Adaptive Fourier analysis based on the Walsh Hadamard transformation

1. Structure and Frequency Adaptation of the Recursive Adaptive Fourier Analyzer Based on the Walsh-Hadamard Transformation 1.3 1.3 1.2 1.2 1.1 1.1 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 1.8 0.3 0.3 1.7 0.2 0.2 0.1 0.1 1.6 0 0 0 32 64 96 128 160 192 224 256 288 320 352 384 416 448 480 512 544 0 32 64 96 128 160 192 224 256 288 320 352 384 416 448 480 512 544 1.5 1.4 1.3 1.2 1.1 1 0.9 544 0 32 64 96 128 160 192 224 256 288 320 352 384 416 448 480 512 1.5 1 0.5 0 -0.5 -1 -1.5 f (n) Memory for the -1 z 1 -2 Frequency 0 32 64 96 128 160 192 224 256 288 320 352 384 416 448 480 512 544 - j j ( n ) weight factors e 1 determination j j ( n ) e 1 Frequency - ( N 1 ) / 2 å Linear - 2 Re( Y ( n )) determination WHT m combination in = m 0 (R)WHT => frequency Y (n) 1 FFT domain Linear combination in WHT domain - - 1 - ( 1 z z ) M 1 Õ = n T ( z ) m - - 1 ( 1 z z ) = ¹ n 0 , n m n m Memory for the f ( n) 1 Frequency weight factors determination Linear RDFT combination Y ( n) 1 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0 32 64 96 128 160 192 224 256 288 320 352 384 416 448 480 512 544 Csaba BENEDECSIK, Annamária R. VÁRKONYI KÓCZY Budapest University of Technology and Economics, Hungary • Abstract. • The measurement of a periodic signal with an unknown fundamental frequency can be well done with an adaptive Fourier analyzer. This paper presents first a further development of the transform domain adaptive Fourier analyzer, based on the Walsh Hadamard transform, which requires less computation. The frequency determination methods are also studied and a new method is presented. • Introduction • Measuring periodic signals with unknown or changing frequency • - problems with DFT : leakage, picket-fence • Important in : active noise control, vibration analysis • Solution offered for the same accuracy : Adaptive Fourier Analysis • Alternatives for the adaptive Fourier analysis : • Sampling controlled by the fundamental frequency • AFA in time domain, adaptive resonator based filter bank • Transform domain Adaptive Fourier Analysis • In this paper • a further development of the transform domain adaptive Fourier analysis based on the Walsh-Hadamard transform • the frequency adaptation method is studied, a new method is presented • Fast implementations of the adaptive Fourier analyzer • The structure of the time-domain adaptive Fourier analyzer • is based on a resonator structure, implementing the recursive Fourier transform Adaptive Fourier analysis based on the Walsh Hadamard transformation is a further development of the transform domain AFA presented Instead of the recursive Fourier transform uses the Walsh Hadamard transform, recursive or filter based implementation The Lagrange polynome used on the Walsh Hadamard domain can be determined by where x(n) is the incoming signal , F(f,k,n) is the Fourier transform matrix, X(k,i) is the Fourier transform, fs is the sampling frequency, L(fs, l, n) is the Lagrange polynome, Lw(fs,l,n) is the Lagrange polynome in the Walsh Hadamard space,i is the current time position, k,l are index from 1 to N, n is an index of the Lagrange matrix from 1 to M, N is the dimension for which the Fourier transform is computed, the incoming signal frequency is f1=N*fs M is the rapport between the incoming frequency and the sampling frequency , and N>M Fig. 1.The structure of the time domain AFA The structure of the transform domain adaptive Fourier analyzer is based on transform domain filtering The given channel’s component is passed, the other components are filtered The transfer function implemented : Fig 3. The general structure of the Walsh Hadamard based AFA Complexity comparison The new structure (WAFA) is M(2*N) less complex than the AFA Using the Fourier transform, total complexity is M(2*N2+2*N)+A(2*N) in time domain the RFT has a complexity of M(2*N)+A(N) in transform domain, the Lagrange matrix’s complexity is M(2*N2)+A(N) Using the Walsh-Hadamard transform, total complexity is M(2*N2)+A(2*N) in the time domain complexity decreases to A(N) , the transform domain complexity remains the same The complexity is expressed by two parameters: the number of multiplication M() and the number of additions A() Frequency adaptation based on ellipse fitting In above the same frequency adaptation method is used, based on the angle difference of two consequent vectors A new method, based on the Fourier transform of a sine signal is proposed The transform in complex plane is an ellipsis The basic frequency can be determined fitting an ellipsis Can be generalized for periodic signals, the image is a filtered sum of the ellipses, can be used if the first component (or any other) is sufficiently prominent relative to his neighbors. The error can be determined The ellipsis determination problem can be reduced to a LSQ linear problem where zn is the n-th root of –1, and Tm denotes the transfer function of the m-th filter bank. The function implements the Lagrange interpolation The components mixed by the leakage are separated The Lagrange polynome is computed offline, stored in a memory Based on the determined frequency the factors are changed. Figure 2. The structure of the transform domain AFA Conclusions The Walsh Hadamard based adaptive Fourier analyzer development to the transform domain adaptive Fourier analysis, needs less computation in the time-domain, can be easily implemented a new frequency determination method has also been proposed, based on the most probable fitting ellipse. Experimental results Test and comparison in three phases : A simple change of the frequency A fractured switch A switch to a noisy input signal (SNR =-15 dB) (3). Fig.4.The incoming signal for which the two methods are compared Fig. 5.The estimated amplitude in case of the first adaptation method based on the angle difference Fig.8.The estimated frequency in case of the new adaptation method based on the ellipsis fitting Fig.6.The estimated amplitude in case of the first adaptation method based on the ellipsis fitting Fig.7.The estimated frequency in case of the first adaptation method based on the angle difference