周波数・時間領域での信号解析論 : 信号および解析信号の帯域フィルターでのサンプリング

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概要

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• The sampling theorem for bandpass filters is devoloped. We assign frequency bands 0-W, W-2W, . . . , (N-1)W-NW to bandpass filters, in order to discuss the signal analysis in the frequency-time domain. The elementary signals, of which examples are illustrated by Fig, 1 and 2, are u^k_n={sinkπ(2Wt-n)-sink-1π(2Wt-n)}/π(2Wt-n), k=1, 2, . . . , N, n=-∞, . . . , 0, 1, . . . , +∞ and make a complete system of orthogonal functions. Then the signal function can be expressed as f(x)=Σ^^N__<k=1>Σ^^<+∞>__<n=-∞>a^k_nu^k_n, where a^k_n=f^k(n/2W):the coefficient is the amplitude sampled at every 1/2W sec of the signal f^k(t)separated by the k^<th>filter. The sequence of sampled amplitudes is the amplitude modulation of pulse train having the repition frequency 1/T_s or 2W and these pulse trains have both side bands about each harmonic of the repitition frequency. When it is required to remake the continuous signal from the sequence of sampled amplitudes, we can realize the signal with any one of limited spectrums stated above by making pulses pass through the corresponding bandpass filter. The physical meaning of the "analytical signal" introduced by D. Gabor is discussed by showing the method which may realize the quadrature component of the analytical signal and by clarifying the independent variables of the analytical signal with limited spectrum. The analytical signals of the elementary signals stated above are obtained, and are expressed as Ψ^k_n(t)=exp{i2k-n)}sinπ(Wt-n)/π(Wt-n), k=1, 2, . . . , N, whose examples are illustrated in Fig. 5 and 7. From the complete orthogonality of the analytical signals mentioned above, a sampling theorem is derived and is expressed as Ψ(t)=Σ^^N__<k-1>Σ^^<+∞>__<n=-∞>C^k_nΨ^k_n(t), where C^k_n=∫^∞_-∞ψ(t)Ψ^k_n(t)dt=ψ^k(n/W), and bar means to take complex conjugate and ψ^k(n/2W)is the n^<th> sampled amplitude of the analytical signal separated by the filter. (It corresponds to the result obtained independently by J. Oswald. )The signal analysis in the frequency-time domain is discussed by using the results described above. We may consider that the sampling theorems stated above contain the so-called uncertainty relation in the spectral analysis, and these theorems may be more convenient to estimate the amount of information of the signal than Gabor's treatment, because our sampling coefficients are independent of each other.

• 社団法人日本音響学会の論文
• 1953-06-30

著者

• 蒲生 秀也

  東京大学理学部

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