209 to the third memory 208 and the weight update block 207. The third memory 208 temporarily stores predetermined set of previous gradients, e.g., α(n-1), α(n-2), a(n-3) and a(n-4), to provide them to the gradient providing block 209.

From Eq. 2, the gradient a(n) may be expressed as: a (n) =ay (n) /W (n) = -[b1Y(n-1)+2b2W(n)y(n-2)+b3y(n-3) +b1W(n) a(n-l) + (b2W2(n) +b'2)α(n-2) +b3W(n)α(n-3)+b4α(n-4)] (Eq. 7) The gradient providing block 209 calculates the gradient a(n) based on Eq. 7, wherein W(n) is provided from the weight update block 207; the delayed output signals y(n-l) to y(n-3) are provided from the second memory 205; and the set of previous gradients a(n-l) to a(n-4) are provided from the third memory 208.

The tracking block 210, responsive to the weight vector W(n) from the weight update block 207, calculates the center frequency fc(n) based on Eq. 3.

It should be noted that the factors aO, az, a4, b1, b2, b' b3, b4, fl, f2, T and W0 are all predetermined values and provided to corresponding function blocks 204, 207, 209 and 210 for the calculation of y(n), W(n), a(n) and fc(n).

While the present invention has been described with respect to certain preferred embodlments only, other modifications and variations may be made without departing from the spirit and scope of the present invention as set forth in the following claims.

Claims

Claims 1. A method for tracking a center frequency fc(n) of a random signal by using a recursive algorithm, comprising the steps of: (a) sampling the random signal to provide a sequence of sample signals [x(n)j; (b) filtering the sequence of sample signals [x(n)] to generate an output signal y(n), n representing time sequence; (c) calculating a gradient ot(n) of the output signal y(n); (d) generating an updated weight vector W(n) based on a previous weight vector W(n-l), a previous output signal y(n-l) and a previous gradient a(n), wherein W(n-l), y(n-l) and cu(n) are stored in a memory; (d) storing the updated weight vector W(n) in the memory; (e) calculating, responsive to the updated weight vector W(n), the center frequency f,(n); (f) updating a new weight vector W(n+1) for-calculating a next center frequency fc(n+l); and (g) repeating the steps (a) to (f) for the rest of the sequence of sample signals [x(n)].
2. The method of claim 1, wherein the new weight vector W(n+l) is defined as: W(n+1)=W(n)-1/2 (-V(J(n))) wherein p is a positive real-valued constant and J(n) is a cost function.
3. The method of claim 1 or 2, wherein the cost function J(n) is defined as: J(n) = E[y2(n)] wherein E denotes a statistical expectation operator.
The method of any of claims 1 to 3, wherein the center frequency fc is calculated as follows: fc = cos1Wcos(nBT)/2,tT], wherein B is a bandwidth; the weight vector W is defined as: W=cost(f2+f1)T]/cos[ir(f2-f1)T] ; and the bandwidth B is defined as: B=f2-f1, wherein f2 is an upper cutoff frequency and f1 is a lower cutoff frequency.
5. An apparatus for tracking a center frequency fc(n) of a random signal, comprising: means for sampling the random signal to provide a sequence of sample signals [x(n)]; means for filtering the sequence of sample signals [x(n)] to provide an output signal y(n); means for updating a weight vector W(n+l) based on the output signal y(n), a gradient a(n) thereof and a previous weight vector W(n); means for computing, responsive to the previous weight vector W(n), the center frequency fc(n) of the random signal; and a memory for storing the sequence of sample signals [x(n)], the output signal y(n) and the gradient a(n) of the output signal.
6. The apparatus of claim 5, wherein the updating means includes means for calculating the gradient cu(n) of the output signal y(n).
7. The apparatus of claim 5 or 6, wherein the weight vector W(n+l) of the recursive algorithm is defined as: W(n+1)=W(n)-1/2 p(-V(J(n))) wherein p is a positive real-valued constant and J(n) is a cost function.
8. The apparatus of any one of claims 5 to 7, wherein the cost function J(n) is defined as: J(n) = wherein E denotes a statistical expectation operator.
9 The apparatus of any one of claims 5 to 8, wherein the center frequency f is calculated as follows: fc = cos~1[Wcos(nBT)/2rT], wherein B is a bandwidth; the weight vector W is defined as: W=cos[#(f2+f1)T]/cos[#(f2-f1)T]; and the bandwidth B is defined as: B=f2-fl, wherein fz is an upper cutoff frequency and f1 is a lower cutoff frequency.
10. A method for tracking a center frequency substantially as herein described with reference to or as shown in Figure 2 of accompanying drawings.
11. An apparatus for tracking a center frequency constructed and arranged substantially as herein described with reference to or as shown in Figure 2 of accompanying drawings.