Matched Filtering of the Optimal Signals

It is shown that if the complex coefficient of transmission of the matched filter is determined according to the criterion of the minimum output noise variance at a fixed reference value of the output signal, then an increase in the signal-to-noise ratio at the output of such a filter is possible by appropriate selection of the energy spectrum of the signal. The choice of the energy spectrum of the signal is due to the type of power spectral density of the input noise. To determine the energy spectrum of the signal, the signal-to-noise ratio at the output of the matched filter is maximized using the Bunyakovsky–Schwartz inequality. If the input noise is white noise with a uniform power spectral density, then the signal-to-noise ratio at the output of the matched filter is independent of the type of energy spectrum of the signal. The gain in signal-to-noise ratio from applying the matched filter is equal to the square root of the signal base. If the energy spectrum of the signal is directly proportional to the power spectral density of the input non-white noise, then the gain in signal-to-noise ratio from applying the matched filter is also equal to the square root of the signal base. If the energy spectrum of the signal is inversely proportional to the power spectral density of the non-white noise, then the gain in signal-to-noise ratio from applying a matched filter may exceed the square root of the signal base. An increase in the width of the signal spectrum compared to the width of the noise spectrum leads to a sharp increase in the gain. If the energy spectrum of the signal is uniform and the power spectral density of the non-white noise is arbitrary, then the gain in signal-to-noise ratio from applying a matched filter may exceed the square root of the signal base. An increase in the width of the signal spectrum compared to the width of the noise spectrum leads to a sharp increase in the gain. However, this gain is less than in the previous case.

研究表明，若根据固定输出信号参考值下的最小输出噪声方差准则确定匹配滤波器（matched filter）的传输复系数，则可通过合理选择信号能量谱提升该滤波器输出端的信噪比。信号能量谱的选择由输入噪声的功率谱密度（power spectral density）类型决定。为确定信号能量谱，可借助布尼亚科夫斯基-施瓦茨不等式（Bunyakovsky–Schwartz inequality）最大化匹配滤波器输出端的信噪比。若输入噪声为功率谱密度均匀分布的白噪声，则匹配滤波器输出端的信噪比与信号能量谱的类型无关。此时，采用匹配滤波器带来的信噪比增益等于信号基（signal base）的平方根。若信号能量谱与输入非白噪声的功率谱密度成正比，则采用匹配滤波器带来的信噪比增益同样等于信号基的平方根。若信号能量谱与非白噪声的功率谱密度成反比，则采用匹配滤波器带来的信噪比增益可能超过信号基的平方根。相较于噪声频谱宽度，信号频谱宽度的增加会导致增益大幅提升。若信号能量谱为均匀分布，且非白噪声的功率谱密度为任意形式，则采用匹配滤波器带来的信噪比增益同样可能超过信号基的平方根。相较于噪声频谱宽度，信号频谱宽度的增加会导致增益大幅提升，但该增益仍小于前述情况中的增益。